Sharp Integrality Gaps in Calibration Distance

2026-10-05

AI Summary

This study addresses the long-standing lack of a tight characterization of the worst-case offline gap between binary sequence calibration distance and its fractional relaxation, for which the existing upper bound of O(√T) remains loose. By integrating combinatorial optimization analysis with a polynomial-time grid-independent algorithm, this work establishes, for the first time, the sharp asymptotic order of this integrality gap and derives exact constants for both the upper and lower bounds. Specifically, it tightens the gap upper bound from O(√T) to Θ(T^(1/3)) and introduces sparsity-based bounds parameterized by the number of predictions m. These results enable efficient approximate computation alongside low-cost exact calibration repair, thereby providing a tight complexity characterization that advances the theoretical foundations of online calibration.

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

Problem

• calibration distance
• fractional relaxation
• integrality gap
• binary sequences
• total absolute-change cost

Innovation

• Calibration Distance
• Integrality Gap
• Fractional Relaxation
• Grid-free Algorithm
• Worst-case Bound