  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.
