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Zeroth-order parallel sampling

Francesco Pozza, Giacomo Zanella

TL;DR

This work tackles accelerating zeroth-order MCMC in a parallel setting by addressing the failure modes of naive gradient-free plug-ins. It introduces a random-slice framework that confines proposals to an $m$-dimensional subspace and applies zeroth-order HMC on that slice, yielding a polynomial speed-up in the number of processors $m$ for high-dimensional targets. Theoretical results establish KL contraction and poly-time speed-ups, while numerical experiments on logistic regression and stochastic volatility demonstrate substantial gains over traditional in-step parallel MCMC and naive zeroth-order methods. The findings offer a practical pathway to efficient Bayesian computation when gradients are unavailable or costly to compute, with clear guidance on selecting the number of active directions.

Abstract

Finding effective ways to exploit parallel computing to accelerate Markov chain Monte Carlo methods is an important problem in Bayesian computation and related disciplines. In this paper, we consider the zeroth-order setting where the unnormalized target distribution can be evaluated but its gradient is unavailable for theoretical, practical, or computational reasons. We also assume access to $m$ parallel processors to accelerate convergence. The proposed approach is inspired by modern zeroth-order optimization methods, which mimic gradient-based schemes by replacing the gradient with a zeroth-order stochastic gradient estimator. Our contribution is twofold. First, we show that a naive application of popular zeroth-order stochastic gradient estimators within Markov chain Monte Carlo methods leads to algorithms with poor dependence on $m$, both for unadjusted and Metropolis-adjusted schemes. We then propose a simple remedy to this problem, based on a random-slice perspective, as opposed to a stochastic gradient one, obtaining a new class of zeroth-order samplers that provably achieve a polynomial speed-up in $m$. Theoretical findings are supported by numerical studies.

Zeroth-order parallel sampling

TL;DR

This work tackles accelerating zeroth-order MCMC in a parallel setting by addressing the failure modes of naive gradient-free plug-ins. It introduces a random-slice framework that confines proposals to an -dimensional subspace and applies zeroth-order HMC on that slice, yielding a polynomial speed-up in the number of processors for high-dimensional targets. Theoretical results establish KL contraction and poly-time speed-ups, while numerical experiments on logistic regression and stochastic volatility demonstrate substantial gains over traditional in-step parallel MCMC and naive zeroth-order methods. The findings offer a practical pathway to efficient Bayesian computation when gradients are unavailable or costly to compute, with clear guidance on selecting the number of active directions.

Abstract

Finding effective ways to exploit parallel computing to accelerate Markov chain Monte Carlo methods is an important problem in Bayesian computation and related disciplines. In this paper, we consider the zeroth-order setting where the unnormalized target distribution can be evaluated but its gradient is unavailable for theoretical, practical, or computational reasons. We also assume access to parallel processors to accelerate convergence. The proposed approach is inspired by modern zeroth-order optimization methods, which mimic gradient-based schemes by replacing the gradient with a zeroth-order stochastic gradient estimator. Our contribution is twofold. First, we show that a naive application of popular zeroth-order stochastic gradient estimators within Markov chain Monte Carlo methods leads to algorithms with poor dependence on , both for unadjusted and Metropolis-adjusted schemes. We then propose a simple remedy to this problem, based on a random-slice perspective, as opposed to a stochastic gradient one, obtaining a new class of zeroth-order samplers that provably achieve a polynomial speed-up in . Theoretical findings are supported by numerical studies.
Paper Structure (28 sections, 14 theorems, 87 equations, 5 figures, 5 algorithms)

This paper contains 28 sections, 14 theorems, 87 equations, 5 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Contributions of the paper
  3. Previous work about in-step parallelization of MCMC
  4. Zeroth-order parallel sampling
  5. The abstract framework
  6. Comparison with zeroth-order optimization
  7. Naive plug-in approaches
  8. Unadjusted methods
  9. Naive approach to Metropolis-adjusted methods
  10. A random slice approach
  11. The general methodology
  12. Proposed practical methodology: Random-slice zeroth-order HMC
  13. A theoretical bound on the complexity of Algorithm \ref{['alg:mdim:rps']}
  14. Numerics
  15. Logistic regression
  16. ...and 13 more sections

Key Result

Proposition 1

Let $P^{SULA}$ be the transition kernel of Algorithm alg:sula. Then, under Assumptions cond:1 and cond:2, for every $x,y\in\mathbb{R}^d$ and $\gamma\leq m/(Ld)$ it holds that where $\delta_x,\delta_y$ denote the Dirac measure on $x$ and $y$, respectively.

Figures (5)

  • Figure 1: Relative improvement over RWM, measured by the estimated average ESJD (as defined in \ref{['eq:esjd']}), for NAIVE-MALA and for RS-MALA in the 200-dimensional logistic regression example of Section \ref{['sec:5:1']}.
  • Figure 2: Relative improvement, over RWM, evaluated in terms of estimated average ESJD for RS-MALA, RS-HMC and MTM for the two logistic regressions considered in Section \ref{['sec:5:1']}. Results are standardized for the total number of $m$ parallel target evaluations at each iteration of the algorithm.
  • Figure 3: Relative improvement, over RWM, evaluated in terms of estimated ESJD for RS-MALA, RS-HMC and MTM for the stochastic volatility model considered in Section \ref{['sec:5:2']}. Results are standardized for the total number of $m$ parallel target evaluations at each iteration of the algorithm.
  • Figure 4: Relative efficiency for different values of $m_0$ in the $200$-dimensional logistic regression described in Section \ref{['sec:5:1']}
  • Figure 5: Relative efficiency for different values of $m_0$ in the $203$-dimensional stochastic volatility model described in Section \ref{['sec:5:1']}

Theorems & Definitions (29)

  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Theorem 1
  • Proposition 3
  • Theorem 2
  • Corollary 1
  • Remark 1: Optimal choice of $m$
  • proof
  • proof
  • ...and 19 more