Certifying Distributionally Robust Decisions from Projection Observations

Pouya Ahadi; Kamran Paynabar

stat.MLOct 25, 2026

Certifying Distributionally Robust Decisions from Projection Observations

Authors: Pouya Ahadi, Kamran Paynabar

Organizations: H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA.

Abstract

A scalar measurement can determine a robust decision without identifying the latent covariance or every decision risk. We study finite Gaussian projection experiments with a fixed mean-plus-standard-deviation target. The observation laws define a set of indistinguishable moment models. We exhibit decisions that agree throughout that set despite unidentified risk values, and give a confidence-set certificate that returns a decision or an unresolved outcome. We prove soundness, necessary abstention for indistinguishable models with incompatible optima, and a fixed-checkpoint certification guarantee. A scalar profile reduces nonlinear risk comparisons to convex inner problems. Across seven synthetic instances, jointly calibrated moment regions give sufficient budgets 21 to 52 times explicit expected-sample information lower bounds. Separately measured stopping times are lower, with zero decision errors in 1,900 simulated streams. These comparisons do not constitute matching bounds, and the guarantees require Gaussian replication.

Figures & tables

Case	$\Delta_{S}$	Observed mean	Sufficient	Info. lower	Suff./lower
Restricted $M=6$	0.300	$1{,}500$	$3{,}000$	73.24	41.0
Restricted $M=8$	0.100	$4{,}505$	$24{,}000$	703	34.1
Restricted $M=8$	0.040	$25{,}670$	$96{,}000$	$4{,}502$	21.3
Restricted $M=6$	0.010	$402{,}240$	$1{,}536{,}000$	$72{,}949$	21.1
Full-query $M=8$	0.00204	$41{,}656{,}320$	$196{,}608{,}000$	$4{,}008{,}117$	49.1
Three decisions $M=6$	0.0494	$7{,}200$	$24{,}000$	766	31.3

Table 1: Decision margin

\Delta_{S}

, observed mean stopping time, sufficient checkpoint budget, and explicit-alternative expected-sample lower bound. All rows use the joint set ( 13 ) on the same 21-checkpoint grid. Observed means use 300 independent streams per row, except the all-unknown row (100). The all-unknown row uses the global contrast profile. The final ratio divides the sufficient budget by the information lower bound; these are not matching theorems.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Set	Inner solves	Lower	Feasible upper	Direct check
Exact fiber, $r=3$	105	$-0.7457943$	$-0.7457843$	$-0.7457845$
Exact fiber, $r=5$	123	$-0.7528310$	$-0.7528210$	$-0.7528212$
Centered set, $r=3$	69	$0.0028067$	$0.0029067$	$0.0028990$
Observed set, $r=3$	63	$0.0208455$	$0.0209455$	$0.0209401$

Appendix

Table 2: Numerical contrast intervals and direct checks. The centered and observed confidence sets also used 844 and 780 linear bound solves for their moment slabs and PSD constraints. Bounds use ordinary floating point, not validated interval arithmetic.

Case	Queries	Balanced over all	Decision-aware uniform
Mixed projections	8	3.22	1.005
Mixed projections	10	4.23	1.005
Three decisions	10	5.00	1.50
Three decisions	12	4.76	1.19

Appendix

Table 3: Optimal information rate divided by each fixed reference rate. Large all-query ratios mainly measure the cost of sampling weak or nuisance queries; they do not measure an adaptive gain.

Case	Box budget	Ratio
Row 1	$6{,}000$	$81.9$
Row 2	$48{,}000$	$68.3$
Row 3	$192{,}000$	$42.7$
Row 4	$3{,}072{,}000$	$42.1$
Row 5	$393{,}216{,}000$	$98.1$
Row 6	$48{,}000$	$62.7$

Appendix

Table 4: Coordinatewise fallback budgets, in the row order of Table 1 .

Decision-Making under Uncertainty

Decision-making under uncertainty studies how to choose actions when outcomes, states, or models are imperfectly known.

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Exact Risk-Complexity Laws for Projective Boundaries in Scenario Optimization and Distribution-Free Certification

Giuseppe C. Calafiore

Department of Electronics and Telecommunications, Politecnico di Torino, Italy

Conformal Prediction Generalization Bounds

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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.

Abstract

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