Added slide on residual errors

Author: faressc

slides/02-machine_learning_fundamentals.md

@@ -107,8 +107,6 @@
     </div>
 </div>
 
-<div class="fragment" data-fragment-index="1"></div>
-
 ---
 
 ## Machine Learning
@@ -225,6 +223,32 @@ The loss function $$\mathcal{L}$$ quantifies the difference between the predicte
 
 ---
 
+## Residual Errors
+
+- Data points are generated by a true function $f^*$ plus noise:
+
+<div class="formula">
+$$
+\mathbf{y}_i = f^*(\mathbf{x}_i) + \epsilon_i
+$$
+</div>
+
+where $\epsilon_i$ represents inherent noise or randomness in the data generation process.
+
+- Residual errors measure the difference between predictions and observations:
+
+<div class="formula">
+$$
+r_i = \mathbf{y}_i - f_{\boldsymbol{\theta}}(\mathbf{x}_i)
+$$
+</div>
+
+<div class="highlight" style="padding: 40px 40px">
+Errors persist due to: (1) inherent noise $\epsilon_i$, and (2) approximation error when $f^* \notin \mathcal{F}_{\Theta}$
+</div>
+
+---
+
 ## Example: Linear Regression
 
 <div style="text-align: center;">