High-accuracy sampling from constrained spaces with the Metropolis-adjusted Preconditioned Langevin Algorithm

• URL: http://arxiv.org/abs/2412.18701v2
• Date: Tue, 31 Dec 2024 00:05:59 GMT
• Title: High-accuracy sampling from constrained spaces with the Metropolis-adjusted Preconditioned Langevin Algorithm
• Authors: Vishwak Srinivasan, Andre Wibisono, Ashia Wilson,
• Abstract summary: We propose a first-order sampling method for approximate sampling from a target distribution whose support is a proper convex subset of $mathbbRd$. Our proposed method is the result of applying a Metropolis-Hastings filter to the Markov chain formed by a single step of the preconditioned Langevin algorithm.
• Score: 12.405427902037971
• License:
• Abstract: In this work, we propose a first-order sampling method called the Metropolis-adjusted Preconditioned Langevin Algorithm for approximate sampling from a target distribution whose support is a proper convex subset of $\mathbb{R}^{d}$. Our proposed method is the result of applying a Metropolis-Hastings filter to the Markov chain formed by a single step of the preconditioned Langevin algorithm with a metric $\mathscr{G}$, and is motivated by the natural gradient descent algorithm for optimisation. We derive non-asymptotic upper bounds for the mixing time of this method for sampling from target distributions whose potentials are bounded relative to $\mathscr{G}$, and for exponential distributions restricted to the support. Our analysis suggests that if $\mathscr{G}$ satisfies stronger notions of self-concordance introduced in Kook and Vempala (2024), then these mixing time upper bounds have a strictly better dependence on the dimension than when is merely self-concordant. We also provide numerical experiments that demonstrates the practicality of our proposed method. Our method is a high-accuracy sampler due to the polylogarithmic dependence on the error tolerance in our mixing time upper bounds.

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