Convergence Rates of Stochastic Zeroth-order Gradient Descent for \L ojasiewicz Functions

TL;DR

This paper establishes convergence rates for Stochastic Zeroth-order Gradient Descent algorithms applied to Lojasiewicz functions, demonstrating faster convergence of function values than iterates, applicable to both smooth and nonsmooth cases.

Contribution

It provides the first convergence rate analysis of SZGD algorithms for Lojasiewicz functions, including non-smooth cases, expanding understanding of zeroth-order optimization.

Findings

Function values converge faster than iterates.

Convergence rates hold for both smooth and nonsmooth functions.

Results apply to a broad class of Lojasiewicz functions.

Abstract

We prove convergence rates of Stochastic Zeroth-order Gradient Descent (SZGD) algorithms for Lojasiewicz functions. The SZGD algorithm iterates as \begin{align*} \mathbf{x}_{t+1} = \mathbf{x}_t - \eta_t \widehat{\nabla} f (\mathbf{x}_t), \qquad t = 0,1,2,3,\cdots , \end{align*} where $f$ is the objective function that satisfies the \L ojasiewicz inequality with \L ojasiewicz exponent $\theta$, $\eta_t$ is the step size (learning rate), and $ \widehat{\nabla} f (\mathbf{x}_t) $ is the approximate gradient estimated using zeroth-order information only. Our results show that $ \{ f (\mathbf{x}_t) - f (\mathbf{x}_\infty) \}_{t \in \mathbb{N} } $ can converge faster than $ \{ \| \mathbf{x}_t - \mathbf{x}_\infty \| \}_{t \in \mathbb{N} }$, regardless of whether the objective $f$ is smooth or nonsmooth.