Hybrid Stochastic-Adversarial On-line Learning
Alessandro Lazaric and Rémi Munos
Most of the research in online learning focused either on the problem of adversarial classiﬁcation (i.e., both inputs and labels are arbitrarily chosen by an adversary) or on the traditional supervised learning problem in which samples are indepen- dently generated from a ﬁxed probability distribution. Nonetheless, in a number of domains the relationship between inputs and labels may be adversarial, whereas input instances are generated according to a constant distribution. This scenario can be formalized as an hybrid classiﬁcation problem in which inputs are stochastic, while labels are adversarial. In this paper, we introduce this hybrid stochastic-adversarial classiﬁcation problem, we propose an online learning algorithm for its so- lution, and we analyze its performance. In particular, we show that, given a hypothesis space H with ﬁnite VC dimension, it is possible to incrementally build a suitable ﬁnite set of hypotheses that can be used as input for an exponentially weighted forecaster achieving a cumulative regret of order O( n VC(H) log n) with overwhelming probability. Finally, we discuss extensions to multi-label classiﬁcation, learning from experts and bandit set- tings with stochastic side information, and application to games.