COLT '90 Proceedings of the third annual workshop on Computational learning theory
A game of prediction with expert advice
COLT '95 Proceedings of the eighth annual conference on Computational learning theory
Loss functions, complexities, and the legendre transformation
Theoretical Computer Science - Special issue: Algorithmic learning theory
Prediction, Learning, and Games
Prediction, Learning, and Games
The Minimum Description Length Principle (Adaptive Computation and Machine Learning)
The Minimum Description Length Principle (Adaptive Computation and Machine Learning)
The weak aggregating algorithm and weak mixability
Journal of Computer and System Sciences
Prediction With Expert Advice For The Brier Game
The Journal of Machine Learning Research
Supermartingales in prediction with expert advice
Theoretical Computer Science
The Journal of Machine Learning Research
Information, Divergence and Risk for Binary Experiments
The Journal of Machine Learning Research
Sequential prediction of individual sequences under general loss functions
IEEE Transactions on Information Theory
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Mixability of a loss characterizes fast rates in the online learning setting of prediction with expert advice. The determination of the mixability constant for binary losses is straight forward but opaque. In the binary case we make this transparent and simpler by characterising mixability in terms of the second derivative of the Bayes risk of proper losses. We then extend this result to multiclass proper losses where there are few existing results. We show that mixability is governed by the maximum eigenvalue of the Hessian of the Bayes risk, relative to the Hessian of the Bayes risk for log loss. We conclude by comparing our result to other work that bounds prediction performance in terms of the geometry of the Bayes risk. Although all calculations are for proper losses, we also show how to carry the results across to improper losses.