From 5cc1604996756bec0d0a4563ec22f5df61357a82 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Batuhan=20Berk=20Ba=C5=9Fo=C4=9Flu?= Date: Sun, 19 Oct 2025 02:07:40 +0300 Subject: [PATCH] Implemented task 3 and 4 --- effect-of-regularization-on-loss.py | 72 +++++++++ regularization-w-cross-validation.py | 149 ++++++++++++++++++ results/task3-bias-decomposition-l1.png | Bin 0 -> 39322 bytes results/task3-bias-decomposition-l2.png | Bin 0 -> 38349 bytes results/task3-train-validation-errors-l1.png | Bin 0 -> 26440 bytes results/task3-train-validation-errors-l2.png | Bin 0 -> 21343 bytes ...fect-of-regularization-on-loss-l1-0.01.png | Bin 0 -> 130357 bytes ...ffect-of-regularization-on-loss-l1-0.1.png | Bin 0 -> 130205 bytes ...ffect-of-regularization-on-loss-l1-1.0.png | Bin 0 -> 131624 bytes ...fect-of-regularization-on-loss-l2-0.01.png | Bin 0 -> 130798 bytes ...ffect-of-regularization-on-loss-l2-0.1.png | Bin 0 -> 132828 bytes ...ffect-of-regularization-on-loss-l2-1.0.png | Bin 0 -> 144412 bytes 12 files changed, 221 insertions(+) create mode 100644 effect-of-regularization-on-loss.py create mode 100644 results/task3-bias-decomposition-l1.png create mode 100644 results/task3-bias-decomposition-l2.png create mode 100644 results/task3-train-validation-errors-l1.png create mode 100644 results/task3-train-validation-errors-l2.png create mode 100644 results/task4-effect-of-regularization-on-loss-l1-0.01.png create mode 100644 results/task4-effect-of-regularization-on-loss-l1-0.1.png create mode 100644 results/task4-effect-of-regularization-on-loss-l1-1.0.png create mode 100644 results/task4-effect-of-regularization-on-loss-l2-0.01.png create mode 100644 results/task4-effect-of-regularization-on-loss-l2-0.1.png create mode 100644 results/task4-effect-of-regularization-on-loss-l2-1.0.png diff --git a/effect-of-regularization-on-loss.py b/effect-of-regularization-on-loss.py new file mode 100644 index 0000000..426f1b1 --- /dev/null +++ b/effect-of-regularization-on-loss.py @@ -0,0 +1,72 @@ +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) \ No newline at end of file diff --git a/regularization-w-cross-validation.py b/regularization-w-cross-validation.py index e69de29..8375744 100644 --- a/regularization-w-cross-validation.py +++ b/regularization-w-cross-validation.py @@ -0,0 +1,149 @@ +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, 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