There is a recent interest on first-order methods for linear programming (LP). In this paper,we propose a stochastic algorithm using variance reduction and restarts for solving sharp primal-dual problems such as LP. We show that the proposed stochastic method exhibits a linear convergence rate for solving sharp instances with a high probability. In addition, we propose an efficient coordinate-based stochastic oracle for unconstrained bilinear problems, which has $\mathcal O(1)$ per iteration cost and improves the complexity of the existing deterministic and stochastic algorithms. Finally, we show that the obtained linear convergence rate is nearly optimal (upto $\log$ terms) for a wide class of stochastic primal dual methods.

翻译：近年来，线性规划的一阶方法引起了广泛关注。本文提出了一种结合方差缩减与重启动技术的随机算法，用于求解诸如线性规划等尖锐原始-对偶问题。我们证明，所提出的随机方法能够以高概率实现尖锐实例的线性收敛速率。此外，针对无约束双线性问题，我们设计了一种高效的基于坐标的随机预言机，其每次迭代的计算复杂度为$\mathcal O(1)$，并提升了现有确定性与随机算法的复杂度。最后，我们证明，对于一大类随机原始-对偶方法，所获得的线性收敛速率（在对数项意义下）是近乎最优的。