← 研踪 PaperTrailarXiv:2610.13799 · v2
Adaptive Variance Reduction across Smoothness and Noise Conditions
首次提交 2026-10-10 00:13
版本更新 2026-11-06 21:14
雷达观测 2026-11-09 11:19
摘要
In this paper, we study adaptive stochastic non-convex optimization under unknown smoothness and noise conditions. Our algorithms require no smoothness constant, noise bound, initial function gap, or oracle-regime label. We first propose a fixed-budget method that combines normalized momentum, recursive variance reduction, and a fresh gradient stage, followed by fresh validation of the candidates. We prove that the same output has expected gradient norm $O(T^{-1/4})$ under objective smoothness and bounded noise variance, $O(T^{-1/3})$ under average smoothness, and $O(T^{-1/2})$ for noiseless gradients, where $T$ counts individual gradient evaluations. We then develop an anytime algorithm whose three optimization paths preserve their endpoints across reporting and budget-extension boundaries. The same three rates hold at every deterministic query budget, although the algorithm receives no information about the budget $T$. The analysis combines a scale potential, a uniform second-moment bound, and accounting for candidates retained through unfinished blocks. An optional sample-mean selector further reduces cumulative validation work from $O(T)$ to $O(T^{2/3})$ while preserving these rates.
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v2 · 2026-11-06 21:14 · 最新观测版本
Adaptive Variance Reduction across Smoothness and Noise Conditions
In this paper, we study adaptive stochastic non-convex optimization under unknown smoothness and noise conditions. Our algorithms require no smoothness constant, noise bound, initial function gap, or oracle-regime label. We first propose a fixed-budget method that combines normalized momentum, recursive variance reduction, and a fresh gradient stage, followed by fresh validation of the candidates. We prove that the same output has expected gradient norm $O(T^{-1/4})$ under objective smoothness and bounded noise variance, $O(T^{-1/3})$ under average smoothness, and $O(T^{-1/2})$ for noiseless gradients, where $T$ counts individual gradient evaluations. We then develop an anytime algorithm whose three optimization paths preserve their endpoints across reporting and budget-extension boundaries. The same three rates hold at every deterministic query budget, although the algorithm receives no information about the budget $T$. The analysis combines a scale potential, a uniform second-moment bound, and accounting for candidates retained through unfinished blocks. An optional sample-mean selector further reduces cumulative validation work from $O(T)$ to $O(T^{2/3})$ while preserving these rates.
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