Classical extragradient schemes and their stochastic counterpart represent a cornerstone for resolving monotone variational inequality problems. Yet, such schemes have a per-iteration complexity of two projections onto a convex set and require two evaluations of the map, the former of which could be relatively expensive. We consider two related avenues where the per-iteration complexity is significantly reduced: (i) A stochastic projected reflected gradient method requiring a single evaluation of the map and a single projection; and (ii) A stochastic subgradient extragradient method that requires two evaluations of the map, a single projection onto the associated feasibility set, and a significantly cheaper projection (onto a halfspace) computable in closed form. Under a variance-reduced framework reliant on a sample-average of the map based on an increasing batch-size, we prove almost sure convergence of the iterates to a random point in the solution set for both schemes. Additionally, non-asymptotic rate guarantees are derived for both schemes in terms of the gap function; notably, both rates match the best-known rates obtained in deterministic regimes. To address feasibility sets given by the intersection of a large number of convex constraints, we adapt both of the aforementioned schemes to a random projection framework. We then show that the random projection analogs of both schemes also display almost sure convergence under a weak-sharpness requirement; furthermore, without imposing the weak-sharpness requirement, both schemes are characterized by the optimal rate in terms of the gap function of the projection of the averaged sequence onto the set as well as the infeasibility of this sequence. Preliminary numerics support theoretical findings and the schemes outperform standard extragradient schemes in terms of the per-iteration complexity.

经典的梯度方案及其随机对应方案是解决单调变分不等式问题的基石。然而，这样的方案在凸集上具有两次投影的逐项迭代复杂性，并且需要对地图进行两次评估，而前者可能相对昂贵。我们考虑了两个相关的途径，可以显着降低每次迭代的复杂度：（i）一种随机投影反射梯度方法，需要对地图进行一次评估并进行一次投影；（ii）一种随机次梯度超梯度方法，该方法需要对地图进行两次评估，即对相关可行性集的单个投影，以及可以以封闭形式计算的便宜得多的投影（在半空间上）。在依赖于基于增加的批处理大小的图的样本均值的减少方差的框架下，我们证明了几乎可以肯定的是，两种方案的解决方案集合中的迭代都收敛到随机点。另外，根据间隙函数得出了两种方案的非渐近速率保证。值得注意的是，这两个比率都与确定性体制中获得的最知名比率相匹配。为了解决由大量凸约束的交集给出的可行性集，我们将上述两种方案都适配于随机投影框架。然后，我们证明两种方案的随机投影类似物在弱锐度要求下也显示出几乎确定的收敛性。此外，在不施加弱锐度要求的情况下，两种方案的特征均在于根据平均序列投影到集合上的缺口函数的最佳速率以及该序列的不可行性。初步的数值支持理论发现，并且在逐次迭代复杂度方面，该方案优于标准的超梯度方案。