Linear Convergence of Accelerated Stochastic Gradient Descent for Nonconvex Nonsmooth Optimization

TL;DR

This paper introduces an accelerated stochastic gradient descent method combining variance reduction and Nesterov's extrapolation, proving its linear convergence to stationary points in nonconvex nonsmooth optimization, supported by numerical experiments.

Contribution

It is the first to establish linear convergence of an accelerated SGD method to local minima in nonconvex nonsmooth problems.

Findings

Proved linear convergence of the proposed method.

Demonstrated effectiveness through numerical experiments.

Established convergence to stationary points.

Abstract

In this paper, we study the stochastic gradient descent (SGD) method for the nonconvex nonsmooth optimization, and propose an accelerated SGD method by combining the variance reduction technique with Nesterov's extrapolation technique. Moreover, based on the local error bound condition, we establish the linear convergence of our method to obtain a stationary point of the nonconvex optimization. In particular, we prove that not only the sequence generated linearly converges to a stationary point of the problem, but also the corresponding sequence of objective values is linearly convergent. Finally, some numerical experiments demonstrate the effectiveness of our method. To the best of our knowledge, it is first proved that the accelerated SGD method converges linearly to the local minimum of the nonconvex optimization.