# The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexity

arXiv: https://arxiv.org/abs/2608.06283 · HTML: https://mlsanity.com/paper/2608.06283/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 83 references that are in the map.
- First posted on arXiv 6 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Nemotron-CLIMB: CLustering-based Iterative Data Mixture Bootstrapping for Language Model Pre-training](https://mlsanity.com/paper/2504.13161/) (2025)
- [A fully data-driven approach to minimizing CVaR for portfolio of assets via SGLD with discontinuous updating](https://mlsanity.com/paper/2007.01672/) (2020)
- [Ergodicity of Langevin Dynamics and its Discretizations for Non-smooth Potentials](https://mlsanity.com/paper/2411.12051/) (2024)
- [kTULA: A Langevin sampling algorithm with improved KL bounds under super-linear log-gradients](https://mlsanity.com/paper/2506.04878/) (2025)
- [Anchored Langevin Algorithms](https://mlsanity.com/paper/2509.19455/) (2025)
- [The ballistic limit of the log-Sobolev constant equals the Polyak-Lojasiewicz constant](https://mlsanity.com/paper/2411.11415/) (2024)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)
- [Error estimates for tamed Euler and Randomized Euler schemes for SDEs with locally Lipschitz drift with applications to non-logconcave sampling and optimization](https://mlsanity.com/paper/2605.24937/) (2026)
- [When Langevin Monte Carlo Meets Randomization: New Sampling Algorithms with Non-asymptotic Error Bounds beyond Log-Concavity and Gradient Lipschitzness](https://mlsanity.com/paper/2509.25630/) (2025)

## Built on by (most cited first, 1 of 1)

- [Muon meets Tamed Langevin: Momentum Preconditioning beyond Convex and gradient-Lipschitz Potentials](https://mlsanity.com/paper/2610.02158/) (2026)

## Abstract (arXiv)

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.