Valid Stopping in Adaptive Generator-Verifier Loops

1 Introduction

• Verifier reuse can accumulate false acceptances as generators adapt to proxy feedback, making oracle-measured accepted-set quality the key stopping problem.A verifier can be reliable on isolated queries yet still be exploited under repeated adaptive search.Source: §1 · p. 1
• The procedure halts after at least κ accepted proposals while controlling accepted-set FDR below user-specified α, without restricting how proposals depend on past history.For κ = 1, the guarantee is the probability of a false first acceptance in the single-return setting.Source: §1 · p. 1
• The approach combines index-betting local e-values, robust-core conformal risk control for non-monotone losses, and weighted online e-BH for stopped-FDR control.Its stated guarantee permits arbitrary within-run dependence.Source: §1 · pp. 1–2
• The protein-design benchmark uses RFdiffusion or Proteina to generate backbones, ProteinMPNN to design sequences, ESMFold as proxy, and AlphaFold 3 as oracle.The benchmark is derived from MotifBench.Source: §1 · p. 2

2 Setting and Overview

• Within a run, proposals, verifier feedback, and scores may depend arbitrarily on past history, while historical runs are i.i.d. at the run level.The deployment run shares the historical run-level distribution; its correctness labels remain hidden.Source: §2 · p. 2
• Each round's decision uses a window of the most recent K+1 verifier scores, and labeled historical runs supply an augmented sample with a pseudo-label of 0 for the deployment window.The convention treats the null hypothesis as the proposal being incorrect.Source: §2 · p. 2
• Index betting normalizes learned nonnegative scores against offline null windows to produce marginally valid e-values without requiring the fitting rule to be more than order-invariant.A direct plug-in likelihood-ratio estimate need not be valid because its null expectation can exceed 1.Source: §2 · p. 3
• Weighted online e-BH aggregates roundwise e-values into nested accepted sets and gives trajectory-level FDR control uniformly over stopping times up to the κ-acceptance time.The guarantee follows from marginal validity at each round and the bound of any stopping-time FDP by the trajectory's supremum FDP.Source: §2 · p. 3
• Historical runs calibrate the largest schedule scale whose expected process loss is at most α, but that loss is non-monotone in the scale.Standard monotone CRC may fail here, motivating robust-core CRC for finite-sample calibration.Source: §2 · pp. 3–4

Exchangeability-based e-values

• Exchangeability-based e-values normalize statistics using null-preserving symmetry; the paper instead constructs a new marginally valid e-value for each generator round from logged null windows.With the true likelihood ratio as score, the local statistic is a Besag–Clifford e-value with i.i.d. reference samples.Source: §3 · p. 4
• The local construction separates estimation error from finite-pool normalization error and establishes matching finite-sample rates under fractional-moment assumptions, with vanishing normalization penalty under finite KL divergence.The stated fractional-moment assumptions cover heavy-tailed likelihood ratios.Source: §3 · p. 4

Sequential testing and online FDR control

• Unlike sequential testing of a single trajectory-level error event, the paper targets the expected fraction of incorrect proposals accepted and uses online e-BH for stopping-time control under arbitrary dependence.Online e-BH provides the stronger bound on the expected supremum of FDP over rounds.Source: §3 · p. 4

Risk control under non-monotone losses

• Standard CRC requires monotone losses, while robust-core CRC lifts calibration to parameter sets to recover finite-sample risk control under exchangeability without algorithmic stability.The method also gives an exact algorithm on finite grids.Source: §3 · pp. 4–5

4 Roundwise Marginally Valid e-Values

• Under the null, exchangeability makes the test window’s hidden batch position uniform, so budgeted batch-symmetric index bets yield valid e-values.The construction treats the proposal as a hidden coordinate among logged null windows.Source: §4 · p. 5
• Under the alternative, the posterior probability of the hidden position is proportional to the likelihood ratio, making likelihood-ratio betting Kelly-optimal among budgeted strategies.This mirrors classical log-optimal betting results [Kelly, 1956, Larsson et al., 2025].Source: §4 · pp. 5–6

4.1 Power decomposition and asymptotic rates

• A bounded (1+δ)-moment of the likelihood ratio makes betting cost vanish at a rate controlled by δ; δ=1 gives O(1/n0,t), while δ<1 covers heavy tails.The upper bound is paired with a matching worst-case lower bound for each δ in (0,1).Source: §4.1 · p. 7
• Finite KL divergence alone guarantees asymptotic vanishing of betting cost, even without the bounded fractional-moment condition used for polynomial rates.The estimation residual is a separate loss controlled by worst-case log-ratio error on the augmented batch.Source: §4.1 · pp. 6–7

4.2 Examples and simulations

• Gaussian shifts have finite likelihood-ratio moments of every order, whereas the Pareto pair has a finite fractional-moment range only for δ < b/(a −b).For the Pareto pair, the supplied result states that this range holds for some δ>0 when a>b.Source: §4.2 · p. 7
• The error-rate simulations compare the true likelihood ratio with oracle and estimated index e-values across reference sizes M and nominal levels α.The reported curves distinguish α=0.05 from α=0.01.Source: §4.2 · pp. 7–8, Figure 1 · p. 8
• The normalized log-growth gap between oracle index e-values and the likelihood ratio narrows as fractional-moment exponent δ and reference batch M increase.The likelihood-ratio benchmark is normalized to one.Source: §4.2 · p. 8, Figure 2 · p. 8

5 Robust-Core Risk Control for Non-Monotone Losses

• When losses are non-monotone, classical CRC can be invalid, and the conservative envelope rule can discard parameters that are safe on the realized curve.The envelope replaces each loss profile with its running supremum, so a local spike can suppress later candidates.Source: §5 · p. 9

5.1 Robust core on ordered spaces

• Lifting calibration to sets makes set loss monotone and gives finite-sample risk control under exchangeability for a symmetric certificate and zero-loss fallback.The most preferred parameter in the robust core is selected when that set is nonempty; otherwise the estimator uses the fallback.Source: §5.1 · pp. 9–10

5.2 Finite-grid specialization

• On finite grids, Algorithm 2 tracks a finite summary of possible hidden loss completions and exactly computes the robust core.Theorem 4 and Lemma 1 together provide a finite-sample calibration rule for non-monotone loss profiles.Source: §5.2 · p. 11

5.3 Non-monotone CRC experiment

• Robust-core CRC approaches the highest fixed safe parameter, while the monotone-envelope and stable-CRC baselines remain below it.The envelope rule treats an upward fluctuation as permanent; stable-CRC subtracts a concentration slack that shrinks slowly with n, while robust core avoids both penalties.Source: §5.3 · pp. 12–13, Figure 3 · p. 12
• Algorithm 2 is exponential in grid size in the worst case, but terminates early and merges coincident states; selection times are reported for grids up to |Λ| = 250.Figure 4 plots median selection time against grid size for different calibration sizes and clipped-Gaussian noise levels.Source: §5.3 · p. 13, Figure 4 · p. 13

6.1 Valid stoppage on the unobserved run

• The procedure constructs roundwise e-values by testing each recent score window with a leave-one-out index-betting method; under the roundwise null, each e-value has expectation at most 1.The pipeline uses separate labeled-data splits for e-value construction and calibration.Source: §6.1 · pp. 13–14
• At λ=1, marginally valid roundwise e-values and online e-BH yield trajectory-level FDR control under arbitrary within-run dependence.The procedure forms nested accepted sets and stops at the κ-acceptance time or the run horizon.Source: §6.1 · p. 14

6.2 Robust-core calibration

• Robust-core CRC selects a preferred grid scale from exchangeable calibration loss profiles, falling back to λ=0 when the robust core is empty.The runs used for calibration are exchangeable with the unlabeled test run and jointly independent of the e-value construction split.Source: §6.2 · p. 14

6.3 End-to-end validation on synthetic data

• At α=0.10, Robust-Core CRC and Envelope CRC have mean stopped FDPs of 0.082 and 0.085; Robust-Core CRC is closer to the target at larger α.Online e-BH lies far below the target diagonal because it fixes λ=1.Source: Figure 5 · p. 15, §6.3 · p. 16
• The synthetic validation makes verifier scores progressively favor a neighboring basin, so high proxy scores can persist while proposals move away from ground-truth correctness.It evaluates stopped FDP over 350 runs and compares Robust-Core CRC, Envelope CRC, and unscaled online e-BH.Source: §6.3 · pp. 15–16

7 Experiments

• The protein task asks whether a generated backbone is designable, assessed by whether a predicted amino-acid sequence plausibly folds into that structure.Each later feedback round combines four fresh explore candidates with four exploit candidates seeded from the previous best-scoring structure’s midpoint state.Source: §7 · pp. 16, 18, Figure 7 · p. 17
• The cheap verifier’s score weights were selected heuristically to balance term scale and importance, rather than tuned to correlate with a specific ground-truth correctness notion.The ground-truth criteria use AlphaFold 3 and were also described as heuristic, with different metrics possible.Source: §7 · pp. 17–18
• Over feedback rounds, both generators’ cheap-verifier scores improve rapidly and then plateau; motif RMSD generally falls while mean pLDDT rises.Proteina reaches lower motif RMSD, whereas RFdiffusion attains higher mean pLDDT.Source: Figure 8 · p. 17, §7 · p. 18
• As α and κ increase, the procedure returns more designs; stopped FDP rises with α, while curves for different κ remain close.Figure 9 reports these patterns for both RFdiffusion and Proteina.Source: §7 · p. 18, Figure 9 · p. 18

8 Conclusions

• The procedure controls stopped FDR up to the κ-acceptance time under arbitrary within-run dependence, but requires logged and deployed runs to be exchangeable or i.i.d.The authors identify non-exchangeable sequential data and adaptive tree or graph construction as open extensions.Source: §8 · p. 19

A.1 Proposition 2

• Under batch symmetry, the maximizing budgeted strategy assigns each position its posterior probability of containing the alternative sample.The proof states that the unique maximizing budget is β = q.Source: §A.1 · p. 23

A.4 Theorem 3

• The index strategy's normalization gap is at least 2(2 log 2 − 1)n^-δ for sufficiently large n, and this lower bound applies to every budgeted batch-symmetric strategy.The passage also states that the index strategy minimizes the gap within that class.Source: §A.4 · p. 25

A.5 Corollary 1

• When the likelihood ratio has a finite (1+δ)-moment, the empirical average A_n satisfies E[A_n log A_n] → 0.The proof uses uniform integrability and almost-sure convergence of A_n to 1.Source: §A.5 · pp. 25–26

A.7 Lemma 1

• The robust-core scan returns the largest member of the core, or the fallback if no point belongs to it.A point is in the core exactly when every completion includes it.Source: §A.7 · p. 26

A.8 Proposition 3 and Theorem 5

• Conditioning on the construction split preserves exchangeability across calibration runs and the unobserved run, so the same permutation relabels their local e-value streams and process losses.Integrating over the split gives marginal exchangeability; the robust-core estimator and Theorem 4 then yield Equation (13).Source: §A.8 · p. 27

B Additional Experiment Details

• Each of 300 motif-scaffolding targets receives a 30-round generator trajectory, followed by AlphaFold 3 evaluation.The passage does not specify the generator identities or report outcomes.Source: B Additional Experiment Details · p. 27

B.1 MotifBench-derived benchmark construction

• The benchmark restricts tasks to motifs that form a single contiguous segment and uses ten MotifBench tasks.Each task specifies the motif and integer ranges for left and right scaffold lengths.Source: §B.1 · p. 27
• Sampling 30 distinct left/right scaffold-length pairs per task yields 300 targets, shared by RFdiffusion and Proteina.The targets are formed from the ten tasks' specified scaffold-length ranges.Source: §B.1 · p. 27
• Table 1 lists the MotifBench source tasks and fixes one pair of left and right scaffold lengths for each.Source: Table 1 · p. 27

• Each 30-round trajectory uses eight backbones per round; after the first round, four continue from the preceding winner’s midpoint state and four come from fresh trajectories.Source: §B.2 · pp. 27–28
• For each backbone, ProteinMPNN samples eight sequences and ESMFold refolds them; the highest proxy-scoring pair wins, and its midpoint state seeds later exploit candidates.Source: §B.2 · p. 28
• RMSDs use separate least-squares rigid alignments; scaffold RMSD excludes motif residues for Proteina but equals backbone RMSD for RFdiffusion.Source: §B.2 · p. 29
• AlphaFold 3 evaluates each round’s winning sequence offline; success requires motif RMSD ≤ 0.75 Å, scaffold-mask Cα RMSD ≤ 1.5 Å, pTM ≥ 0.80, and mean pLDDT ≥ 85.Source: §B.2 · pp. 28–29
• At each round, separate likelihood-ratio estimators use the previous and current proxy scores, standardizing feature coordinates from positive-labeled evidence trajectories.Source: §B.2 · p. 29
• The 100 calibration trajectories select the online e-BH scale through robust-core CRC; held-out labels evaluate stopped FDP across the tested α and κ values.Source: §B.2 · p. 29