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This research narrows the theoretical gap between deterministic calibration distance and its fractional relaxation, establishing a sharp worst-case order of Theta(T^(1/3)) for binary unit-weight sequences. By providing tight upper and lower bounds alongside an efficient polynomial-time procedure, the study offers concrete improvements to understanding fundamental limits in probabilistic forecasting and model calibration.
Sharp Integrality Gaps in Calibration Distance
arXiv:2610.05679v1 Announce Type: new Abstract: We study the offline gap between deterministic calibration distance C and its fractional relaxation L for binary unit-weight sequences under total absolute-change cost. We sharpen the offline comparison C = 216. The upper bound holds for every input, while each T >= 216 has a rational lower-bound input. For every input with m distinct forecasts, C <= L + m, and the unrestricted-sample worst-case sparse order is Theta(m). For rational forecasts and accuracy, with binary-encoded multiplicities of separately assignable unit identities, a grid-free polynomial-bit-time procedure returns B <= L <= U, U - B < eta, and an exactly calibrated compact repair of cost at most U + m <= L + m + eta.
Abstract:We study the offline gap between deterministic calibration distance C and its fractional relaxation L for binary unit-weight sequences under total absolute-change cost. We sharpen the offline comparison C <= L + O(sqrt(T)) (Qiao and Zheng, 2024, Theorem 2) to the sharp worst-case order Theta(T^(1/3)). If Delta_T is the supremum of C - L over length-T inputs, then T^(1/3)/1000 <= Delta_T <= 41T^(1/3) for T >= 216. The upper bound holds for every input, while each T >= 216 has a rational lower-bound input. For every input with m distinct forecasts, C <= L + m, and the unrestricted-sample worst-case sparse order is Theta(m). For rational forecasts and accuracy, with binary-encoded multiplicities of separately assignable unit identities, a grid-free polynomial-bit-time procedure returns B <= L <= U, U - B < eta, and an exactly calibrated compact repair of cost at most U + m <= L + m + eta.
Machine Learning (cs.LG); Probability (math.PR)
arXiv:2610.05679 [cs.LG]
(or arXiv:2610.05679v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.05679 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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