  In this paper, we study the learning rate of generalized Bayes estimators in
a general setting where the hypothesis class can be uncountable and have an
irregular shape, the loss function can have heavy tails, and the optimal
hypothesis may not be unique. We prove that under the multi-scale Bernstein's
condition, the generalized posterior distribution concentrates around the set
of optimal hypotheses and the generalized Bayes estimator can achieve fast
learning rate. Our results are applied to show that the standard Bayesian
linear regression is robust to heavy-tailed distributions.
