Adaptive Norm-Based Regularization for Neural Networks

In this paper, we study norm-based regularization methods for neural networks. We compare existing penalization approaches and introduce two regularization strategies that extend classical ridge- and lasso-type penalties to neural network models. The first strategy modifies weight decay by incorporating the covariance structure of the input features into a ridge-type $\ell_2$ penalty, allowing regularization to account for feature dependence. The second combines an $\ell_1$ sparsity penalty with covariance-aware $\ell_2$ regularization, producing neural network weights that are both sparse and structurally informed. Monte Carlo simulations are used to evaluate these methods under different data-generating settings, followed by two real-data applications on building cooling-load prediction and leukemia cell-type classification from high-dimensional gene expression data. Across simulated and real-data examples, the proposed regularizers improve predictive performance on unseen data and provide more effective complexity control than standard norm-based penalties, particularly when features are correlated or high-dimensional.