Coverage Cliffs in Off-Policy Quantile Certification

arXiv:2610.22565v1 Announce Type: new Abstract: We study when logged contextual-bandit data can certify an upper tail quantile for a policy selected from the same sample. We construct a PAC-Bayes Bernstein lower-confidence envelope for the target cumulative distribution function (CDF), simultaneous over a predeclared threshold grid and all data-dependent posteriors. The resulting procedure, TailCert, uses the envelope's first crossing of level p to return an atom-safe upper confidence bound on the selected posterior's pooled p-quantile (or inconclusive). It can also be used to select a policy under a certified tail constraint. The concentration ingredients are standard; the main theoretical contribution is a matched finite-class certification boundary comparing full logs with their joint weighted CDF indicators at a fixed threshold. On an explicit rare-action class, the required sample size grows linearly with the overlap factor W, logarithmically with policy-search and confidence complexity, and quadratically with the inverse CDF safety margin; the upper and lower bounds agree up to constants in the stated regime. On this class, the CDF reduction hides which actions exceeded the budget; retaining action counts restores the full-log rate. Simulations verify the predicted search-complexity shift, winner's-curse failure of naive reuse, and the power cost of splitting.

Abstract:We study when logged contextual-bandit data can certify an upper tail quantile for a policy selected from the same sample. We construct a PAC-Bayes Bernstein lower-confidence envelope for the target cumulative distribution function (CDF), simultaneous over a predeclared threshold grid and all data-dependent posteriors. The resulting procedure, TailCert, uses the envelope's first crossing of level p to return an atom-safe upper confidence bound on the selected posterior's pooled p-quantile (or inconclusive). It can also be used to select a policy under a certified tail constraint. The concentration ingredients are standard; the main theoretical contribution is a matched finite-class certification boundary comparing full logs with their joint weighted CDF indicators at a fixed threshold. On an explicit rare-action class, the required sample size grows linearly with the overlap factor W, logarithmically with policy-search and confidence complexity, and quadratically with the inverse CDF safety margin; the upper and lower bounds agree up to constants in the stated regime. On this class, the CDF reduction hides which actions exceeded the budget; retaining action counts restores the full-log rate. Simulations verify the predicted search-complexity shift, winner's-curse failure of naive reuse, and the power cost of splitting.

Machine Learning (cs.LG)

arXiv:2610.22565 [cs.LG]

(or arXiv:2610.22565v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.22565 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Coverage Cliffs in Off-Policy Quantile Certification

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