Improved Oracle Complexity of Variance Reduced Methods for Nonsmooth Convex Stochastic Composition Optimization

TL;DR

This paper introduces improved variance reduced algorithms for nonsmooth convex stochastic composition problems, achieving faster convergence rates than previous methods and demonstrating superior empirical performance.

Contribution

The paper provides the first non-asymptotic IFO complexity analysis for SCVRG in nonsmooth convex composition, surpassing existing methods in efficiency.

Findings

SCVRG achieves IFO complexity of O((m+n) log(1/ε) + 1/ε^3)

SCVRG outperforms SCGD and AGD in theoretical complexity

Experimental results show SCVRG's superior performance on sparse mean-variance optimization

Abstract

We consider the nonsmooth convex composition optimization problem where the objective is a composition of two finite-sum functions and analyze stochastic compositional variance reduced gradient (SCVRG) methods for them. SCVRG and its variants have recently drawn much attention given their edge over stochastic compositional gradient descent (SCGD); but the theoretical analysis exclusively assumes strong convexity of the objective, which excludes several important examples such as Lasso, logistic regression, principle component analysis and deep neural nets. In contrast, we prove non-asymptotic incremental first-order oracle (IFO) complexity of SCVRG or its novel variants for nonsmooth convex composition optimization and show that they are provably faster than SCGD and gradient descent. More specifically, our method achieves the total IFO complexity of $O\left((m+n)\log\left(1/\epsilon\right)+1/\epsilon^3\right)$ which improves that of $O\left(1/\epsilon^{3.5}\right)$ and $O\left((m+n)/\sqrt{\epsilon}\right)$ obtained by SCGD and accelerated gradient descent (AGD) respectively. Experimental results confirm that our methods outperform several existing methods, e.g., SCGD and AGD, on sparse mean-variance optimization problem.