Bounded projection distance for MultiQT with general ordered base forecasts

Prove that, for the Multi-Level Quantile Tracker (MultiQT) run with arbitrary ordered base quantile forecasts b_t ∈ R^{|A|} (i.e., b_t^{α_1} ≤ b_t^{α_2} ≤ ··· ≤ b_t^{α_{|A|}}), the Euclidean distance between the played offset θ_t = Π_{K−b_t}(\tilde θ_t) and the hidden offset \tilde θ_t remains uniformly bounded over time; specifically, show the existence of a constant B (independent of t) such that ||θ_t − \tilde θ_t||_2 ≤ B for all t.

Background

The paper proves that MultiQT achieves an O(1/√T) calibration error rate in general, and shows that a faster O(1/T) rate holds when the projection distance ||θ_t − \tilde θ_t||_2 remains bounded. This bounded-distance property is established in Lemma 7.1 for the special case of point forecasts (b_t = k_t·1), enabling the improved calibration rate.

The authors conjecture that the same bounded-distance property holds when the base forecasts b_t are arbitrary ordered vectors, which would extend the O(1/T) calibration rate beyond the point-forecast setting. They note that their experiments provide empirical evidence for this behavior but that a theoretical proof requires techniques beyond those used for the point-forecast case.

References

We conjecture that the MultiQT iterates also maintain bounded projection distance in the general setting where $b_t$ is an arbitrary ordered vector. Although our experiments provide empirical evidence of this, establishing it theoretically requires techniques beyond those used in \Cref{lemma:point_forecasts_bounded_distance} and thus we leave it as an open problem.

— Calibrated Multi-Level Quantile Forecasting  (2512.23671 - Ding et al., 29 Dec 2025) in Section 6 (Discussion), Subsection "1/T rate for calibration error"