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
本文提出并分析Polyak型外梯度方法，用于求解确定性和随机单调求根问题，统一了收敛性分析，并针对无共同解情形提出DecPolyakSEG算法，实现次线性残差收敛。
AI 中文摘要
我们研究了用于求解确定性和随机单调求根问题的外梯度方法中的Polyak型步长选择。我们表明，确定性外梯度的已知投影型修正源于对到解的距离上界的最小化，这与经典的Polyak步长构造相平行。利用这一观点，我们基于控制算子$F$沿外推方向变化的局部临界条件，提供了Polyak型外梯度方法（PolyakEG）的统一确定性分析。该分析不要求全局Lipschitz连续性，并在更广泛的条件下（如Hölder连续性或$(L_0, L_1)$-Lipschitz性）涵盖次线性收敛，在额外强单调性下涵盖线性收敛，所有这些都通过单一框架实现。然后我们研究了该方法的随机扩展。我们首先证明了直接随机变体PolyakSEG在所有随机分量算子共享一个共同解时的收敛性。我们还表明，在没有此条件的情况下，使用非消失步长的PolyakSEG可能无法收敛到均值算子的零点。为解决这一局限性，我们提出了DecPolyakSEG，它结合了递减步长与Polyak型更新，并在不需要分量算子间共同解的情况下建立了次线性残差收敛结果。这些结果与凸最小化文献中随机Polyak步长的最新进展相平行，并在更广泛的求根领域中建立了类似的研究途径。
英文摘要
We study Polyak-type step-size selection for extragradient methods for solving deterministic and stochastic monotone root-finding problems. We show that the known projection-type correction for deterministic extragradient arises from minimizing an upper bound on the distance to a solution, paralleling the classical Polyak step-size construction. Using this viewpoint, we provide a unified deterministic analysis of the Polyak-type Extragradient Method (PolyakEG), based on a local critical condition controlling the variation of operator $F$ along the extrapolation direction. This analysis does not require global Lipschitz continuity, and covers sublinear convergence under broader conditions such as Hölder continuity or $(L_0, L_1)$-Lipschitzness and linear convergence under additional strong monotonicity, all through a single framework. We then study the stochastic extensions of this approach. We first prove convergence of a direct stochastic variant, PolyakSEG, when all stochastic component operators share a common solution. We also show that, without this condition, PolyakSEG with nonvanishing step-sizes may fail to converge to a zero of the mean operator. To address this limitation, we propose DecPolyakSEG, which combines decreasing step-sizes with Polyak-type updates, and establish a sublinear residual convergence result without requiring a common solution across the component operators. These results parallel recent developments in stochastic Polyak step-sizes from the convex minimization literature and establish an analogous research avenue in the broader root-finding regime.
