We present new excess risk bounds for general unbounded loss functions including log loss and squared loss, where the distribution of the losses may be heavy-tailed. The bounds hold for general estimators, but they are optimized when applied to η-generalized Bayesian, MDL, and empirical risk minimization estimators. In the case of log loss, the bounds imply convergence rates for generalized Bayesian inference under misspecification in terms of a generalization of the Hellinger metric as long as the learning rate η is set correctly. For general loss functions, our bounds rely on two separate conditions: the v-GRIP (generalized reversed information projection) conditions, which control the lower tail of the excess loss; and the newly introduced witness condition, which controls the upper tail. The parameter v in the v-GRIP conditions determines the achievable rate and is akin to the exponent in the Tsybakov margin condition and the Bernstein condition for bounded losses, which the v-GRIP conditions generalize; favorable v in combination with small model complexity leads to Õ(1/n) rates. The witness condition allows us to connect the excess risk to an “annealed” version thereof, by which we generalize several previous results connecting Hellinger and Rényi divergence to KL divergence.

, , , ,
Journal of Machine Learning Research
Safe Bayesian Inference: A Theory of Misspecification based on Statistical Learning
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Grünwald, P.D, & Mehta, N.A. (2020). Fast rates for general unbounded loss functions: From ERM to generalized Bayes. Journal of Machine Learning Research, 21, 1–80.