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onlinerisk - Risk-Sensitive Online Learning Eyal Even-Dar...

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Risk-Sensitive Online Learning Eyal Even-Dar, Michael Kearns, and Jennifer Wortman Department of Computer and Information Science University of Pennsylvania, Philadelphia, PA 19104 { evendar,wortmanj } @seas.upenn.edu, [email protected] Abstract. We consider the problem of online learning in settings in which we want to compete not simply with the rewards of the best ex- pert or stock, but with the best trade-oF between rewards and risk . Motivated by ±nance applications, we consider two common measures balancing returns and risk: the Sharpe ratio [7] and the mean-variance criterion of Markowitz [6]. We ±rst provide negative results establishing the impossibility of no-regret algorithms under these measures, thus pro- viding a stark contrast with the returns-only setting. We then show that the recent algorithm of Cesa-Bianchi et al. [3] achieves nontrivial perfor- mance under a modi±ed bicriteria risk-return measure, and also give a no-regret algorithm for a “localized” version of the mean-variance crite- rion. To our knowledge this paper initiates the investigation of explicit risk considerations in the standard models of worst-case online learning. 1 Introduction Despite the large literature on online learning, and the rich collection of al- gorithms with guaranteed worst-case regret bounds, virtually no attention has been given to the risk incurred by such algorithms 1 . Especially in Fnance-related applications [4], where consideration of various measures of the volatility of a portfolio are often given equal footing with the returns themselves, this omission is particularly glaring. The Fnance literature on balancing risk and return, and the proposed met- rics for doing so, are far too large to survey here (see [1], chapter 4 for a nice overview). But among the two most common methods are the Sharpe ratio [7], and the mean-variance (MV) criterion of which Markowitz was the Frst pro- ponent [6]. Let r t [ 1 , ] be the return of any given Fnancial instrument (a stock, bond, portfolio, trading strategy, etc.) during time period t .Thu s ,i f v t represents the dollar value of the instrument immediately after period t ,wehave v t =(1+ r t ) v t 1 . Negative values of r t (down to -1, representing the limiting case of the instrument losing all of its value) are losses, and positive values are gains. ±or a sequence of returns r =( r 1 ,...,r T )w eu s e µ ( r ) to denote the (arithmetic) mean or average value, and σ ( r ) to denote the standard deviation. Then the Sharpe ratio of the instrument on the sequence is simply µ ( r ) ( r ), 1 A partial exception is the recent work of [3], which we analyze in our framework.
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2 while the MV is µ ( r ) σ ( r ). (Note that the term mean-variance is slightly mis- leading since the risk is actually measured by the standard deviation, but we use this term to adhere to convention.) A common alternative is to use the mean and standard deviation not of the r t but of the log(1 + r t ), which corresponds to geometric rather than arithmetic averaging of returns (see Section 2); we shall refer to the resulting measures the
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This note was uploaded on 12/08/2011 for the course CIS 625 taught by Professor Michaelkearns during the Spring '12 term at Penn State.

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onlinerisk - Risk-Sensitive Online Learning Eyal Even-Dar...

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