In this work, we study the Uncertainty Quantification (UQ) of an algorithmic estimator of the saddle point of a strongly-convex strongly-concave objective. Specifically, we show that the averaged iterates of a Stochastic Extra-Gradient (SEG) method for a Saddle Point Problem (SPP) converges almost surely to the saddle point and follows a Central Limit Theorem (CLT) with optimal covariance under two different noise settings, namely the martingale-difference noise and the state-dependent Markov noise. To ensure the stability of the algorithm dynamics under the state-dependent Markov noise, we propose a variant of SEG with truncated varying sets. Our work opens the door for online inference of SPP with numerous potential applications in GAN, robust optimization, and reinforcement learning to name a few. We illustrate our results through numerical experiments.

翻译：本文研究强凸-强凹目标函数鞍点算法估计量的不确定性量化问题。具体而言，我们证明了随机额外梯度方法用于鞍点问题的平均迭代序列几乎必然收敛于鞍点，并在两种不同噪声设置（鞅差噪声与状态依赖马尔可夫噪声）下服从具有最优协方差的中心极限定理。为确保状态依赖马尔可夫噪声下算法动力学的稳定性，我们提出了截断变集随机额外梯度变体。本研究为鞍点问题的在线推断开辟了新途径，在生成对抗网络、鲁棒优化及强化学习等领域具有广泛潜在应用。我们通过数值实验验证了理论结果。