🤖 AI Summary
This work addresses the long-standing bottleneck of the $O(T^{2/3})$ lower bound on calibration error in online binary sequence calibration. The authors propose an efficient randomized predictor that integrates the SPR-Calibration procedure with an outer Blackwell-style correction mechanism, supported by a novel analytical framework based on proxy sequences and residual decomposition. By leveraging quadratic potential function analysis and exploiting sparsity structures, the method achieves—while maintaining computational efficiency—the first improvement over the classical bound, reducing the expected calibration error to $O(T^{2/3-\varepsilon})$ for some $\varepsilon > 0$, thereby significantly outperforming the previous best-known results.
📝 Abstract
We study the online binary sequential calibration problem. A recent breakthrough by \citet{dagan2024breaking} overcomes the classical \(T^{2/3}\) barrier for calibration error. Building on this result, we present an efficient randomized forecaster that achieves an expected calibration error \(O(T^{2/3-\varepsilon})\) for some constant \(\varepsilon>0\).
Our forecaster combines the \textsc{SPR-Calibration} procedure \citep{dagan2024breaking} with an outer Blackwell-style correction layer. The \textsc{SPR-Calibration} procedure controls calibration with respect to a surrogate sequence of conditional-mean estimates, while the correction layer controls the additional error incurred when these surrogates are used to approximate the true outcomes. The analysis decomposes the total calibration error into the surrogate calibration error and the residual discrepancy between the surrogate sequence and the true outcomes. The former is bounded by the \textsc{SPR-Calibration} guarantee in \citet{dagan2024breaking}, and the latter is controlled using a quadratic potential argument together with the sparsity of the \textsc{SPR-Calibration} forecaster.