[Paper Review] Asymptotically Optimal Exact Minibatch Metropolis-Hastings

Ruqi Zhang, A. Feder Cooper+1 more|arXiv (Cornell University)|Jun 20, 2020

Markov Chains and Monte Carlo Methods27 references4 citations

TL;DR

This paper proposes TunaMH, an asymptotically optimal exact minibatch Metropolis-Hastings algorithm that maintains the true posterior distribution while enabling a tunable trade-off between batch size and convergence rate. It proves the first lower bound on batch size for exact minibatch MH methods and shows TunaMH achieves asymptotic optimality in minimizing expected batch size while preserving convergence speed.

ABSTRACT

Metropolis-Hastings (MH) is a commonly-used MCMC algorithm, but it can be intractable on large datasets due to requiring computations over the whole dataset. In this paper, we study minibatch MH methods, which instead use subsamples to enable scaling. We observe that most existing minibatch MH methods are inexact (i.e. they may change the target distribution), and show that this inexactness can cause arbitrarily large errors in inference. We propose a new exact minibatch MH method, TunaMH, which exposes a tunable trade-off between its batch size and its theoretically guaranteed convergence rate. We prove a lower bound on the batch size that any minibatch MH method must use to retain exactness while guaranteeing fast convergence-the first such bound for minibatch MH-and show TunaMH is asymptotically optimal in terms of the batch size. Empirically, we show TunaMH outperforms other exact minibatch MH methods on robust linear regression, truncated Gaussian mixtures, and logistic regression.

Motivation & Objective

• To address the critical issue of inexactness in existing minibatch MH methods, which can lead to arbitrarily large inference errors.
• To develop an exact minibatch MH method that maintains the true posterior distribution without relying on strong assumptions or sacrificing efficiency.
• To formally characterize the trade-off between batch size and convergence rate in exact minibatch MH, which prior work had not done.
• To prove a theoretical lower bound on the expected batch size required for any exact minibatch MH method to maintain a given convergence rate.
• To demonstrate that TunaMH achieves asymptotic optimality in balancing batch size and convergence speed.

Proposed method

• TunaMH uses a Poisson-distributed sampling mechanism to stochastically select minibatches, ensuring exactness while allowing control over expected batch size via a tunable hyperparameter χ.
• The algorithm constructs an acceptance probability that exactly matches the full-data Metropolis-Hastings ratio using unbiased estimators of the log-target density difference.
• It employs local Lipschitz bounds on the potential functions U_i(θ) to ensure the validity of the acceptance ratio under subsampling.
• The method introduces a novel coupling mechanism between the full-data MH and minibatch MH chains to preserve detailed balance and exact stationarity.
• It derives convergence rate guarantees based on the spectral gap, showing that the convergence rate of TunaMH is within a constant factor of standard MH.
• The algorithm is designed to be applicable under mild local regularity conditions, avoiding the need for global bounds or Bernstein-von Mises approximations.

Experimental results

Research questions

• RQ1 Can inexact minibatch MH methods be arbitrarily inaccurate, and if so, what are the implications for practical inference?
• RQ2 Is it possible to design an exact minibatch MH method that maintains both exactness and efficient convergence without sacrificing scalability?
• RQ3 What is the theoretical minimum expected batch size required for any exact minibatch MH method to achieve a given convergence rate?
• RQ4 Can a minibatch MH algorithm be asymptotically optimal in balancing batch size and convergence rate?
• RQ5 How does the proposed method compare empirically to state-of-the-art exact minibatch MH methods across diverse models?

Key findings

• The paper demonstrates that any inexact minibatch MH method can have a stationary distribution arbitrarily far from the true posterior, making such methods unreliable for accurate inference.
• TunaMH achieves a convergence rate (spectral gap) that is within a constant factor of standard, full-data Metropolis-Hastings, ensuring strong sampling efficiency.
• The authors prove a lower bound on the expected batch size required for any exact minibatch MH method to maintain a given convergence rate, establishing a theoretical foundation for optimality.
• TunaMH is asymptotically optimal: it uses the smallest possible expected batch size among all exact minibatch MH methods with the same convergence rate guarantee.
• Empirically, TunaMH outperforms other exact minibatch MH methods on robust linear regression, truncated Gaussian mixtures, and logistic regression, producing more accurate density estimates under time constraints.
• In a 1-second runtime comparison on a 2D problem, TunaMH produced a density estimate closest to the true posterior, while all other methods failed to converge accurately.