Fast rates by transferring from auxiliary hypotheses

In this work we consider the learning setting where, in addition to the training set, the learner receives a collection of auxiliary hypotheses originating from other tasks. We focus on a broad class of ERM-based linear algorithms that can be instantiated with any non-negative smooth loss function and any strongly convex regularizer. We establish generalization and excess risk bounds, showing that, if the algorithm is fed with a good combination of source hypotheses, generalization happens at the fast rate $$\mathcal {O}(1/m)$$O(1/m) instead of the usual $$\mathcal {O}(1/\sqrt{m})$$O(1/m). On the other hand, if the source hypotheses combination is a misfit for the target task, we recover the usual learning rate. As a byproduct of our study, we also prove a new bound on the Rademacher complexity of the smooth loss class under weaker assumptions compared to previous works.