Asymptotic Convergence in Online Learning with Unbounded Delays

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.