发表机构
Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对传统锚点类加速方法因误差累积无法直接用于随机求根的问题，提出双锚点加速机制，在无方差缩减、固定批量规模下实现了对应复杂度，强单调场景下接近下界。
AI 中文摘要
近年来，确定性求根问题的加速方法已得到广泛研究；具体而言，基于锚点的方法（或称Halpern型方法）在算子范数意义下可达到最优收敛速率。但由于误差累积效应，这类方法的加速机制无法直接迁移到随机场景，除非通过增大批量规模或采用方差缩减技术来迫使方差递减。本研究表明，另一类加速机制——双锚点机制可推广至随机场景且不会出现此类误差累积，这与基于锚点的算法形成鲜明对比。据此，我们针对期望意义下具有余强制性（对应不动点问题为平方非扩张性）的随机求根（对应不动点）问题，在批量规模不随迭代次数变化、无需任何方差缩减或双循环递归正则化的条件下，简洁地实现了$O(ε^{-3})$的复杂度。对于强单调算子，同一算法可达到更优的$\tilde{O}(ε^{-2})$复杂度，其$ε$依赖关系几乎匹配下界。
英文摘要
Acceleration for deterministic root-finding problems has been extensively studied in recent years; specifically, the anchor-based, or Halpern-type methods achieve optimal convergence rates with respect to the operator norm. However, acceleration via these methods does not directly carry over to stochastic setting due to accumulation of errors, unless one enforces diminishing variance via increasing batch sizes or variance reduction techniques. In this work, we show that another class of acceleration, namely the dual-anchor mechanism, extends to the stochastic setting without such error accumulation, in contrast to anchor-based algorithms. Consequently, we cleanly achieve $O(ε^{-3})$ complexity with iteration-independent batch size, without any variance reduction or double-loop recursive regularization, for stochastic root-finding (resp. fixed-point) problems with cocoercivity (resp. square-nonexpansivity) in expectation. For strongly monotone operators, the same algorithm attains a sharper $\widetilde{O} (ε^{-2})$ complexity, nearly matching the lower bound in terms of $ε$-dependence.
