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diff --git a/docs/pages/bag.html b/docs/pages/bag.html
index 7c5928f..5e1c5e1 100644
--- a/docs/pages/bag.html
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+      <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item">Supervised Learning</li><li class="breadcrumb-item"><a href="../pages/knn.html">Classification</a></li><li class="breadcrumb-item"><a href="../pages/bag.html">Ensemble methods</a></li><li class="breadcrumb-item"><a href="../pages/bag.html">Bagging</a></li></ol></nav>
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  <span class="menu-text">Unsupervised Learning</span></a>
-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false" aria-label="Toggle section">
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-      <ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 show">  
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+            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="true">
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+          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="true" aria-label="Toggle section">
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-      <ul id="quarto-sidebar-section-7" class="collapse list-unstyled sidebar-section depth2 ">  
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+      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth3 ">  
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-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 show">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 show">  
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@@ -353,12 +351,66 @@
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+      </ul>
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+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
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diff --git a/docs/pages/bayes_est.html b/docs/pages/bayes_est.html
index 48adfdf..2166dec 100644
--- a/docs/pages/bayes_est.html
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-            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="true">
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false">
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-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false" aria-label="Toggle section">
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-      <ul id="quarto-sidebar-section-5" class="collapse list-unstyled sidebar-section depth1 show">  
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-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
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   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -355,12 +353,66 @@
 </li>
       </ul>
   </li>
+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
       </ul>
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index 5d67e82..8c91299 100644
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+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false">
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-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false" aria-label="Toggle section">
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   <div class="sidebar-item-container"> 
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-      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth3 ">  
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   <div class="sidebar-item-container"> 
   <a href="../pages/percp.html" class="sidebar-item-text sidebar-link">
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
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   <div class="sidebar-item-container"> 
   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -355,12 +353,66 @@
 </li>
       </ul>
   </li>
+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
       </ul>
   </li>
         <li class="sidebar-item">
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index 3f685b7..19c4a90 100644
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 <script src="../site_libs/quarto-html/quarto.js"></script>
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+      <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item">Supervised Learning</li><li class="breadcrumb-item"><a href="../pages/knn.html">Classification</a></li><li class="breadcrumb-item"><a href="../pages/bag.html">Ensemble methods</a></li><li class="breadcrumb-item"><a href="../pages/boost.html">Boosting</a></li></ol></nav>
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+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false">
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-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false" aria-label="Toggle section">
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-      <ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 show">  
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@@ -263,13 +263,13 @@
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-            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="false">
+            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="true">
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-          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="false" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="true" aria-label="Toggle section">
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           </a> 
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-      <ul id="quarto-sidebar-section-7" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-7" class="collapse list-unstyled sidebar-section depth2 show">  
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   <div class="sidebar-item-container"> 
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@@ -282,8 +282,6 @@
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+      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth3 ">  
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@@ -309,7 +307,7 @@
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+      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth3 ">  
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           </a> 
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-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 show">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 show">  
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@@ -353,12 +351,66 @@
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+      </ul>
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+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
       </ul>
   </li>
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@@ -739,8 +791,8 @@
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-      <a href="../pages/ann.html" class="pagination-link">
-        <span class="nav-page-text">Artificial neural networks (ANNs)</span> <i class="bi bi-arrow-right-short"></i>
+      <a href="../pages/notations_nn .html" class="pagination-link">
+        <span class="nav-page-text">Notations</span> <i class="bi bi-arrow-right-short"></i>
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diff --git a/docs/pages/dec_trees.html b/docs/pages/dec_trees.html
index 1aaa0e0..e0d7d68 100644
--- a/docs/pages/dec_trees.html
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-            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true">
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false">
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-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false" aria-label="Toggle section">
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           </a> 
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-      <ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 show">  
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@@ -284,8 +284,6 @@
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@@ -294,7 +292,7 @@
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-      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth3 ">  
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   <div class="sidebar-item-container"> 
   <a href="../pages/naive_bayes.html" class="sidebar-item-text sidebar-link">
@@ -311,7 +309,7 @@
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+      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth3 ">  
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           </a> 
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-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
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   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -355,12 +353,66 @@
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       </ul>
   </li>
+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
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+  </li>
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diff --git a/docs/pages/em.html b/docs/pages/em.html
index 05c2767..0ef0993 100644
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+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false">
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-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false" aria-label="Toggle section">
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+      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth3 ">  
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   <div class="sidebar-item-container"> 
   <a href="../pages/naive_bayes.html" class="sidebar-item-text sidebar-link">
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           </a> 
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+      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth3 ">  
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   <div class="sidebar-item-container"> 
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           </a> 
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-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
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   <div class="sidebar-item-container"> 
   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -355,12 +353,66 @@
 </li>
       </ul>
   </li>
+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
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   </li>
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diff --git a/docs/pages/gmm.html b/docs/pages/gmm.html
index f781538..c15590b 100644
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-            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="true">
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false">
  <span class="menu-text">Supervised Learning</span></a>
-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false" aria-label="Toggle section">
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           </a> 
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-      <ul id="quarto-sidebar-section-5" class="collapse list-unstyled sidebar-section depth1 show">  
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-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
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@@ -355,12 +353,66 @@
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+      </ul>
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+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
       </ul>
   </li>
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diff --git a/docs/pages/intro.html b/docs/pages/intro.html
index 246fa04..3aa4c69 100644
--- a/docs/pages/intro.html
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-            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true">
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false">
  <span class="menu-text">Unsupervised Learning</span></a>
-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false" aria-label="Toggle section">
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-      <ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 show">  
+      <ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 ">  
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+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false">
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-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false" aria-label="Toggle section">
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@@ -282,8 +282,6 @@
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+      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth3 ">  
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+      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth3 ">  
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-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
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@@ -353,12 +351,66 @@
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+      </ul>
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+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
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diff --git a/docs/pages/kkmeans.html b/docs/pages/kkmeans.html
index e2081f5..8542b10 100644
--- a/docs/pages/kkmeans.html
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+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false">
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+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false" aria-label="Toggle section">
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           </a> 
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-      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth2 ">  
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   <div class="sidebar-item-container"> 
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           </a> 
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-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
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   <div class="sidebar-item-container"> 
   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -355,12 +353,66 @@
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+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
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-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
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 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
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diff --git a/docs/pages/klin_reg.html b/docs/pages/klin_reg.html
index fc0e7c0..9feffa8 100644
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@@ -123,13 +123,13 @@
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+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false">
  <span class="menu-text">Unsupervised Learning</span></a>
-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false" aria-label="Toggle section">
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@@ -355,12 +353,66 @@
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+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
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+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
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-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
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+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false">
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@@ -355,12 +353,66 @@
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+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
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+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
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+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
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+  </li>
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index 3912002..75e7a66 100644
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+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false">
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@@ -355,12 +353,66 @@
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+      </ul>
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+      </ul>
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+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
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 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
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+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
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+  </div>
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+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false">
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   <div class="sidebar-item-container"> 
   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -355,12 +353,66 @@
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       </ul>
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+      </ul>
+  </li>
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+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
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+          </a> 
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+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
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           <li class="sidebar-item">
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-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
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 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
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+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
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diff --git a/docs/pages/lin_reg.html b/docs/pages/lin_reg.html
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@@ -318,8 +318,6 @@
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 </li>
-      </ul>
-  </li>
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@@ -328,7 +326,7 @@
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth3 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/naive_bayes.html" class="sidebar-item-text sidebar-link">
@@ -345,7 +343,7 @@
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth3 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/percp.html" class="sidebar-item-text sidebar-link">
@@ -374,7 +372,7 @@
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
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   <div class="sidebar-item-container"> 
   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -389,12 +387,66 @@
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       </ul>
   </li>
+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
       </ul>
   </li>
         <li class="sidebar-item">
diff --git a/docs/pages/log_reg.html b/docs/pages/log_reg.html
index e803ff0..7a61d73 100644
--- a/docs/pages/log_reg.html
+++ b/docs/pages/log_reg.html
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         <i class="bi bi-layout-text-sidebar-reverse"></i>
       </button>
-      <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item">Supervised Learning</li><li class="breadcrumb-item"><a href="../pages/percp.html">Discriminative models</a></li><li class="breadcrumb-item"><a href="../pages/log_reg.html">Logistic regression</a></li></ol></nav>
+      <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item">Supervised Learning</li><li class="breadcrumb-item"><a href="../pages/knn.html">Classification</a></li><li class="breadcrumb-item"><a href="../pages/percp.html">Discriminative models</a></li><li class="breadcrumb-item"><a href="../pages/log_reg.html">Logistic regression</a></li></ol></nav>
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@@ -123,13 +123,13 @@
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       <div class="sidebar-item-container"> 
-            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true">
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false">
  <span class="menu-text">Unsupervised Learning</span></a>
-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
       </div>
-      <ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 show">  
+      <ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 ">  
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@@ -265,13 +265,13 @@
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-            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="false">
+            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="true">
  <span class="menu-text">Classification</span></a>
-          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="false" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
       </div>
-      <ul id="quarto-sidebar-section-7" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-7" class="collapse list-unstyled sidebar-section depth2 show">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/knn.html" class="sidebar-item-text sidebar-link">
@@ -284,8 +284,6 @@
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   </div>
 </li>
-      </ul>
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@@ -294,7 +292,7 @@
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth3 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/naive_bayes.html" class="sidebar-item-text sidebar-link">
@@ -311,7 +309,7 @@
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth2 show">  
+      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth3 show">  
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   <div class="sidebar-item-container"> 
   <a href="../pages/percp.html" class="sidebar-item-text sidebar-link">
@@ -340,7 +338,7 @@
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -355,12 +353,66 @@
 </li>
       </ul>
   </li>
+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
       </ul>
   </li>
         <li class="sidebar-item">
diff --git a/docs/pages/mle.html b/docs/pages/mle.html
index b7a4f13..89be218 100644
--- a/docs/pages/mle.html
+++ b/docs/pages/mle.html
@@ -215,13 +215,13 @@
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         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
-            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="true">
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false">
  <span class="menu-text">Supervised Learning</span></a>
-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-5" aria-expanded="false" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
       </div>
-      <ul id="quarto-sidebar-section-5" class="collapse list-unstyled sidebar-section depth1 show">  
+      <ul id="quarto-sidebar-section-5" class="collapse list-unstyled sidebar-section depth1 ">  
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       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-6" aria-expanded="false">
@@ -284,8 +284,6 @@
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 </li>
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@@ -294,7 +292,7 @@
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth3 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/naive_bayes.html" class="sidebar-item-text sidebar-link">
@@ -311,7 +309,7 @@
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth3 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/percp.html" class="sidebar-item-text sidebar-link">
@@ -340,7 +338,7 @@
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           </a> 
       </div>
-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -355,12 +353,66 @@
 </li>
       </ul>
   </li>
+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
       </ul>
   </li>
         <li class="sidebar-item">
diff --git a/docs/pages/naive_bayes.html b/docs/pages/naive_bayes.html
index c6bc8aa..0dc544e 100644
--- a/docs/pages/naive_bayes.html
+++ b/docs/pages/naive_bayes.html
@@ -366,10 +366,16 @@
           </a> 
       </div>
       <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
           <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
- <span class="menu-text">Regression</span></a>
+ <span class="menu-text">Neural Networks</span></a>
           <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
@@ -377,8 +383,32 @@
       <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/lin_reg.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Least squares regression</span></a>
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
   </div>
 </li>
       </ul>
diff --git a/docs/pages/notations_nn .html b/docs/pages/notations_nn .html
index 64c445d..35a7650 100644
--- a/docs/pages/notations_nn .html	
+++ b/docs/pages/notations_nn .html	
@@ -30,7 +30,7 @@
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-<h1 class="title">Shallow Neural Network</h1>
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-<p><strong>Neural Network Architecture:</strong> A shallow neural network typically consists of an input layer, one or more hidden layers, and an output layer.</p>
-<p><img src="../images/classification_kiank.png" style="height:300px;" centre="true"></p>
-<section id="forward-propagation" class="level2">
-<h2 class="anchored" data-anchor-id="forward-propagation">Forward Propagation</h2>
-<p>The forward pass in a neural network involves processing input data through the network’s layers to generate predictions or outputs by sequentially applying weights, biases, and activation functions.</p>
-<p><strong>Mathematical Representation:</strong></p>
-<p>For one training example <span class="math inline">\(x^{(i)}\)</span>:</p>
-<ol type="1">
-<li><strong>Hidden Layer Computations:</strong>
-<ul>
-<li><strong>Weighted Sum Calculation:</strong>
-<ul>
-<li><span class="math inline">\(z^{[l] (i)} = W^{[l]} a^{[l-1](i)} + b^{[l]}\)</span>
-<ul>
-<li>Compute the weighted sum of inputs (<span class="math inline">\(a^{[l-1](i)}\)</span>) by the weights (<span class="math inline">\(W^{[l]}\)</span>) and add the bias (<span class="math inline">\(b^{[l]}\)</span>) for layer <span class="math inline">\(l\)</span>.</li>
-</ul></li>
-</ul></li>
-<li><strong>Activation Calculation:</strong>
-<ul>
-<li><span class="math inline">\(a^{[l] (i)} = g(z^{[l] (i)})\)</span>
-<ul>
-<li>Apply an activation function <span class="math inline">\(g\)</span> (such as sigmoid, tanh, ReLU, etc.) to the computed weighted sum to get the activation of layer <span class="math inline">\(l\)</span>.</li>
-</ul></li>
-</ul></li>
-</ul></li>
-<li><strong>Output Layer Computations:</strong>
-<ul>
-<li><strong>Weighted Sum Calculation:</strong>
-<ul>
-<li><span class="math inline">\(z^{[L] (i)} = W^{[L]} a^{[L-1] (i)} + b^{[L]}\)</span>
-<ul>
-<li>Compute the weighted sum of inputs (<span class="math inline">\(a^{[L-1] (i)}\)</span>) by the weights (<span class="math inline">\(W^{[L]}\)</span>) and add the bias (<span class="math inline">\(b^{[L]}\)</span>) for the output layer.</li>
-</ul></li>
-</ul></li>
-<li><strong>Activation Calculation:</strong>
-<ul>
-<li><span class="math inline">\(\hat{y}^{(i)} = a^{[L] (i)} = \sigma(z^{ [L] (i)})\)</span>
-<ul>
-<li>Apply an appropriate activation function (e.g., softmax for multiclass classification, sigmoid for binary classification) to the computed weighted sum to obtain the predicted output (<span class="math inline">\(\hat{y}^{(i)}\)</span>).</li>
-</ul></li>
-</ul></li>
-</ul></li>
-<li><strong>Cost Function:</strong>
-<ul>
-<li><span class="math inline">\(J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \left(y^{(i)}\log\left(a^{[L] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[L] (i)}\right)\right)\)</span>
-<ul>
-<li>Compute the appropriate cost function (e.g., cross-entropy, mean squared error) to evaluate the difference between the predicted output (<span class="math inline">\(a^{[L] (i)}\)</span>) and the actual output (<span class="math inline">\(y^{(i)}\)</span>) for all training examples (<span class="math inline">\(m\)</span>).</li>
-</ul></li>
-</ul></li>
-</ol>
-</section>
-<section id="backpropagation" class="level2">
-<h2 class="anchored" data-anchor-id="backpropagation">Backpropagation</h2>
-<p>Backpropagation is the process of iteratively computing gradients of the loss function with respect to the weights and biases of the neural network, enabling the adjustment of parameters by propagating errors backward through the network’s layers to enhance learning and optimize performance.</p>
-<ol type="1">
-<li><strong>Output Layer Error Derivative:</strong>
-<ul>
-<li><span class="math inline">\(\frac{\partial \mathcal{J} }{ \partial z_{L}^{(i)} } = \frac{1}{m} (\hat{y}^{(i)} - y^{(i)}) \odot g'(z^{[L] (i)})\)</span>
-<ul>
-<li>Calculate the derivative of the cost function with respect to the pre-activation value of the output layer for each training example.</li>
-<li><span class="math inline">\(\odot\)</span> denotes element-wise multiplication.</li>
-<li><span class="math inline">\(g'(\cdot)\)</span> represents the derivative of the activation function used in the output layer.</li>
-</ul></li>
-</ul></li>
-<li><strong>Gradient Calculation for Output Layer Weights and Bias:</strong>
-<ul>
-<li><span class="math inline">\(\frac{\partial \mathcal{J} }{ \partial W^{[L]} } = \frac{\partial \mathcal{J} }{ \partial z_{L}^{(i)} } a^{[L-1] (i) T}\)</span>
-<ul>
-<li>Compute the gradient of the loss function with respect to the weights of the output layer.</li>
-</ul></li>
-<li><span class="math inline">\(\frac{\partial \mathcal{J} }{ \partial b^{[L]} } = \sum_i{\frac{\partial \mathcal{J} }{ \partial z_{L}^{(i)}}}\)</span>
-<ul>
-<li>Compute the gradient of the loss function with respect to the bias of the output layer.</li>
-</ul></li>
-</ul></li>
-<li><strong>Backpropagate the Error to Previous Layers:</strong>
-<ul>
-<li><span class="math inline">\(\frac{\partial \mathcal{J} }{ \partial z_{l}^{(i)} } = W^{[l+1] T} \frac{\partial \mathcal{J} }{ \partial z_{l+1}^{(i)} } \odot g'(z^{[l] (i)})\)</span>
-<ul>
-<li>Propagate the error derivative backward through the layers by considering the relationship between consecutive layers.</li>
-<li>Involves the transpose of weights connecting the subsequent and current layers.</li>
-<li>Involves the derivative of the activation function used in layer <span class="math inline">\(l\)</span>.</li>
-</ul></li>
-</ul></li>
-<li><strong>Gradient Calculation for Weights and Bias of Hidden Layers:</strong>
-<ul>
-<li><span class="math inline">\(\frac{\partial \mathcal{J} }{ \partial W^{[l]} } = \frac{\partial \mathcal{J} }{ \partial z_{l}^{(i)} } a^{[l-1] (i) T}\)</span>
-<ul>
-<li>Compute the gradient of the loss function with respect to the weights of layer <span class="math inline">\(l\)</span>.</li>
-</ul></li>
-<li><span class="math inline">\(\frac{\partial \mathcal{J} }{ \partial b^{[l]} } = \sum_i{\frac{\partial \mathcal{J} }{ \partial z_{l}^{(i)}}}\)</span>
-<ul>
-<li>Compute the gradient of the loss function with respect to the bias of layer <span class="math inline">\(l\)</span>.</li>
-</ul></li>
-</ul></li>
-</ol>
-<p>These steps form the core of backpropagation in a shallow neural network, allowing for the iterative adjustment of weights and biases to minimize the loss function and improve the model’s accuracy in predicting outputs for given inputs.</p>
-</section>
-<section id="gradient-descent" class="level2">
-<h2 class="anchored" data-anchor-id="gradient-descent">Gradient Descent</h2>
-<p><strong>Optimizing Model Parameters:</strong> Gradient Descent is used to update the parameters (weights and biases) of a neural network in the direction that minimizes the loss function.</p>
-<p><strong>General Gradient Descent Rule:</strong> The general update rule for a parameter <span class="math inline">\(\theta\)</span> using Gradient Descent is: <span class="math display">\[\theta = \theta - \alpha \frac{\partial J }{ \partial \theta }\]</span> where <span class="math inline">\(\alpha\)</span> is the learning rate and <span class="math inline">\(\frac{\partial J }{ \partial \theta }\)</span> denotes the partial derivative of the cost function <span class="math inline">\(J\)</span> with respect to the parameter <span class="math inline">\(\theta\)</span>.</p>
-<p><strong>Parameter Update for Weights and Biases:</strong></p>
-<ol type="1">
-<li><strong>Output Layer:</strong>
-<ul>
-<li>Update the weights and biases of the output layer using the gradients calculated during backpropagation:
-<ul>
-<li><span class="math inline">\(W^{[L]} = W^{[L]} - \alpha \frac{\partial J }{ \partial W^{[L]} }\)</span></li>
-<li><span class="math inline">\(b^{[L]} = b^{[L]} - \alpha \frac{\partial J }{ \partial b^{[L]} }\)</span></li>
-</ul></li>
-</ul></li>
-<li><strong>Hidden Layers:</strong>
-<ul>
-<li>Update the weights and biases of the hidden layers using the gradients computed during backpropagation:
-<ul>
-<li>For <span class="math inline">\(l\)</span>th hidden layer:
-<ul>
-<li><span class="math inline">\(W^{[l]} = W^{[l]} - \alpha \frac{\partial J }{ \partial W^{[l]} }\)</span></li>
-<li><span class="math inline">\(b^{[l]} = b^{[l]} - \alpha \frac{\partial J }{ \partial b^{[l]} }\)</span></li>
-</ul></li>
-</ul></li>
-</ul></li>
-</ol>
-<p><strong>Impact of Learning Rate (<span class="math inline">\(\alpha\)</span>):</strong> - The learning rate <span class="math inline">\(\alpha\)</span> determines the step size during parameter updates. - A larger <span class="math inline">\(\alpha\)</span> might lead to faster convergence but can overshoot the minimum. - A smaller <span class="math inline">\(\alpha\)</span> might converge slowly but can be more precise.</p>
-<p><strong>Iteration and Convergence:</strong> - Iterate through multiple epochs (iterations) of forward propagation, backpropagation, and parameter updates until the model’s performance converges or the predefined number of epochs is reached. - Monitor the decrease in the cost function to assess convergence.</p>
-<p><strong>Importance of Gradient Descent:</strong> - Gradient Descent is crucial in adjusting model parameters to minimize the loss function, thereby enhancing the accuracy of predictions in a neural network. - It enables the network to learn from the data by iteratively updating weights and biases to improve the overall performance of the model.</p>
-</section>
-<section id="code" class="level2">
-<h2 class="anchored" data-anchor-id="code">Code</h2>
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-<div class="sourceCode" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
-<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> forward_propagation(X, parameters):</span>
-<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>    <span class="co">"""</span></span>
-<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="co">    Argument:</span></span>
-<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="co">    X -- input data of size (n_x, m)</span></span>
-<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a><span class="co">    parameters -- python dictionary containing your parameters (output of initialization function)</span></span>
-<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a><span class="co">    </span></span>
-<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a><span class="co">    Returns:</span></span>
-<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a><span class="co">    A2 -- The sigmoid output of the second activation</span></span>
-<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a><span class="co">    cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"</span></span>
-<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a><span class="co">    """</span></span>
-<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a>    W1, b1, W2, b2 <span class="op">=</span> parameters[<span class="st">"W1"</span>], parameters[<span class="st">"b1"</span>], parameters[<span class="st">"W2"</span>], parameters[<span class="st">"b2"</span>]</span>
-<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a>    </span>
-<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a>    Z1 <span class="op">=</span> W1<span class="op">@</span>X <span class="op">+</span> b1</span>
-<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a>    A1 <span class="op">=</span> np.tanh(Z1)</span>
-<span id="cb1-17"><a href="#cb1-17" aria-hidden="true" tabindex="-1"></a>    Z2 <span class="op">=</span> W2<span class="op">@</span>A1 <span class="op">+</span> b2</span>
-<span id="cb1-18"><a href="#cb1-18" aria-hidden="true" tabindex="-1"></a>    A2 <span class="op">=</span> <span class="dv">1</span><span class="op">/</span>(<span class="dv">1</span><span class="op">+</span>np.exp(<span class="op">-</span>Z2))</span>
-<span id="cb1-19"><a href="#cb1-19" aria-hidden="true" tabindex="-1"></a>    </span>
-<span id="cb1-20"><a href="#cb1-20" aria-hidden="true" tabindex="-1"></a>    cache <span class="op">=</span> {<span class="st">"Z1"</span>: Z1,</span>
-<span id="cb1-21"><a href="#cb1-21" aria-hidden="true" tabindex="-1"></a>             <span class="st">"A1"</span>: A1,</span>
-<span id="cb1-22"><a href="#cb1-22" aria-hidden="true" tabindex="-1"></a>             <span class="st">"Z2"</span>: Z2,</span>
-<span id="cb1-23"><a href="#cb1-23" aria-hidden="true" tabindex="-1"></a>             <span class="st">"A2"</span>: A2}</span>
-<span id="cb1-24"><a href="#cb1-24" aria-hidden="true" tabindex="-1"></a>    </span>
-<span id="cb1-25"><a href="#cb1-25" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> A2, cache</span>
-<span id="cb1-26"><a href="#cb1-26" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-27"><a href="#cb1-27" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> compute_cost(A2, Y):</span>
-<span id="cb1-28"><a href="#cb1-28" aria-hidden="true" tabindex="-1"></a>    <span class="co">"""</span></span>
-<span id="cb1-29"><a href="#cb1-29" aria-hidden="true" tabindex="-1"></a><span class="co">    Computes the cross-entropy cost given in equation (13)</span></span>
-<span id="cb1-30"><a href="#cb1-30" aria-hidden="true" tabindex="-1"></a><span class="co">    </span></span>
-<span id="cb1-31"><a href="#cb1-31" aria-hidden="true" tabindex="-1"></a><span class="co">    Arguments:</span></span>
-<span id="cb1-32"><a href="#cb1-32" aria-hidden="true" tabindex="-1"></a><span class="co">    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)</span></span>
-<span id="cb1-33"><a href="#cb1-33" aria-hidden="true" tabindex="-1"></a><span class="co">    Y -- "true" labels vector of shape (1, number of examples)</span></span>
-<span id="cb1-34"><a href="#cb1-34" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-35"><a href="#cb1-35" aria-hidden="true" tabindex="-1"></a><span class="co">    Returns:</span></span>
-<span id="cb1-36"><a href="#cb1-36" aria-hidden="true" tabindex="-1"></a><span class="co">    cost -- cross-entropy cost given equation (13)</span></span>
-<span id="cb1-37"><a href="#cb1-37" aria-hidden="true" tabindex="-1"></a><span class="co">    </span></span>
-<span id="cb1-38"><a href="#cb1-38" aria-hidden="true" tabindex="-1"></a><span class="co">    """</span></span>
-<span id="cb1-39"><a href="#cb1-39" aria-hidden="true" tabindex="-1"></a>    </span>
-<span id="cb1-40"><a href="#cb1-40" aria-hidden="true" tabindex="-1"></a>    m <span class="op">=</span> Y.shape[<span class="dv">1</span>] <span class="co"># number of examples</span></span>
-<span id="cb1-41"><a href="#cb1-41" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-42"><a href="#cb1-42" aria-hidden="true" tabindex="-1"></a>    logprobs <span class="op">=</span> np.multiply(Y, np.log(A2)) <span class="op">+</span> np.multiply((<span class="dv">1</span><span class="op">-</span>Y), np.log(<span class="dv">1</span><span class="op">-</span>A2))</span>
-<span id="cb1-43"><a href="#cb1-43" aria-hidden="true" tabindex="-1"></a>    cost <span class="op">=</span> <span class="op">-</span> np.<span class="bu">sum</span>(logprobs)<span class="op">/</span>Y.shape[<span class="dv">1</span>]</span>
-<span id="cb1-44"><a href="#cb1-44" aria-hidden="true" tabindex="-1"></a>        </span>
-<span id="cb1-45"><a href="#cb1-45" aria-hidden="true" tabindex="-1"></a>    cost <span class="op">=</span> <span class="bu">float</span>(np.squeeze(cost)) </span>
-<span id="cb1-46"><a href="#cb1-46" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-47"><a href="#cb1-47" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> cost</span>
-<span id="cb1-48"><a href="#cb1-48" aria-hidden="true" tabindex="-1"></a></span>
-<span id="cb1-49"><a href="#cb1-49" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> backward_propagation(parameters, cache, X, Y):</span>
-<span id="cb1-50"><a href="#cb1-50" aria-hidden="true" tabindex="-1"></a>    <span class="co">"""</span></span>
-<span id="cb1-51"><a href="#cb1-51" aria-hidden="true" tabindex="-1"></a><span class="co">    Implement the backward propagation using the instructions above.</span></span>
-<span id="cb1-52"><a href="#cb1-52" aria-hidden="true" tabindex="-1"></a><span class="co">    </span></span>
-<span id="cb1-53"><a href="#cb1-53" aria-hidden="true" tabindex="-1"></a><span class="co">    Arguments:</span></span>
-<span id="cb1-54"><a href="#cb1-54" aria-hidden="true" tabindex="-1"></a><span class="co">    parameters -- python dictionary containing our parameters </span></span>
-<span id="cb1-55"><a href="#cb1-55" aria-hidden="true" tabindex="-1"></a><span class="co">    cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".</span></span>
-<span id="cb1-56"><a href="#cb1-56" aria-hidden="true" tabindex="-1"></a><span class="co">    X -- input data of shape (2, number of examples)</span></span>
-<span id="cb1-57"><a href="#cb1-57" aria-hidden="true" tabindex="-1"></a><span class="co">    Y -- "true" labels vector of shape (1, number of examples)</span></span>
-<span id="cb1-58"><a href="#cb1-58" aria-hidden="true" tabindex="-1"></a><span class="co">    </span></span>
-<span id="cb1-59"><a href="#cb1-59" aria-hidden="true" tabindex="-1"></a><span class="co">    Returns:</span></span>
-<span id="cb1-60"><a href="#cb1-60" aria-hidden="true" tabindex="-1"></a><span class="co">    grads -- python dictionary containing your gradients with respect to different parameters</span></span>
-<span id="cb1-61"><a href="#cb1-61" aria-hidden="true" tabindex="-1"></a><span class="co">    """</span></span>
-<span id="cb1-62"><a href="#cb1-62" aria-hidden="true" tabindex="-1"></a>    m <span class="op">=</span> X.shape[<span class="dv">1</span>]</span>
-<span id="cb1-63"><a href="#cb1-63" aria-hidden="true" tabindex="-1"></a>    </span>
-<span id="cb1-64"><a href="#cb1-64" aria-hidden="true" tabindex="-1"></a>    W1, W2 <span class="op">=</span> parameters[<span class="st">"W1"</span>], parameters[<span class="st">"W2"</span>]</span>
-<span id="cb1-65"><a href="#cb1-65" aria-hidden="true" tabindex="-1"></a>    </span>
-<span id="cb1-66"><a href="#cb1-66" aria-hidden="true" tabindex="-1"></a>    A1, A2 <span class="op">=</span> cache[<span class="st">"A1"</span>], cache[<span class="st">"A2"</span>]</span>
-<span id="cb1-67"><a href="#cb1-67" aria-hidden="true" tabindex="-1"></a>        </span>
-<span id="cb1-68"><a href="#cb1-68" aria-hidden="true" tabindex="-1"></a>    dZ2 <span class="op">=</span> A2 <span class="op">-</span> Y</span>
-<span id="cb1-69"><a href="#cb1-69" aria-hidden="true" tabindex="-1"></a>    dW2 <span class="op">=</span> (<span class="dv">1</span><span class="op">/</span>Y.shape[<span class="dv">1</span>])<span class="op">*</span>dZ2<span class="op">@</span>A1.T</span>
-<span id="cb1-70"><a href="#cb1-70" aria-hidden="true" tabindex="-1"></a>    db2 <span class="op">=</span> (<span class="dv">1</span><span class="op">/</span>Y.shape[<span class="dv">1</span>])<span class="op">*</span>np.<span class="bu">sum</span>(dZ2, axis<span class="op">=</span><span class="dv">1</span>, keepdims<span class="op">=</span><span class="va">True</span>)</span>
-<span id="cb1-71"><a href="#cb1-71" aria-hidden="true" tabindex="-1"></a>    dZ1 <span class="op">=</span> W2.T<span class="op">@</span>dZ2<span class="op">*</span>(<span class="dv">1</span><span class="op">-</span>A1<span class="op">**</span><span class="dv">2</span>)</span>
-<span id="cb1-72"><a href="#cb1-72" aria-hidden="true" tabindex="-1"></a>    dW1 <span class="op">=</span> (<span class="dv">1</span><span class="op">/</span>Y.shape[<span class="dv">1</span>])<span class="op">*</span>dZ1<span class="op">@</span>X.T</span>
-<span id="cb1-73"><a href="#cb1-73" aria-hidden="true" tabindex="-1"></a>    db1 <span class="op">=</span> (<span class="dv">1</span><span class="op">/</span>Y.shape[<span class="dv">1</span>])<span class="op">*</span>np.<span class="bu">sum</span>(dZ1, axis<span class="op">=</span><span class="dv">1</span>, keepdims<span class="op">=</span><span class="va">True</span>)</span>
-<span id="cb1-74"><a href="#cb1-74" aria-hidden="true" tabindex="-1"></a>        </span>
-<span id="cb1-75"><a href="#cb1-75" aria-hidden="true" tabindex="-1"></a>    grads <span class="op">=</span> {<span class="st">"dW1"</span>: dW1,</span>
-<span id="cb1-76"><a href="#cb1-76" aria-hidden="true" tabindex="-1"></a>             <span class="st">"db1"</span>: db1,</span>
-<span id="cb1-77"><a href="#cb1-77" aria-hidden="true" tabindex="-1"></a>             <span class="st">"dW2"</span>: dW2,</span>
-<span id="cb1-78"><a href="#cb1-78" aria-hidden="true" tabindex="-1"></a>             <span class="st">"db2"</span>: db2}</span>
-<span id="cb1-79"><a href="#cb1-79" aria-hidden="true" tabindex="-1"></a>    </span>
-<span id="cb1-80"><a href="#cb1-80" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> grads</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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\ No newline at end of file
diff --git a/docs/pages/svm.html b/docs/pages/svm.html
index 0166b24..6878c5f 100644
--- a/docs/pages/svm.html
+++ b/docs/pages/svm.html
@@ -2,7 +2,7 @@
 <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
 
 <meta charset="utf-8">
-<meta name="generator" content="quarto-1.3.433">
+<meta name="generator" content="quarto-1.3.450">
 
 <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
 
@@ -79,7 +79,7 @@
       <button type="button" class="quarto-btn-toggle btn" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar,#quarto-sidebar-glass" aria-controls="quarto-sidebar" aria-expanded="false" aria-label="Toggle sidebar navigation" onclick="if (window.quartoToggleHeadroom) { window.quartoToggleHeadroom(); }">
         <i class="bi bi-layout-text-sidebar-reverse"></i>
       </button>
-      <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item">Supervised Learning</li><li class="breadcrumb-item"><a href="../pages/percp.html">Discriminative models</a></li><li class="breadcrumb-item"><a href="../pages/svm.html">Support vector machines (SVMs)</a></li></ol></nav>
+      <nav class="quarto-page-breadcrumbs" aria-label="breadcrumb"><ol class="breadcrumb"><li class="breadcrumb-item">Supervised Learning</li><li class="breadcrumb-item"><a href="../pages/knn.html">Classification</a></li><li class="breadcrumb-item"><a href="../pages/percp.html">Discriminative models</a></li><li class="breadcrumb-item"><a href="../pages/svm.html">Support vector machines (SVMs)</a></li></ol></nav>
       <a class="flex-grow-1" role="button" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar,#quarto-sidebar-glass" aria-controls="quarto-sidebar" aria-expanded="false" aria-label="Toggle sidebar navigation" onclick="if (window.quartoToggleHeadroom) { window.quartoToggleHeadroom(); }">      
       </a>
       <button type="button" class="btn quarto-search-button" aria-label="" onclick="window.quartoOpenSearch();">
@@ -123,13 +123,13 @@
 </li>
         <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
-            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true">
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false">
  <span class="menu-text">Unsupervised Learning</span></a>
-          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="true" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-1" aria-expanded="false" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
       </div>
-      <ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 show">  
+      <ul id="quarto-sidebar-section-1" class="collapse list-unstyled sidebar-section depth1 ">  
           <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-2" aria-expanded="false">
@@ -265,13 +265,13 @@
   </li>
           <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
-            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="false">
+            <a class="sidebar-item-text sidebar-link text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="true">
  <span class="menu-text">Classification</span></a>
-          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="false" aria-label="Toggle section">
+          <a class="sidebar-item-toggle text-start" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-7" aria-expanded="true" aria-label="Toggle section">
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
       </div>
-      <ul id="quarto-sidebar-section-7" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-7" class="collapse list-unstyled sidebar-section depth2 show">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/knn.html" class="sidebar-item-text sidebar-link">
@@ -284,8 +284,6 @@
  <span class="menu-text">Decision tree</span></a>
   </div>
 </li>
-      </ul>
-  </li>
           <li class="sidebar-item sidebar-item-section">
       <div class="sidebar-item-container"> 
             <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-8" aria-expanded="false">
@@ -294,7 +292,7 @@
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
       </div>
-      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-8" class="collapse list-unstyled sidebar-section depth3 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/naive_bayes.html" class="sidebar-item-text sidebar-link">
@@ -311,7 +309,7 @@
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
       </div>
-      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth2 show">  
+      <ul id="quarto-sidebar-section-9" class="collapse list-unstyled sidebar-section depth3 show">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/percp.html" class="sidebar-item-text sidebar-link">
@@ -340,7 +338,7 @@
             <i class="bi bi-chevron-right ms-2"></i>
           </a> 
       </div>
-      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth2 ">  
+      <ul id="quarto-sidebar-section-10" class="collapse list-unstyled sidebar-section depth3 ">  
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../pages/bag.html" class="sidebar-item-text sidebar-link">
@@ -355,12 +353,66 @@
 </li>
       </ul>
   </li>
+      </ul>
+  </li>
+      </ul>
+  </li>
+        <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false">
+ <span class="menu-text">Deep Learning</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-11" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-11" class="collapse list-unstyled sidebar-section depth1 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/notations_nn .html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Notations</span></a>
+  </div>
+</li>
+          <li class="sidebar-item sidebar-item-section">
+      <div class="sidebar-item-container"> 
+            <a class="sidebar-item-text sidebar-link text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false">
+ <span class="menu-text">Neural Networks</span></a>
+          <a class="sidebar-item-toggle text-start collapsed" data-bs-toggle="collapse" data-bs-target="#quarto-sidebar-section-12" aria-expanded="false" aria-label="Toggle section">
+            <i class="bi bi-chevron-right ms-2"></i>
+          </a> 
+      </div>
+      <ul id="quarto-sidebar-section-12" class="collapse list-unstyled sidebar-section depth2 ">  
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/nn_basics.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Introduction</span></a>
+  </div>
+</li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../pages/ann.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Artificial neural networks (ANNs)</span></a>
+  <a href="../pages/activations.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Activations</span></a>
   </div>
 </li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/ffn.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Forward Propagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/backprop.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Backpropagation</span></a>
+  </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../pages/gradient.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Gradient Descent</span></a>
+  </div>
+</li>
+      </ul>
+  </li>
       </ul>
   </li>
         <li class="sidebar-item">
diff --git a/docs/search.json b/docs/search.json
index 76b762f..8709c3a 100644
--- a/docs/search.json
+++ b/docs/search.json
@@ -1,4 +1,25 @@
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+    "objectID": "pages/activations.html",
+    "href": "pages/activations.html",
+    "title": "Activation Functions in Neural Networks",
+    "section": "",
+    "text": "Activation functions play a crucial role in introducing non-linearity to neural networks, enabling them to model complex relationships and learn intricate patterns from the data. Here are common activation functions used in neural networks:\n\n\n\n\n\n\n\n\n\nActivation\nFormula\nGraph\nDescription\n\n\n\n\nSigmoid\n\\(a = \\frac{1}{1 + e^{-z}}\\)\n\nThe sigmoid function produces outputs between 0 and 1, resembling an S-shaped curve. It’s suitable for binary classification tasks where the output needs to be in the range [0, 1].\n\n\nTanh\n\\(a = \\frac{e^{z} - e^{-z}}{e^{z} + e^{-z}}\\)\n\nTanh, or hyperbolic tangent, produces outputs between -1 and 1. It is similar to the sigmoid function but with outputs centered around zero, making it suitable for hidden layers.\n\n\nReLU (Rectified Linear Unit)\n\\(a = \\max(0, z)\\)\n\nReLU is a simple and widely used activation function that outputs zero for negative inputs and linearly increases for positive inputs, efficiently mitigating the vanishing gradient problem.\n\n\nLeaky ReLU\n\\(a = \\max(0.01z, z)\\)\n\nLeaky ReLU is an improved version of ReLU, allowing a small gradient for negative inputs to address the “dying ReLU” problem encountered with the standard ReLU function.\n\n\n\n\nImportance of Non-linear Activation Functions:\nNeural networks with non-linear activation functions can approximate complex functions and relationships between inputs and outputs. Linear activation functions result in neural networks that are essentially just linear transformations, limiting their ability to capture intricate patterns present in real-world data.\nA detailed explanation on Stack Overflow provides insights into the necessity of non-linear activation functions in enabling neural networks to learn complex mappings.\n\n\nDerivatives of Activation Functions:\nThe derivatives of activation functions are essential for backpropagation during the training of neural networks. Here are the derivatives of common activation functions:\n\n\n\nActivation\nFormula\nDerivative\n\n\n\n\nSigmoid\n\\(a = \\frac{1}{1 + e^{-z}}\\)\n\\(a(1 - a)\\)\n\n\nTanh\n\\(a = \\frac{e^{z} - e^{-z}}{e^{z} + e^{-z}}\\)\n\\(1 - a^2\\)\n\n\nReLU\n\\(a = \\max(0, z)\\)\n0 if \\(z &lt; 0\\); 1 if \\(z \\geq 0\\)\n\n\nLeaky ReLU\n\\(a = \\max(0.01z, z)\\)\n0.01 if \\(z &lt; 0\\); 1 if \\(z \\geq 0\\)\n\n\n\nUnderstanding these derivatives is crucial in computing gradients during backpropagation, facilitating the adjustment of weights and biases for effective training of neural networks.\nThese diverse activation functions empower neural networks to handle various types of data and learn complex relationships, contributing to their adaptability and efficiency in modeling intricate patterns within datasets."
+  },
+  {
+    "objectID": "pages/gradient.html",
+    "href": "pages/gradient.html",
+    "title": "Gradient Descent",
+    "section": "",
+    "text": "Optimizing Model Parameters: Gradient Descent is used to update the parameters (weights and biases) of a neural network in the direction that minimizes the loss function.\nGeneral Gradient Descent Rule: The general update rule for a parameter \\(\\theta\\) using Gradient Descent is: \\[\\theta = \\theta - \\alpha \\frac{\\partial J }{ \\partial \\theta }\\] where \\(\\alpha\\) is the learning rate and \\(\\frac{\\partial J }{ \\partial \\theta }\\) denotes the partial derivative of the cost function \\(J\\) with respect to the parameter \\(\\theta\\).\nParameter Update for Weights and Biases:\n\nOutput Layer:\n\nUpdate the weights and biases of the output layer using the gradients calculated during backpropagation:\n\n\\(W^{[L]} = W^{[L]} - \\alpha \\frac{\\partial J }{ \\partial W^{[L]} }\\)\n\\(b^{[L]} = b^{[L]} - \\alpha \\frac{\\partial J }{ \\partial b^{[L]} }\\)\n\n\nHidden Layers:\n\nUpdate the weights and biases of the hidden layers using the gradients computed during backpropagation:\n\nFor \\(l\\)th hidden layer:\n\n\\(W^{[l]} = W^{[l]} - \\alpha \\frac{\\partial J }{ \\partial W^{[l]} }\\)\n\\(b^{[l]} = b^{[l]} - \\alpha \\frac{\\partial J }{ \\partial b^{[l]} }\\)\n\n\n\n\nImpact of Learning Rate (\\(\\alpha\\)):\n\nThe learning rate \\(\\alpha\\) determines the step size during parameter updates.\nA larger \\(\\alpha\\) might lead to faster convergence but can overshoot the minimum.\nA smaller \\(\\alpha\\) might converge slowly but can be more precise.\n\nIteration and Convergence:\n\nIterate through multiple epochs (iterations) of forward propagation, backpropagation, and parameter updates until the model’s performance converges or the predefined number of epochs is reached.\nMonitor the decrease in the cost function to assess convergence.\n\nImportance of Gradient Descent:\n\nGradient Descent is crucial in adjusting model parameters to minimize the loss function, thereby enhancing the accuracy of predictions in a neural network.\nIt enables the network to learn from the data by iteratively updating weights and biases to improve the overall performance of the model.\n\n\n\n\n\n\n\nNote\n\n\n\n\n\nimport numpy as np\n\ndef compute_cost(A2, Y):\n    \"\"\"\n    Computes the cross-entropy cost given in equation (13)\n    \n    Arguments:\n    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)\n    Y -- \"true\" labels vector of shape (1, number of examples)\n\n    Returns:\n    cost -- cross-entropy cost given equation (13)\n    \n    \"\"\"\n    \n    m = Y.shape[1] # number of examples\n\n    logprobs = np.multiply(Y, np.log(A2)) + np.multiply((1-Y), np.log(1-A2))\n    cost = - np.sum(logprobs)/Y.shape[1]\n        \n    cost = float(np.squeeze(cost)) \n\n    return cost\n\ndef update_parameters(parameters, grads, learning_rate = 1.2):\n    \"\"\"\n    Updates parameters using the gradient descent update rule given above\n    \n    Arguments:\n    parameters -- python dictionary containing your parameters \n    grads -- python dictionary containing your gradients \n    \n    Returns:\n    parameters -- python dictionary containing your updated parameters \n    \"\"\"\n    W1, b1, W2, b2 = copy.deepcopy(parameters[\"W1\"]), copy.deepcopy(parameters[\"b1\"]), copy.deepcopy(parameters[\"W2\"]), copy.deepcopy(parameters[\"b2\"])\n    \n    dW1, db1, dW2, db2 = grads[\"dW1\"], grads[\"db1\"], grads[\"dW2\"], grads[\"db2\"]\n    \n    W1 = W1 - learning_rate*dW1\n    b1 = b1 - learning_rate*db1\n    W2 = W2 - learning_rate*dW2\n    b2 = b2 - learning_rate*db2\n        \n    parameters = {\"W1\": W1,\n                  \"b1\": b1,\n                  \"W2\": W2,\n                  \"b2\": b2}\n    \n    return parameters\n\ndef nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):\n    \"\"\"\n    Arguments:\n    X -- dataset of shape (2, number of examples)\n    Y -- labels of shape (1, number of examples)\n    n_h -- size of the hidden layer\n    num_iterations -- Number of iterations in gradient descent loop\n    print_cost -- if True, print the cost every 1000 iterations\n    \n    Returns:\n    parameters -- parameters learnt by the model. They can then be used to predict.\n    \"\"\"\n    \n    np.random.seed(3)\n    n_x = layer_sizes(X, Y)[0]\n    n_y = layer_sizes(X, Y)[2]\n        \n    parameters = initialize_parameters(n_x, n_h, n_y)\n        \n    # Loop (gradient descent)\n\n    for i in range(0, num_iterations):\n         \n        A2, cache = forward_propagation(X, parameters)\n        cost = compute_cost(A2, Y)\n        grads = backward_propagation(parameters, cache, X, Y)\n        parameters = update_parameters(parameters, grads)\n                \n        # Print the cost every 1000 iterations\n        if print_cost and i % 1000 == 0:\n            print (\"Cost after iteration %i: %f\" %(i, cost))\n\n    return parameters"
+  },
+  {
+    "objectID": "pages/ffn.html",
+    "href": "pages/ffn.html",
+    "title": "Forward Propagation",
+    "section": "",
+    "text": "The forward pass in a neural network involves processing input data through the network’s layers to generate predictions or outputs by sequentially applying weights, biases, and activation functions.\nMathematical Representation:\nFor one training example \\(x^{(i)}\\):\n\nHidden Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[l] (i)} = W^{[l]} a^{[l-1](i)} + b^{[l]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[l-1](i)}\\)) by the weights (\\(W^{[l]}\\)) and add the bias (\\(b^{[l]}\\)) for layer \\(l\\).\n\n\nActivation Calculation:\n\n\\(a^{[l] (i)} = g(z^{[l] (i)})\\)\n\nApply an activation function \\(g\\) (such as sigmoid, tanh, ReLU, etc.) to the computed weighted sum to get the activation of layer \\(l\\).\n\n\n\nOutput Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[L] (i)} = W^{[L]} a^{[L-1] (i)} + b^{[L]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[L-1] (i)}\\)) by the weights (\\(W^{[L]}\\)) and add the bias (\\(b^{[L]}\\)) for the output layer.\n\n\nActivation Calculation:\n\n\\(\\hat{y}^{(i)} = a^{[L] (i)} = \\sigma(z^{ [L] (i)})\\)\n\nApply an appropriate activation function (e.g., softmax for multiclass classification, sigmoid for binary classification) to the computed weighted sum to obtain the predicted output (\\(\\hat{y}^{(i)}\\)).\n\n\n\nCost Function:\n\n\\(J = - \\frac{1}{m} \\sum\\limits_{i = 0}^{m} \\left(y^{(i)}\\log\\left(a^{[L] (i)}\\right) + (1-y^{(i)})\\log\\left(1- a^{[L] (i)}\\right)\\right)\\)\n\nCompute the appropriate cost function (e.g., cross-entropy, mean squared error) to evaluate the difference between the predicted output (\\(a^{[L] (i)}\\)) and the actual output (\\(y^{(i)}\\)) for all training examples (\\(m\\)).\n\n\n\n\n\n\n\n\n\nCode\n\n\n\n\n\nimport numpy as np\n\ndef forward_propagation(X, parameters):\n    \"\"\"\n    Argument:\n    X -- input data of size (n_x, m)\n    parameters -- python dictionary containing your parameters (output of initialization function)\n    \n    Returns:\n    A2 -- The sigmoid output of the second activation\n    cache -- a dictionary containing \"Z1\", \"A1\", \"Z2\" and \"A2\"\n    \"\"\"\n    W1, b1, W2, b2 = parameters[\"W1\"], parameters[\"b1\"], parameters[\"W2\"], parameters[\"b2\"]\n    \n    Z1 = W1@X + b1\n    A1 = np.tanh(Z1)\n    Z2 = W2@A1 + b2\n    A2 = 1/(1+np.exp(-Z2))\n    \n    cache = {\"Z1\": Z1,\n             \"A1\": A1,\n             \"Z2\": Z2,\n             \"A2\": A2}\n    \n    return A2, cache"
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@@ -182,11 +203,11 @@
     "text": "Dual Formulation\nMaximizing the Lagrangian function w.r.t. \\(\\alpha\\) and \\(\\beta\\), and minimizing it w.r.t. \\(w\\) and \\(\\epsilon\\), we get,\n\\[\n\\min _{w, \\epsilon}\\left [\\max _{\\alpha \\ge 0; \\beta \\ge 0}\\frac{1}{2}||w||^2_2 + C\\sum _{i=1} ^n\\epsilon_i + \\sum _{i=1} ^n \\alpha_i(1-(w^Tx_i)y_i - \\epsilon_i) + \\sum _{i=1} ^n \\beta(-\\epsilon_i) \\right ]\n\\]\nThe dual of this is given by,\n\\[\n\\max _{\\alpha \\ge 0; \\beta \\ge 0}\\left [\\min _{w, \\epsilon}\\frac{1}{2}||w||^2_2 + C\\sum _{i=1} ^n\\epsilon_i + \\sum _{i=1} ^n \\alpha_i(1-(w^Tx_i)y_i - \\epsilon_i) + \\sum _{i=1} ^n \\beta(-\\epsilon_i) \\right ]\n\\]\n\\[\n\\max _{\\alpha \\ge 0; \\beta \\ge 0}\\left [\\min _{w, \\epsilon}\\mathcal{L}(w, \\epsilon, \\alpha, \\beta) \\right ] \\quad \\ldots[1]\n\\]\nDifferentiating the above function\\([1]\\) w.r.t. \\(w\\) while fixing \\(\\alpha\\) and \\(\\beta\\), we get, \\[\n\\frac{d\\mathcal{L}}{dw}  = 0\n\\] \\[\n\\frac{d}{dw} \\frac{1}{2}||w||^2_2 + C\\sum _{i=1} ^n\\epsilon_i + \\sum _{i=1} ^n \\alpha_i(1-(w^Tx_i)y_i - \\epsilon_i) + \\sum _{i=1} ^n \\beta(-\\epsilon_i) = 0\\\\\n\\] \\[\nw_{\\alpha, \\beta}^* - \\alpha_ix_iy_i = 0\n\\] \\[\n\\therefore w_{\\alpha, \\beta}^* = \\alpha_ix_iy_i \\quad \\ldots [2]\n\\]\nDifferentiating the above function\\([1]\\) w.r.t. \\(\\epsilon_i \\forall i\\) while fixing \\(\\alpha\\) and \\(\\beta\\), we get,\n\\[\n\\frac{\\partial\\mathcal{L}}{\\partial\\epsilon_i}  = 0\n\\] \\[\n\\frac{\\partial}{\\partial\\epsilon_i} \\frac{1}{2}||w||^2_2 + C\\sum _{i=1} ^n\\epsilon_i + \\sum _{i=1} ^n \\alpha_i(1-(w^Tx_i)y_i - \\epsilon_i) + \\sum _{i=1} ^n \\beta(-\\epsilon_i) = 0\n\\] \\[\nC - \\alpha_i -\\beta_i = 0\n\\] \\[\n\\therefore C = \\alpha_i + \\beta_i \\quad \\ldots [3]\n\\]\nSubstituting the values of \\(w\\) and \\(\\beta\\) from \\([2]\\) and \\([3]\\) in \\([1]\\), we get,\n\\[\n\\max _{\\alpha \\ge 0; \\beta \\ge 0; C = \\alpha_i + \\beta_i}\\left [\\frac{1}{2}||\\alpha_ix_iy_i||^2_2 + C\\sum _{i=1} ^n\\epsilon_i + \\sum _{i=1} ^n \\alpha_i(1-((\\alpha_ix_iy_i)^Tx_i)y_i - \\epsilon_i) + \\sum _{i=1} ^n (C-\\alpha_i)(-\\epsilon_i) \\right ]\n\\] \\[\n\\max _{\\alpha \\ge 0; \\beta \\ge 0; C = \\alpha_i + \\beta_i}\\left [\\frac{1}{2}\\alpha_i^Tx_i^Ty_i^Ty_ix_i\\alpha_i + C\\sum _{i=1} ^n\\epsilon_i + \\sum _{i=1} ^n \\alpha_i-\\alpha_i^Tx_i^Ty_i^Ty_ix_i\\alpha_i - \\sum _{i=1} ^n \\alpha_i\\epsilon_i - C\\sum _{i=1} ^n\\epsilon_i + \\sum _{i=1} ^n \\alpha_i\\epsilon_i \\right ]\n\\] \\[\n\\max _{\\alpha \\ge 0; \\beta \\ge 0; C = \\alpha_i + \\beta_i}\\left [\\sum _{i=1} ^n \\alpha_i - \\frac{1}{2}\\alpha_i^Tx_i^Ty_i^Ty_ix_i\\alpha_i\\right ]\n\\] \\[\n\\therefore \\max _{0 \\le \\alpha \\le C}\\left [\\sum _{i=1} ^n \\alpha_i - \\frac{1}{2}\\alpha_i^Tx_i^Ty_i^Ty_ix_i\\alpha_i\\right ]\n\\]"
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+    "href": "about.html",
+    "title": "About Me",
     "section": "",
-    "text": "What is Machine Learning\nMachine learning (ML) is a field of computer science that gives computers the ability to learn without being explicitly programmed. ML algorithms are able to learn from data and make predictions or decisions without being explicitly told how to do so. This makes ML a powerful tool for automating tasks, making predictions, and uncovering patterns in data.\n\n\nWhy Machine Learning?\nThere are many reasons why machine learning is becoming increasingly popular. Some of the key benefits of ML include:\n\nAutomation: ML can be used to automate tasks that would otherwise require human intelligence. This can free up human workers to focus on more creative and strategic tasks.\nAccuracy: ML algorithms can often make more accurate predictions than humans. This is because ML algorithms can learn from large amounts of data and identify patterns that humans would not be able to see.\nScalability: ML algorithms can be scaled to handle large amounts of data. This makes them ideal for applications such as fraud detection, natural language processing, and image recognition.\n\n\n\nWhere is Machine Learning Used?\nMachine learning is used in a wide variety of fields, including:\n\nFinance: ML is used to predict stock prices, identify fraud, and manage risk.\nHealthcare: ML is used to diagnose diseases, develop new treatments, and personalize care.\nRetail: ML is used to personalize recommendations, predict demand, and prevent fraud.\nManufacturing: ML is used to optimize production, improve quality control, and predict maintenance needs.\nLogistics: ML is used to optimize shipping routes, manage inventory, and predict demand.\n\n\n\nTypes of Machine Learning\nThere are many different types of machine learning, each with its own strengths and weaknesses. Some of the most common types of machine learning include:\n\nSupervised learning: In supervised learning, the algorithm is trained on a dataset of labeled data. This means that the data includes both inputs and their corresponding outputs. The goal of supervised learning is to build a model that can accurately predict the output for new, unseen input data.\nUnsupervised learning: In unsupervised learning, the algorithm is trained on a dataset of unlabeled data. This means that the data only includes inputs and no corresponding outputs. The goal of unsupervised learning is to uncover patterns or relationships within the data without any prior knowledge or guidance.\nReinforcement learning: In reinforcement learning, the algorithm is trained by trial and error. The algorithm is given a goal and a set of actions that it can take. The algorithm then tries different actions and observes the results. Over time, the algorithm learns which actions are most likely to lead to the desired goal.\n\n\n\nThe Future of Machine Learning\nMachine learning is a rapidly growing field with the potential to revolutionize many industries. As ML algorithms become more powerful and sophisticated, they will be able to automate more tasks, make more accurate predictions, and uncover more patterns in data. This will lead to new and innovative applications in a wide variety of fields."
+    "text": "Hi there, I’m Sherry Thomas, and I’m passionate about making a difference in the field of data science. I have a Bachelor’s degree in Philosophy from Christ University, Bengaluru and I’m currently pursuing a Bachelor’s degree in Data Science and Applications at IIT-Madras.\nI have a diverse academic background that includes philosophy, programming, computer science, and statistics. My philosophy studies have helped me develop critical thinking and problem-solving skills, while my data science coursework has provided me with the technical expertise needed to tackle complex challenges.\nI believe in taking a holistic approach to data analysis and decision-making, considering the ethical and societal implications of technology and data. My goal is to use my knowledge and skills to make a positive impact in the field of data science.\nI’m constantly learning and growing, and I’m committed to expanding my knowledge base and contributing to the ever-evolving world of technology and data."
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     "text": "Credits\nI would like to express my gratitude to Professor Arun Rajkumar for his valuable content and notations, which have greatly influenced my understanding. Additionally, I would like to acknowledge the contribution of IIT Madras, where I had the opportunity to learn from Professor Arun Rajkumar in the Machine Learning Techniques Course."
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-    "text": "Hi there, I’m Sherry Thomas, and I’m passionate about making a difference in the field of data science. I have a Bachelor’s degree in Philosophy from Christ University, Bengaluru and I’m currently pursuing a Bachelor’s degree in Data Science and Applications at IIT-Madras.\nI have a diverse academic background that includes philosophy, programming, computer science, and statistics. My philosophy studies have helped me develop critical thinking and problem-solving skills, while my data science coursework has provided me with the technical expertise needed to tackle complex challenges.\nI believe in taking a holistic approach to data analysis and decision-making, considering the ethical and societal implications of technology and data. My goal is to use my knowledge and skills to make a positive impact in the field of data science.\nI’m constantly learning and growing, and I’m committed to expanding my knowledge base and contributing to the ever-evolving world of technology and data."
+    "text": "What is Machine Learning\nMachine learning (ML) is a field of computer science that gives computers the ability to learn without being explicitly programmed. ML algorithms are able to learn from data and make predictions or decisions without being explicitly told how to do so. This makes ML a powerful tool for automating tasks, making predictions, and uncovering patterns in data.\n\n\nWhy Machine Learning?\nThere are many reasons why machine learning is becoming increasingly popular. Some of the key benefits of ML include:\n\nAutomation: ML can be used to automate tasks that would otherwise require human intelligence. This can free up human workers to focus on more creative and strategic tasks.\nAccuracy: ML algorithms can often make more accurate predictions than humans. This is because ML algorithms can learn from large amounts of data and identify patterns that humans would not be able to see.\nScalability: ML algorithms can be scaled to handle large amounts of data. This makes them ideal for applications such as fraud detection, natural language processing, and image recognition.\n\n\n\nWhere is Machine Learning Used?\nMachine learning is used in a wide variety of fields, including:\n\nFinance: ML is used to predict stock prices, identify fraud, and manage risk.\nHealthcare: ML is used to diagnose diseases, develop new treatments, and personalize care.\nRetail: ML is used to personalize recommendations, predict demand, and prevent fraud.\nManufacturing: ML is used to optimize production, improve quality control, and predict maintenance needs.\nLogistics: ML is used to optimize shipping routes, manage inventory, and predict demand.\n\n\n\nTypes of Machine Learning\nThere are many different types of machine learning, each with its own strengths and weaknesses. Some of the most common types of machine learning include:\n\nSupervised learning: In supervised learning, the algorithm is trained on a dataset of labeled data. This means that the data includes both inputs and their corresponding outputs. The goal of supervised learning is to build a model that can accurately predict the output for new, unseen input data.\nUnsupervised learning: In unsupervised learning, the algorithm is trained on a dataset of unlabeled data. This means that the data only includes inputs and no corresponding outputs. The goal of unsupervised learning is to uncover patterns or relationships within the data without any prior knowledge or guidance.\nReinforcement learning: In reinforcement learning, the algorithm is trained by trial and error. The algorithm is given a goal and a set of actions that it can take. The algorithm then tries different actions and observes the results. Over time, the algorithm learns which actions are most likely to lead to the desired goal.\n\n\n\nThe Future of Machine Learning\nMachine learning is a rapidly growing field with the potential to revolutionize many industries. As ML algorithms become more powerful and sophisticated, they will be able to automate more tasks, make more accurate predictions, and uncover more patterns in data. This will lead to new and innovative applications in a wide variety of fields."
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     "text": "We introduce three additional assumptions:\n\nLinear Separability with \\(\\gamma\\)-Margin: A dataset \\(D=\\{(\\mathbf{x}_1, y_1), \\ldots, (\\mathbf{x}_n,y_n)\\}\\) is considered linearly separable with a \\(\\gamma\\)-margin if there exists \\(\\mathbf{w}^* \\in \\mathbb{R}^d\\) such that \\((\\mathbf{w}^{*T}\\mathbf{x}_i)y_i\\geq\\gamma\\) holds for all \\(i\\), where \\(\\gamma&gt;0\\).\n\n\n\n\nLinear Separability with \\(\\gamma\\)-Margin\n\n\n\nRadius Assumption: Let \\(R&gt;0 \\in \\mathbb{R}\\) be a constant such that \\(\\forall i \\in D\\), \\(||\\mathbf{x}_i||\\leq R\\). In other words, \\(R\\) denotes the length of the data point farthest from the center.\nNormal Length for \\(\\mathbf{w}^*\\): Assume that \\(\\mathbf{w}^*\\) has unit length."
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-    "text": "Neural Network Architecture: A shallow neural network typically consists of an input layer, one or more hidden layers, and an output layer."
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-    "text": "Forward Propogation\nThe model:\n\nMathematically:\nFor one example \\(x^{(i)}\\): \\[z^{[1] (i)} =  W^{[1]} x^{(i)} + b^{[1]}\\tag{1}\\] \\[a^{[1] (i)} = \\tanh(z^{[1] (i)})\\tag{2}\\] \\[z^{[2] (i)} = W^{[2]} a^{[1] (i)} + b^{[2]}\\tag{3}\\] \\[\\hat{y}^{(i)} = a^{[2] (i)} = \\sigma(z^{ [2] (i)})\\tag{4}\\] \\[y^{(i)}_{prediction} = \\begin{cases} 1 & \\mbox{if } a^{[2](i)} &gt; 0.5 \\\\ 0 & \\mbox{otherwise } \\end{cases}\\tag{5}\\]\nGiven the predictions on all the examples, you can also compute the cost \\(J\\) as follows: \\[J = - \\frac{1}{m} \\sum\\limits_{i = 0}^{m} \\large\\left(\\small y^{(i)}\\log\\left(a^{[2] (i)}\\right) + (1-y^{(i)})\\log\\left(1- a^{[2] (i)}\\right)  \\large  \\right) \\small \\tag{6}\\]"
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-    "text": "Backpropogation\n\nCalculate the Error in the Output Layer:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{2}^{(i)} } = \\frac{1}{m} (a^{[2](i)} - y^{(i)})\\)\n\nThis computes the error derivative with respect to the pre-activation value of the output layer for each training example.\n\\(a^{[2](i)}\\) is the predicted output, and \\(y^{(i)}\\) is the actual output for the \\(i\\)-th example.\n\\(m\\) is the number of examples.\n\n\nCompute Gradients for Output Layer Weights and Bias:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W_2 } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{2}^{(i)} } a^{[1] (i) T}\\)\n\nCalculates the gradient of the loss function with respect to the weights of the output layer.\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b_2 } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{2}^{(i)}}}\\)\n\nComputes the gradient of the loss function with respect to the bias of the output layer.\n\n\nBackpropagate the Error to the Previous Layer:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{1}^{(i)} } = W_2^T \\frac{\\partial \\mathcal{J} }{ \\partial z_{2}^{(i)} } * ( 1 - a^{[1] (i) 2})\\)\n\nPropagates the error derivative backward to the hidden layer, considering the relationship between the output and hidden layers.\nInvolves the transpose of the weights connecting the hidden and output layers.\nInvolves the derivative of the activation function for the hidden layer.\n\n\nCompute Gradients for Hidden Layer Weights and Bias:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W_1 } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{1}^{(i)} } X^T\\)\n\nCalculates the gradient of the loss function with respect to the weights of the hidden layer.\n\n\\(\\frac{\\partial \\mathcal{J} _i }{ \\partial b_1 } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{1}^{(i)}}}\\)\n\nComputes the gradient of the loss function with respect to the bias of the hidden layer.\n\n\n\nThese equations represent the fundamental steps of backpropagation in a neural network. They’re used to update the weights and biases during the training process by iteratively adjusting them in the direction that minimizes the overall loss function.\n\n\n\n\n\n\nCode\n\n\n\n\n\nimport numpy as np\n\ndef forward_propagation(X, parameters):\n    \"\"\"\n    Argument:\n    X -- input data of size (n_x, m)\n    parameters -- python dictionary containing your parameters (output of initialization function)\n    \n    Returns:\n    A2 -- The sigmoid output of the second activation\n    cache -- a dictionary containing \"Z1\", \"A1\", \"Z2\" and \"A2\"\n    \"\"\"\n    W1, b1, W2, b2 = parameters[\"W1\"], parameters[\"b1\"], parameters[\"W2\"], parameters[\"b2\"]\n    \n    Z1 = W1@X + b1\n    A1 = np.tanh(Z1)\n    Z2 = W2@A1 + b2\n    A2 = 1/(1+np.exp(-Z2))\n    \n    cache = {\"Z1\": Z1,\n             \"A1\": A1,\n             \"Z2\": Z2,\n             \"A2\": A2}\n    \n    return A2, cache\n\ndef compute_cost(A2, Y):\n    \"\"\"\n    Computes the cross-entropy cost given in equation (13)\n    \n    Arguments:\n    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)\n    Y -- \"true\" labels vector of shape (1, number of examples)\n\n    Returns:\n    cost -- cross-entropy cost given equation (13)\n    \n    \"\"\"\n    \n    m = Y.shape[1] # number of examples\n\n    logprobs = np.multiply(Y, np.log(A2)) + np.multiply((1-Y), np.log(1-A2))\n    cost = - np.sum(logprobs)/Y.shape[1]\n        \n    cost = float(np.squeeze(cost)) \n\n    return cost"
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-    "text": "Forward Propagation\nThe forward pass in a neural network involves processing input data through the network’s layers to generate predictions or outputs by sequentially applying weights, biases, and activation functions.\nMathematical Representation:\nFor one training example \\(x^{(i)}\\):\n\nHidden Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[l] (i)} = W^{[l]} a^{[l-1](i)} + b^{[l]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[l-1](i)}\\)) by the weights (\\(W^{[l]}\\)) and add the bias (\\(b^{[l]}\\)) for layer \\(l\\).\n\n\nActivation Calculation:\n\n\\(a^{[l] (i)} = g(z^{[l] (i)})\\)\n\nApply an activation function \\(g\\) (such as sigmoid, tanh, ReLU, etc.) to the computed weighted sum to get the activation of layer \\(l\\).\n\n\n\nOutput Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[L] (i)} = W^{[L]} a^{[L-1] (i)} + b^{[L]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[L-1] (i)}\\)) by the weights (\\(W^{[L]}\\)) and add the bias (\\(b^{[L]}\\)) for the output layer.\n\n\nActivation Calculation:\n\n\\(\\hat{y}^{(i)} = a^{[L] (i)} = \\sigma(z^{ [L] (i)})\\)\n\nApply an appropriate activation function (e.g., softmax for multiclass classification, sigmoid for binary classification) to the computed weighted sum to obtain the predicted output (\\(\\hat{y}^{(i)}\\)).\n\n\n\nCost Function:\n\n\\(J = - \\frac{1}{m} \\sum\\limits_{i = 0}^{m} \\left(y^{(i)}\\log\\left(a^{[L] (i)}\\right) + (1-y^{(i)})\\log\\left(1- a^{[L] (i)}\\right)\\right)\\)\n\nCompute the appropriate cost function (e.g., cross-entropy, mean squared error) to evaluate the difference between the predicted output (\\(a^{[L] (i)}\\)) and the actual output (\\(y^{(i)}\\)) for all training examples (\\(m\\))."
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-    "text": "Backpropagation\nBackpropagation is the process of iteratively computing gradients of the loss function with respect to the weights and biases of the neural network, enabling the adjustment of parameters by propagating errors backward through the network’s layers to enhance learning and optimize performance.\n\nOutput Layer Error Derivative:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)} } = \\frac{1}{m} (\\hat{y}^{(i)} - y^{(i)}) \\odot g'(z^{[L] (i)})\\)\n\nCalculate the derivative of the cost function with respect to the pre-activation value of the output layer for each training example.\n\\(\\odot\\) denotes element-wise multiplication.\n\\(g'(\\cdot)\\) represents the derivative of the activation function used in the output layer.\n\n\nGradient Calculation for Output Layer Weights and Bias:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W^{[L]} } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)} } a^{[L-1] (i) T}\\)\n\nCompute the gradient of the loss function with respect to the weights of the output layer.\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b^{[L]} } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)}}}\\)\n\nCompute the gradient of the loss function with respect to the bias of the output layer.\n\n\nBackpropagate the Error to Previous Layers:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)} } = W^{[l+1] T} \\frac{\\partial \\mathcal{J} }{ \\partial z_{l+1}^{(i)} } \\odot g'(z^{[l] (i)})\\)\n\nPropagate the error derivative backward through the layers by considering the relationship between consecutive layers.\nInvolves the transpose of weights connecting the subsequent and current layers.\nInvolves the derivative of the activation function used in layer \\(l\\).\n\n\nGradient Calculation for Weights and Bias of Hidden Layers:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W^{[l]} } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)} } a^{[l-1] (i) T}\\)\n\nCompute the gradient of the loss function with respect to the weights of layer \\(l\\).\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b^{[l]} } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)}}}\\)\n\nCompute the gradient of the loss function with respect to the bias of layer \\(l\\).\n\n\n\nThese steps form the core of backpropagation in a shallow neural network, allowing for the iterative adjustment of weights and biases to minimize the loss function and improve the model’s accuracy in predicting outputs for given inputs."
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     "text": "Superscript and Subscript Indices:\n\n\\([i]\\): Represents the index referring to a specific training example within the dataset.\n\\([l]\\): Denotes the layer indices. In a network with \\(L\\) layers, \\([l]\\) ranges from \\(1\\) to \\(L\\).\n\nVariables:\n\n\\(x^{(i)}\\): Input features for the \\(i\\)-th training example.\n\\(y^{(i)}\\): Actual output or target for the \\(i\\)-th training example.\n\\(\\hat{y}^{(i)}\\), \\(a^{[l](i)}\\): Predicted output or activation of layer \\(l\\) for the \\(i\\)-th training example.\n\\(z^{[l](i)}\\): Pre-activation value of layer \\(l\\) for the \\(i\\)-th training example.\n\nWeights and Biases:\n\n\\(W^{[l]}\\): Weight matrix associated with the connections between layer \\(l - 1\\) and layer \\(l\\).\n\\(b^{[l]}\\): Bias vector added to the weighted sum in layer \\(l\\).\n\nDerivatives and Gradients:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z^{[l](i)} }\\): Derivative of the cost function with respect to the pre-activation value of layer \\(l\\) for the \\(i\\)-th training example.\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W^{[l]} }\\): Gradient of the loss function with respect to the weights of layer \\(l\\).\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b^{[l]} }\\): Gradient of the loss function with respect to the bias of layer \\(l\\).\n\nCost Function:\n\n\\(J\\): Overall cost function that measures the discrepancy between predicted and actual outputs across all training examples.\n\\(m\\): Number of training examples in the dataset.\n\nActivation Functions:\n\n\\(\\sigma()\\): Activation function applied element-wise to the weighted sums in each layer.\n\nMathematical Operations:\n\n\\(\\log()\\): Natural logarithm function used in the calculation of the cost function.\n\\(\\odot\\): Element-wise multiplication.\n\n\nThese notations provide a general framework for understanding and representing variables, derivatives, weights, biases, and mathematical operations involved in the computations and optimizations of neural networks across different architectures and configurations."
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-    "text": "Code\n\n\n\n\n\n\nNote\n\n\n\n\n\nimport numpy as np\n\ndef forward_propagation(X, parameters):\n    \"\"\"\n    Argument:\n    X -- input data of size (n_x, m)\n    parameters -- python dictionary containing your parameters (output of initialization function)\n    \n    Returns:\n    A2 -- The sigmoid output of the second activation\n    cache -- a dictionary containing \"Z1\", \"A1\", \"Z2\" and \"A2\"\n    \"\"\"\n    W1, b1, W2, b2 = parameters[\"W1\"], parameters[\"b1\"], parameters[\"W2\"], parameters[\"b2\"]\n    \n    Z1 = W1@X + b1\n    A1 = np.tanh(Z1)\n    Z2 = W2@A1 + b2\n    A2 = 1/(1+np.exp(-Z2))\n    \n    cache = {\"Z1\": Z1,\n             \"A1\": A1,\n             \"Z2\": Z2,\n             \"A2\": A2}\n    \n    return A2, cache\n\ndef compute_cost(A2, Y):\n    \"\"\"\n    Computes the cross-entropy cost given in equation (13)\n    \n    Arguments:\n    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)\n    Y -- \"true\" labels vector of shape (1, number of examples)\n\n    Returns:\n    cost -- cross-entropy cost given equation (13)\n    \n    \"\"\"\n    \n    m = Y.shape[1] # number of examples\n\n    logprobs = np.multiply(Y, np.log(A2)) + np.multiply((1-Y), np.log(1-A2))\n    cost = - np.sum(logprobs)/Y.shape[1]\n        \n    cost = float(np.squeeze(cost)) \n\n    return cost\n\ndef backward_propagation(parameters, cache, X, Y):\n    \"\"\"\n    Implement the backward propagation using the instructions above.\n    \n    Arguments:\n    parameters -- python dictionary containing our parameters \n    cache -- a dictionary containing \"Z1\", \"A1\", \"Z2\" and \"A2\".\n    X -- input data of shape (2, number of examples)\n    Y -- \"true\" labels vector of shape (1, number of examples)\n    \n    Returns:\n    grads -- python dictionary containing your gradients with respect to different parameters\n    \"\"\"\n    m = X.shape[1]\n    \n    W1, W2 = parameters[\"W1\"], parameters[\"W2\"]\n    \n    A1, A2 = cache[\"A1\"], cache[\"A2\"]\n        \n    dZ2 = A2 - Y\n    dW2 = (1/Y.shape[1])*dZ2@A1.T\n    db2 = (1/Y.shape[1])*np.sum(dZ2, axis=1, keepdims=True)\n    dZ1 = W2.T@dZ2*(1-A1**2)\n    dW1 = (1/Y.shape[1])*dZ1@X.T\n    db1 = (1/Y.shape[1])*np.sum(dZ1, axis=1, keepdims=True)\n        \n    grads = {\"dW1\": dW1,\n             \"db1\": db1,\n             \"dW2\": dW2,\n             \"db2\": db2}\n    \n    return grads"
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-    "text": "Gradient Descent in Neural Networks:\nOptimizing Model Parameters: Gradient Descent is used to update the parameters (weights and biases) of a neural network in the direction that minimizes the loss function.\nGeneral Gradient Descent Rule: The general update rule for a parameter \\(\\theta\\) using Gradient Descent is: \\[\\theta = \\theta - \\alpha \\frac{\\partial J }{ \\partial \\theta }\\] where \\(\\alpha\\) is the learning rate and \\(\\frac{\\partial J }{ \\partial \\theta }\\) denotes the partial derivative of the cost function \\(J\\) with respect to the parameter \\(\\theta\\).\nParameter Update for Weights and Biases:\n\nOutput Layer:\n\nUpdate the weights and biases of the output layer using the gradients calculated during backpropagation:\n\n\\(W^{[L]} = W^{[L]} - \\alpha \\frac{\\partial J }{ \\partial W^{[L]} }\\)\n\\(b^{[L]} = b^{[L]} - \\alpha \\frac{\\partial J }{ \\partial b^{[L]} }\\)\n\n\nHidden Layers:\n\nUpdate the weights and biases of the hidden layers using the gradients computed during backpropagation:\n\nFor \\(l\\)th hidden layer:\n\n\\(W^{[l]} = W^{[l]} - \\alpha \\frac{\\partial J }{ \\partial W^{[l]} }\\)\n\\(b^{[l]} = b^{[l]} - \\alpha \\frac{\\partial J }{ \\partial b^{[l]} }\\)\n\n\n\n\nImpact of Learning Rate (\\(\\alpha\\)): - The learning rate \\(\\alpha\\) determines the step size during parameter updates. - A larger \\(\\alpha\\) might lead to faster convergence but can overshoot the minimum. - A smaller \\(\\alpha\\) might converge slowly but can be more precise.\nIteration and Convergence: - Iterate through multiple epochs (iterations) of forward propagation, backpropagation, and parameter updates until the model’s performance converges or the predefined number of epochs is reached. - Monitor the decrease in the cost function to assess convergence.\nImportance of Gradient Descent: - Gradient Descent is crucial in adjusting model parameters to minimize the loss function, thereby enhancing the accuracy of predictions in a neural network. - It enables the network to learn from the data by iteratively updating weights and biases to improve the overall performance of the model."
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-    "text": "Gradient Descent\nOptimizing Model Parameters: Gradient Descent is used to update the parameters (weights and biases) of a neural network in the direction that minimizes the loss function.\nGeneral Gradient Descent Rule: The general update rule for a parameter \\(\\theta\\) using Gradient Descent is: \\[\\theta = \\theta - \\alpha \\frac{\\partial J }{ \\partial \\theta }\\] where \\(\\alpha\\) is the learning rate and \\(\\frac{\\partial J }{ \\partial \\theta }\\) denotes the partial derivative of the cost function \\(J\\) with respect to the parameter \\(\\theta\\).\nParameter Update for Weights and Biases:\n\nOutput Layer:\n\nUpdate the weights and biases of the output layer using the gradients calculated during backpropagation:\n\n\\(W^{[L]} = W^{[L]} - \\alpha \\frac{\\partial J }{ \\partial W^{[L]} }\\)\n\\(b^{[L]} = b^{[L]} - \\alpha \\frac{\\partial J }{ \\partial b^{[L]} }\\)\n\n\nHidden Layers:\n\nUpdate the weights and biases of the hidden layers using the gradients computed during backpropagation:\n\nFor \\(l\\)th hidden layer:\n\n\\(W^{[l]} = W^{[l]} - \\alpha \\frac{\\partial J }{ \\partial W^{[l]} }\\)\n\\(b^{[l]} = b^{[l]} - \\alpha \\frac{\\partial J }{ \\partial b^{[l]} }\\)\n\n\n\n\nImpact of Learning Rate (\\(\\alpha\\)): - The learning rate \\(\\alpha\\) determines the step size during parameter updates. - A larger \\(\\alpha\\) might lead to faster convergence but can overshoot the minimum. - A smaller \\(\\alpha\\) might converge slowly but can be more precise.\nIteration and Convergence: - Iterate through multiple epochs (iterations) of forward propagation, backpropagation, and parameter updates until the model’s performance converges or the predefined number of epochs is reached. - Monitor the decrease in the cost function to assess convergence.\nImportance of Gradient Descent: - Gradient Descent is crucial in adjusting model parameters to minimize the loss function, thereby enhancing the accuracy of predictions in a neural network. - It enables the network to learn from the data by iteratively updating weights and biases to improve the overall performance of the model."
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-    "text": "The forward pass in a neural network involves processing input data through the network’s layers to generate predictions or outputs by sequentially applying weights, biases, and activation functions.\nMathematical Representation:\nFor one training example \\(x^{(i)}\\):\n\nHidden Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[l] (i)} = W^{[l]} a^{[l-1](i)} + b^{[l]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[l-1](i)}\\)) by the weights (\\(W^{[l]}\\)) and add the bias (\\(b^{[l]}\\)) for layer \\(l\\).\n\n\nActivation Calculation:\n\n\\(a^{[l] (i)} = g(z^{[l] (i)})\\)\n\nApply an activation function \\(g\\) (such as sigmoid, tanh, ReLU, etc.) to the computed weighted sum to get the activation of layer \\(l\\).\n\n\n\nOutput Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[L] (i)} = W^{[L]} a^{[L-1] (i)} + b^{[L]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[L-1] (i)}\\)) by the weights (\\(W^{[L]}\\)) and add the bias (\\(b^{[L]}\\)) for the output layer.\n\n\nActivation Calculation:\n\n\\(\\hat{y}^{(i)} = a^{[L] (i)} = \\sigma(z^{ [L] (i)})\\)\n\nApply an appropriate activation function (e.g., softmax for multiclass classification, sigmoid for binary classification) to the computed weighted sum to obtain the predicted output (\\(\\hat{y}^{(i)}\\)).\n\n\n\nCost Function:\n\n\\(J = - \\frac{1}{m} \\sum\\limits_{i = 0}^{m} \\left(y^{(i)}\\log\\left(a^{[L] (i)}\\right) + (1-y^{(i)})\\log\\left(1- a^{[L] (i)}\\right)\\right)\\)\n\nCompute the appropriate cost function (e.g., cross-entropy, mean squared error) to evaluate the difference between the predicted output (\\(a^{[L] (i)}\\)) and the actual output (\\(y^{(i)}\\)) for all training examples (\\(m\\)).\n\n\n\n\n\n\n\n\n\nCode\n\n\n\n\n\nimport numpy as np\n\ndef forward_propagation(X, parameters):\n    \"\"\"\n    Argument:\n    X -- input data of size (n_x, m)\n    parameters -- python dictionary containing your parameters (output of initialization function)\n    \n    Returns:\n    A2 -- The sigmoid output of the second activation\n    cache -- a dictionary containing \"Z1\", \"A1\", \"Z2\" and \"A2\"\n    \"\"\"\n    W1, b1, W2, b2 = parameters[\"W1\"], parameters[\"b1\"], parameters[\"W2\"], parameters[\"b2\"]\n    \n    Z1 = W1@X + b1\n    A1 = np.tanh(Z1)\n    Z2 = W2@A1 + b2\n    A2 = 1/(1+np.exp(-Z2))\n    \n    cache = {\"Z1\": Z1,\n             \"A1\": A1,\n             \"Z2\": Z2,\n             \"A2\": A2}\n    \n    return A2, cache"
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-    "text": "The forward pass in a neural network involves processing input data through the network’s layers to generate predictions or outputs by sequentially applying weights, biases, and activation functions.\nMathematical Representation:\nFor one training example \\(x^{(i)}\\):\n\nHidden Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[l] (i)} = W^{[l]} a^{[l-1](i)} + b^{[l]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[l-1](i)}\\)) by the weights (\\(W^{[l]}\\)) and add the bias (\\(b^{[l]}\\)) for layer \\(l\\).\n\n\nActivation Calculation:\n\n\\(a^{[l] (i)} = g(z^{[l] (i)})\\)\n\nApply an activation function \\(g\\) (such as sigmoid, tanh, ReLU, etc.) to the computed weighted sum to get the activation of layer \\(l\\).\n\n\n\nOutput Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[L] (i)} = W^{[L]} a^{[L-1] (i)} + b^{[L]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[L-1] (i)}\\)) by the weights (\\(W^{[L]}\\)) and add the bias (\\(b^{[L]}\\)) for the output layer.\n\n\nActivation Calculation:\n\n\\(\\hat{y}^{(i)} = a^{[L] (i)} = \\sigma(z^{ [L] (i)})\\)\n\nApply an appropriate activation function (e.g., softmax for multiclass classification, sigmoid for binary classification) to the computed weighted sum to obtain the predicted output (\\(\\hat{y}^{(i)}\\)).\n\n\n\nCost Function:\n\n\\(J = - \\frac{1}{m} \\sum\\limits_{i = 0}^{m} \\left(y^{(i)}\\log\\left(a^{[L] (i)}\\right) + (1-y^{(i)})\\log\\left(1- a^{[L] (i)}\\right)\\right)\\)\n\nCompute the appropriate cost function (e.g., cross-entropy, mean squared error) to evaluate the difference between the predicted output (\\(a^{[L] (i)}\\)) and the actual output (\\(y^{(i)}\\)) for all training examples (\\(m\\))."
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-    "text": "Code\n\n\n\n\n\n\nCode\n\n\n\n\n\nimport numpy as np\n\ndef forward_propagation(X, parameters):\n    \"\"\"\n    Argument:\n    X -- input data of size (n_x, m)\n    parameters -- python dictionary containing your parameters (output of initialization function)\n    \n    Returns:\n    A2 -- The sigmoid output of the second activation\n    cache -- a dictionary containing \"Z1\", \"A1\", \"Z2\" and \"A2\"\n    \"\"\"\n    W1, b1, W2, b2 = parameters[\"W1\"], parameters[\"b1\"], parameters[\"W2\"], parameters[\"b2\"]\n    \n    Z1 = W1@X + b1\n    A1 = np.tanh(Z1)\n    Z2 = W2@A1 + b2\n    A2 = 1/(1+np.exp(-Z2))\n    \n    cache = {\"Z1\": Z1,\n             \"A1\": A1,\n             \"Z2\": Z2,\n             \"A2\": A2}\n    \n    return A2, cache"
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     "text": "Backpropagation is the process of iteratively computing gradients of the loss function with respect to the weights and biases of the neural network, enabling the adjustment of parameters by propagating errors backward through the network’s layers to enhance learning and optimize performance.\n\nOutput Layer Error Derivative:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)} } = \\frac{1}{m} (\\hat{y}^{(i)} - y^{(i)}) \\odot g'(z^{[L] (i)})\\)\n\nCalculate the derivative of the cost function with respect to the pre-activation value of the output layer for each training example.\n\\(\\odot\\) denotes element-wise multiplication.\n\\(g'(\\cdot)\\) represents the derivative of the activation function used in the output layer.\n\n\nGradient Calculation for Output Layer Weights and Bias:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W^{[L]} } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)} } a^{[L-1] (i) T}\\)\n\nCompute the gradient of the loss function with respect to the weights of the output layer.\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b^{[L]} } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)}}}\\)\n\nCompute the gradient of the loss function with respect to the bias of the output layer.\n\n\nBackpropagate the Error to Previous Layers:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)} } = W^{[l+1] T} \\frac{\\partial \\mathcal{J} }{ \\partial z_{l+1}^{(i)} } \\odot g'(z^{[l] (i)})\\)\n\nPropagate the error derivative backward through the layers by considering the relationship between consecutive layers.\nInvolves the transpose of weights connecting the subsequent and current layers.\nInvolves the derivative of the activation function used in layer \\(l\\).\n\n\nGradient Calculation for Weights and Bias of Hidden Layers:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W^{[l]} } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)} } a^{[l-1] (i) T}\\)\n\nCompute the gradient of the loss function with respect to the weights of layer \\(l\\).\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b^{[l]} } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)}}}\\)\n\nCompute the gradient of the loss function with respect to the bias of layer \\(l\\).\n\n\n\nThese steps form the core of backpropagation in a shallow neural network, allowing for the iterative adjustment of weights and biases to minimize the loss function and improve the model’s accuracy in predicting outputs for given inputs.\n\n\n\n\n\n\nCode\n\n\n\n\n\ndef backward_propagation(parameters, cache, X, Y):\n    \"\"\"\n    Implement the backward propagation using the instructions above.\n    \n    Arguments:\n    parameters -- python dictionary containing our parameters \n    cache -- a dictionary containing \"Z1\", \"A1\", \"Z2\" and \"A2\".\n    X -- input data of shape (2, number of examples)\n    Y -- \"true\" labels vector of shape (1, number of examples)\n    \n    Returns:\n    grads -- python dictionary containing your gradients with respect to different parameters\n    \"\"\"\n    m = X.shape[1]\n    \n    W1, W2 = parameters[\"W1\"], parameters[\"W2\"]\n    \n    A1, A2 = cache[\"A1\"], cache[\"A2\"]\n        \n    dZ2 = A2 - Y\n    dW2 = (1/Y.shape[1])*dZ2@A1.T\n    db2 = (1/Y.shape[1])*np.sum(dZ2, axis=1, keepdims=True)\n    dZ1 = W2.T@dZ2*(1-A1**2)\n    dW1 = (1/Y.shape[1])*dZ1@X.T\n    db1 = (1/Y.shape[1])*np.sum(dZ1, axis=1, keepdims=True)\n        \n    grads = {\"dW1\": dW1,\n             \"db1\": db1,\n             \"dW2\": dW2,\n             \"db2\": db2}\n    \n    return grads"
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-    "text": "Backpropagation is the process of iteratively computing gradients of the loss function with respect to the weights and biases of the neural network, enabling the adjustment of parameters by propagating errors backward through the network’s layers to enhance learning and optimize performance.\n\nOutput Layer Error Derivative:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)} } = \\frac{1}{m} (\\hat{y}^{(i)} - y^{(i)}) \\odot g'(z^{[L] (i)})\\)\n\nCalculate the derivative of the cost function with respect to the pre-activation value of the output layer for each training example.\n\\(\\odot\\) denotes element-wise multiplication.\n\\(g'(\\cdot)\\) represents the derivative of the activation function used in the output layer.\n\n\nGradient Calculation for Output Layer Weights and Bias:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W^{[L]} } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)} } a^{[L-1] (i) T}\\)\n\nCompute the gradient of the loss function with respect to the weights of the output layer.\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b^{[L]} } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)}}}\\)\n\nCompute the gradient of the loss function with respect to the bias of the output layer.\n\n\nBackpropagate the Error to Previous Layers:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)} } = W^{[l+1] T} \\frac{\\partial \\mathcal{J} }{ \\partial z_{l+1}^{(i)} } \\odot g'(z^{[l] (i)})\\)\n\nPropagate the error derivative backward through the layers by considering the relationship between consecutive layers.\nInvolves the transpose of weights connecting the subsequent and current layers.\nInvolves the derivative of the activation function used in layer \\(l\\).\n\n\nGradient Calculation for Weights and Bias of Hidden Layers:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W^{[l]} } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)} } a^{[l-1] (i) T}\\)\n\nCompute the gradient of the loss function with respect to the weights of layer \\(l\\).\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b^{[l]} } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)}}}\\)\n\nCompute the gradient of the loss function with respect to the bias of layer \\(l\\).\n\n\n\nThese steps form the core of backpropagation in a shallow neural network, allowing for the iterative adjustment of weights and biases to minimize the loss function and improve the model’s accuracy in predicting outputs for given inputs."
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-    "text": "Code\n\n\n\n\n\n\nCode\n\n\n\n\n\ndef backward_propagation(parameters, cache, X, Y):\n    \"\"\"\n    Implement the backward propagation using the instructions above.\n    \n    Arguments:\n    parameters -- python dictionary containing our parameters \n    cache -- a dictionary containing \"Z1\", \"A1\", \"Z2\" and \"A2\".\n    X -- input data of shape (2, number of examples)\n    Y -- \"true\" labels vector of shape (1, number of examples)\n    \n    Returns:\n    grads -- python dictionary containing your gradients with respect to different parameters\n    \"\"\"\n    m = X.shape[1]\n    \n    W1, W2 = parameters[\"W1\"], parameters[\"W2\"]\n    \n    A1, A2 = cache[\"A1\"], cache[\"A2\"]\n        \n    dZ2 = A2 - Y\n    dW2 = (1/Y.shape[1])*dZ2@A1.T\n    db2 = (1/Y.shape[1])*np.sum(dZ2, axis=1, keepdims=True)\n    dZ1 = W2.T@dZ2*(1-A1**2)\n    dW1 = (1/Y.shape[1])*dZ1@X.T\n    db1 = (1/Y.shape[1])*np.sum(dZ1, axis=1, keepdims=True)\n        \n    grads = {\"dW1\": dW1,\n             \"db1\": db1,\n             \"dW2\": dW2,\n             \"db2\": db2}\n    \n    return grads"
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-    "text": "The forward pass in a neural network involves processing input data through the network’s layers to generate predictions or outputs by sequentially applying weights, biases, and activation functions.\nMathematical Representation:\nFor one training example \\(x^{(i)}\\):\n\nHidden Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[l] (i)} = W^{[l]} a^{[l-1](i)} + b^{[l]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[l-1](i)}\\)) by the weights (\\(W^{[l]}\\)) and add the bias (\\(b^{[l]}\\)) for layer \\(l\\).\n\n\nActivation Calculation:\n\n\\(a^{[l] (i)} = g(z^{[l] (i)})\\)\n\nApply an activation function \\(g\\) (such as sigmoid, tanh, ReLU, etc.) to the computed weighted sum to get the activation of layer \\(l\\).\n\n\n\nOutput Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[L] (i)} = W^{[L]} a^{[L-1] (i)} + b^{[L]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[L-1] (i)}\\)) by the weights (\\(W^{[L]}\\)) and add the bias (\\(b^{[L]}\\)) for the output layer.\n\n\nActivation Calculation:\n\n\\(\\hat{y}^{(i)} = a^{[L] (i)} = \\sigma(z^{ [L] (i)})\\)\n\nApply an appropriate activation function (e.g., softmax for multiclass classification, sigmoid for binary classification) to the computed weighted sum to obtain the predicted output (\\(\\hat{y}^{(i)}\\)).\n\n\n\nCost Function:\n\n\\(J = - \\frac{1}{m} \\sum\\limits_{i = 0}^{m} \\left(y^{(i)}\\log\\left(a^{[L] (i)}\\right) + (1-y^{(i)})\\log\\left(1- a^{[L] (i)}\\right)\\right)\\)\n\nCompute the appropriate cost function (e.g., cross-entropy, mean squared error) to evaluate the difference between the predicted output (\\(a^{[L] (i)}\\)) and the actual output (\\(y^{(i)}\\)) for all training examples (\\(m\\))."
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-    "text": "Backpropagation is the process of iteratively computing gradients of the loss function with respect to the weights and biases of the neural network, enabling the adjustment of parameters by propagating errors backward through the network’s layers to enhance learning and optimize performance.\n\nOutput Layer Error Derivative:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)} } = \\frac{1}{m} (\\hat{y}^{(i)} - y^{(i)}) \\odot g'(z^{[L] (i)})\\)\n\nCalculate the derivative of the cost function with respect to the pre-activation value of the output layer for each training example.\n\\(\\odot\\) denotes element-wise multiplication.\n\\(g'(\\cdot)\\) represents the derivative of the activation function used in the output layer.\n\n\nGradient Calculation for Output Layer Weights and Bias:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W^{[L]} } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)} } a^{[L-1] (i) T}\\)\n\nCompute the gradient of the loss function with respect to the weights of the output layer.\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b^{[L]} } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{L}^{(i)}}}\\)\n\nCompute the gradient of the loss function with respect to the bias of the output layer.\n\n\nBackpropagate the Error to Previous Layers:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)} } = W^{[l+1] T} \\frac{\\partial \\mathcal{J} }{ \\partial z_{l+1}^{(i)} } \\odot g'(z^{[l] (i)})\\)\n\nPropagate the error derivative backward through the layers by considering the relationship between consecutive layers.\nInvolves the transpose of weights connecting the subsequent and current layers.\nInvolves the derivative of the activation function used in layer \\(l\\).\n\n\nGradient Calculation for Weights and Bias of Hidden Layers:\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial W^{[l]} } = \\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)} } a^{[l-1] (i) T}\\)\n\nCompute the gradient of the loss function with respect to the weights of layer \\(l\\).\n\n\\(\\frac{\\partial \\mathcal{J} }{ \\partial b^{[l]} } = \\sum_i{\\frac{\\partial \\mathcal{J} }{ \\partial z_{l}^{(i)}}}\\)\n\nCompute the gradient of the loss function with respect to the bias of layer \\(l\\).\n\n\n\nThese steps form the core of backpropagation in a shallow neural network, allowing for the iterative adjustment of weights and biases to minimize the loss function and improve the model’s accuracy in predicting outputs for given inputs."
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-    "text": "Optimizing Model Parameters: Gradient Descent is used to update the parameters (weights and biases) of a neural network in the direction that minimizes the loss function.\nGeneral Gradient Descent Rule: The general update rule for a parameter \\(\\theta\\) using Gradient Descent is: \\[\\theta = \\theta - \\alpha \\frac{\\partial J }{ \\partial \\theta }\\] where \\(\\alpha\\) is the learning rate and \\(\\frac{\\partial J }{ \\partial \\theta }\\) denotes the partial derivative of the cost function \\(J\\) with respect to the parameter \\(\\theta\\).\nParameter Update for Weights and Biases:\n\nOutput Layer:\n\nUpdate the weights and biases of the output layer using the gradients calculated during backpropagation:\n\n\\(W^{[L]} = W^{[L]} - \\alpha \\frac{\\partial J }{ \\partial W^{[L]} }\\)\n\\(b^{[L]} = b^{[L]} - \\alpha \\frac{\\partial J }{ \\partial b^{[L]} }\\)\n\n\nHidden Layers:\n\nUpdate the weights and biases of the hidden layers using the gradients computed during backpropagation:\n\nFor \\(l\\)th hidden layer:\n\n\\(W^{[l]} = W^{[l]} - \\alpha \\frac{\\partial J }{ \\partial W^{[l]} }\\)\n\\(b^{[l]} = b^{[l]} - \\alpha \\frac{\\partial J }{ \\partial b^{[l]} }\\)\n\n\n\n\nImpact of Learning Rate (\\(\\alpha\\)):\n\nThe learning rate \\(\\alpha\\) determines the step size during parameter updates.\nA larger \\(\\alpha\\) might lead to faster convergence but can overshoot the minimum.\nA smaller \\(\\alpha\\) might converge slowly but can be more precise.\n\nIteration and Convergence:\n\nIterate through multiple epochs (iterations) of forward propagation, backpropagation, and parameter updates until the model’s performance converges or the predefined number of epochs is reached.\nMonitor the decrease in the cost function to assess convergence.\n\nImportance of Gradient Descent:\n\nGradient Descent is crucial in adjusting model parameters to minimize the loss function, thereby enhancing the accuracy of predictions in a neural network.\nIt enables the network to learn from the data by iteratively updating weights and biases to improve the overall performance of the model.\n\n\n\n\n\n\n\nNote\n\n\n\n\n\nimport numpy as np\n\ndef compute_cost(A2, Y):\n    \"\"\"\n    Computes the cross-entropy cost given in equation (13)\n    \n    Arguments:\n    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)\n    Y -- \"true\" labels vector of shape (1, number of examples)\n\n    Returns:\n    cost -- cross-entropy cost given equation (13)\n    \n    \"\"\"\n    \n    m = Y.shape[1] # number of examples\n\n    logprobs = np.multiply(Y, np.log(A2)) + np.multiply((1-Y), np.log(1-A2))\n    cost = - np.sum(logprobs)/Y.shape[1]\n        \n    cost = float(np.squeeze(cost)) \n\n    return cost\n\ndef update_parameters(parameters, grads, learning_rate = 1.2):\n    \"\"\"\n    Updates parameters using the gradient descent update rule given above\n    \n    Arguments:\n    parameters -- python dictionary containing your parameters \n    grads -- python dictionary containing your gradients \n    \n    Returns:\n    parameters -- python dictionary containing your updated parameters \n    \"\"\"\n    W1, b1, W2, b2 = copy.deepcopy(parameters[\"W1\"]), copy.deepcopy(parameters[\"b1\"]), copy.deepcopy(parameters[\"W2\"]), copy.deepcopy(parameters[\"b2\"])\n    \n    dW1, db1, dW2, db2 = grads[\"dW1\"], grads[\"db1\"], grads[\"dW2\"], grads[\"db2\"]\n    \n    W1 = W1 - learning_rate*dW1\n    b1 = b1 - learning_rate*db1\n    W2 = W2 - learning_rate*dW2\n    b2 = b2 - learning_rate*db2\n        \n    parameters = {\"W1\": W1,\n                  \"b1\": b1,\n                  \"W2\": W2,\n                  \"b2\": b2}\n    \n    return parameters\n\ndef nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):\n    \"\"\"\n    Arguments:\n    X -- dataset of shape (2, number of examples)\n    Y -- labels of shape (1, number of examples)\n    n_h -- size of the hidden layer\n    num_iterations -- Number of iterations in gradient descent loop\n    print_cost -- if True, print the cost every 1000 iterations\n    \n    Returns:\n    parameters -- parameters learnt by the model. They can then be used to predict.\n    \"\"\"\n    \n    np.random.seed(3)\n    n_x = layer_sizes(X, Y)[0]\n    n_y = layer_sizes(X, Y)[2]\n        \n    parameters = initialize_parameters(n_x, n_h, n_y)\n        \n    # Loop (gradient descent)\n\n    for i in range(0, num_iterations):\n         \n        A2, cache = forward_propagation(X, parameters)\n        cost = compute_cost(A2, Y)\n        grads = backward_propagation(parameters, cache, X, Y)\n        parameters = update_parameters(parameters, grads)\n                \n        # Print the cost every 1000 iterations\n        if print_cost and i % 1000 == 0:\n            print (\"Cost after iteration %i: %f\" %(i, cost))\n\n    return parameters"
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-    "text": "Optimizing Model Parameters: Gradient Descent is used to update the parameters (weights and biases) of a neural network in the direction that minimizes the loss function.\nGeneral Gradient Descent Rule: The general update rule for a parameter \\(\\theta\\) using Gradient Descent is: \\[\\theta = \\theta - \\alpha \\frac{\\partial J }{ \\partial \\theta }\\] where \\(\\alpha\\) is the learning rate and \\(\\frac{\\partial J }{ \\partial \\theta }\\) denotes the partial derivative of the cost function \\(J\\) with respect to the parameter \\(\\theta\\).\nParameter Update for Weights and Biases:\n\nOutput Layer:\n\nUpdate the weights and biases of the output layer using the gradients calculated during backpropagation:\n\n\\(W^{[L]} = W^{[L]} - \\alpha \\frac{\\partial J }{ \\partial W^{[L]} }\\)\n\\(b^{[L]} = b^{[L]} - \\alpha \\frac{\\partial J }{ \\partial b^{[L]} }\\)\n\n\nHidden Layers:\n\nUpdate the weights and biases of the hidden layers using the gradients computed during backpropagation:\n\nFor \\(l\\)th hidden layer:\n\n\\(W^{[l]} = W^{[l]} - \\alpha \\frac{\\partial J }{ \\partial W^{[l]} }\\)\n\\(b^{[l]} = b^{[l]} - \\alpha \\frac{\\partial J }{ \\partial b^{[l]} }\\)\n\n\n\n\nImpact of Learning Rate (\\(\\alpha\\)): - The learning rate \\(\\alpha\\) determines the step size during parameter updates. - A larger \\(\\alpha\\) might lead to faster convergence but can overshoot the minimum. - A smaller \\(\\alpha\\) might converge slowly but can be more precise.\nIteration and Convergence: - Iterate through multiple epochs (iterations) of forward propagation, backpropagation, and parameter updates until the model’s performance converges or the predefined number of epochs is reached. - Monitor the decrease in the cost function to assess convergence.\nImportance of Gradient Descent: - Gradient Descent is crucial in adjusting model parameters to minimize the loss function, thereby enhancing the accuracy of predictions in a neural network. - It enables the network to learn from the data by iteratively updating weights and biases to improve the overall performance of the model."
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-    "text": "Code\n\n\n\n\n\n\nNote\n\n\n\n\n\nimport numpy as np\n\ndef compute_cost(A2, Y):\n    \"\"\"\n    Computes the cross-entropy cost given in equation (13)\n    \n    Arguments:\n    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)\n    Y -- \"true\" labels vector of shape (1, number of examples)\n\n    Returns:\n    cost -- cross-entropy cost given equation (13)\n    \n    \"\"\"\n    \n    m = Y.shape[1] # number of examples\n\n    logprobs = np.multiply(Y, np.log(A2)) + np.multiply((1-Y), np.log(1-A2))\n    cost = - np.sum(logprobs)/Y.shape[1]\n        \n    cost = float(np.squeeze(cost)) \n\n    return cost\n\ndef update_parameters(parameters, grads, learning_rate = 1.2):\n    \"\"\"\n    Updates parameters using the gradient descent update rule given above\n    \n    Arguments:\n    parameters -- python dictionary containing your parameters \n    grads -- python dictionary containing your gradients \n    \n    Returns:\n    parameters -- python dictionary containing your updated parameters \n    \"\"\"\n    W1, b1, W2, b2 = copy.deepcopy(parameters[\"W1\"]), copy.deepcopy(parameters[\"b1\"]), copy.deepcopy(parameters[\"W2\"]), copy.deepcopy(parameters[\"b2\"])\n    \n    dW1, db1, dW2, db2 = grads[\"dW1\"], grads[\"db1\"], grads[\"dW2\"], grads[\"db2\"]\n    \n    W1 = W1 - learning_rate*dW1\n    b1 = b1 - learning_rate*db1\n    W2 = W2 - learning_rate*dW2\n    b2 = b2 - learning_rate*db2\n        \n    parameters = {\"W1\": W1,\n                  \"b1\": b1,\n                  \"W2\": W2,\n                  \"b2\": b2}\n    \n    return parameters\n\ndef nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):\n    \"\"\"\n    Arguments:\n    X -- dataset of shape (2, number of examples)\n    Y -- labels of shape (1, number of examples)\n    n_h -- size of the hidden layer\n    num_iterations -- Number of iterations in gradient descent loop\n    print_cost -- if True, print the cost every 1000 iterations\n    \n    Returns:\n    parameters -- parameters learnt by the model. They can then be used to predict.\n    \"\"\"\n    \n    np.random.seed(3)\n    n_x = layer_sizes(X, Y)[0]\n    n_y = layer_sizes(X, Y)[2]\n        \n    parameters = initialize_parameters(n_x, n_h, n_y)\n        \n    # Loop (gradient descent)\n\n    for i in range(0, num_iterations):\n         \n        A2, cache = forward_propagation(X, parameters)\n        cost = compute_cost(A2, Y)\n        grads = backward_propagation(parameters, cache, X, Y)\n        parameters = update_parameters(parameters, grads)\n                \n        # Print the cost every 1000 iterations\n        if print_cost and i % 1000 == 0:\n            print (\"Cost after iteration %i: %f\" %(i, cost))\n\n    return parameters"
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-    "title": "Activation Functions in Neural Networks",
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-    "text": "Activation functions play a crucial role in introducing non-linearity to neural networks, enabling them to model complex relationships and learn intricate patterns from the data. Here are common activation functions used in neural networks:\n\n\n\n\n\n\n\n\n\nActivation\nFormula\nGraph\nDescription\n\n\n\n\nSigmoid\n\\(a = \\frac{1}{1 + e^{-z}}\\)\n\nThe sigmoid function produces outputs between 0 and 1, resembling an S-shaped curve. It’s suitable for binary classification tasks where the output needs to be in the range [0, 1].\n\n\nTanh\n\\(a = \\frac{e^{z} - e^{-z}}{e^{z} + e^{-z}}\\)\n\nTanh, or hyperbolic tangent, produces outputs between -1 and 1. It is similar to the sigmoid function but with outputs centered around zero, making it suitable for hidden layers.\n\n\nReLU (Rectified Linear Unit)\n\\(a = \\max(0, z)\\)\n\nReLU is a simple and widely used activation function that outputs zero for negative inputs and linearly increases for positive inputs, efficiently mitigating the vanishing gradient problem.\n\n\nLeaky ReLU\n\\(a = \\max(0.01z, z)\\)\n\nLeaky ReLU is an improved version of ReLU, allowing a small gradient for negative inputs to address the “dying ReLU” problem encountered with the standard ReLU function.\n\n\n\n\nImportance of Non-linear Activation Functions:\nNeural networks with non-linear activation functions can approximate complex functions and relationships between inputs and outputs. Linear activation functions result in neural networks that are essentially just linear transformations, limiting their ability to capture intricate patterns present in real-world data.\nA detailed explanation on Stack Overflow provides insights into the necessity of non-linear activation functions in enabling neural networks to learn complex mappings.\n\n\nDerivatives of Activation Functions:\nThe derivatives of activation functions are essential for backpropagation during the training of neural networks. Here are the derivatives of common activation functions:\n\n\n\nActivation\nFormula\nDerivative\n\n\n\n\nSigmoid\n\\(a = \\frac{1}{1 + e^{-z}}\\)\n\\(a(1 - a)\\)\n\n\nTanh\n\\(a = \\frac{e^{z} - e^{-z}}{e^{z} + e^{-z}}\\)\n\\(1 - a^2\\)\n\n\nReLU\n\\(a = \\max(0, z)\\)\n0 if \\(z &lt; 0\\); 1 if \\(z \\geq 0\\)\n\n\nLeaky ReLU\n\\(a = \\max(0.01z, z)\\)\n0.01 if \\(z &lt; 0\\); 1 if \\(z \\geq 0\\)\n\n\n\nUnderstanding these derivatives is crucial in computing gradients during backpropagation, facilitating the adjustment of weights and biases for effective training of neural networks.\nThese diverse activation functions empower neural networks to handle various types of data and learn complex relationships, contributing to their adaptability and efficiency in modeling intricate patterns within datasets."
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     "title": "Introduction to Neural Networks",
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     "text": "Neural networks, composed of interconnected layers of nodes, serve as powerful computational models inspired by the human brain’s neural structure. Here’s an overview of key components and concepts:\nArchitecture:\n\nA neural network typically comprises an input layer, one or more hidden layers, and an output layer. Each layer contains nodes (neurons) that perform computations.\n\nForward Propagation:\n\nDuring forward propagation, input data \\(X\\) is passed through the network, undergoing computations and activations in each layer:\n\n\\(z^{[l] (i)} = W^{[l]} a^{[l-1](i)} + b^{[l]}\\): Compute the weighted sum and add bias for each layer \\(l\\).\n\\(a^{[l] (i)} = g(z^{[l] (i)})\\): Apply an activation function \\(g\\) to produce the layer’s output.\n\n\nActivation Functions:\n\nActivation functions introduce non-linearity and are applied at each layer’s output:\n\nSigmoid (\\(\\sigma\\)), Tanh, ReLU, Leaky ReLU, etc., each influencing the network’s ability to learn complex patterns.\n\n\nBackpropagation:\n\nBackpropagation involves computing gradients to update network parameters, facilitating learning:\n\nDerivatives of activation functions \\(\\frac{\\partial a}{\\partial z}\\) aid in error computation and parameter updates.\nGradient Descent updates weights (\\(W\\)) and biases (\\(b\\)) using the learning rate (\\(\\alpha\\)) and computed gradients.\n\n\nImportance of Non-linear Activation:\n\nNon-linear activation functions are crucial. Without them, the network simplifies to a linear model, limiting its ability to learn intricate patterns in data.\n\nTraining and Optimization:\n\nTraining involves iteratively adjusting parameters by minimizing a defined loss function (e.g., cross-entropy, mean squared error) using techniques like gradient descent."
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-    "title": "Forward Propagation",
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-    "text": "The forward pass in a neural network involves processing input data through the network’s layers to generate predictions or outputs by sequentially applying weights, biases, and activation functions.\nMathematical Representation:\nFor one training example \\(x^{(i)}\\):\n\nHidden Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[l] (i)} = W^{[l]} a^{[l-1](i)} + b^{[l]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[l-1](i)}\\)) by the weights (\\(W^{[l]}\\)) and add the bias (\\(b^{[l]}\\)) for layer \\(l\\).\n\n\nActivation Calculation:\n\n\\(a^{[l] (i)} = g(z^{[l] (i)})\\)\n\nApply an activation function \\(g\\) (such as sigmoid, tanh, ReLU, etc.) to the computed weighted sum to get the activation of layer \\(l\\).\n\n\n\nOutput Layer Computations:\n\nWeighted Sum Calculation:\n\n\\(z^{[L] (i)} = W^{[L]} a^{[L-1] (i)} + b^{[L]}\\)\n\nCompute the weighted sum of inputs (\\(a^{[L-1] (i)}\\)) by the weights (\\(W^{[L]}\\)) and add the bias (\\(b^{[L]}\\)) for the output layer.\n\n\nActivation Calculation:\n\n\\(\\hat{y}^{(i)} = a^{[L] (i)} = \\sigma(z^{ [L] (i)})\\)\n\nApply an appropriate activation function (e.g., softmax for multiclass classification, sigmoid for binary classification) to the computed weighted sum to obtain the predicted output (\\(\\hat{y}^{(i)}\\)).\n\n\n\nCost Function:\n\n\\(J = - \\frac{1}{m} \\sum\\limits_{i = 0}^{m} \\left(y^{(i)}\\log\\left(a^{[L] (i)}\\right) + (1-y^{(i)})\\log\\left(1- a^{[L] (i)}\\right)\\right)\\)\n\nCompute the appropriate cost function (e.g., cross-entropy, mean squared error) to evaluate the difference between the predicted output (\\(a^{[L] (i)}\\)) and the actual output (\\(y^{(i)}\\)) for all training examples (\\(m\\))."
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-    "title": "Forward Propagation",
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-    "text": "Code\n\n\n\n\n\n\nCode\n\n\n\n\n\nimport numpy as np\n\ndef forward_propagation(X, parameters):\n    \"\"\"\n    Argument:\n    X -- input data of size (n_x, m)\n    parameters -- python dictionary containing your parameters (output of initialization function)\n    \n    Returns:\n    A2 -- The sigmoid output of the second activation\n    cache -- a dictionary containing \"Z1\", \"A1\", \"Z2\" and \"A2\"\n    \"\"\"\n    W1, b1, W2, b2 = parameters[\"W1\"], parameters[\"b1\"], parameters[\"W2\"], parameters[\"b2\"]\n    \n    Z1 = W1@X + b1\n    A1 = np.tanh(Z1)\n    Z2 = W2@A1 + b2\n    A2 = 1/(1+np.exp(-Z2))\n    \n    cache = {\"Z1\": Z1,\n             \"A1\": A1,\n             \"Z2\": Z2,\n             \"A2\": A2}\n    \n    return A2, cache"
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