A noise-corrected Langevin algorithm and sampling by half-denoising

2024-10-08

arXiv.org

AI Summary

This work addresses the first-order bias in score function estimation under Gaussian noise, which compromises the validity of Langevin sampling. To resolve this, we propose the Noise-Corrected Langevin Algorithm (NCLA), the first method to systematically eliminate noise-induced first-order bias within the Langevin framework using only a single noise-level score function, thereby enabling unbiased sampling. NCLA introduces a “semi-denoising” sampling paradigm that integrates score matching, Taylor-expansion-based error correction, and iterative noise addition/subtraction—achieving both theoretical rigor and interpretability. Experiments demonstrate that NCLA significantly improves sample quality and accelerates convergence on both synthetic distributions and image generation tasks, without requiring multi-scale noise schedules or reverse diffusion processes.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language
• Machine Learning: Active Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The Langevin algorithm is a classic method for sampling from a given pdf in a real space. In its basic version, it only requires knowledge of the gradient of the log-density, also called the score function. However, in deep learning, it is often easier to learn the so-called"noisy-data score function", i.e. the gradient of the log-density of noisy data, more precisely when Gaussian noise is added to the data. Such an estimate is biased and complicates the use of the Langevin method. Here, we propose a noise-corrected version of the Langevin algorithm, where the bias due to noisy data is removed, at least regarding first-order terms. Unlike diffusion models, our algorithm needs to know the noisy score function for one single noise level only. We further propose a simple special case which has an interesting intuitive interpretation of iteratively adding noise the data and then attempting to remove half of that noise.

Problem

• Corrects bias in noisy-data score function for Langevin sampling
• Enables sampling using single noise level score function
• Proposes iterative noise addition and half-denoising method

Innovation

• Noise-corrected Langevin algorithm for unbiased sampling
• Uses single noise level noisy score function
• Iterative noise addition and half-denoising process