Reverse-BSDE Monte Carlo

2025-05-11

AI Summary

Sampling from high-dimensional Bayesian posterior distributions is challenging due to unknown normalization constants and intractable, noisy gradient estimates of the log-density. Method: This paper proposes a novel diffusion sampling paradigm that avoids explicit computation of log-density gradients. It reformulates diffusion-based generative modeling as a forward-backward stochastic differential equation (FBSDE) solving problem, proves uniqueness of the solution, and develops an end-to-end sampling framework integrating backward SDE simulation, Monte Carlo estimation, and a deep BSDE solver. Contribution/Results: The method enables stable and efficient sampling solely from the unnormalized density—without gradient estimation—by leveraging the FBSDE formulation. Experiments demonstrate significant improvements in accuracy and robustness for multimodal Bayesian posterior sampling in high dimensions, outperforming gradient-dependent diffusion baselines.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Bayesian Learning

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Recently, there has been a growing interest in generative models based on diffusions driven by the empirical robustness of these methods in generating high-dimensional photorealistic images and the possibility of using the vast existing toolbox of stochastic differential equations. %This remarkable ability may stem from their capacity to model and generate multimodal distributions. In this work, we offer a novel perspective on the approach introduced in Song et al. (2021), shifting the focus from a"learning"problem to a"sampling"problem. To achieve this, we reformulate the equations governing diffusion-based generative models as a Forward-Backward Stochastic Differential Equation (FBSDE), which avoids the well-known issue of pre-estimating the gradient of the log target density. The solution of this FBSDE is proved to be unique using non-standard techniques. Additionally, we propose a numerical solution to this problem, leveraging on Deep Learning techniques. This reformulation opens new pathways for sampling multidimensional distributions with densities known up to a normalization constant, a problem frequently encountered in Bayesian statistics.

Problem

• Reformulating diffusion models as FBSDE to avoid gradient estimation
• Proving unique solution for FBSDE using non-standard techniques
• Enabling sampling of multidimensional distributions in Bayesian statistics

Innovation

• Reformulates diffusion models as FBSDE
• Avoids pre-estimating log density gradient
• Uses Deep Learning for numerical solution