Owing to their stability and convergence speed, extragradient methods have become a staple for solving large-scale saddle-point problems in machine learning. The basic premise of these algorithms is the use of an extrapolation step before performing an update; thanks to this exploration step, extra-gradient methods overcome many of the non-convergence issues that plague gradient descent/ascent schemes. On the other hand, as we show in this paper, running vanilla extragradient with stochastic gradients may jeopardize its convergence, even in simple bilinear models. To overcome this failure, we investigate a double stepsize extragradient algorithm where the exploration step evolves at a more aggressive time-scale compared to the update step. We show that this modification allows the method to converge even with stochastic gradients, and we derive sharp convergence rates under an error bound condition.

中文翻译：

---

积极探索，保守更新：具有可变步长缩放的随机超梯度方法

由于其稳定性和收敛速度，超梯度方法已成为解决机器学习中大规模鞍点问题的主要方法。这些算法的基本前提是在执行更新之前使用外推步骤；由于这个探索步骤，超梯度方法克服了许多困扰梯度下降/上升方案的非收敛问题。另一方面，正如我们在本文中所展示的，即使在简单的双线性模型中，运行具有随机梯度的 vanilla extragradient 也可能会危及它的收敛性。为了克服这种失败，我们研究了一种双步长的超梯度算法，与更新步骤相比，探索步骤在更激进的时间尺度上演化。我们表明，这种修改使该方法即使使用随机梯度也能收敛，