The Geometry of Mixability
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Mixable loss functions are of fundamental importance in the context of prediction with expert advice in the online setting since they characterize fast learning rates. By re-interpreting properness from the point of view of differential geometry, we provide a simple geometric characterization of mixability for the binary and multi-class cases: a proper loss function ℓ is η-mixable if and only if the superpredition set spr(ηℓ) of the scaled loss function ηℓ slides freely inside the superprediction set spr(ℓ_log) of the log loss ℓ_log, under fairly general assumptions on the differentiability of ℓ. Our approach provides a way to treat some concepts concerning loss functions (like properness) in a ”coordinate-free” manner and reconciles previous results obtained for mixable loss functions for the binary and the multi-class cases.
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