Asymptotics of Stochastic Gradient Descent with Dropout Regularization in Linear Models
This work addresses the need for rigorous statistical inference in machine learning algorithms using dropout regularization, offering incremental theoretical advancements for researchers in optimization and statistics.
This paper tackles the problem of providing asymptotic theory for stochastic gradient descent with dropout regularization in linear models, establishing geometric-moment contraction and central limit theorems for the iterates, and demonstrates that confidence intervals for averaged SGD with dropout achieve near-nominal coverage probability in large samples.
This paper proposes an asymptotic theory for online inference of the stochastic gradient descent (SGD) iterates with dropout regularization in linear regression. Specifically, we establish the geometric-moment contraction (GMC) for constant step-size SGD dropout iterates to show the existence of a unique stationary distribution of the dropout recursive function. By the GMC property, we provide quenched central limit theorems (CLT) for the difference between dropout and $\ell^2$-regularized iterates, regardless of initialization. The CLT for the difference between the Ruppert-Polyak averaged SGD (ASGD) with dropout and $\ell^2$-regularized iterates is also presented. Based on these asymptotic normality results, we further introduce an online estimator for the long-run covariance matrix of ASGD dropout to facilitate inference in a recursive manner with efficiency in computational time and memory. The numerical experiments demonstrate that for sufficiently large samples, the proposed confidence intervals for ASGD with dropout nearly achieve the nominal coverage probability.