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Counterdiabatic Hamiltonian Monte Carlo

Reuben Cohn-Gordon, Uroš Seljak, Dries Sels

TL;DR

This work proposes Counterdiabatic Hamiltonian Monte Carlo (CHMC), which can be viewed as an SMC sampler with a more efficient kernel, and establishes its relationship to recent proposals for accelerating gradient-based sampling with learned drift terms, and demonstrates on simple benchmark problems.

Abstract

Hamiltonian Monte Carlo (HMC) is a state of the art method for sampling from distributions with differentiable densities, but can converge slowly when applied to challenging multimodal problems. Running HMC with a time varying Hamiltonian, in order to interpolate from an initial tractable distribution to the target of interest, can address this problem. In conjunction with a weighting scheme to eliminate bias, this can be viewed as a special case of Sequential Monte Carlo (SMC) sampling \cite{doucet2001introduction}. However, this approach can be inefficient, since it requires slow change between the initial and final distribution. Inspired by \cite{sels2017minimizing}, where a learned \emph{counterdiabatic} term added to the Hamiltonian allows for efficient quantum state preparation, we propose \emph{Counterdiabatic Hamiltonian Monte Carlo} (CHMC), which can be viewed as an SMC sampler with a more efficient kernel. We establish its relationship to recent proposals for accelerating gradient-based sampling with learned drift terms, and demonstrate on simple benchmark problems.

Counterdiabatic Hamiltonian Monte Carlo

TL;DR

This work proposes Counterdiabatic Hamiltonian Monte Carlo (CHMC), which can be viewed as an SMC sampler with a more efficient kernel, and establishes its relationship to recent proposals for accelerating gradient-based sampling with learned drift terms, and demonstrates on simple benchmark problems.

Abstract

Hamiltonian Monte Carlo (HMC) is a state of the art method for sampling from distributions with differentiable densities, but can converge slowly when applied to challenging multimodal problems. Running HMC with a time varying Hamiltonian, in order to interpolate from an initial tractable distribution to the target of interest, can address this problem. In conjunction with a weighting scheme to eliminate bias, this can be viewed as a special case of Sequential Monte Carlo (SMC) sampling \cite{doucet2001introduction}. However, this approach can be inefficient, since it requires slow change between the initial and final distribution. Inspired by \cite{sels2017minimizing}, where a learned \emph{counterdiabatic} term added to the Hamiltonian allows for efficient quantum state preparation, we propose \emph{Counterdiabatic Hamiltonian Monte Carlo} (CHMC), which can be viewed as an SMC sampler with a more efficient kernel. We establish its relationship to recent proposals for accelerating gradient-based sampling with learned drift terms, and demonstrate on simple benchmark problems.
Paper Structure (30 sections, 1 theorem, 16 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 30 sections, 1 theorem, 16 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Technical Background
  4. Hamiltonian dynamics
  5. Poisson bracket
  6. Canonical distribution
  7. Hamiltonian Monte Carlo
  8. Counterdiabatic driving
  9. Learning the counterdiabatic gauge potential
  10. Counterdiabatic Hamiltonian Monte Carlo (CHMC)
  11. CHMC in terms of Sequential Monte Carlo
  12. SMCS with physical dynamics: using the nonequilibrium fluctuation theorem
  13. Discretization
  14. Related work
  15. Where CHMC differs from related work
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let $\mathcal{L}_H(A) = \mathbb{E}[\left\|\{A, H\} - \partial_\lambda H\right\|^2]$, where the expectation is taken over $\rho_H$. Then equation eq:prop is satisfied by

Figures (2)

  • Figure 1: Illustration of lag induced by a time-varying Hamiltonian (top row), and the correction introduced by a learned counterdiabatic term (bottom row). The lefthand column shows a Gaussian with time-varying mean, and the righthand column an interpolation between a Gaussian and a mixture of two Gaussians. In the simple case, parametrizing $A$ as a sum of polynomials with learned coefficients suffices, and in the more complex case, $A$ is parameterized by a neural network.
  • Figure 2: Here, the path of distributions is a geometric path between a standard normal and a mixture of two

Theorems & Definitions (1)

  • Lemma 2.1