High-Dimensional Calibration from Swap Regret

TL;DR

This work studies online calibration of high-dimensional forecasts over a convex set P and reveals a tight connection to online linear optimization via swap regret. By reframing calibration as a swap-regret minimization problem and instantiating a TreeSwap-based approach, the authors introduce TreeCal, a universal calibration algorithm that attains Cal_T^{||·||^2} ≤ ε T once T is large enough, with bounds scaling as (diam(P)/√ε)^{O(Rate(P, ||·||)/ε). The analysis shows that the optimal regularizer for online linear optimization governs the calibration error, enabling simultaneous guarantees for any norm and any convex P without relying on external OLO subroutines. For special cases, such as the simplex with l1 norm, the results recover and generalize existing bounds to yield ε-calibration in d^{O(1/ε^2)} rounds, while a lower bound demonstrates the necessity of exponential dependence on 1/ε in general. Overall, the paper bridges calibration and OLO theory, providing a unifying, norm-agnostic framework with strong high-dimensional guarantees and clear limitations.

Abstract

We study the online calibration of multi-dimensional forecasts over an arbitrary convex set $\mathcal{P} \subset \mathbb{R}^d$ relative to an arbitrary norm $\Vert\cdot\Vert$. We connect this with the problem of external regret minimization for online linear optimization, showing that if it is possible to guarantee $O(\sqrt{ρT})$ worst-case regret after $T$ rounds when actions are drawn from $\mathcal{P}$ and losses are drawn from the dual $\Vert \cdot \Vert_*$ unit norm ball, then it is also possible to obtain $ε$-calibrated forecasts after $T = \exp(O(ρ/ε^2))$ rounds. When $\mathcal{P}$ is the $d$-dimensional simplex and $\Vert \cdot \Vert$ is the $\ell_1$-norm, the existence of $O(\sqrt{T\log d})$-regret algorithms for learning with experts implies that it is possible to obtain $ε$-calibrated forecasts after $T = \exp(O(\log{d}/ε^2)) = d^{O(1/ε^2)}$ rounds, recovering a recent result of Peng (2025). Interestingly, our algorithm obtains this guarantee without requiring access to any online linear optimization subroutine or knowledge of the optimal rate $ρ$ -- in fact, our algorithm is identical for every setting of $\mathcal{P}$ and $\Vert \cdot \Vert$. Instead, we show that the optimal regularizer for the above OLO problem can be used to upper bound the above calibration error by a swap regret, which we then minimize by running the recent TreeSwap algorithm with Follow-The-Leader as a subroutine. Finally, we prove that any online calibration algorithm that guarantees $εT$ $\ell_1$-calibration error over the $d$-dimensional simplex requires $T \geq \exp(\mathrm{poly}(1/ε))$ (assuming $d \geq \mathrm{poly}(1/ε)$). This strengthens the corresponding $d^{Ω(\log{1/ε})}$ lower bound of Peng, and shows that an exponential dependence on $1/ε$ is necessary.

Figures (3)

• Figure 1: Geometric depiction of the Bregman divergence from $p$ to $y$.
• Figure 2: [Proof of Lemma \ref{['lem:bias-variance-decomp']}] the average Bregman divergence (orange + purple) decomposes into the Jensen error (orange) and the Bregman divergence to the mean (purple). For example, when $R(p) = \left\|p\right\|_2^2$, $D_R(y|p) = \left\|y-p\right\|_2^2$ and we recover the bias-variance decomposition.
• Figure 3: Visualization of the state of $\mathtt{TreeCal}$/$\mathtt{TreeSwap}$ at time step $t$ (about half-way through the algorithm). For $H=3$, we depict the intervals $\Gamma$ of the first three non-root levels of the tree $(l=1,2,3)$. Each rectangular node represents an interval, with sibling nodes separated by red lines. We represent the specific time step $t$ via the vertical dashed green line. The yellow intervals it intersects at each level correspond to the nodes on the root-to-leaf-$t$ path. Accordingly, $\mathbf{x}_t$ will be the uniform distribution over the labels $p$ of these yellow intervals. We see that the algorithm has committed to the labels of all intervals that started at or before time $t$, and has yet to label the future intervals.

Theorems & Definitions (40)

• Theorem 1.1: Informal restatement of Corollary \ref{['cor:cauchy']}
• Corollary 1.2: Informal restatement of Corollary \ref{['cor:olo-rate']}
• Theorem 1.3: Informal restatement of Theorem \ref{['thm:treecal']}
• Theorem 1.4: Informal restatement of Theorem \ref{['thm:cal-lb']}
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• proof
• Lemma 2.4
• Theorem 3.1: Main theorem