Fast Rates for General Unbounded Loss Functions: from ERM to Generalized Bayes
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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 ERM estimators. When applied with 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 well-known 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.
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