Convex Calibrated Surrogates for Hierarchical Classification Convex Calibrated Surrogates for Hierarchical Classification

A. Additional Notation and Setup Let μ be the marginal distribution induced by D over X , and let p(x) be the distribution over [n] conditioned on X = x. For every function ` : [n]⇥ [k]!R+ and t 2 [k] let `t = [`(1, t), . . . , `(n, t)]> 2 R+. For every surrogate : [n]⇥ R!R+ let : R!R+ be a vector function such that y(u) = (y,u) for y 2 [n],u 2 Rd. For any integer d0 2 Z+ and pair of vectors u,v 2 Rd , their inner product is denoted as hu,vi = Pd0 i=1 uivi. For a vector u 2 Rn and a positive integer a  n, the vector u 1:a 2 Ra gives the first a components of u. Define the conditional regrets R` H p , R` ?,n p and R p as the regrets incurred for a singleton instance space X , with conditional probability p 2 n. In particular, we have that R H p [b y] = hp, ` H b y i inf y02[n] hp, `y0i, 8b y 2 [n] R ?,n p [b y] = hp, ` ?,n b y i inf y02[n][{?} hp, ` y0 i, 8b y 2 [n] [ {?} R p [u] = hp, (u)i inf u02Rd hp, (u0)i, 8u 2 R .