A noise-corrected Langevin algorithm and sampling by half-denoising (2410.05837v2)

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.

• The paper demonstrates that adjusting step size and noise variance in Langevin dynamics counteracts biases from noisy score functions.
• The paper introduces half-denoising, an iterative method inspired by the Tweedie-Miyasawa theorem, which simplifies the denoising process.
• Experimental results on Gaussian mixtures and high-dimensional models confirm the method's efficiency and competitive mixing speed compared to oracle approaches.

Noise-Corrected Langevin Algorithm and Half-Denoising Sampling

This paper introduces a noise-corrected variant of the Langevin algorithm for effective sampling in the presence of a noisy score function, typically encountered in deep learning. The challenge addressed here arises from the bias introduced by using the noisy score function—derived from data with added Gaussian noise—in standard Langevin dynamics. This paper proposes a novel method to correct this bias, thereby enabling sampling from the original, noise-free data distribution.

Central Contributions

The paper’s main contributions are twofold:

• Noise-Corrected Langevin Algorithm: The authors present a modified Langevin algorithm that effectively utilizes the noisy score function to sample from the noise-free data distribution. This is achieved by adjusting the Langevin dynamics' step size and noise variance parameters, countering the bias introduced by using noisy score functions. The convergence of this method is demonstrated up to first-order terms, providing theoretical assurance of its efficacy.
• Sampling by Half-Denoising: A special case of the proposed algorithm, termed "half-denoising," is introduced. This procedure involves iteratively adding Gaussian noise and then removing only half of it, inspired by the Tweedie-Miyasawa theorem. This intuitive approach simplifies the denoising process and serves as a computationally efficient sampling method.

Theoretical and Practical Implications

The theoretical contribution lies in extending the Langevin algorithm such that it operates accurately using only one noise level's score function, which differs from existing methods like diffusion models that require multiple noise levels. This single-noise-level approach not only simplifies the model complexity but also decreases computational demands, making it a compelling option for practical applications.

Practically, this proposed algorithm promises enhancement in generative modeling tasks by allowing efficient sampling without extensive parameter tuning across different noise levels, a limitation evident in techniques such as annealed Langevin dynamics.

Experimental Validation

The paper presents empirical validation through experiments with Gaussian mixture models and high-dimensional Gaussian distributions. The results indicate that the proposed method reliably matches the performance of the Oracle Langevin method—known for using the true, noise-free score function—demonstrating its proficiency in bias reduction. Moreover, the mixing speed of the half-denoising approach is shown to be comparable to traditional methods, further validating its practical utility.

Future Directions

The proposed method opens avenues for further exploration, including:

• Integration with Annealing Schedules: Investigating annealing strategies, particularly ones utilizing the single-level noise characteristic, can potentially optimize the sampling process further.
• Exploration of Underdamped Versions: Modifying the approach to include underdamped dynamics could enhance sampling efficiency in complex, multi-modal distributions.
• Relation to Existing Diffusion and Walk-Jump Models: Understanding how this novel noise correction and half-denoising method compares and potentially collaborates with existing methods like diffusion models or heuristic Walk-Jump approaches.

Conclusion

This paper contributes a significant advancement in the domain of generative models and MCMC methods by rectifying the biases induced by noisy score functions, proposing a computationally efficient and theoretically sound alternative to existing diffusion-based methods. The noise-corrected Langevin algorithm and half-denoising sampling present robust tools for future exploration and implementation in generative AI systems.