Add solutions to sample exam

Author: carlportz

src/codes/07-summary/exam_preparation.py

@@ -33,6 +33,50 @@ def euler_method(
     return x, y
 ### ANCHOR_END: euler_errors
 
+### ANCHOR: gradient_descent_errors
+def double_well_gradient(x):  # x: float
+    grad = 6 * x**5 - 2 * x - 0.25
+    return grad
+
+def gradient_descent(func_grad, x0, tau=0.01, maxgrad=1e-6, maxiter=500)
+    x = 0
+    converged = False
+    
+    for _ in range(0, maxiter):
+        grad = func_grad(x)
+        x = x - tau * grad
+        if grad < maxgrad:
+            converged = True
+            break
+    
+        return x, converged
+
+x_opt, converged = gradient_descent(double_well_gradient, -1.2)
+print("Ein lokales Minimum liegt bei x = " , x_opt)  # x_opt = -0.8872
+### ANCHOR_END: gradient_descent_errors
+
+### ANCHOR: gradient_descent_correct
+def double_well_gradient(x):  # x: float
+    grad = 6 * x**5 - 4 * x - 0.25  # Error 1: -2*x should be -4*x (derivative of -2x^2 is -4x)
+    return grad
+
+def gradient_descent(func_grad, x0, tau=0.01, maxgrad=1e-6, maxiter=500):
+    x = x0  # Error 2: Should initialize x with x0, not 0
+    converged = False
+
+    for _ in range(0, maxiter):
+        grad = func_grad(x)
+        x = x - tau * grad
+        if abs(grad) < maxgrad:  # Error 3: Should use abs(grad) for convergence check
+            converged = True
+            break
+
+    return x, converged  # Error 4: return must be outside the for loop
+
+x_opt, converged = gradient_descent(double_well_gradient, -1.2)
+print("Ein lokales Minimum liegt bei x = " , x_opt)  # x_opt = -0.8872
+### ANCHOR_END: gradient_descent_correct
+
 ### ANCHOR: knn_incomplete
 import numpy as np
 import matplotlib.pyplot as plt
@@ -92,6 +136,22 @@ def predict(self, X, y, xi):
         return y_pred
 ### ANCHOR_END: knn_complete
 
+### ANCHOR: knn_weighted_complete
+class kNNWeightedClassifier:
+    def __init__(self, k, sigma=1.0):
+        self.k = k
+        self.sigma = sigma
+
+    def predict(self, X, y, xi):
+        distances = np.linalg.norm(X - xi, axis=1)
+        weights = np.exp(-distances**2 / (2 * self.sigma**2))
+        nearest = np.argsort(distances)[:self.k]
+        y_nearest = y[nearest]
+        unique_labels, label_counts = np.unique(y_nearest, return_counts=True)
+        y_pred = unique_labels[np.argmax(label_counts)]
+        return y_pred
+### ANCHOR_END: knn_weighted_complete
+
 ### ANCHOR: knn_example
 N = 20
 X = np.random.randn(N, 2)

src/psets/exam_preparation.md

@@ -35,6 +35,11 @@ print(y)
 <div style="max-width: 300px; margin: 1em auto; 
     padding: 1em; border: 1px solid #ccc; border-radius: 4px; text-align: left;"
 
+<!-- 
+**Lösung:**
+`[6, 4, 2]` 
+-->
+
 **(b)**
 
 ```python
@@ -57,6 +62,11 @@ print(z)
 <div style="max-width: 300px; margin: 1em auto; 
     padding: 1em; border: 1px solid #ccc; border-radius: 4px; text-align: left;"
 
+<!-- 
+**Lösung:**
+`[5, 7, 9]`
+-->
+
 **(c)**
 
 ```python
@@ -77,6 +87,11 @@ print(z)
 
 </div>
 
+<!-- 
+**Lösung:**
+`32`
+-->
+
 <!--- ANCHOR_END: aufgabe_1 --->
 
 ## Aufgabe 2: Gradientenverfahren
@@ -121,6 +136,13 @@ x_opt, converged = gradient_descent(double_well_gradient, -1.2)
 print("Ein lokales Minimum liegt bei x = " , x_opt)  # x_opt = -0.8872
 ```
 
+<!-- 
+**Lösung:**
+```
+{{#include ../codes/07-summary/exam_preparation.py:gradient_descent_correct}}
+```
+-->
+
 <!--- ANCHOR_END: aufgabe_2 --->
 
 ## Aufgabe 3: $k$-Nearest Neighbors