Gradient-free stochastic optimization for additive models

• URL: http://arxiv.org/abs/2503.02131v2
• Date: Sat, 05 Apr 2025 13:02:50 GMT
• Title: Gradient-free stochastic optimization for additive models
• Authors: Arya Akhavan, Alexandre B. Tsybakov,
• Abstract summary: We address the problem of zero-order optimization from noisy observations for an objective function satisfying the Polyak-Lojasiewicz or the strong convexity condition.<n>We assume that the objective function has an additive structure and satisfies a higher-order smoothness property, characterized by the H"older family of functions.
• Score: 56.42455605591779
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We address the problem of zero-order optimization from noisy observations for an objective function satisfying the Polyak-{\L}ojasiewicz or the strong convexity condition. Additionally, we assume that the objective function has an additive structure and satisfies a higher-order smoothness property, characterized by the H\"older family of functions. The additive model for H\"older classes of functions is well-studied in the literature on nonparametric function estimation, where it is shown that such a model benefits from a substantial improvement of the estimation accuracy compared to the H\"older model without additive structure. We study this established framework in the context of gradient-free optimization. We propose a randomized gradient estimator that, when plugged into a gradient descent algorithm, allows one to achieve minimax optimal optimization error of the order $dT^{-(\beta-1)/\beta}$, where $d$ is the dimension of the problem, $T$ is the number of queries and $\beta\ge 2$ is the H\"older degree of smoothness. We conclude that, in contrast to nonparametric estimation problems, no substantial gain of accuracy can be achieved when using additive models in gradient-free optimization.

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