发表机构
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院 H. 米尔顿·斯图尔特工业与系统工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带Lipschitz连续算子的单调变分不等式，提出基于累积正则化的AR算法，在不同噪声模型下实现了近最优的随机预言机复杂度，改进了现有残差缩减的复杂度结果。
AI 中文摘要
本文研究带有Lipschitz连续算子的单调变分不等式（VIs）强解的计算问题。基于累积正则化的思想，我们为变分不等式开发了一个通用框架，尤其关注随机情形。在具有一致有界方差σ²的无偏随机预言机下，AR算法（累积正则化算法）使用最多Õ(LD₀/ε + σ²/ε² (log(LD₀/ε))³)次随机预言机调用，计算出预期算子残差界为ε的近似解，其中L为Lipschitz常数，D₀为初始点到解的距离的界。这大幅改进了现有残差缩减的𝒪(σ²/ε⁴)复杂度，且在对数因子内达到下界。对于强单调变分不等式，以到解的距离衡量，当强单调模已知时，AR达到最优预言机复杂度；若仅将问题视为单调，无需知晓该模，AR仍能达到近最优复杂度。我们进一步引入适用于一般单调变分不等式（可能存在非唯一解）的状态依赖噪声模型，将状态依赖噪声分析扩展至强单调情形之外；在该模型下，结合增强型随机算子外推（SOE）方法的AR算法，达到近最优复杂度，其中随机项依赖于解处的方差。
英文摘要
This paper studies the computation of strong solutions of monotone variational inequalities (VIs) with Lipschitz continuous operators. Building on the idea of accumulative regularization (AR), we develop a general framework for VIs, with particular emphasis on stochastic settings. Under unbiased stochastic oracles with uniformly bounded variance $ σ^2$, AR computes an approximate solution with expected operator residual bounded by $\varepsilon$ using at most $ \widetilde{\mathcal{O}}\left(\tfrac{LD_0}{\varepsilon}+\tfrac{ σ^2}{\varepsilon^2}(\log\tfrac{LD_0}{\varepsilon})^3\right) $ stochastic oracle calls, where $L$ is the Lipschitz constant and $D_0$ bounds the initial distance to the solution. This substantially improves the existing $\mathcal{O}( σ^2/\varepsilon^4)$ complexity for residual reduction and matches the lower bound up to logarithmic factors. For strongly monotone VIs, measured by the distance to the solution, AR achieves the optimal oracle complexity when the strong monotonicity modulus is known. By treating the problem as merely monotone, AR still achieves nearly optimal complexity without knowledge of this modulus. We further introduce a state-dependent noise model applicable to general monotone VIs with potentially nonunique solutions, extending state-dependent noise analysis beyond the strongly monotone setting. We apply these results to policy evaluation in reinforcement learning, where we show that estimators of the projected Bellman operator satisfy our state-dependent noise condition. To the best of our knowledge, the AR framework, the improved complexity under uniformly bounded noise, the state-dependent noise model and its guarantees, the adaptivity to an unknown strong monotonicity modulus, and the improved guarantees for FTD learning are all new in the VI literature.
