In this paper we consider convex feasibility problems where the feasible set is given as the intersection of a collection of closed convex sets. We assume that each set is specified algebraically as a convex inequality, where the associated convex function is general (possibly non-differentiable). For finding a point satisfying all the convex inequalities we design and analyze random projection algorithms using special subgradient iterations and extrapolated stepsizes. Moreover, the iterate updates are performed based on parallel random observations of several constraint components. For these minibatch stochastic subgradient-based projection methods we prove sublinear convergence results and, under some linear regularity condition for the functional constraints, we prove linear convergence rates. We also derive sufficient conditions under which these rates depend explicitly on the minibatch size. To the best of our knowledge, this work is the first deriving conditions that show theoretically when minibatch stochastic subgradient-based projection updates have a better complexity than their single-sample variants when parallel computing is used to implement the minibatch. Numerical results also show a better performance of our minibatch scheme over its non-minibatch counterpart.

在本文中，我们考虑凸可行性问题，其中可行集被给出为封闭凸集集合的交集。我们假设每个集合在代数上被指定为凸不等式，其中相关的凸函数是一般的（可能是不可微的）。为了找到满足所有凸不等式的点，我们使用特殊的次梯度迭代和外推步长来设计和分析随机投影算法。此外，迭代更新是基于几个约束组件的并行随机观察来执行的。对于这些基于小批量随机次梯度的投影方法，我们证明了亚线性收敛结果，并且在函数约束的某些线性规律条件下，我们证明了线性收敛速度。我们还推导出这些速率明确取决于小批量大小的充分条件。据我们所知，这项工作是第一个推导条件，当使用并行计算来实现小批量时，当小批量随机次梯度投影更新具有比它们的单样本变体更好的复杂性时，理论上表明。数值结果还表明，我们的小批量方案比非小批量方案具有更好的性能。