Stochastic Nested Variance Reduction for Nonconvex Optimization

TL;DR

This paper introduces a nested variance reduction stochastic gradient descent algorithm for nonconvex optimization, achieving faster convergence and lower gradient complexity than existing methods, supported by theoretical analysis and experiments.

Contribution

The paper proposes a novel K+1 nested reference point approach that improves variance reduction and convergence rates in nonconvex stochastic optimization.

Findings

Achieves faster convergence to stationary points than SVRG and SCSG.

Reduces gradient evaluation complexity for nonconvex problems.

Experimental results confirm theoretical improvements.

Abstract

We study finite-sum nonconvex optimization problems, where the objective function is an average of $n$ nonconvex functions. We propose a new stochastic gradient descent algorithm based on nested variance reduction. Compared with conventional stochastic variance reduced gradient (SVRG) algorithm that uses two reference points to construct a semi-stochastic gradient with diminishing variance in each iteration, our algorithm uses $K+1$ nested reference points to build a semi-stochastic gradient to further reduce its variance in each iteration. For smooth nonconvex functions, the proposed algorithm converges to an $\epsilon$-approximate first-order stationary point (i.e., $\|\nabla F(\mathbf{x})\|_2\leq \epsilon$) within $\tilde O(n\land \epsilon^{-2}+\epsilon^{-3}\land n^{1/2}\epsilon^{-2})$ number of stochastic gradient evaluations. This improves the best known gradient complexity of SVRG $O(n+n^{2/3}\epsilon^{-2})$ and that of SCSG $O(n\land \epsilon^{-2}+\epsilon^{-10/3}\land n^{2/3}\epsilon^{-2})$. For gradient dominated functions, our algorithm also achieves better gradient complexity than the state-of-the-art algorithms. Thorough experimental results on different nonconvex optimization problems back up our theory.