Log-loss games with bounded adversaries

Worst-case analysis of the game assumes, by definition, that the adversary is trying to minimize the learner’s regret without any restrictions on the resources it uses while doing so. In practice, however, it may not be necessary (or indeed desirable) to get bounds of this kind—real-world data are typically generated by processes of bounded computational power, memory, etc., and it would be useful to have a model in which it is possible to describe guarantees against such bounded adversaries. In this paper, we describe a generalization of the log-loss game which allows explicit penalization of “expensive” adversarial behavior. Our main result is a description of a new estimator which outperforms the NML estimator against adversaries penalized in this way, subject to only very weak assumptions about the penalty function and reference class of models.