Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this shortcoming, in this work, we establish an asymptotic normality result for the constant step size stochastic gradient descent (SGD) algorithm--a widely used algorithm in practice. Specifically, based on the relationship between SGD and Markov Chains [DDB19], we show that the average of SGD iterates is asymptotically normally distributed around the expected value of their unique invariant distribution, as long as the non-convex and non-smooth objective function satisfies a dissipativity property. We also characterize the bias between this expected value and the critical points of the objective function under various local regularity conditions. Together, the above two results could be leveraged to construct confidence intervals for non-convex problems that are trained using the SGD algorithm.

本研究探讨了非凸非光滑目标函数中常数步长随机梯度下降算法的渐近正态结果，结果表明只要非凸和非光滑目标函数满足耗散性特性，SGD算法的迭代平均值就会渐近正态分布，该结果可用于构建对于使用SGD算法的非凸问题的置信区间。同时，本文通过对其与马尔可夫链的关系进行了详细地分析，还对目标函数的临界点与其期望值之间的偏差进行了表征。

非凸区域中恒定步长随机梯度下降的分析：渐近正态性和偏差