This paper develops new variance-reduction techniques for the forward-reflected-backward splitting (FRBS) method to solve a class of possibly nonmonotone stochastic composite inclusions. Unlike unbiased estimators such as mini-batching, developing stochastic biased variants faces a fundamental technical challenge and has not been utilized before for inclusions and fixed-point problems. We fill this gap by designing a new framework that can handle both unbiased and biased estimators. Our main idea is to construct stochastic variance-reduced estimators for the forward-reflected direction and use them to perform iterate updates. First, we propose a class of unbiased variance-reduced estimators and show that increasing mini-batch SGD, loopless-SVRG, and SAGA estimators fall within this class. For these unbiased estimators, we establish a $\mathcal{O}(1/k)$ best-iterate convergence rate for the expected squared residual norm, together with almost-sure convergence of the iterate sequence to a solution. Consequently, we prove that the best oracle complexities for the $n$-finite-sum and expectation settings are $\mathcal{O}(n^{2/3}ε^{-2})$ and $\mathcal{O}(ε^{-10/3})$, respectively, when employing loopless-SVRG or SAGA, where $ε$ is a desired accuracy. Second, we introduce a new class of biased variance-reduced estimators for the forward-reflected direction, which includes SARAH, Hybrid SGD, and Hybrid SVRG as special instances. While the convergence rates remain valid for these biased estimators, the resulting oracle complexities are $\mathcal{O}(n^{3/4}ε^{-2})$ and $\mathcal{O}(ε^{-5})$ for the $n$-finite-sum and expectation settings, respectively. Finally, we conduct two numerical experiments on AUC optimization for imbalanced classification and policy evaluation in reinforcement learning.

翻译：本文针对求解一类可能非单调的随机复合包含问题，为前向反射后向分裂方法开发了新的方差缩减技术。与诸如小批量采样等无偏估计器不同，开发随机有偏变体面临根本性的技术挑战，此前从未在包含问题与不动点问题中得到应用。我们通过设计一个能同时处理无偏与有偏估计器的新框架填补了这一空白。核心思想是为前向反射方向构建随机方差缩减估计器，并利用其执行迭代更新。首先，我们提出一类无偏方差缩减估计器，证明递增小批量随机梯度下降、无循环随机方差缩减梯度及随机增量梯度算法估计器均属此类。针对这些无偏估计器，我们建立了期望残差范数平方的$\mathcal{O}(1/k)$最优迭代收敛率，同时证明迭代序列以概率1收敛至解。由此证得：当采用无循环随机方差缩减梯度或随机增量梯度算法时，$n$有限和情形与期望情形的最优Oracle复杂度分别为$\mathcal{O}(n^{2/3}ε^{-2})$与$\mathcal{O}(ε^{-10/3})$，其中$ε$为期望精度。其次，我们为前向反射方向引入新型有偏方差缩减估计器类，其特例包含随机递归梯度、混合随机梯度下降及混合随机方差缩减梯度算法。虽然收敛率对这些有偏估计器依然成立，但所得Oracle复杂度在$n$有限和情形与期望情形下分别为$\mathcal{O}(n^{3/4}ε^{-2})$与$\mathcal{O}(ε^{-5})$。最后，我们在不平衡分类的AUC优化与强化学习策略评估两个问题上进行了数值实验。