Intermediate group family sizes for prediction-independent multicalibration

Determine the minimax online multicalibration rates for prediction-independent binary group families whose cardinality |G| grows with T between constant and Θ(T), including the regime |G| = polylog(T). Ascertain whether there exists a sharp threshold in |G| at which the statistical complexity separates from marginal calibration, or whether the multicalibration rates interpolate smoothly across intermediate |G|.

Background

The authors prove tight multicalibration lower bounds for prediction-independent groups at |G| = Θ(T) and note an upper-bound reduction to marginal calibration for constant |G|. This establishes two endpoints in the dependence of multicalibration complexity on the group-family size.

However, it remains unclear how the complexity behaves for intermediate-sized families, particularly |G| growing polylogarithmically with T. Although Appendix C provides an oracle lower bound obstructing certain black-box reductions when |G| = Θ(log T), it does not resolve the overall rate landscape or whether a sharp threshold exists. Addressing this question would refine our understanding of multicalibration’s hardness spectrum between constant and polynomially large group families.