Asymptotically Optimal Exact Minibatch Metropolis-Hastings
This addresses scaling MCMC for large datasets, offering a theoretically sound solution with practical improvements, though it is incremental in the context of minibatch MCMC methods.
The paper tackles the intractability of Metropolis-Hastings on large datasets by proposing TunaMH, an exact minibatch method that avoids errors from inexact approaches, and shows it is asymptotically optimal in batch size and outperforms other methods in experiments on tasks like robust linear regression.
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.