PreprintCC BY 4.0arXiv
Tight Heavy-Tailed Complexity Bounds for Optimizing Polyak--Łojasiewicz Objectives
Abstract
We study optimizing L-smooth and μ-Polyak-Lojasiewicz (PL) objectives with unbiased stochastic gradients under α-heavy-tailed noise (1<α≤ 2). In the unrestricted high-dimensional setting, we establish the noise-adaptive lower bound for attaining ε-suboptimal function value. Under the generalized mirror-PL condition, we propose a centered-clipped mirror-descent method that achieves a high-probability upper bound matching the lower bound up to logarithmic factors. We further provide the improved tight stochastic complexity in fixed dimensions.
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BibTeX
@misc{ou2026tight,
title = {{Tight Heavy-Tailed Complexity Bounds for Optimizing Polyak--Łojasiewicz Objectives}},
author = {Weiming Ou and Xiao Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03990},
eprint = {2609.03990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03990},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tight Heavy-Tailed Complexity Bounds for Optimizing Polyak--Łojasiewicz Objectives
AU - Weiming Ou
AU - Xiao Wang
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.03990
UR - https://arxiv.org/abs/2609.03990
AB - We study optimizing L-smooth and μ-Polyak-Lojasiewicz (PL) objectives with unbiased stochastic gradients under α-heavy-tailed noise (1<α≤ 2). In the unrestricted high-dimensional setting, we establish the noise-adaptive lower bound for attaining ε-suboptimal function value. Under the generalized mirror-PL condition, we propose a centered-clipped mirror-descent method that achieves a high-probability upper bound matching the lower bound up to logarithmic factors. We further provide the improved tight stochastic complexity in fixed dimensions.
ER -