When Should Adaptive Generator-Verifier Loops Stop While Controlling False Discoveries?

A cheap verifier can accumulate false acceptances

Generator-verifier loops are common in inference-time systems: a generator proposes candidates, a cheap verifier scores or filters them, and the search stops when something looks acceptable. The verifier stands in for a more expensive ground-truth oracle, which is the whole point, since the verifier is cheap and fast to query repeatedly.

The paper's central concern is that this substitution is not statistically neutral. Once the generator adapts to past verifier outputs, it can search for candidates that exploit the proxy's weaknesses. A verifier that looks reliable on isolated queries can accumulate false acceptances under reuse, so proposals pass the proxy but fail the costlier ground-truth check. The question the paper answers is therefore not how to score candidates, but when to stop a loop while controlling the quality of the accepted set as measured by the oracle, not the proxy.

FDR counts oracle failures, not low proxy scores

The paper's target quantity is the false discovery rate of the accepted set: the expected fraction of accepted proposals that the oracle would reject, held below a user-specified level alpha. This is a different quantity from a verifier score threshold or a per-query error rate. A verifier can be accurate marginally and still produce a bad accepted set once the generator has adapted against it.

Formally, at round t the generator proposes X_t, the verifier returns a score S_t and feedback V_t, and the oracle label Y_t in {0,1} stays unobserved. The procedure halts after collecting kappa >= 1 accepted proposals and returns a set of accepted proposals among X_1 through X_t with cardinality smaller than kappa. No restriction is placed on how X_t depends on the past history, so the generator may adapt aggressively to prior failures and verifier corrections. When kappa = 1, the guarantee controls the FDR of the first nonempty returned set, which in the common single-return setting is exactly the probability of a false first acceptance.

Three ingredients turn stopping into online testing

The construction has three parts. First, the paper builds marginally valid local e-values through an index-betting normalization, wrapping nonnegative permutation-equivariant scoring rules so that validity survives learning the score from data. A direct plug-in likelihood-ratio estimate is not automatically valid, because its expectation under the null can exceed 1.

Second, the paper introduces robust-core conformal risk control, a finite-sample calibration method for conformal risk control with non-monotone losses. Standard conformal risk control assumes the loss is monotone in the calibrated parameter, and its guarantee can fail otherwise; here the per-run loss, defined as the maximum false discovery proportion up to the kappa-acceptance time, is non-monotone in the scale parameter lambda.

Third, the local e-values are aggregated with the weighted online e-Benjamini-Hochberg procedure, which yields stopped FDR control under arbitrary within-run dependence. Online e-BH spends the error budget across rounds through a schedule gamma_t, and the paper calibrates that schedule by scaling it with lambda, choosing the largest scale whose historical losses still certify the target level.

Synthetic checks and a protein-design benchmark

The approach is validated in synthetic settings and on a protein-design benchmark derived from MotifBench. In the protein setting, RFdiffusion or Proteina generates candidate backbones, ProteinMPNN designs amino-acid sequences, ESMFold provides a cheap structural proxy, and AlphaFold 3 serves as the more expensive oracle that defines the benchmark label.

A selected proposal counts as successful only when the AlphaFold 3 prediction for its ProteinMPNN sequence matches the prescribed motif within 0.75 Angstrom RMSD, matches the proposed backbone within 1.5 Angstrom C-alpha RMSD over the scaffold mask, has pTM at least 0.80, and has mean pLDDT at least 85. The paper notes these criteria were heuristically selected and that the pipeline could be reimplemented with different ground-truth metrics.

The cheap verifier combines pLDDT mean, pTM, and three RMSD terms into a single proxy score, with the largest penalty on motif disagreement. The paper states this score was not tuned to correlate with a specific notion of ground-truth correctness; its weights were chosen heuristically to balance the scale and importance of the terms.

Feedback improves proxy scores and risks proxy drift

In a 30-round feedback loop, both generators show rapid gains in early rounds followed by a gradual plateau. The cumulative-best score curves are nondecreasing by construction, because each trajectory retains its best-scoring candidate. Motif RMSD generally decreases over the same rounds while mean pLDDT increases. Proteina reaches a lower motif RMSD, whereas RFdiffusion attains a higher mean pLDDT.

These trends are measured through the cheap verification, and the paper reads them as evidence that feedback finds proposals with better motif preservation and higher confidence. The same trends also create the risk the paper is built around: as rounds pass, the loop may favor proposals that the cheap verifier scores well rather than properties associated with ground-truth correctness.

Stopped FDP tracks alpha while kappa adds utility

For the held-out evaluation, the authors generated 300 labeled trajectories for each of Proteina and RFdiffusion, where one trajectory is a 30-round feedback run for a fixed motif-scaffolding task. For each held-out trajectory, the remaining 299 trajectories from the same generator are split into 199 for e-value construction and 100 for conformal calibration. The stopping rule is replayed with AlphaFold 3 labels withheld, and stopped FDP and returned designs are scored afterward.

The robust-core calibrated online e-BH rule is evaluated over target levels alpha in {0.05, 0.10, 0.15, 0.20} and target acceptance counts kappa in {1, 5, 10}. Stopped FDP increases as alpha increases, while curves for different kappa values remain close, so allowing more accepted designs does not substantially raise the observed error rate. Returned designs increase with both alpha and kappa, with counts capped at kappa.

In the synthetic comparison, online e-BH sits far below the diagonal because it fixes lambda = 1. At alpha = 0.10, robust-core CRC and envelope CRC have similar mean FDPs of 0.082 and 0.085. Robust-core CRC is closer to the diagonal for larger alpha levels, while the difference between the two is very minor at smaller alpha values.

The guarantee assumes exchangeable logged and deployed runs

The current guarantee requires the logged and deployed runs to be exchangeable or even i.i.d. The paper flags this as a limitation, since in many real-world examples data arrives sequentially and may not be exchangeable. A weighted form of conformal risk control could extend the results to the non-exchangeable setting, but combining weighted conformal prediction with the non-monotone construction remains an open problem.

A second extension concerns structure: many inference-time systems organize proposals as a tree or directed acyclic graph rather than a single trajectory. Extending the framework to adaptive branching, pruning, and merging would require FDR control under those operations.

Within the paper's own setting, the practical priority is the pair (alpha, kappa). The evidence shows that raising kappa returns more designs without substantially raising stopped FDP, while raising alpha moves stopped FDP toward the target. The choice that matters most is therefore how much error budget to spend, not how many accepted designs to allow.

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