## AbstractMost 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 independently 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 solution, and we analyze its performance. In particular, we show that, given a hypothesis space $\hypSpace$ 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 $\bigO(\sqrt{n\VCdim(\hypSpace)\log n})$ with overwhelming probability. Finally, we discuss extensions to multi-label classification, learning from experts and bandit settings with stochastic side information, and application to games.
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