Exact convex calibration dimension for the multi-label Jaccard loss

Determine the exact convex calibration dimension of the multi-label Jaccard loss, or improve the constants in the bounds 2^{s-1} ≤ CCdim(L^{Jac}) ≤ 2^s−1.

Background

The paper proves that the convex calibration dimension of the instance-wise multi-label Jaccard loss lies between 2^{s-1} and 2^s−1, establishing exponential scaling in the number s of labels. The lower bound is obtained from a factorially weighted conditional distribution whose Bayes-optimal reports form a tied family of size 2^{s-1}+1, while the upper bound follows from the loss matrix's affine dimension, 2^s−1.

The remaining uncertainty concerns the precise value of the calibration dimension and, more generally, whether either endpoint can be sharpened. Closing this gap would determine the exact prediction dimension required by an exactly calibrated convex surrogate for the Jaccard loss, rather than only its exponential order.