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

Simon Weissmann (University of Mannheim) · Leif Döring (Universität Mannheim) · Sebastian Kassing (University of Wuppertal)

boundedness assumptionsconvex optimizationill-posed inverse problemsiterative algorithmsl-smooth functionsminimum-norm solutionnumerical experimentsrate of convergenceregularization decayregularized stochastic gradient descentseparable hilbert spacestable learning dynamicsstep-sizes optimizationstrong convergencetikhonov regularizationvanishing regularization parameter

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.