发表机构
University of Manchester; University of Southern California（曼彻斯特大学; 南加利福尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将确定性校准距离与分数松弛的离线差距从 O(√T) 改进为尖锐的 Θ(T^(1/3))，并给出多项式时间算法，实现近似最优的精确校准修复。
AI 中文摘要
我们研究了在总绝对变化成本下，二元单位权重序列的确定性校准距离 C 与其分数松弛 L 之间的离线差距。我们将离线比较 C ≤ L + O(√T)（Qiao 和 Zheng，2024，定理 2）改进为尖锐的最坏情况阶 Θ(T^(1/3))。若 Δ_T 是长度为 T 的输入上 C - L 的上确界，则对于 T ≥ 216，有 T^(1/3)/1000 ≤ Δ_T ≤ 41T^(1/3)。上界对所有输入成立，而每个 T ≥ 216 都存在一个有理数的下界输入。对于具有 m 个不同预测的每个输入，C ≤ L + m，且无限制样本的最坏情况稀疏阶为 Θ(m)。对于有理预测和精度，通过二元编码的可分别分配单位身份的多重性，一种无网格的多项式位时间过程返回 B ≤ L ≤ U，U - B < η，以及一个成本至多为 U + m ≤ L + m + η 的精确校准的紧凑修复。
英文摘要
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.
Comments38 pages
