Online Bootstrap Inference with Nonconvex Stochastic Gradient Descent Estimator

Abstract: In this paper, we investigate the theoretical properties of stochastic gradient descent (SGD) for statistical inference in the context of nonconvex optimization problems, which have been relatively unexplored compared to convex settings. Our study is the first to establish provable inferential procedures using the SGD estimator for general nonconvex objective functions, which may contain multiple local minima. We propose two novel online inferential procedures that combine SGD and the multiplier bootstrap technique. The first procedure employs a consistent covariance matrix estimator, and we establish its error convergence rate. The second procedure approximates the limit distribution using bootstrap SGD estimators, yielding asymptotically valid bootstrap confidence intervals. We validate the effectiveness of both approaches through numerical experiments. Furthermore, our analysis yields an intermediate result: the in-expectation error convergence rate for the original SGD estimator in nonconvex settings, which is comparable to existing results for convex problems. We believe this novel finding holds independent interest and enriches the literature on optimization and statistical inference.