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

The first text provides a high-level overview of a comprehensive research paper focusing on relaxing convergence guarantees for Stochastic Gradient Langevin Monte Carlo (SG-LMC) in non-logconcave settings, introducing novel oracle models (BG p) and three corresponding sampling methods (RLMC, A-LMC, RA-LMC). The second text appears to be a more technical excerpt detailing the specific mathematical derivations, finite horizon bounds, and experimental setup for one of these robust methods under state-dependent heavy-tailed noise.

My task is to synthesize these into a single, long, and detailed summary that captures the full scope of the work.

Comprehensive Research Summary: Robust Langevin Sampling Under State-Dependent Heavy-Tailed Noise

This research addresses the limitations of existing convergence guarantees for Stochastic Gradient Langevin Monte Carlo (SG-LMC) methods when applied to complex sampling problems, particularly those arising in practical applications like Bayesian fine-tuning of Large Language Models (LLMs). The core innovation lies in developing a flexible framework that relaxes traditional assumptions—namely, globally Lipschitz gradients and uniformly bounded noise variance—through the introduction of a state-dependent heavy-tailed oracle and subsequent robust sampling strategies.

Core Problem and Motivation

Traditional convergence proofs for LMC methods often rely on stringent assumptions like global Lipschitz continuity of the gradient or bounded noise variance. These assumptions are frequently violated in real-world scenarios, especially when fine-tuning massive models, where gradients can exhibit complex behavior (e.g., state dependence) and stochastic noise can be heavy-tailed (i.e., possessing moments higher than two). Relaxing these assumptions is crucial for accurately modeling practical optimization landscapes.

Key Theoretical Contributions: The BGp Oracle

The central theoretical contribution is the proposal of a state-dependent heavy-tailed oracle, denoted as BG p. This oracle generalizes the standard Blum–Gladyshev (BG 0) condition by incorporating a state-independent bound on the p-th moment of the gradient noise, where 1 < p at most 2.

• Heavy-Tailed Noise Handling: The introduction of BG p allows for handling noise models with infinite variance when p < 2, which is a significant advancement over standard assumptions.

• Robustness: This oracle enables the establishment of non-asymptotic Fisher Information (FI) guarantees for Robust Langevin Monte Carlo (R-LMC), allowing the analysis to accommodate both state dependence and heavy-tailed noise simultaneously.

Proposed Robust Sampling Methods

The research proposes three distinct sampling methodologies designed to leverage the BG p oracle and address the various relaxations:

1. Robust Langevin Monte Carlo (RLMC): This method utilizes geometric-median centering for its state estimation and employs residual clipping based on the oracle's bounds. Theorem 3.2 establishes non-asymptotic FI guarantees for R-LMC under the BG p oracle, yielding complexity bounds dependent on B, G, p, and epsilon.

2. Adaptive LMC (A-LMC): This approach focuses on sampling under generalized smoothness (Assumption 2.3) while assuming access to exact gradients. Theorem 3.3 establishes FI guarantees for A-LMC under symmetric generalized smoothness, showing that the complexity bounds recover the standard globally smooth LMC rate in the special case of global smoothness.

3. Robust Adaptive LMC (RA-LMC): This is the most comprehensive method, designed to tackle both relaxations simultaneously. RA-LMC integrates robust estimation with a noise-aware step size, effectively controlling the dependence between gradient estimation errors and adaptive step sizes. Theorem 3.4 provides non-asymptotic FI guarantees for RA-LMC under generalized smoothness and BG p noise.

Detailed Analysis and Mathematical Derivations

The analysis proceeds in a three-stage framework:

1. Oracle Proposal: Defining the BG p oracle (Equation 3.1) to capture state dependence and heavy tails.

2. Adaptive Scheme Development: Developing an adaptive step-size scheme under generalized smoothness (Assumption 2.3) when exact gradients are available (Theorem 3.3).

3. Robust Integration: Combining robust estimation with the adaptive step size to form RA-LMC and deriving its FI bounds (Theorem 3.4).

The technical derivations provided in the supplementary material detail the rigorous bounding process:

• Specific equations (e.g., Equations 104 through 111) are used to bound estimation errors (e) and drift norms (g).

• These bounds lead to deterministic coefficients (JT, h and chi T, h) which are then substituted back into the main FI inequality (Equation 57).

• The final result demonstrates that the FI bound for RA-LMC under generalized smoothness and BG p noise is bounded by:

FI(T | pi) at most FIT at most 2K 0 T + 9D gamma(h) + epsilon squared

where the term 9D gamma(h) is shown to be bounded by epsilon 4 + 9 epsilon 64 + epsilon squared, ultimately yielding a bound of 57 epsilon 64 < epsilon.

Empirical Validation and Findings

The theoretical framework is strongly supported by both synthetic experiments and practical applications:

• Synthetic Experiments: These verify the convergence of FI for all three methods (RLMC, A-LMC, RA-LMC) and quantitatively demonstrate the benefit derived from each mechanism. Figure 1 illustrates that robust estimation and adaptive steps individually improve FI convergence under heavy-tailed noise and generalized smoothness, while RA-LMC excels when both challenges coexist.

• Bayesian Fine-Tuning: Experiments involving Bayesian LoRA fine-tuning on LLMs and Vision-Language Models (VLMs) provide empirical validation for the proposed BG p oracle model. Crucially, the results show that ** RA-LMC matches or outperforms leading state-of-the-art baselines in both predictive accuracy and uncertainty quantification.**

• Error Analysis: Histograms (Figure 2(a) and (b)) confirm heavy-tailed behavior in stochastic gradient errors during LoRA fine-tuning, while plots (Figure 3) support the necessity of the state-dependent term within the BG p oracle. Furthermore, Figure 4 highlights that RA-LMC closely tracks exact-gradient A-LMC and exhibits lower estimated FI than fixed-step LMC or even standard adaptive LMC (A-SG-LMC) when both state dependence and growing curvature are present.

Conclusion

The paper successfully extends the Fisher Information (FI) framework for nonasymptotic convergence in Langevin sampling beyond classical assumptions. The introduction of the BG p oracle provides a powerful tool to handle state-dependent heavy-tailed noise, and the proposed RA-LMC method effectively controls the interaction between robust estimation and adaptive step sizes under generalized smoothness. This unified approach establishes rigorous nonasymptotic guarantees for time-averaged laws without requiring assumptions like dissipativity or functional inequalities. The empirical evidence from LLM fine-tuning strongly suggests that RA-LMC is a superior methodology for achieving high accuracy and reliable uncertainty quantification in complex Bayesian optimization tasks.

1. Robust LLM Fine-Tuning under Heavy-Tailed Noise: The RA-LMC method can generate approximate posterior samples for LLMs and vision-language models that match or outperform state-of-the-art baselines in accuracy and uncertainty quantification, as shown in Bayesian fine-tuning experiments.

2. Handling State Dependence and Infinite Variance: The proposed BG-p oracle allows the sampler to accommodate "infinite variance when p < 2" while including the Blum–Gladyshev (BG–0) condition, enabling sampling from non-log-concave targets where existing methods fail due to state-dependent noise growth.

3. Adaptive Step Size for Non-Log-Concavity: The Robust Adaptive LMC (RA-LMC) can control discretization error under generalized smoothness by using a noise-aware step size defined by equation (8), which addresses the coupling between estimation error and gradient-dependent local curvature.

4. Enhanced Gradient Estimation for Non-Convex Optimization: By employing robust gradient estimation via geometric median centering and residual clipping, the system can achieve nonasymptotic Fisher information (FI) convergence guarantees even when facing state-dependent heavy-tailed noise and generalized smoothness.

5. Improved Uncertainty Quantification: The RA-LMC framework provides explicit iteration and oracle complexities, allowing for precise prediction of computational cost in Bayesian fine-tuning tasks, which is crucial for efficiently scaling LLM training with LoRA.

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 at most 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.