High Confidence Level Inference is Almost Free using Parallel Stochastic Optimization

Uncertainty quantification for estimation through stochastic optimization solutions in an online setting has gained popularity recently. This paper introduces a novel inference method focused on constructing confidence intervals with efficient computation and fast convergence to the nominal level. Specifically, we propose to use a small number of independent multi-runs to acquire distribution information and construct a t-based confidence interval. Our method requires minimal additional computation and memory beyond the standard updating of estimates, making the inference process almost cost-free. We provide a rigorous theoretical guarantee for the confidence interval, demonstrating that the coverage is approximately exact with an explicit convergence rate and allowing for high confidence level inference. In particular, a new Gaussian approximation result is developed for the online estimators to characterize the coverage properties of our confidence intervals in terms of relative errors. Additionally, our method also allows for leveraging parallel computing to further accelerate calculations using multiple cores. It is easy to implement and can be integrated with existing stochastic algorithms without the need for complicated modifications.

Weighted Averaged Stochastic Gradient Descent: Asymptotic Normality and Optimality

Stochastic Gradient Descent (SGD) is one of the simplest and most popular algorithms in modern statistical and machine learning due to its computational and memory efficiency. Various averaging schemes have been proposed to accelerate the convergence of SGD in different settings. In this paper, we explore a general averaging scheme for SGD. Specifically, we establish the asymptotic normality of a broad range of weighted averaged SGD solutions and provide asymptotically valid online inference approaches. Furthermore, we propose an adaptive averaging scheme that exhibits both optimal statistical rate and favorable non-asymptotic convergence, drawing insights from the optimal weight for the linear model in terms of non-asymptotic mean squared error (MSE).