Implemented task 3 and 4

effect-of-regularization-on-loss.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+# Generating Synthetic Data
+def generate_linear_data(n):
+    x = np.random.uniform(0, 10, n) # initialize x
+    eps = np.random.normal(0, 1, n) # initialize epsilon
+    y = -3 * x + 8 + 2 * eps # y = −3x + 8 + 2ϵ
+    return x.reshape(-1, 1), y
+
+# Gradient Descent with L1/L2
+def gradient_descent(x, y, lam, reg_type, lr, iters):
+    x_b = np.hstack([np.ones_like(x), x]) # initialize x
+    w = np.zeros(x_b.shape[1]) # initialize weight
+    path = [w.copy()]
+
+    for i in range(iters):
+        pred = x_b @ w # linear regression prediction
+        error = pred - y # error
+        grad = x_b.T @ error / len(y) # gradient formula
+
+        if reg_type == 'l2':
+            grad += lam * w # L2 formula
+        elif reg_type == 'l1':
+            grad += lam * np.sign(w) # L1 formula
+
+        w -= lr * grad # loss calculation
+        path.append(w.copy())
+
+    return w, np.array(path)
+
+# Plotting the loss
+def plot_contour(x, y, reg_type, lam):
+    x_b = np.hstack([np.ones_like(x), x]) # initialize x
+    w0, w1 = np.meshgrid(np.linspace(-10, 10, 100), np.linspace(-10, 10, 100)) # initialize intercept and slope
+    loss = np.zeros_like(w0) # initialize loss
+
+    for i in range(w0.shape[0]):
+        for j in range(w0.shape[1]):
+            w = np.array([w0[i, j], w1[i, j]])
+            error = y - x_b @ w # error
+            mse = np.mean(error ** 2) # mean square error
+            reg = lam * (np.sum(w ** 2) if reg_type == 'l2' else np.sum(np.abs(w))) # regularization
+            loss[i, j] = mse + reg # regularization and mse for the loss
+
+    _, path = gradient_descent(x, y, lam, reg_type, 0.01, 500)
+
+    # plotting the figure
+    plt.figure(figsize=(6, 5))
+    plt.contour(w0, w1, loss, levels=50, cmap='viridis')
+    plt.plot(path[:, 0], path[:, 1], 'ro-', markersize=2, label='Gradient Descent Path')
+    plt.title(f"{reg_type.upper()} Regularization (λ={lam})")
+    plt.xlabel("w0 (intercept)")
+    plt.ylabel("w1 (slope)")
+    plt.grid(True)
+    plt.legend()
+    plt.tight_layout()
+    plt.savefig('results/task4-effect-of-regularization-on-loss-' + reg_type + '-'  + str(lam) + '.png')
+
+if __name__ == "__main__":
+    print("Running Task 4: Effect of L1 and L2 Regularization on Loss Landscape")
+
+    # Generate dataset
+    x, y = generate_linear_data(30)
+
+    # Values of lambda to visualize
+    lambda_values = [0.01, 0.1, 1.0]
+
+    # Plot for both L1 and L2 regularization
+    for reg_type in ['l1', 'l2']:
+        for lam in lambda_values:
+            plot_contour(x, y, reg_type, lam)

regularization-w-cross-validation.py

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+import numpy as np
+import matplotlib.pyplot as plt
+from numpy.linalg import inv
+
+# Linear Regression Data Generation
+def generate_data(N, noise_var):
+    x = np.random.uniform(0, 1, N)
+    noise = np.random.normal(0, np.sqrt(noise_var), N)
+    y = np.sin(2 * np.pi * x) + noise
+    return x.reshape(-1, 1), y
+
+# Gaussian Basis is used for the linear regression
+def gaussian_basis(x, D):
+    mus = np.linspace(0, 1, D) # means initialized.
+    s = 0.1 # standard deviation
+    basis = np.exp(-(x - mus)**2 / (2 * s**2)) # Gaussian Formula
+    return basis
+
+# Ridge Regression Formula
+def l2_ridge_regression(x, y, lam):
+    I = np.eye(x.shape[1]) # regularization term
+    return inv(x.T @ x + lam * I) @ x.T @ y # (X^T*X+λ*I)^−1*X^T*y
+
+# Lasso Regression Formula
+def l1_lasso_regression(x, y, lam, lr=0.01, iterations=1000):
+    w = np.zeros(x.shape[1]) # w initialized the weight
+    for i in range(iterations): # iterations
+        grad = -x.T @ (y - x @ w) # gradient formula
+        w -= lr * grad # gradient descent
+        w = np.sign(w) * np.maximum(0, np.abs(w) - lr * lam) # soft thresholding
+    return w
+
+# Cross Validation
+def cross_validate(x, y, lam_values, reg_type, k):
+    fold_size = len(x) // k # fold size is defined as 10
+    train_errors, val_errors = [], []
+
+    for lam in lam_values:
+        train_mse, val_mse = [], []
+        for i in range(k): # trained for 9 folds and validated in the remaining with each λ.
+            val_idx = np.arange(i * fold_size, (i + 1) * fold_size)
+            train_idx = np.setdiff1d(np.arange(len(x)), val_idx)
+            x_train, y_train = x[train_idx], y[train_idx]
+            x_val, y_val = x[val_idx], y[val_idx]
+
+            # apply the regression
+            if reg_type == 'l2':
+                w = l2_ridge_regression(x_train, y_train, lam)
+            else:
+                w = l1_lasso_regression(x_train, y_train, lam)
+
+            train_pred = x_train @ w
+            val_pred = x_val @ w
+            train_mse.append(np.mean((y_train - train_pred)**2)) # train mse added for each fold
+            val_mse.append(np.mean((y_val - val_pred)**2)) # validation mse added for each fold
+
+        train_errors.append(np.mean(train_mse)) # train error added for each λ
+        val_errors.append(np.mean(val_mse)) # validation error added for each λ
+    return train_errors, val_errors
+
+# Train and Validation Errors
+def train_validation_err(reg_type, lam_values, num_datasets, N, D):
+    if lam_values is None:
+        lam_values = np.logspace(-3, 1, 10) # λ is defined
+
+    train_all, val_all = [], []
+
+    for i in range(num_datasets): # inside given 50 datasets
+        x, y = generate_data(N, 1.0) # data generated
+        Phi = gaussian_basis(x, D) # linear regression with gaussian basis
+        train_err, val_err = cross_validate(Phi, y, lam_values, reg_type, 10) # cross-validation training
+        train_all.append(train_err)
+        val_all.append(val_err)
+
+    mean_train = np.mean(train_all, 0) # mean training
+    mean_val = np.mean(val_all, 0) # mean validation
+
+    # plotting the figure
+    plt.figure()
+    plt.plot(lam_values, mean_train, label='Train Error')
+    plt.plot(lam_values, mean_val, label='Validation Error')
+    plt.xscale('log')
+    plt.xlabel("λ")
+    plt.ylabel("MSE")
+    plt.title(f"{reg_type.upper()} Regularization")
+    plt.legend()
+    plt.grid(True)
+    plt.savefig('results/task3-train-validation-errors-' + reg_type + '.png')
+
+    return mean_train, mean_val, lam_values
+
+# Bias-Variance Decomposition
+def bias_variance_decomp(reg_type, lam_values, num_datasets, N, D):
+    if lam_values is None:
+        lam_values = np.logspace(-3, 1, 10) # λ is defined
+
+    x_test = np.linspace(0, 1, 100).reshape(-1, 1) # x test value defined
+    Phi_test = gaussian_basis(x_test, D) # linear regression with gaussian basis
+    y_true = np.sin(2 * np.pi * x_test).ravel() # sin(2*π*x)
+
+    biases, variances, total_mse = [], [], []
+
+    for lam in lam_values:
+        preds = []
+        for i in range(num_datasets):
+            x_train, y_train = generate_data(N, 1.0) # data generated
+            Phi_train = gaussian_basis(x_train, D) # linear regression with gaussian basis
+            # apply the regression
+            if reg_type == 'l2':
+                w = l2_ridge_regression(Phi_train, y_train, lam)
+            else:
+                w = l1_lasso_regression(Phi_train, y_train, lam)
+            preds.append(Phi_test @ w)
+
+        preds = np.array(preds)
+        mean_pred = np.mean(preds, 0) # mean of predictions
+        bias2 = np.mean((mean_pred - y_true) ** 2) # squared bias formula
+        # bias is defined as mean of difference between predicted mean and true y value.
+        var = np.mean(np.var(preds, 0)) # mean variance of predictions
+        mse = bias2 + var + 1  # noise variance
+
+        biases.append(bias2) # add bias for each λ
+        variances.append(var) # add variance for each λ
+        total_mse.append(mse) # add total mse for each λ
+
+    # plotting the figure
+    plt.figure()
+    plt.plot(lam_values, biases, label='Bias^2')
+    plt.plot(lam_values, variances, label='Variance')
+    plt.plot(lam_values, np.array(biases) + np.array(variances), label='Bias^2 + Var')
+    plt.plot(lam_values, total_mse, label='Bias^2 + Var + Noise')
+    plt.xscale('log')
+    plt.xlabel("λ")
+    plt.ylabel("Error")
+    plt.title(f"{reg_type.upper()} Bias-Variance Decomposition")
+    plt.legend()
+    plt.grid(True)
+    plt.savefig('results/task3-bias-decomposition-' + reg_type + '.png')
+
+if __name__ == "__main__":
+    print("Running Task 3: Regularization with Cross-Validation")
+
+    lam_values = np.logspace(-3, 1, 10) # initial λ values -3, 1, 10
+
+    train_validation_err('l2', lam_values, 50, 20, 45)
+    train_validation_err('l1', lam_values, 50, 20, 45)
+
+    bias_variance_decomp('l2', lam_values, 50, 20, 45)
+    bias_variance_decomp('l1', lam_values, 50, 20, 45)