  Current approaches to amortizing Bayesian inference focus solely on
approximating the posterior distribution. Typically, this approximation is, in
turn, used to calculate expectations for one or more target functions - a
computational pipeline which is inefficient when the target function(s) are
known upfront. In this paper, we address this inefficiency by introducing AMCI,
a method for amortizing Monte Carlo integration directly. AMCI operates
similarly to amortized inference but produces three distinct amortized
proposals, each tailored to a different component of the overall expectation
calculation. At runtime, samples are produced separately from each amortized
proposal, before being combined to an overall estimate of the expectation. We
show that while existing approaches are fundamentally limited in the level of
accuracy they can achieve, AMCI can theoretically produce arbitrarily small
errors for any integrable target function using only a single sample from each
proposal at runtime. We further show that it is able to empirically outperform
the theoretically optimal self-normalized importance sampler on a number of
example problems. Furthermore, AMCI allows not only for amortizing over
datasets but also amortizing over target functions.
