Zeroth-Order Algorithms for Stochastic Distributed Nonconvex Optimization

TL;DR

This paper introduces two distributed zeroth-order algorithms for stochastic nonconvex optimization, achieving the first linear speedup convergence rates under general variance assumptions, with applications demonstrated in neural network adversarial example generation.

Contribution

The paper presents the first distributed ZO algorithms with proven linear speedup convergence rates under general variance conditions, extending the applicability of ZO methods in distributed machine learning.

Findings

Achieved linear speedup convergence rate of (\u221a{p/(nT)}) for smooth functions.

Established convergence rate of (p/(nT)) under Polyak-ojasiewicz condition.

Demonstrated efficiency in neural network adversarial example generation.

Abstract

In this paper, we consider a stochastic distributed nonconvex optimization problem with the cost function being distributed over $n$ agents having access only to zeroth-order (ZO) information of the cost. This problem has various machine learning applications. As a solution, we propose two distributed ZO algorithms, in which at each iteration each agent samples the local stochastic ZO oracle at two points with a time-varying smoothing parameter. We show that the proposed algorithms achieve the linear speedup convergence rate $\mathcal{O}(\sqrt{p/(nT)})$ for smooth cost functions under the state-dependent variance assumptions which are more general than the commonly used bounded variance and Lipschitz assumptions, and $\mathcal{O}(p/(nT))$ convergence rate when the global cost function additionally satisfies the Polyak--{\L}ojasiewicz (P--{\L}) condition in addition, where $p$ and $T$ are the dimension of the decision variable and the total number of iterations, respectively. To the best of our knowledge, this is the first linear speedup result for distributed ZO algorithms, which enables systematic processing performance improvements by adding more agents. We also show that the proposed algorithms converge linearly under the relative bounded second moment assumptions and the P--{\L} condition. We demonstrate through numerical experiments the efficiency of our algorithms on generating adversarial examples from deep neural networks in comparison with baseline and recently proposed centralized and distributed ZO algorithms.