The minimax rate of estimating second-order calibration error is Õ(1/√n) with a matching Ω(1/√n) lower bound, enabled by analyticity from the sech kernel and yielding the first finite-sample guarantee for second-order Platt scaling.
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Risk-controlled post-processing yields a threshold-structured policy that follows the baseline except where an oracle fallback sharply reduces conditional violation risk, achieving O(log n/n) expected excess risk in i.i.d. settings and exact risk control under exchangeability.
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The Minimax Rate of Second-Order Calibration
The minimax rate of estimating second-order calibration error is Õ(1/√n) with a matching Ω(1/√n) lower bound, enabled by analyticity from the sech kernel and yielding the first finite-sample guarantee for second-order Platt scaling.
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Risk-Controlled Post-Processing of Decision Policies
Risk-controlled post-processing yields a threshold-structured policy that follows the baseline except where an oracle fallback sharply reduces conditional violation risk, achieving O(log n/n) expected excess risk in i.i.d. settings and exact risk control under exchangeability.