Fractional Safe Certification: Exact Multiple-Risk Control and a Sharp Boundary Separation

Jeff Pickhardt

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Fractional Safe Certification: Exact Multiple-Risk Control and a Sharp Boundary Separation

Abstract

Can a calibration rule satisfy exact expected-risk constraints while retaining the utility of a feasible policy at an atomic risk boundary? We study this question for bounded policy losses under exchangeability. Given a candidate selected on separate data and a pointwise zero-risk fallback, fractional safe certification deploys the candidate with a closed-form probability computed on a held-out sample. One common mixture controls every expected test loss at its original budget for every sample size, including policies with nonmonotone losses and multidimensional thresholds. The guarantee averages over calibration data and deployment randomization. On a one-risk binary instance, we characterize all permutation-invariant rules that are valid for every exchangeable law. The fractional rule is pointwise maximal. At the active i.i.d. boundary, every valid deterministic symmetric rule has asymptotic objective regret at least 1/2, while the optimal randomized regret is asymptotic to sqrt((1 − β)/(2πβn)). Under i.i.d. sampling, uniform convergence, and approximate empirical optimization, we also derive a one-sided oracle inequality against the best feasible deterministic policy. For threshold-comparison DAGs, the bound depends on the coordinates influencing each loss. These results connect safe-policy regulation with exact randomized inference and quantify a sharp cost of determinism for boundary efficiency.

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