A Dimension-Insensitive Algorithm for Stochastic Zeroth-Order Optimization

TL;DR

This paper introduces a new stochastic zeroth-order optimization algorithm that achieves nearly dimension-free query complexity without requiring gradient sparsity or compressibility, improving efficiency in high-dimensional problems.

Contribution

The paper presents the first dimension-insensitive algorithm for stochastic zeroth-order optimization that does not rely on gradient sparsity or compressibility assumptions.

Findings

Achieves dimension-free query complexity up to a logarithmic factor.

Works for both convex and strongly convex problems.

Numerical results confirm theoretical predictions.

Abstract

This paper concerns a convex, stochastic zeroth-order optimization (S-ZOO) problem. The objective is to minimize the expectation of a cost function whose gradient is not directly accessible. For this problem, traditional optimization algorithms mostly yield query complexities that grow polynomially with dimensionality (the number of decision variables). Consequently, these methods may not perform well in solving massive-dimensional problems arising in many modern applications. Although more recent methods can be provably dimension-insensitive, almost all of them require arguably more stringent conditions such as everywhere sparse or compressible gradient. In this paper, we propose a sparsity-inducing stochastic gradient-free (SI-SGF) algorithm, which provably yields a dimension-free (up to a logarithmic term) query complexity in both convex and strongly convex cases. Such insensitivity to the dimensionality growth is proven, for the first time, to be achievable when neither gradient sparsity nor gradient compressibility is satisfied. Our numerical results demonstrate a consistency between our theoretical prediction and the empirical performance.