Statistical Inference for Online Algorithms

TL;DR

This paper introduces HulC, a computationally efficient method for constructing valid confidence intervals for online algorithms without explicit variance estimation, demonstrated with SGD variants.

Contribution

The paper presents HulC, a novel wrapper method that provides asymptotically valid confidence regions for any online algorithm with asymptotic normality, bypassing variance estimation.

Findings

HulC achieves rate-optimal confidence intervals for online estimators.

Numerical simulations show HulC performs well with SGD, implicit-SGD, and ROOT-SGD.

HulC is computationally efficient and bypasses traditional variance estimation challenges.

Abstract

The construction of confidence intervals and hypothesis tests for functionals is a cornerstone of statistical inference. Traditionally, the most efficient procedures - such as the Wald interval or the Likelihood Ratio Test - require both a point estimator and a consistent estimate of its asymptotic variance. However, when estimators are derived from online or sequential algorithms, computational constraints often preclude multiple passes over the data, complicating variance estimation. In this article, we propose a computationally efficient, rate-optimal wrapper method (HulC) that wraps around any online algorithm to produce asymptotically valid confidence regions bypassing the need for explicit asymptotic variance estimation. The method is provably valid for any online algorithm that yields an asymptotically normal estimator. We evaluate the practical performance of the proposed method primarily using Stochastic Gradient Descent (SGD) with Polyak-Ruppert averaging. Furthermore, we provide extensive numerical simulations comparing the performance of our approach (HulC) when used with other online algorithms, including implicit-SGD and ROOT-SGD.