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') # 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 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) 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)