Calibeating Made Simple
The paper studies calibeating—post-processing external forecasts online to minimize cumulative losses while matching an informativeness-based benchmark. Unlike prior work that used loss-specific arguments, the authors reduce calibeating to standard online learning primitives, showing it is minimax-equivalent to regret minimization. This yields optimal rates for general proper losses and improves bounds for simultaneous calibration and calibeating.
The paper presents a clean reduction-based framework that unifies and generalizes prior results on calibeating. The core contribution is establishing minimax equivalences between calibeating and no-regret learning (Theorem 3.1), and between multi-calibeating and the combination of calibeating plus the expert problem (Theorem 4.1). For binary predictions with Brier loss, the authors achieve the optimal $O(|Q|\log T)$ calibeating rate simultaneously with $\tilde{O}(\sqrt{T})$ calibration, improving polynomial dependencies in prior work.
The reduction framework is elegant and modular. Theorem 3.1 shows that any no-regret learner with regret $\alpha(T)$ yields a calibeating algorithm with rate $|Q|\alpha(T/|Q|)$ by running independent copies for each distinct forecast value. The lower bound in Theorem 3.5 matches this up to constants, establishing minimax equivalence. The multi-calibeating result (Theorem 4.1) cleanly decomposes the problem into per-forecaster calibeating plus expert aggregation, yielding $O(\log N+|Q|\log T)$ for mixable losses—exponentially improving the polynomial dependence on $N$ in prior work.
The simultaneous calibeating and calibration results (Section 5) are restricted to the Brier loss, and the extension to general losses remains open. The calibration error bound of $\tilde{O}(\sqrt{T})$ for the binary case, while sublinear, is significantly larger than the $O(\log T)$ calibeating rate, and the authors note that matching the best-known $O(T^{1/3})$ calibration bound simultaneously with $O(\log T)$ calibeating remains an open question. The analysis relies heavily on discretization with size $O(\varepsilon^{-K+1})$, which suffers from the curse of dimensionality as the number of outcomes $K$ grows.
The comparisons to prior work are fair and well-documented in Table 1. For calibeating, the authors recover the $O(|Q|\log T)$ rates of Foster and Hart (2023) for Brier and log losses via the reduction, but also extend to general mixable losses and bounded losses with tight lower bounds (Corollaries 3.6 and 3.7). For multi-calibeating, they improve upon Foster and Hart (2023) and Lee et al. (2022) by achieving logarithmic dependence on both $N$ and $T$ rather than polynomial. The lower bound constructions (Theorems 3.5 and 4.4) rigorously establish minimax optimality by reducing to known hard instances for regret minimization and the expert problem.
The algorithms are described clearly with pseudocode (Algorithms 1, 2, 3), and the analysis relies on standard online learning primitives (Follow-the-Leader, Exponentially Weighted Online Optimization, Hedge) with well-documented regret bounds. However, no code or data is provided in the paper, and some experimental validation or empirical demonstration of the bounds is absent. The hyperparameters (e.g., discretization level $\varepsilon$, learning rates) are specified in terms of asymptotic rates, but practical instantiation would require additional tuning. The reduction-based approach should be reproducible in principle given the clear algorithmic descriptions, but the absence of implementation details or simulated experiments limits empirical verification.
We study calibeating, the problem of post-processing external forecasts online to minimize cumulative losses and match an informativeness-based benchmark. Unlike prior work, which analyzed calibeating for specific losses with specific arguments, we reduce calibeating to existing online learning techniques and obtain results for general proper losses. More concretely, we first show that calibeating is minimax-equivalent to regret minimization. This recovers the $O(\log T)$ calibeating rate of Foster and Hart [FH23] for the Brier and log losses and its optimality, and yields new optimal calibeating rates for mixable losses and general bounded losses. Second, we prove that multi-calibeating is minimax-equivalent to the combination of calibeating and the classical expert problem. This yields new optimal multi-calibeating rates for mixable losses, including Brier and log losses, and general bounded losses. Finally, we obtain new bounds for achieving calibeating and calibration simultaneously for the Brier loss. For binary predictions, our result gives the first calibrated algorithm that at the same time also achieves the optimal $O(\log T)$ calibeating rate.
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