Stochastic Localization via Iterative Posterior Sampling (2402.10758v2)

Abstract: Building upon score-based learning, new interest in stochastic localization techniques has recently emerged. In these models, one seeks to noise a sample from the data distribution through a stochastic process, called observation process, and progressively learns a denoiser associated to this dynamics. Apart from specific applications, the use of stochastic localization for the problem of sampling from an unnormalized target density has not been explored extensively. This work contributes to fill this gap. We consider a general stochastic localization framework and introduce an explicit class of observation processes, associated with flexible denoising schedules. We provide a complete methodology, $\textit{Stochastic Localization via Iterative Posterior Sampling}$ (SLIPS), to obtain approximate samples of this dynamics, and as a by-product, samples from the target distribution. Our scheme is based on a Markov chain Monte Carlo estimation of the denoiser and comes with detailed practical guidelines. We illustrate the benefits and applicability of SLIPS on several benchmarks of multi-modal distributions, including Gaussian mixtures in increasing dimensions, Bayesian logistic regression and a high-dimensional field system from statistical-mechanics.

• The paper presents an efficient SLIPS algorithm that refines stochastic localization for robust sample generation using iterative posterior sampling.
• It leverages a denoising schedule and SNR-adapted observation processes to outperform traditional MCMC methods in convergence and accuracy.
• Empirical validations across benchmarks, including Gaussian mixtures and Bayesian logistic regression, demonstrate its effectiveness in high-dimensional settings.

Exploring the Depths of Stochastic Localization for Sample Generation

Insights into the Methodology

Stochastic Localization via Iterative Posterior Sampling (SLIPS) emerges against the backdrop of continuous innovation in sampling techniques, a crucial facet of probabilistic modeling and Bayesian inference. This methodology is delineated within a framework that reaches beyond traditional Markov Chain Monte Carlo (MCMC) algorithms, exploring a generalized observation process framework associated with flexible denoising schedules. It particularly emphasizes the inception of an explicit class of observation processes and the pertinent role of signal-to-noise ratio (SNR) scheduling in efficient sampling.

SLIPS encapsulates the refinement of stochastic localization (SL) practices, utilizing a Markov chain estimation of the denoiser to navigate through the distribution space with a minimized necessity for extensive manual tuning. The proposed algorithm proficiently broaches distributions of increasing complexity and dimensionality, exhibiting commendable robustness and highlighting the potential superiority over contemporary methodologies.

Theoretical Underpinnings and Numerical Validation

The research delineates a comprehensive exploration into the SL framework, unraveling the intrinsic relationships between the observation dynamics, denoising trajectories, and the ensued sample generation process. A theoretical analysis affirms that under specified conditions concerning the target distributions, the SLIPS scheme achieves a convergence rate that aligns with, or excels, that of prevalent sampling algorithms.

Empirically, SLIPS' effectiveness is showcased across various benchmarks, including Gaussian mixtures and Bayesian logistic regression, with performance metrics substantiating its capacity to capture the underlying distributional characteristics accurately. These insights are pivotal, suggesting the algorithm's applicability across a broad spectrum of statistical and machine learning tasks where sampling from complex distributions is imperative.

Implications and Future Trajectories

The research implicates a significant stride towards resolving long-standing challenges in sampling from unnormalized densities, particularly in high-dimensional settings where traditional methods falter. By elucidating the duality of log-concavity and its implications for posterior sampling, the paper paves the way for more nuanced understanding and strategies in constructing efficient sampling algorithms.

Moreover, the exploration of SNR-adapted discretization and the Langevin-within-Langevin initialization protocol provides fertile ground for future exploration. These notions invite broader contemplation on the interplay between denoising strategies and the efficiency of sampling processes, hinting at uncharted territories in the field of generative modeling and beyond.

Concluding Reflections

This research stands as a testament to the dynamic evolution within the field of generative AI and probabilistic modeling, marking a notable advancement in our quest for efficient and robust sampling methodologies. As we move forward, the insights garnered from the SLIPS framework will undoubtedly influence future endeavors in statistical inference, machine learning model development, and the broader pursuit of understanding complex distributions. The convergence of theoretical rigor and practical efficacy in SLIPS not only enriches the current landscape but also sets a precedent for transformative research directions in the science of sample generation.