[Paper Review] Speeding Up MCMC by Efficient Data Subsampling

Matias Quiroz, Robert Kohn+2 more|UTS ePRESS (University of Technology Sydney)|Apr 16, 2014

Markov Chains and Monte Carlo Methods4 citations

TL;DR

This paper proposes Subsampling MCMC, a scalable Bayesian inference framework that uses control variates to efficiently estimate the log-likelihood from a small random subset of data, enabling fast and accurate MCMC sampling. By combining bias-corrected likelihood estimation with correlated pseudo-marginal schemes, the method achieves near-exact posterior approximation with only m = O(√n) samples, reducing computational cost while maintaining high sampling efficiency and negligible error.

ABSTRACT

We propose Subsampling MCMC, a Markov Chain Monte Carlo (MCMC) framework where the likelihood function for $n$ observations is estimated from a random subset of $m$ observations. We introduce a highly efficient unbiased estimator of the log-likelihood based on control variates, such that the computing cost is much smaller than that of the full log-likelihood in standard MCMC. The likelihood estimate is bias-corrected and used in two dependent pseudo-marginal algorithms to sample from a perturbed posterior, for which we derive the asymptotic error with respect to $n$ and $m$, respectively. We propose a practical estimator of the error and show that the error is negligible even for a very small $m$ in our applications. We demonstrate that Subsampling MCMC is substantially more efficient than standard MCMC in terms of sampling efficiency for a given computational budget, and that it outperforms other subsampling methods for MCMC proposed in the literature.

Motivation & Objective

• Address the computational bottleneck of MCMC in large-scale Bayesian inference where full-data likelihood evaluation is expensive.
• Develop a scalable MCMC framework that subsamples data at each iteration without sacrificing posterior accuracy.
• Reduce the variance of likelihood estimators in pseudo-marginal MCMC through control variates, enabling faster mixing and higher acceptance rates.
• Ensure the perturbed posterior induced by subsampling remains close to the true posterior, even with small subsample sizes.
• Demonstrate superior performance compared to existing subsampling MCMC methods in terms of computational efficiency and accuracy.

Proposed method

• Estimate the log-likelihood using a random subset of m observations, where m ≪ n, to reduce computational cost.
• Apply two types of control variates: parameter-expanded (around MLE) and data-expanded (around data centroids) to reduce variance in the log-likelihood estimator.
• Construct an unbiased estimator of the log-likelihood by combining control variates with a residual term based on a small subsample.
• Use a bias-correction step to ensure the estimated log-likelihood is approximately unbiased for the full-data log-likelihood.
• Implement a correlated pseudo-marginal scheme using block proposals and shared random numbers to improve Metropolis-Hastings acceptance rates.
• Employ a two-stage strategy: initially use larger m for exploration, then reduce m after obtaining a good estimate of the MLE to improve efficiency.

Experimental results

Research questions

• RQ1 Can we achieve near-exact posterior approximation in MCMC using only a small fraction of the data, while maintaining computational efficiency?
• RQ2 How does the use of control variates reduce the variance of the likelihood estimator in subsampled MCMC, and what is the resulting impact on mixing and acceptance rates?
• RQ3 What is the theoretical error bound of the perturbed posterior induced by subsampling, and how does it scale with n and m?
• RQ4 How does the proportional error in the posterior compare to the error in the likelihood estimator, especially in high-posterior-concentration regions?
• RQ5 Can the proposed method outperform existing subsampling MCMC approaches in terms of sampling efficiency and accuracy under a fixed computational budget?

Key findings

• With m = O(√n), the total variation distance between the perturbed posterior and the true posterior is O(n⁻²) when using the full-data MLE for control variates, or O(n⁻¹/²) when using a subset of size O(√n).
• The absolute proportional error in the perturbed posterior is extremely small: for example, 1.418 × 10⁻⁶ (mean) on the Bankruptcy dataset, 8.594 × 10⁻⁸ on HIGGS, and 5.136 × 10⁻⁸ on Covtype.
• The method achieves significant speedups over standard MCMC, with Relative Computational Time (RCT) reductions of up to several orders of magnitude in large datasets.
• The error is substantially smaller in high-posterior-concentration regions, indicating that the method performs best where posterior inference is most critical.
• Switching to parameter-expanded control variates after a training phase reduces m from a large initial value to m = 1,000, improving efficiency without loss of accuracy.
• The method outperforms other subsampling MCMC approaches in both computational efficiency and posterior accuracy, as validated across multiple real-world datasets.