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The self-normalized Cramér-type moderate deviation of the empirical measure of stochastic gradient Langevin dynamics is established, along with the derivation of the Berry–Esseen bound for SGLD. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

March 4, 2026Open Access

Self-normalized Cramér-type moderate deviation of stochastic gradient Langevin dynamics

HDHongsheng DaiNewcastle University XFXiequan FanUniversidad del Noreste JLJianya LuUniversity of Essex

Key Points

To investigate the self-normalized Cramér-type moderate deviation of the empirical measure in SGLD.
Constructed a stochastic differential equation to approximate SGLD
Applied Stein’s method to decompose the empirical measure
Analyzed the deviation properties of the empirical measure
Derivation of the Berry-Esseen bound for the SGLD
Characterization of the empirical measure as a martingale difference series
Identification of a negligible remainder term in the approximation

Abstract

Abstract In this paper, we study the self-normalized Cramér-type moderate deviation of the empirical measure of the stochastic gradient Langevin dynamics (SGLD). Consequently, we also derive the Berry–Esseen bound for the SGLD. Our approach is by constructing a stochastic differential equation to approximate the SGLD and then applying Stein’s method to decompose the empirical measure into a martingale difference series sum and a negligible remainder term.

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