A Variance-Reduced and Stabilized Proximal Stochastic Gradient Method with Support Identification Guarantees for Structured Optimization

TL;DR

This paper presents S-PStorm, a variance-reduced and stabilized proximal stochastic gradient method that guarantees support identification in structured optimization without needing exact gradient evaluations or gradient history, showing strong convergence and efficiency.

Contribution

Introduces S-PStorm, a novel stabilized variance-reduced proximal stochastic gradient algorithm with support identification guarantees that does not require gradient evaluation or storage of historical gradients.

Findings

S-PStorm achieves support identification with high probability in finite iterations.

The method outperforms existing algorithms in support recovery efficiency.

Supports structured sparsity optimization with strong convergence guarantees.

Abstract

This paper introduces a new proximal stochastic gradient method with variance reduction and stabilization for minimizing the sum of a convex stochastic function and a group sparsity-inducing regularization function. Since the method may be viewed as a stabilized version of the recently proposed algorithm PStorm, we call our algorithm S-PStorm. Our analysis shows that S-PStorm has strong convergence results. In particular, we prove an upper bound on the number of iterations required by S-PStorm before its iterates correctly identify (with high probability) an optimal support (i.e., the zero and nonzero structure of an optimal solution). Most algorithms in the literature with such a support identification property use variance reduction techniques that require either periodically evaluating an exact gradient or storing a history of stochastic gradients. Unlike these methods, S-PStorm achieves variance reduction without requiring either of these, which is advantageous. Moreover, our support-identification result for S-PStorm shows that, with high probability, an optimal support will be identified correctly in all iterations with the index above a threshold. We believe that this type of result is new to the literature since the few existing other results prove that the optimal support is identified with high probability at each iteration with a sufficiently large index (meaning that the optimal support might be identified in some iterations, but not in others). Numerical experiments on regularized logistic loss problems show that S-PStorm outperforms existing methods in various metrics that measure how efficiently and robustly iterates of an algorithm identify an optimal support.