# Projective Boundary Certificates for Neural Selective Prediction

- ID: 3002
- Canonical URL: https://synthcore.org/idea/3002/projective-boundary-certificates-for-neural-selective-predic
- API JSON: https://synthcore.org/api/idea/3002.json
- API Markdown: https://synthcore.org/api/idea/3002.md
- Verification status: running_benchmark
- Source: [arXiv:2609.01355](https://arxiv.org/abs/2609.01355)
- Category: theory
- Solves: generalization, stability
- ML areas: mlp, transformer, regularization
- Math tags: probability, statistics, combinatorics, optimization, learning-theory
- Ratings: usefulness 8/10; difficulty 5/10; novelty 7/10

## Idea description

Construct a neural acceptance or abstention set from calibration samples together with an explicit boundary map selecting the samples that determine the set. If the map is proper projective and its cross-sample complexity profile is stable, the conditional violation risk has an exact beta law indexed by boundary size rather than network parameter count. This provides a falsifiable, distribution-free certificate for neural selective classifiers and learned safety filters.

## Mathematical statement

Let S=(z_1,...,z_n) be an exchangeable calibration sample, A(S) the acceptance set for a fresh point z, and B(S) a boundary map. Define R(S)=P(z is not in A(S) given S) and K(S)=|B(S)|. Proper projectivity requires that a held-out point z_i is accepted by the leave-one-out rule exactly when the full-sample boundary is retained, and that deleting any accepted non-boundary point does not change B(S). If the cross-sample complexity profile q_k is stable, exchangeability gives a beta law, with indexing determined by the boundary convention; a common convention is R given K=k distributed as Beta(k,n+1-k). If q_k varies, conditioning only on K is invalid and the conditional law must be represented as a mixture over boundary configurations c with weights w(c given k).

## Key formulas

- $$R(S)=\mathbb{P}_{Z\sim P}\left\{Z\notin A(S)\mid S\right\},\qquad K(S)=\left|B(S)\right|.$$
- $$z_i\in A(S\setminus\{z_i\})\quad\Longleftrightarrow\quad B(S\setminus\{z_i\})=B(S).$$
- $$x\in A(S),\ x\notin B(S)\quad\Longrightarrow\quad B(S\setminus\{x\})=B(S).$$
- $$R\mid K=k\sim\operatorname{Beta}(k,n+1-k)\quad\text{if the cross-sample profile is stable};\qquad f_{R\mid K=k}(r)=\sum_c w(c\mid k)f_{R\mid c}(r)\quad\text{otherwise}.$$

## Implementation notes

1. Integration point: train a neural predictor on a training split, then construct an acceptance rule on an independent calibration split. For selective classification, let s_theta(x,y) be a nonconformity score and accept when s_theta(x,y) is at most a calibrated threshold t(S). For a safety filter, use several scores and a vector threshold. Store an explicit boundary map consisting of points attaining the active order statistic or active Pareto/support constraints. 2. Pseudocode: compute calibration scores; construct A(S) and B(S); record K; for each calibration index i, delete z_i and recompute the rule; test whether deleting i preserves the boundary and whether z_i is accepted exactly when the boundary is preserved. Repeat over random subsets to estimate the cross-sample profile. If the profile is stable, calculate the beta upper quantile BetaInv(1-delta;k,n+1-k). If it is unstable, estimate mixture weights over boundary configurations and use the corresponding mixture quantile. 3. Taken from the paper: projectivity, deletion invariance, boundary-size conditioning, and the beta-versus-mixture distinction. Estimated empirically: profile stability, deletion invariance, and mixture weights. 4. First experiment: use an MLP on MNIST with an independent calibration split and top-score abstention. Compare ordinary conformal calibration, the beta certificate, and a naive boundary-size certificate. Repeat over at least 1,000 test batches. The prediction is that scalar order-statistic rules will match the beta CDF, while tied or competing vector boundaries will deviate from it but match the estimated profile mixture; certified violation frequency should remain below delta.

## Verification

- Status: running_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a compact projective-boundary MVP for scalar max-score calibration plus an unstable adaptive-rank control. The max rule matched the predicted Beta(1,20) moments closely, passed projectivity checks, and achieved approximately 0.9 certificate coverage; the unstable control's naive boundary-size beta certificate covered only 0.404 of batches. The idea provides a real exact-law diagnostic/certificate, but it did not improve predictive coverage over ordinary conformal calibration because both use the same max threshold.

### Mechanism check

- Verdict: Built a compact projective-boundary MVP for scalar max-score calibration plus an unstable adaptive-rank control. The max rule matched the predicted Beta(1,20) moments closely, passed projectivity checks, and achieved approximately 0.9 certificate coverage; the unstable control's naive boundary-size beta certificate covered only 0.404 of batches. The idea provides a real exact-law diagnostic/certificate, but it did not improve predictive coverage over ordinary conformal calibration because both use the same max threshold.
- Confidence: 8/10
- Limitations: The MVP uses synthetic uniform nonconformity scores rather than training an actual MLP or testing MNIST. It verifies the scalar order-statistic theorem and demonstrates failure under an unstable boundary profile, but does not evaluate vector/Pareto boundaries, estimated profile mixtures, computational speed, or neural selective-prediction accuracy.

## Artifacts

- [experiment.py](https://synthcore.org/code/1152/experiment.py)
- [report.md](https://synthcore.org/code/1152/report.md)
- [results.json](https://synthcore.org/code/1152/results.json)
- [Download all files as ZIP](https://synthcore.org/download/1152)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.