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# Exponential Convergence Rates in Classification

## Abstract

Let (X,Y) be a random couple, X being an observable instance and Y∈ {–1,1} being a binary label to be predicted based on an observation of the instance. Let (X i , Y i ), i = 1, . . . , n be training data consisting of n independent copies of (X,Y). Consider a real valued classifier $${\hat{f}_{n}}$$ that minimizes the following penalized empirical riskover a Hilbert space $${\mathcal H}$$ of functions with norm || ·||, ℓ being a convex loss function and λ >0 being a regularization parameter. In particular, $${\mathcal H}$$ might be a Sobolev space or a reproducing kernel Hilbert space. We provide some conditions under which the generalization error of the corresponding binary classifier sign $$({\hat{f}_{n}})$$ converges to the Bayes risk exponentially fast.

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