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Weight decay is one of the most widely used forms of regularization in deeplearning, and has been shown to improve generalization and robustness. Theoptimization objective driving weight decay is a sum of losses plus a termproportional to the sum of squared weights. This paper argues that stochasticgradient descent (SGD) may be an inefficient algorithm for this objective. Forneural networks with ReLU activations, solutions to the weight decay objectiveare equivalent to those of a different objective in which the regularizationterm is instead a sum of products of $\ell_2$ (not squared) norms of the inputand output weights associated with each ReLU neuron. This alternative (andeffectively equivalent) regularization suggests a novel proximal gradientalgorithm for network training. Theory and experiments support the new trainingapproach, showing that it can converge much faster to the sparse solutions itshares with standard weight decay training.

PathProx: A Proximal Gradient Algorithm for Weight Decay Regularized Deep Neural Networks