Mirror Descent Under Generalized Smoothness

• URL: http://arxiv.org/abs/2502.00753v1
• Date: Sun, 02 Feb 2025 11:23:10 GMT
• Title: Mirror Descent Under Generalized Smoothness
• Authors: Dingzhi Yu, Wei Jiang, Yuanyu Wan, Lijun Zhang,
• Abstract summary: We introduce a new $ell*$-smoothness concept that measures the norm of Hessian terms of a general norm and its dual. We establish convergence for mirror-descent-type algorithms, matching the rates under the classic smoothness.
• Score: 23.5387392871236
• License:
• Abstract: Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progress, existing generalizations of smoothness are restricted to Euclidean geometry with $\ell_2$-norm and only have theoretical guarantees for optimization in the Euclidean space. In this paper, we address this limitation by introducing a new $\ell*$-smoothness concept that measures the norm of Hessian in terms of a general norm and its dual, and establish convergence for mirror-descent-type algorithms, matching the rates under the classic smoothness. Notably, we propose a generalized self-bounding property that facilitates bounding the gradients via controlling suboptimality gaps, serving as a principal component for convergence analysis. Beyond deterministic optimization, we establish an anytime convergence for stochastic mirror descent based on a new bounded noise condition that encompasses the widely adopted bounded or affine noise assumptions.

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