Theoretical Analysis of Sparse Optimization with Reparameterization, Weight Decay, and Adaptive Learning Rate

Abstract

Sparse optimization is a fundamental challenge in various practical applications. A popular approach to sparse optimization is ${\ell }_p$ regularization. However, it may encounter optimization instability due to the unbounded gradients when $0<p<1$. In this paper, we introduce a novel approach to sparse optimization termed ReWA, based on Reparameterization, Weight decay, and Adaptive learning rate. ReWA is closely connected to ${\ell }_p$-regularization, yet it unveils a distinct optimization landscape that helps mitigate instability issues. Experiments on CIFAR-10 and ImageNet with ResNets demonstrate that ReWA leads to significant sparsity improvements over the ${\ell }_1$-regularization approach while preserving test accuracy.

32 pages, 5 figures. Submitted to ICML 2026