Faster high-accuracy log-concave sampling via algorithmic warm starts

• URL: http://arxiv.org/abs/2302.10249v1
• Date: Mon, 20 Feb 2023 19:27:21 GMT
• Title: Faster high-accuracy log-concave sampling via algorithmic warm starts
• Authors: Jason M. Altschuler and Sinho Chewi
• Abstract summary: In practice, high-accuracy samplers such as the classical Metropolis-adjusted Langevin algorithm (MALA) remain the de facto gold standard. We improve the dimension dependence of this sampling problem to $tildeO(d1/2)$, whereas the previous best result for MALA was $tildeO(d)$. Our main technical contribution settles this problem by establishing the first $tildeO(d1/2)$ R'enyi mixing rates for the discretized underdamped Langevin diffusion.
• Score: 6.117084972237769
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Understanding the complexity of sampling from a strongly log-concave and log-smooth distribution $\pi$ on $\mathbb{R}^d$ to high accuracy is a fundamental problem, both from a practical and theoretical standpoint. In practice, high-accuracy samplers such as the classical Metropolis-adjusted Langevin algorithm (MALA) remain the de facto gold standard; and in theory, via the proximal sampler reduction, it is understood that such samplers are key for sampling even beyond log-concavity (in particular, for distributions satisfying isoperimetric assumptions). In this work, we improve the dimension dependence of this sampling problem to $\tilde{O}(d^{1/2})$, whereas the previous best result for MALA was $\tilde{O}(d)$. This closes the long line of work on the complexity of MALA, and moreover leads to state-of-the-art guarantees for high-accuracy sampling under strong log-concavity and beyond (thanks to the aforementioned reduction). Our starting point is that the complexity of MALA improves to $\tilde{O}(d^{1/2})$, but only under a warm start (an initialization with constant R\'enyi divergence w.r.t. $\pi$). Previous algorithms took much longer to find a warm start than to use it, and closing this gap has remained an important open problem in the field. Our main technical contribution settles this problem by establishing the first $\tilde{O}(d^{1/2})$ R\'enyi mixing rates for the discretized underdamped Langevin diffusion. For this, we develop new differential-privacy-inspired techniques based on R\'enyi divergences with Orlicz--Wasserstein shifts, which allow us to sidestep longstanding challenges for proving fast convergence of hypocoercive differential equations.

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