Consistent Multiclass Algorithms for Complex Performance Measures Consistent Multiclass Algorithms for Complex Performance Measures

Notations. Let λ be the base measure over ∆n given by the uniform random variable (say U ) over ∆n. Hence, for all measurable A ⊆ ∆n, λ(A) = P(U ∈ A). Also, η : X→∆n is the mapping that gives the conditional probability vector η(x) = [P(Y = 1 |X = x), . . . ,P(Y = n |X = x)]> ∈ ∆n for a given instance x ∈ X . Let ν be the probability measure over the simplex induced by the random variable η(X); in particular, for all measurable A ⊆ ∆n, ν(A) = PX∼μ(η(X) ∈ A). For a matrix L ∈ [0, 1]n×n we let `1, `2, . . . , `n be the columns of L. For any set A ⊆ R, the set A denotes its closure. For any vector v ∈ R, we let (v)(i) denote the i element when the components of v are sorted in ascending order. For any y ∈ [n], we shall denote rand(y) = [1(y = 1), . . . ,1(y = n)]>.