P1: Regularization Techniques in Deep Neural Networks

Regularization in Deep Neural Networks

• Definition: Regularization is a technique used to prevent overfitting in models, particularly deep neural networks, by improving their generalization to unseen data.

• Types of Regularization:

  • Dropout: Involves randomly setting a portion of neurons to zero during training, which helps prevent co-adaptation of neurons and promotes better weight distribution among them.
  • Weight Adjustment: Directly modifies the weights using specific algorithms to ensure that they don’t become excessively large, which can lead to overfitting.

• Overfitting:

  • Occurs when a model learns the noise in the training data instead of the actual distribution. This can be detected by a significant discrepancy between training and testing error rates.
  • A model fitting a polynomial of degree 9 to data that follows a degree 3 distribution is a common example, where the weights become excessively large as they adapt to noise rather than the underlying trend.

• Loss Function:

  • Commonly used loss function is Mean Squared Error (MSE): E = rac{1}{n} ext{Sum} (y - ar{y})^2 where ar{y} represents the predictions.
  • Loss can also be represented in vector form: E = (y - ar{y})^T(y - ar{y}) where T denotes matrix transposition.

• Weights Optimization:

  • Optimal weights are found by minimizing the loss function: set the derivative of the loss function with respect to weights w equal to zero.
  • The solution for optimal weights can be expressed as: ar{w} = (X^T X)^{-1} X^T y where X is the feature matrix and y is the target vector.

• Regularization Techniques:

  • L1 Regularization (Lasso): Adds the sum of the absolute values of the coefficients multiplied by a lambda factor to the loss function, promoting sparsity in the weights.
    E_{L1} = E + rac{ heta}{n} ext{Sum}(|w|)
  • L2 Regularization (Ridge): Adds the sum of the squares of the coefficients multiplied by a lambda factor to the loss function, discouraging large weights.
    E_{L2} = E + rac{ heta}{n} ext{Sum}(w^2)
  • Elastic Net: A combination of L1 and L2 regularizations.

• Effect of Regularization:

  • Regularization helps in balancing the weights, addressing overfitting by penalizing large weights and helping in stabilizing models that can be otherwise sensitive to noise.
  • The choice of lambda (regularization strength) can drastically affect the fit and the generalization performance of the model.

• Matrix Determinants:

  • When working with sparse matrices (many zero values), calculating the determinant can become problematic; adding a small value to the diagonal (identity matrix) can resolve this issue and make inverses computable.

• Conclusion:

  • Regularization is a crucial aspect of training deep learning models, fundamentally altering how models treat weight values in relation to input data. By controlling the optimization of weights, one can prevent overfitting and enhance a model's performance on new samples.