Controlling the Flow: Stability and Convergence for Stochastic Gradient Descent with Decaying Regularization

TL;DR

This paper investigates a variant of stochastic gradient descent with decaying Tikhonov regularization, proving its convergence to the minimum-norm solution and analyzing the optimal decay strategies for stable learning in convex problems.

Contribution

It introduces a regularized SGD method with time-dependent regularization, providing convergence proofs and insights into stability and rate optimization for convex minimization.

Findings

Proves strong convergence of reg-SGD to minimum-norm solution.

Quantifies convergence rates and optimal regularization decay.

Validates theoretical results with numerical experiments on inverse problems.

Abstract

The present article studies the minimization of convex, L-smooth functions defined on a separable real Hilbert space. We analyze regularized stochastic gradient descent (reg-SGD), a variant of stochastic gradient descent that uses a Tikhonov regularization with time-dependent, vanishing regularization parameter. We prove strong convergence of reg-SGD to the minimum-norm solution of the original problem without additional boundedness assumptions. Moreover, we quantify the rate of convergence and optimize the interplay between step-sizes and regularization decay. Our analysis reveals how vanishing Tikhonov regularization controls the flow of SGD and yields stable learning dynamics, offering new insights into the design of iterative algorithms for convex problems, including those that arise in ill-posed inverse problems. We validate our theoretical findings through numerical experiments on image reconstruction and ODE-based inverse problems.