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arXiv · 2407.03507

Improved Iteration Complexity in Black-Box Optimization Problems under Higher Order Smoothness Function Condition

Aleksandr Lobanov
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Abstract

This paper is devoted to the study (common in many applications) of the black-box optimization problem, where the black-box represents a gradient-free oracle $\tilde{f} = f(x) + \xi$ providing the objective function value with some stochastic noise. Assuming that the objective function is $\mu$-strongly convex, and also not just $L$-smooth, but has a higher order of smoothness ($\beta \geq 2$) we provide a novel optimization method: Zero-Order Accelerated Batched Stochastic Gradient Descent, whose theoretical analysis closes the question regarding the iteration complexity, achieving optimal estimates. Moreover, we provide a thorough analysis of the maximum noise level, and show under which condition the maximum noise level will take into account information about batch size $B$ as well as information about the smoothness order of the function $\beta$.

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Aleksandr Lobanov. 2024-07-03. Improved Iteration Complexity in Black-Box Optimization Problems under Higher Order Smoothness Function Condition. https://arxiv.org/abs/2407.03507

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