提出一种无需参数调整的随机优化方法，实现最优收敛速度。
Stochastic Auto-conditioned Fast Gradient Methods with Optimal Rates
- 基于自适应条件快速梯度思想，自动调节步长和小批量大小。
- 在标准假设下达到最优迭代复杂度 $O(1/\\/sqrt{\varepsilon})$ 与样本复杂度 $O(1/\varepsilon^2)$。
- 适用于无先验知识的场景，适合实际应用中参数未知的优化问题。
在不依赖问题参数先验的情况下，实现随机复合凸优化的最优收敛率，仍是核心挑战。尽管确定性情形下已有自适应条件快速梯度法（AC-FGM）可实现无需线搜索或光滑常数先验的加速，但在随机设置下的推广面临技术难题且长期未解。现有无参数随机方法要么无法获得加速率，要么依赖有界域、有界梯度、已知迭代轮数或严格亚高斯噪声等强假设。本文提出一种随机自适应条件快速梯度方法（stochastic AC-FGM），完全自适应于光滑常数、迭代轮数与噪声水平，实现无需线搜索的自适应步长与小批量选择。在标准有界条件方差假设下，证明该方法达到最优迭代复杂度 $O(1/\sqrt{\varepsilon})$ 与最优样本复杂度 $O(1/\varepsilon^2)$。
原文摘要 · Abstract (English)
Achieving optimal rates for stochastic composite convex optimization without prior knowledge of problem parameters remains a central challenge. In the deterministic setting, the auto-conditioned fast gradient method has recently been proposed to attain optimal accelerated rates without line-search procedures or prior knowledge of the Lipschitz smoothness constant, providing a natural prototype for parameter-free acceleration. However, extending this approach to the stochastic setting has proven technically challenging and remains open. Existing parameter-free stochastic methods either fail to achieve accelerated rates or rely on restrictive assumptions, such as bounded domains, bounded gradients, prior knowledge of the iteration horizon, or strictly sub-Gaussian noise. To address these limitations, we propose a stochastic variant of the auto-conditioned fast gradient method, referred to as stochastic AC-FGM. The proposed method is fully adaptive to the Lipschitz constant, the iteration horizon, and the noise level, enabling both adaptive stepsize selection and adaptive mini-batch sizing without line-search procedures. Under standard bounded conditional variance assumptions, we show that stochastic AC-FGM achieves the optimal iteration complexity of $O(1/\sqrt{\varepsilon})$ and the optimal sample complexity of $O(1/\varepsilon^2)$.
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