Certifying Distributionally Robust Decisions from Projection Observations

Scenario optimization, conformal prediction, and related distribution-free certification methods use finite samples to construct decisions or prediction sets with violation-risk guarantees for fresh observations. In several classical settings, the conditional violation risk follows an exact beta law, whose tail has a beta-binomial representation and whose parameter is a support, calibration, or compression dimension. This paper identifies the deterministic boundary mechanism behind these formulas and derives the corresponding law when the observed boundary size is random. A decision rule is represented by an acceptance set for future observations, together with a boundary map selecting the sample points responsible for that set. The resulting pair is called a {\em proper projective boundary scheme} when held-out samples are accepted precisely if the full-sample boundary is retained, and accepted non-boundary samples can be deleted without changing that boundary. For every such scheme, the conditional law of the violation risk given the observed boundary size is determined by the boundary's cross-sample complexity profile. A stable profile yields the usual beta law, whereas a varying profile produces an exact profile correction. The framework covers scalar order-statistic calibration, support-reconstructive scenario programs, cascaded support-removal certificates, coordinatewise envelopes, and Pareto-frontier calibration with vector scores. It also yields conditional probabilistic certificates and a no-go result explaining why observed complexity alone is insufficient.

Adaptive procedures must work without nuisance information an oracle may use, such as a gradient scale or smoothness index, and robust procedures may have to answer queries whose coordinate and inspection time are chosen only after the data are seen. Such comparisons are meaningful only when the oracle advantage and validity contract are stated explicitly. We formalize nuisance adaptation via a slice-normalized minimax ratio retaining the worst-case instance within each nuisance slice, and separately define the robustness cost of expanding from one preannounced Gaussian query to arbitrary post-hoc inspection. Our main result is a finite-horizon composition law for Gaussian certification: from M independent coordinates, a familywise certifier protecting every coordinate and time up to T pays optimal normalized squared half-width of order log(eM) + log log(e^eT), within the sample-mean-centered rectangular class. Epoch stitching gives the upper bound; independent Gaussian block increments across coordinates and geometric time scales give a matching lower bound, already holding on a geometric checkpoint grid, forcing quantiles of the realized maximum width so selection and stopping taxes add. Two benchmark regimes complete the picture: unknown gradient scale in online convex optimization has constant cost, while pointwise adaptation over nested Holder classes costs order (log n / log log n)^(s1/(2s1+1)). Cast as model monitoring, the law lets an analyst inspect any of M slice metrics at any data-dependent time: the naive fixed-query band's selected coverage degrades sharply, to 0.30 at M=1 and to zero for M>=10, while the epoch-stitched certifier holds familywise coverage at an additive iterated-logarithm width cost. Experiments put both sharp predictions at risk of refutation; both survive.

A sound runtime admission gate executes only actions it can certify, and certifies only what its observations support. This paper asks how observations should be designed to maximize the set of actions that can be safely admitted, and shows the question is not a re-vocabulary of classical design problems. First, a projection--estimability separation: decomposing a constraint normal as $c=c_{\mathrm{Range}}+c_{\ker}$ relative to an information matrix, two-point discrimination along $c$ becomes arbitrarily reliable as the budget grows whenever $c_{\mathrm{Range}}\neq 0$, while robust admission of an action with normal $c$ is impossible at every budget whenever $c_{\ker}\neq 0$; such mixed directions are generic at any deficient rank, and at full rank the decoupling is bounded by the Kantorovich ratio and diverges with the condition number. Discrimination-optimal designs, being corner solutions of a linear criterion, land in exactly this regime. Second, an active-face equivalence theorem: the permissiveness-optimal design is $L$-optimal for a target matrix generated endogenously by the action faces that become certification bottlenecks, weighted inversely by their remaining slack; single-face collapse recovers $c$-optimal and goal-oriented design exactly, and a Caratheodory argument yields a bottleneck certificate of at most $r(r+1)/2+1$ faces. Around these we assemble exact certifiability per convex contract mode, for which $κ\approx 3.29$ is derived rather than calibrated, the $\sqrt{r}$ price of contract-agnostic design, an information-to-slack transfer theorem with a curvature-budget corollary, and a two-part audit in which a design meeting every margin requirement still leaves an action face at a certification cost above ten times its testing cost, in every probe library tested.