  We study the problem of predicting the results of computations that are too
expensive to run, via the observation of the results of smaller computations.
We model this as an online learning problem with delayed feedback, where the
length of the delay is unbounded, which we study mainly in a stochastic
setting. We show that in this setting, consistency is not possible in general,
and that optimal forecasters might not have average regret going to zero.
However, it is still possible to give algorithms that converge asymptotically
to Bayes-optimal predictions, by evaluating forecasters on specific sparse
independent subsequences of their predictions. We give an algorithm that does
this, which converges asymptotically on good behavior, and give very weak
bounds on how long it takes to converge. We then relate our results back to the
problem of predicting large computations in a deterministic setting.
