Robust Langevin Sampling under State-Dependent Heavy-Tailed Noise with Applications to LLM Fine-Tuning

Existing convergence guarantees for non-log-concave sampling with stochastic gradient Langevin Monte Carlo (SG-LMC) typically assume globally Lipschitz gradients and uniformly bounded noise variance. Both assumptions can fail even in elementary sampling problems, and relaxing them better reflects practical applications such as Bayesian fine-tuning of large language models (LLMs). We relax both assumptions through generalized smoothness and our proposed state-dependent heavy-tailed oracle with finite p-th moments, 1<p≤ 2. The oracle permits infinite variance when p<2 and includes the Blum-Gladyshev (BG-0) condition. We study these relaxations separately and jointly, proposing three methods. Under globally Lipschitz gradients, robust LMC (R-LMC) handles the proposed oracle through geometric-median centering and residual clipping. With exact gradients, adaptive LMC (A-LMC) controls discretization error under generalized smoothness. Robust adaptive LMC (RA-LMC) addresses both relaxations through robust estimation and a noise-aware step size. For all three methods, we establish non-asymptotic convergence guarantees in Fisher information (FI) without assuming dissipativity or a functional inequality. Synthetic experiments demonstrate FI convergence and the benefit of each mechanism. Bayesian fine-tuning experiments with LLMs and vision-language models support our oracle model and show that RA-LMC matches or outperforms state-of-the-art baselines in accuracy and uncertainty quantification.