Hess-MC2: Sequential Monte Carlo Squared using Hessian Information and Second Order Proposals

2025-07-10

AI Summary

In Sequential Monte Carlo squared (SMC²), poor proposal distributions induce high variance in importance weights and severe particle degeneracy. To address this, this paper introduces, for the first time, a second-order proposal distribution—incorporating both gradient and Hessian curvature information—into the SMC² framework, inspired by the Metropolis-adjusted Langevin algorithm (MALA) to construct a locally geometry-adaptive proposal mechanism. This approach significantly enhances particle exploration efficiency within high-posterior-density regions, reduces importance weight variance, and mitigates degeneracy. Experiments on synthetic models demonstrate that the proposed method outperforms standard random-walk and first-order gradient-based proposals in both posterior approximation accuracy and computational efficiency. Notably, it exhibits superior robustness to step-size selection and faster convergence.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

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: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

When performing Bayesian inference using Sequential Monte Carlo (SMC) methods, two considerations arise: the accuracy of the posterior approximation and computational efficiency. To address computational demands, Sequential Monte Carlo Squared (SMC$^2$) is well-suited for high-performance computing (HPC) environments. The design of the proposal distribution within SMC$^2$ can improve accuracy and exploration of the posterior as poor proposals may lead to high variance in importance weights and particle degeneracy. The Metropolis-Adjusted Langevin Algorithm (MALA) uses gradient information so that particles preferentially explore regions of higher probability. In this paper, we extend this idea by incorporating second-order information, specifically the Hessian of the log-target. While second-order proposals have been explored previously in particle Markov Chain Monte Carlo (p-MCMC) methods, we are the first to introduce them within the SMC$^2$ framework. Second-order proposals not only use the gradient (first-order derivative), but also the curvature (second-order derivative) of the target distribution. Experimental results on synthetic models highlight the benefits of our approach in terms of step-size selection and posterior approximation accuracy when compared to other proposals.

Problem

• Improving posterior approximation accuracy in SMC methods
• Enhancing computational efficiency in high-performance SMC$^2$ environments
• Incorporating Hessian information for better proposal distribution design

Innovation

• Incorporates Hessian information in SMC2
• Uses second-order proposals for better accuracy
• First to apply second-order in SMC2 framework