The Information Cost of Exposure-Weighted Risk Certification

A risk certificate supports deployment only if it can approve a safe policy with the available data. This requirement is difficult to assess when inputs generate different numbers of predictions or amounts of financial exposure. We study the sample cost of certifying the ratio of expected erroneous exposure to expected exposure. For fixed mean exposure and a specified safe alternative, we reduce the finite-sample minimax problem over bounded observation laws to an exactly solvable three-point experiment. The resulting sample requirement has order $\log (1/\delta )/[\mu \mathrm{kl}(r,\alpha )]$, where $\mu$ is normalized mean exposure, $r$ is risk, $\alpha >r$ is the risk cap, and $\delta$ bounds false certification, with $\alpha$ and target power fixed. This benchmark distinguishes necessary data from conservative test design and loose sufficient bounds. In one zero-risk setting, the optimum requires 27 observations, an implementable sampling test requires 29, and its sufficient bound gives 360. We also connect sampling likelihood ratios to direct product betting and identify the same information scale in worst-case log growth. Controlled experiments and multilabel prediction tasks show when direct tests use population structure that sampling discards, and when sampling remains more effective. Together, the analysis and experiments explain how a reduction can preserve worst-case power while leaving room for improvement on particular populations.