  Optimization objectives in the form of a sum of intractable expectations are
rising in importance (e.g., diffusion models, variational autoencoders, and
many more), a setting also known as "finite sum with infinite data." For these
problems, a popular strategy is to employ SGD with doubly stochastic gradients
(doubly SGD): the expectations are estimated using the gradient estimator of
each component, while the sum is estimated by subsampling over these
estimators. Despite its popularity, little is known about the convergence
properties of doubly SGD, except under strong assumptions such as bounded
variance. In this work, we establish the convergence of doubly SGD with
independent minibatching and random reshuffling under general conditions, which
encompasses dependent component gradient estimators. In particular, for
dependent estimators, our analysis allows fined-grained analysis of the effect
correlations. As a result, under a per-iteration computational budget of $b
\times m$, where $b$ is the minibatch size and $m$ is the number of Monte Carlo
samples, our analysis suggests where one should invest most of the budget in
general. Furthermore, we prove that random reshuffling (RR) improves the
complexity dependence on the subsampling noise.
