Fast Rates for General Unbounded Loss Functions: from ERM to Generalized Bayes

TL;DR

This paper derives new excess risk bounds for unbounded loss functions like log and squared loss, applicable to heavy-tailed distributions, and introduces conditions that enable fast convergence rates for generalized Bayesian and ERM estimators.

Contribution

It introduces the $v$-GRIP and witness conditions, extending existing theories to unbounded, heavy-tailed losses and providing convergence rates under misspecification.

Findings

Bounds hold for heavy-tailed loss distributions.

Fast $\tilde{O}(1/n)$ rates are achievable with favorable parameters.

Bounds apply to generalized Bayesian, MDL, and ERM estimators.

Abstract

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 $\eta$-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 $\eta$ 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 $\tilde{O}(1/n)$ 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 R\'enyi divergence to KL divergence.