Abstract

Abstract. We consider the problem of online learning in settings in which we want to compete not simply with the rewards of the best expert or stock, but with the best trade-off between rewards and risk. Motivated by finance applications, we consider two common measures balancing returns and risk: the Sharpe ratio [7] and the mean-variance criterion of Markowitz [6]. We first provide negative results establishing the impossibility of no-regret algorithms under these measures, thus providing a stark contrast with the returns-only setting. We then show that the recent algorithm of Cesa-Bianchi et al. [3] achieves nontrivial performance under a modified bicriteria risk-return measure, and also give a no-regret algorithm for a “localized ” version of the mean-variance criterion. To our knowledge this paper initiates the investigation of explicit risk considerations in the standard models of worst-case online learning. 1