[Paper Review] Rapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions

Oren Mangoubi, Aaron Smith|arXiv (Cornell University)|Aug 23, 2017

Markov Chains and Monte Carlo Methods21 references71 citations

TL;DR

The paper proves non-asymptotic mixing bounds for Hamiltonian Monte Carlo (HMC) on strongly log-concave targets, showing a dimension-free mixing rate for the ideal HMC and near-optimal gradient-cost scaling O(d^{1/4}) for practical leapfrog implementations.

ABSTRACT

We obtain several quantitative bounds on the mixing properties of the Hamiltonian Monte Carlo (HMC) algorithm for a strongly log-concave target distribution $π$ on $\mathbb{R}^{d}$, showing that HMC mixes quickly in this setting. One of our main results is a dimension-free bound on the mixing of an "ideal" HMC chain, which is used to show that the usual leapfrog implementation of HMC can sample from $π$ using only $\mathcal{O}(d^{\frac{1}{4}})$ gradient evaluations. This dependence on dimension is sharp, and our results significantly extend and improve previous quantitative bounds on the mixing of HMC.

Motivation & Objective

• Motivate and quantify the mixing properties of HMC for strongly log-concave distributions in Rd.
• Derive dimension-free mixing bounds for an ideal HMC dynamics and translate them to practical numerical implementations.
• Compare HMC’s performance to Langevin dynamics and ball walk under strong convexity assumptions.
• Provide gradient/evaluation cost bounds for unadjusted and Metropolis-adjusted HMC across integrator orders.

Proposed method

• Model HMC as a random-mapping with momentum resampling and Hamiltonian flow.
• Establish strong convexity assumptions on the potential U and a drift condition to obtain mixing bounds.
• Prove contraction in Wasserstein distances and spectral gap bounds for the ideal HMC kernel with integration time T proportional to (m2/M2)^{-1/2}.
• Use ODE comparison and coupling arguments to analyze the effect of momentum on step sizes and exploration.
• Extend to approximate HMC dynamics using first- and higher-order integrators under separability and regularity assumptions.
• Quantify computational costs in terms of gradient evaluations for unadjusted and Metropolis-adjusted schemes.

Experimental results

Research questions

• RQ1 How fast does HMC mix for strongly log-concave target distributions in high dimensions?
• RQ2 What are the dimension-dependence and constants in non-asymptotic mixing bounds for ideal HMC and its numerical implementations?
• RQ3 How do numerical integrators (order k and separability) affect mixing and computational cost for HMC?
• RQ4 How do HMC bounds compare with Langevin dynamics and ball walks under similar convexity assumptions?

Key findings

• For ideal HMC with integration time T = (1/2) sqrt(2) sqrt(m2/M2), the contraction in Wasserstein distance satisfies Wk(K(x, ·), K(y, ·)) / ||x−y|| ≤ 1 − (m2/M2)^2 / 64.
• The spectral relaxation time τrel(K) is bounded by (M2/m2)^2 / 64 for the ideal dynamics.
• Unadjusted HMC with a first-order Euler integrator achieves sampling from π with gradient-cost Od(d^{1/2}) under suitable θ; for higher-order integrators, cost improves to Od(d^{1/2k}) under separability.
• For higher-order HMC (k ≥ 2) under mild assumptions, the Prokhorov distance to π can be made ≤ ε with cost N(Qθ, I) = O∗(d^{1/(2k)} ε^{−2/k}); Metropolis-adjusted HMC achieves similar guarantees with comparable cost bounds.
• Under strong convexity, preconditioning can reduce the m2/M2 ratio and improve practical performance (Remark on preconditioning).
• The results show HMC can outperform Langevin (d^{1/2} or d^{1/3}–type costs) and ball walk bounds in similar strongly log-concave settings.