一种不精确黎曼邻近动量方差缩减方法：复杂度界与KL序列收敛性
An Inexact Riemannian Proximal Momentum Variance-Reduced Method: Complexity Bounds and KL Sequential Convergence
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中文总结 AI 辅助
针对紧嵌入子流形上的有限和非光滑复合问题，提出不精确黎曼邻近动量方差缩减方法，建立其复杂度界，并基于普通逐点KL性质推导带记忆的条件期望下降抽象KL原理，证明序列收敛性与速率。
中文摘要 AI 辅助
针对紧嵌入子流形上的有限和非光滑复合问题，我们对不精确随机黎曼邻近优化开展了统一分析。该框架在单一条件误差耗散条件下，可容纳方差缩减梯度估计器、投影动量及不精确切空间邻近求解，已在基于投影的SVRG、SARAH/SPIDER、SAGA及SAG上得到验证。可计算的Fenchel对偶残差准则，其容差在采样和内迭代前预设，可实现对内工作量的显式控制。我们建立了条件期望下降、子序列平稳性及\
英文摘要
We develop a unified analysis of inexact stochastic Riemannian proximal optimization for finite-sum nonsmooth composite problems over compact embedded submanifolds. The framework accommodates variance-reduced gradient estimators, projected momentum, and inexact tangent-space proximal solves under a single conditional error-dissipation condition, verified for projection-based SVRG, SARAH/SPIDER, SAGA, and SAG. A computable Fenchel-dual residual criterion, with tolerance prescribed before sampling and inner iterations, enables explicit control of the inner work. We establish conditional expected descent, subsequential stationarity, and an \(O(ε^{-2})\) outer complexity. With SARAH/SPIDER and accumulative regularization, iRPMVR attains \(O(n+\sqrt n\,ε^{-2})\) component-gradient and \(O(ε^{-3})\) proximal-operator complexities. We further develop an abstract KL principle for conditional expected descent with memory and summable tails using only the ordinary pointwise KL property. A counterexample shows that a power-type expected-KL implication used in earlier stochastic analyses can fail. The principle yields almost-sure finite length, whole-sequence convergence, and deterministic KL rates.
