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Neural network regularization

Core Techniques

  1. L1/L2 Regularization

    Loss = Original_Loss + λ * Regularization_term
    • L1: Sum of absolute weights $$ L1 = \lambda \sum_{i} |w_i| $$
    • L2: Sum of squared weights $$ L2 = \lambda \sum_{i} w_i^2 $$

  2. Dropout

    • Randomly “turns off” neurons during training
    • Typical dropout rates: 0.2-0.5
    • Acts as ensemble learning

  3. Early Stopping

    • Monitors validation performance
    • Stops when validation error starts increasing

  4. Data Augmentation

    • Transforms training data
    • Common in image tasks: rotation, scaling, flipping

  5. Batch Normalization $$ \hat{x}^{(k)} = \frac{x^{(k)} - \mu_B}{\sqrt{\sigma_B^2 + \epsilon}} $$

    • Normalizes layer inputs
    • Reduces internal covariate shift

Benefits

  1. Prevents Overfitting

    • Limits model complexity
    • Reduces overfit to training data

  2. Improves Generalization

    • Better performance on new data
    • Reduces test error

  3. Enhances Robustness

    • Reduces noise sensitivity
    • Improves model stability

2. Connection with Lipschitz Continuity

Mathematical Foundation

  1. Gradient Clipping and Lipschitz Constraint

    • For L-Lipschitz functions: $$ |\nabla f(x)| \leq L $$
    # Gradient clipping implementation
max_grad_norm = L
torch.nn.utils.clip_grad_norm_(model.parameters(), max_grad_norm)

  2. Wasserstein GAN Application $$W(p_r, p_g) = \sup_{f \in \mathcal{F}_L} \mathbb{E}_r[f(x)] - \mathbb{E}_g[f(x)]$$

    • Requires 1-Lipschitz continuous discriminator

  3. Robustness Guarantee $$ |f(x + \delta) - f(x)| \leq L|\delta| $$

    • L bounds output change for input perturbation

Practical Implications

  1. Optimization Convergence

    • Learning rate condition: $$ \eta < \frac{2}{L} $$

  2. Implementation Example

    class LipschitzLayer(nn.Module):
    def __init__(self, L):
        super().__init__()
        self.L = L

    def forward(self, x):
        W = self.get_normalized_weights()
        return F.linear(x, W)

    def get_normalized_weights(self):
        with torch.no_grad():
            W = self.weight
            W_norm = torch.norm(W, p=2)
            if W_norm > self.L:
                W = W * (self.L / W_norm)
        return W

  3. Theoretical Bounds

    • Generalization error bound: $$ \text{Generalization Error} \leq O\left(\frac{L}{\sqrt{n}}\right) $$ where n is the training sample size