bbc92a647199b832ec90d7cf57074e9e-Supplemental.pdf

Before defining our algorithm at each iterationt we first lighten our notation with a shorthandba(X) = b(ˆp(t 1)(X),a) (at different iterationt, ba denotes different functions), andb(X) is the vector of (b1(X),,bK(X)). For the intuition of the algorithm, consider the t-th iteration where the current prediction function is ˆp(t 1). Thestatement of the theorem is identical; the proof is also essentially the same except for the use of some new technicaltools. Conversely, if ˆp is LB decision calibrated, then kE[p (X) ˆp(X)|U]k1 = 0 almost surely (because if the expectation of a non-negative random variable is zero, the random variable must be zero almost surely), which implies thatˆp is distributioncalibrated. For BKa we use the VC dimension approach.