Paper #418

Title:

Worst-case bounds for the logarithmic loss of predictors

Authors:

Nicolò Cesa Bianchi and Gábor Lugosi

Date:

October 1999

Abstract:

We investigate on-line prediction of individual sequences. Given a class of predictors, the goal is to predict as well as the best predictor in the class, where the loss is measured by the self information (logarithmic) loss function. The excess loss (regret) is closely related to the redundancy of the associated lossless universal code. Using Shtarkov's theorem and tools from empirical process theory, we prove a general upper bound on the best possible (minimax) regret. The bound depends on certain metric properties of the class of predictors. We apply the bound to both parametric and nonparametric classes of predictors. Finally, we point out a suboptimal behavior of the popular Bayesian weighted average algorithm.

Keywords:

Universal prediction, universal coding, empirical processes, on-line learning, metric entropy

JEL codes:

C1, C13

Area of Research:

Statistics, Econometrics and Quantitative Methods

Published in:

Machine Learning, 43, 3, (2001), pp. 247-264

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