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Regularization is a critical component in deep learning. The most commonlyused approach, weight decay, applies a constant penalty coefficient uniformlyacross all parameters. This may be overly restrictive for some parameters,while insufficient for others. To address this, we present ConstrainedParameter Regularization (CPR) as an alternative to traditional weight decay.Unlike the uniform application of a single penalty, CPR enforces an upper boundon a statistical measure, such as the L2-norm, of individual parametermatrices. Consequently, learning becomes a constraint optimization problem,which we tackle using an adaptation of the augmented Lagrangian method. CPRintroduces only a minor runtime overhead and only requires setting an upperbound. We propose simple yet efficient mechanisms for initializing this bound,making CPR rely on no hyperparameter or one, akin to weight decay. Ourempirical studies on computer vision and language modeling tasks demonstrateCPR's effectiveness. The results show that CPR can outperform traditionalweight decay and increase performance in pre-training and fine-tuning.

Improving Deep Learning Optimization through Constrained Parameter Regularization