Fast Rates for General Unbounded Loss Functions: from ERM to Generalized Bayes
This work provides theoretical guarantees for machine learning estimators like generalized Bayesian and empirical risk minimization, addressing challenges in misspecification and heavy-tailed data, though it is incremental in extending existing bounded loss results to unbounded cases.
The paper tackles the problem of deriving excess risk bounds for general unbounded loss functions, including log loss and squared loss, under heavy-tailed distributions, achieving rates as fast as ̃O(1/n) under favorable conditions.
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 $\tilde{O}(1/n)$ 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 Rényi divergence to KL divergence.