General Stability Analysis for Zeroth-Order Optimization Algorithms

Xinyue Liu, Hualin Zhang, Bin Gu, Hong Chen

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Keywords: Stability Analysis; Zeroth-Order Optimization; Black-Box Learning

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Abstract: Zeroth-order optimization algorithms are widely used for black-box optimization problems, such as those in machine learning and prompt engineering, where the gradients are approximated using function evaluations. Recently, a generalization result was provided for zeroth-order stochastic gradient descent (SGD) algorithms through stability analysis. However, this result was limited to the vanilla 2-point zeroth-order estimate of Gaussian distribution used in SGD algorithms. To address these limitations, we propose a general proof framework for stability analysis that applies to convex, strongly convex, and non-convex conditions, and yields results for popular zeroth-order optimization algorithms, including SGD, GD, and SVRG, as well as various zeroth-order estimates, such as 1-point and 2-point with different distributions and coordinate estimates. Our general analysis shows that coordinate estimation can lead to tighter generalization bounds for SGD, GD, and SVRG versions of zeroth-order optimization algorithms, due to the smaller expansion brought by coordinate estimates to stability analysis.

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Primary Area: learning theory

Submission Number: 5497