HiGrad: Uncertainty Quantification for Online Learning and Stochastic Approximation

TL;DR

This paper introduces HiGrad, a novel hierarchical inference procedure for online learning with stochastic gradient descent, providing statistically valid confidence intervals without extra computational cost.

Contribution

HiGrad is a new hierarchical method that enables statistical inference for SGD-based online learning, using a decorrelation technique and covariance structures for confidence intervals.

Findings

HiGrad achieves asymptotically exact coverage probability.

The method performs well in simulations and real data applications.

An R package 'higrad' is developed for implementation.

Abstract

Stochastic gradient descent (SGD) is an immensely popular approach for online learning in settings where data arrives in a stream or data sizes are very large. However, despite an ever-increasing volume of work on SGD, much less is known about the statistical inferential properties of SGD-based predictions. Taking a fully inferential viewpoint, this paper introduces a novel procedure termed HiGrad to conduct statistical inference for online learning, without incurring additional computational cost compared with SGD. The HiGrad procedure begins by performing SGD updates for a while and then splits the single thread into several threads, and this procedure hierarchically operates in this fashion along each thread. With predictions provided by multiple threads in place, a $t$-based confidence interval is constructed by decorrelating predictions using covariance structures given by a Donsker-style extension of the Ruppert--Polyak averaging scheme, which is a technical contribution of independent interest. Under certain regularity conditions, the HiGrad confidence interval is shown to attain asymptotically exact coverage probability. Finally, the performance of HiGrad is evaluated through extensive simulation studies and a real data example. An R package \texttt{higrad} has been developed to implement the method.