We study variance reduction methods for extragradient (EG) algorithms for a class of variational inequalities satisfying a classical error-bound condition. Previously, linear convergence was only known to hold under strong monotonicity. The error-bound condition is much weaker than strong monotonicity and captures a larger class of problems, including bilinear saddle-point problems such as those arising from two-player zero-sum Nash equilibrium computation. We show that EG algorithms with SVRG-style variance reduction (SVRG-EG) achieve linear convergence under the error-bound condition. In addition, motivated by the empirical success of increasing iterate averaging techniques in solving saddle-point problems, we also establish new convergence results for variance-reduced EG with increasing iterate averaging. Finally, we conduct numerical experiments to demonstrate the advantage of SVRG-EG, with and without increasing iterate averaging, over deterministic EG.

翻译：我们研究满足经典误差界条件的一类变分不等式的梯度外推（EG）算法的方差缩减方法。此前，线性收敛性仅在强单调性假设下成立。误差界条件远弱于强单调性，可涵盖更大一类问题，包括双线性鞍点问题（如两人零和纳什均衡计算中的问题）。我们证明，采用SVRG风格方差缩减的EG算法（SVRG-EG）在误差界条件下可实现线性收敛。此外，受递增迭代平均技术在求解鞍点问题中的实证成功启发，我们为具有递增迭代平均的方差缩减EG建立了新的收敛性结果。最后，通过数值实验展示了SVRG-EG（无论是否结合递增迭代平均）相对于确定性EG的优越性。