Hybrid Stochastic-Adversarial On-Line Learning

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 i.i.d. according to a probability distribution. Nonetheless, in a number of domains the relationship between inputs and labels may be adversarial, whereas inputs are generated according to a fixed distribution. This scenario can be formalized as an hybrid classification problem in which inputs are i.i.d., while labels are adversarial. In this paper we introduce the hybrid stochastic-adversarial problem, we propose an online learning algorithm for its solution, 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 over n rounds of order O( p nV C(H) log n) with overwhelming probability.