On the Convergence of HalpernSGD

TL;DR

This paper introduces HalpernSGD, a simple stochastic gradient method with convergence guarantees for convex smooth functions, achieving almost sure convergence without variance reduction or complex techniques.

Contribution

It provides the first almost sure convergence proof for a Halpern-type stochastic gradient method without requiring variance reduction or multi-point oracles.

Findings

Almost sure convergence to the minimizer under standard stepsize assumptions

Sublinear asymptotic estimate for the expected optimality gap

No full last iterate rate estimate achievable in this setting

Abstract

We study a stochastic anchored gradient scheme, namely HalpernSGD, which combines the classical Halpern iteration for finding a minimizer of a convex and $L$-smooth objective function with a stochastic {first-order} oracle. The algorithm is simple and does not require projections, line-search, or similar techniques. This provides, to the best of our knowledge, the first almost sure convergence guarantee for a Halpern-type stochastic gradient scheme, without requiring variance reduction or multi-point oracle mechanisms. Under standard stepsize assumptions, we prove that the iterates converge almost surely to the anchor-selected minimizer $x^*=P_S(u)$. In addition, for a natural choice of the step sequences, we derive a sublinear asymptotic estimate for the expected optimality gap, namely \( \liminf_{n\to\infty}\sqrt{n+1}\,\mathbb{E}\bigl[f(X_n)-f(x^*)\bigr]=0. \) As shown, a full last iterate rate estimate cannot be reached in the present setting.