The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexity

Abstract:We study the problem of sampling from target distributions whose potentials are simultaneously non-smooth, subject to superlinear gradient growth, and non-convex. We introduce the Subgradient Tamed Unadjusted Langevin Algorithm (SG-TULA), a discretisation of the Langevin diffusion that operates directly on subgradients, without relying on computationally demanding smoothing procedures. To handle the superlinear regime, taming techniques are employed to produce a stable, explicit scheme. We derive non-asymptotic convergence bounds in Wasserstein-2 distance, with all constants tracked explicitly in terms of dimension and inverse temperature, improving upon the currently known rates for subgradient-based Langevin algorithms. We further provide excess risk estimates for the associated optimisation problem. We verify the assumptions, with explicit constants, for the regularized pretraining potential of a LLM in the GPT-2 lineage and the boosted coordinate-wise variant of SG-TULA pretrains the former competitively against finetuned AdamW and Muon, for which no comparable non-asymptotic guarantees are presently available.

Machine Learning (cs.LG); Optimization and Control (math.OC); Probability (math.PR); Machine Learning (stat.ML)

arXiv:2608.06283 [cs.LG]

(or arXiv:2608.06283v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.06283 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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