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On the fundamental limitations of multiproposal Markov chain Monte Carlo algorithms

Francesco Pozza, Giacomo Zanella

Abstract

We study multiproposal Markov chain Monte Carlo algorithms, such as Multiple-try or generalised Metropolis-Hastings schemes, which have recently received renewed attention due to their amenability to parallel computing. First, we prove that no multiproposal scheme can speed-up convergence relative to the corresponding single proposal scheme by more than a factor of $K$, where $K$ denotes the number of proposals at each iteration. This result applies to arbitrary target distributions and it implies that serial multiproposal implementations are always less efficient than single proposal ones. Secondly, we consider log-concave distributions over Euclidean spaces, proving that, in this case, the speed-up is at most logarithmic in $K$, which implies that even parallel multiproposal implementations are fundamentally limited in the computational gain they can offer. Crucially, our results apply to arbitrary multiproposal schemes and purely rely on the two-step structure of the associated kernels (i.e. first generate $K$ candidate points, then select one among those). Our theoretical findings are validated through numerical simulations.

On the fundamental limitations of multiproposal Markov chain Monte Carlo algorithms

Abstract

We study multiproposal Markov chain Monte Carlo algorithms, such as Multiple-try or generalised Metropolis-Hastings schemes, which have recently received renewed attention due to their amenability to parallel computing. First, we prove that no multiproposal scheme can speed-up convergence relative to the corresponding single proposal scheme by more than a factor of , where denotes the number of proposals at each iteration. This result applies to arbitrary target distributions and it implies that serial multiproposal implementations are always less efficient than single proposal ones. Secondly, we consider log-concave distributions over Euclidean spaces, proving that, in this case, the speed-up is at most logarithmic in , which implies that even parallel multiproposal implementations are fundamentally limited in the computational gain they can offer. Crucially, our results apply to arbitrary multiproposal schemes and purely rely on the two-step structure of the associated kernels (i.e. first generate candidate points, then select one among those). Our theoretical findings are validated through numerical simulations.

Paper Structure

This paper contains 18 sections, 10 theorems, 49 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Transition kernels based on multiple proposals
  3. Multiproposal MCMC and parallel computing
  4. Upper bounds on the spectral gap of multiproposal MCMC algorithms
  5. A general Peskun ordering result
  6. Spectral gap behaviour for continuous target distributions
  7. Analysis for random walk multiproposal schemes under log-concavity
  8. A simulation study on high-dimensional logistic regression
  9. Concise overview of multiproposal MCMC methods
  10. Multiple-try Metropolis methods
  11. Generalised Metropolis-Hastings methods
  12. General remarks
  13. Proofs and additional technical results
  14. Proof of Theorem \ref{['theorem:peskun']}
  15. Proof of Corollary \ref{['coroll:peskun_gap']}
  16. ...and 3 more sections

Key Result

Theorem 1

Let $P^{(K)}$ be a $\pi$-reversible Markov transition kernel as in kernel:meta:1. Then for $\pi$-a.e. $x$ and all measurable $A\subseteq { \mathbb{ X}}$, where $\tilde{P}$ is the (single proposal) MH kernel with proposal distribution $\tilde{Q}= K^{-1}\sum_{i=1}^K Q_i$ and target distribution $\pi$.

Figures (1)

  • Figure 1: Estimated ESJD as a function of $K$, for MP-MCMC schemes with random walk (A) and Langevin (B) proposal, as described in Section \ref{['sec:sim']}. Black solid lines indicate the functions $c_1\lbrace 1 + \log(K)\rbrace$ with $c_1 = 2.3 E_{\textsc{rw}}$ for (A) and $c_2 \sqrt{1+\log(K)}$ with $c_2 = E_{\textsc{mala}}$ for (B), where $E_{\textsc{rw}}$ and $E_{\textsc{mala}}$ denote the ESJD of MH with $K=1$ and, respectively, random walk and Langevin proposal.

Theorems & Definitions (25)

  • Example 1: Multiple-try Metropolis
  • Example 2: Tjelmeland's proposal
  • Theorem 1
  • Corollary 1
  • Remark 1: Related results
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Proposition 1
  • proof
  • ...and 15 more