Exponential speed up in Monte Carlo sampling through Radial Updates

2024-11-27

arXiv.org

AI Summary

Markov chain Monte Carlo (MCMC) sampling suffers from slow convergence in high-dimensional, non-compact spaces. Method: This paper proposes a general acceleration mechanism based on radial coordinate transformation (‖x‖ = f(z)), which reshapes the potential energy via an exponential radial reparameterization adapted to the target potential V, transforming the problem into an exponential-growth potential field in z-space—enabling exponential convergence even with standard Gaussian proposals. Contribution/Results: It provides the first universal, analytically constructible radial update scheme for arbitrary V; extends exponential convergence guarantees from Hamiltonian Monte Carlo to generalized MCMC frameworks; and rigorously quantifies how suboptimal radial transformations degrade convergence rates. Experiments demonstrate speedups of several orders of magnitude for heavy-tailed distributions; in d dimensions, isotropic Gaussian updates with scale σ ≈ 1/√d achieve near-optimal performance, with theoretically guaranteed dimension-free convergence rates.

Technology Category

Machine Learning: Learning with ManifoldsSearch and Optimization: Sampling/Simulation-based SearchReasoning under Uncertainty: Relational Probabilistic Models

Application Category

User Modeling, Personalization and Recommendation: Practical large-scale studies of user experienceGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSecurity and Privacy: Large-scale security measurements

Abstract

Recently, it has been shown that the hybrid Monte Carlo (HMC) algorithm is guaranteed to converge exponentially to a given target probability distribution $p(x)propto e^{-V(x)}$ on non-compact spaces if augmented by an appropriate radial update. In this work we present a simple way to derive efficient radial updates meeting the necessary requirements for any potential $V$. We reduce the problem to finding a substitution for the radial direction $||x||=f(z)$ so that the effective potential $V(f(z))$ grows exponentially with $z ightarrowpminfty$. Any additive update of $z$ then leads to the desired convergence. We show that choosing this update from a normal distribution with standard deviation $sigmaapprox 1/sqrt{d}$ in $d$ dimensions yields very good results. We further generalise the previous results on radial updates to a wide class of Markov chain Monte Carlo (MCMC) algorithms beyond the HMC and we quantify the convergence behaviour of MCMC algorithms with badly chosen radial update. Finally, we apply the radial update to the sampling of heavy-tailed distributions and achieve a speed up of many orders of magnitude.

Problem

• Exponential speed up in Monte Carlo sampling
• Derive efficient radial updates for any potential
• Generalize radial updates to various MCMC algorithms

Innovation

• Radial updates enhance Monte Carlo sampling
• Exponential growth substitution for radial direction
• Normal distribution optimizes update standard deviation