On the Convergence of HalpernSGD

Vittorio Colao; Katherine Rossella Foglia

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arXiv · 2601.18906

On the Convergence of HalpernSGD

Vittorio Colao·Katherine Rossella Foglia

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Abstract

We study HalpernSGD for minimizing a convex Fr\'echet differentiable function with Lipschitz gradient on a real Hilbert space. Its deterministic orbit converges strongly to the anchor-selected minimizer $P_S(u)$ under vanishing steps that need not be square summable. For stochastic oracles, weighted summability of the scaled martingale fluctuation and the $\F_n$-measurable bias yields almost sure strong convergence, while expected last-iterate bounds require suffix regularity of the error budget. In the unbiased case, for the critical choice $\varepsilon_n=n^{-1/2}$, $\alpha_n=(\log n)/n$ eventually, a uniform conditional moment bound of order $p>4$ gives both conclusions. We also discuss finite-difference and zeroth-order applications for which scaled, rather than raw, noise is the natural quantity.

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BibTeX RIS

Vittorio Colao, Katherine Rossella Foglia. 2026-01-26. On the Convergence of HalpernSGD. https://arxiv.org/abs/2601.18906

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