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Replicable Dueling Bandits

Rohan Deb, Aadirupa Saha, Udaya Ghai, Karan Singh, Arindam Banerjee

arXiv:2610.20908v1cs.LGcs.AI

TL;DR

Dueling-bandit runs can choose different comparisons because each decision depends on earlier noisy outcomes, raising whether agreement across runs can coexist with low regret. The paper develops trajectory-replicable algorithms for finite-arm and linear preference settings, reporting sublinear regret and a K^2 H_p/ρ^2 bound under strong stochastic transitivity.

  • Problem

    No prior work studied replicability under preference feedback, where a learner observes comparison outcomes rather than numerical rewards, despite the importance of reproducing its sequence of comparisons.

    Source: §1 · pp. 1–2

  • Method

    The paper develops replicable dueling-bandit algorithms, adapting Beat-the-Mean techniques with shared-grid rounding and a reference execution, and using simultaneous elimination under strong stochastic transitivity.

    Source: §1 · p. 2

  • Results

    The algorithms achieve sublinear regret for finite-arm and linear preference models; under strong stochastic transitivity, RepBTM9SST has expected regret e O(K^2 H_p/ρ^2).

    Source: §1 · p. 2, §6 · p. 8

  • Takeaways & Limitations

    Agreement on the entire comparison sequence is compatible with sublinear regret in finite-arm and linear preference models.

    Source: §6 · p. 8

    Whether the K^2/ρ^2 overhead is achievable under relaxed transitivity remains open.

    Source: §1 · p. 3, §6 · p. 8

Abstract

from arXiv · show

Dueling bandits choose each comparison based on earlier noisy outcomes, so two runs on the same preferences can compare different pairs. We study replicable dueling bandits, where two executions with shared internal randomness and independent comparison outcomes must select the same ordered pair in every round with probability at least $1-\rho$. Building on Beat-the-Mean, whose scores change as arms are eliminated and need not rank the best arm highest, we design the first such algorithms with sublinear regret. Under relaxed stochastic transitivity with parameter $\gamma\ge 1$ and the stochastic triangle inequality, we design an algorithm that achieves expected regret $\widetilde{O}(\gamma^8 K^4 H_{\mathrm{p}}/\rho^2)$, where $H_{\mathrm{p}}=\sum_{i=2}^{K}(P_{1i}-1/2)^{-1}$. Under strong stochastic transitivity, we design an algorithm based on simultaneous elimination that improves the dependence on $K$ and achieves $\widetilde{O}(K^2 H_{\mathrm{p}}/\rho^2)$ regret. For linear dueling bandits, we design an algorithm that achieves regret $\widetilde{O}(d^3\sqrt{T}/(c_\mu\rho))$. Experiments on synthetic and language-model preferences show replicability, with sublinear regret.

1 INTRODUCTION

This work introduces trajectory replicability for dueling bandits, where noisy pairwise feedback can make repeated runs choose different comparisons. It gives replicable algorithms with sublinear regret under stated preference assumptions, alongside techniques for handling changing active-set scores.

1 INTRODUCTION

  • Under strong stochastic transitivity, RepBTM9SST achieves expected regret e O(K^2 H_p/ρ^2), improving the dependence on K by a factor of K^2 over the relaxed-regime guarantee.The bound matches the K^2/ρ^2 replicability overhead of elimination algorithms for reward bandits.Source: §1 · p. 2
  • For linear preferences generated by d-dimensional utilities through a known increasing link, RepMaxInP achieves regret e O(d^3√T/(c_μρ)) with high probability.c_μ lower-bounds the derivative of the link.Source: §1 · p. 2
  • Experiments on synthetic and language-model preferences show that the algorithms keep identical-trajectory fractions near 1−ρ and still commit to the best arm, unlike non-replicable baselines that diverge early.Source: §1 · p. 2
  • Shared-grid rounding and a reference execution couple runs, while filtered prefixes of fixed sampling sequences enable concentration uniformly over candidate active sets.BTM scores change as arms are eliminated, and reused comparisons make the active set depend on samples used to estimate those scores.Source: §1 · p. 2
  • RepBTM handles relaxed stochastic transitivity by removing one minimum-score arm at a time, since the best arm need not maximize the active-set score.Its replicability requires O(K^2 log T) rounded estimates and leads to a K^4/ρ^2 overhead.Source: §1 · p. 2
  • Under strong stochastic transitivity, RepBTM9SST eliminates several arms at once and uses doubling budgets, fresh samples, and a rank-sensitive survivor bound to control exploration cost.The method reduces rounded estimates to O(K log T); the survivor bound avoids an additional K^2 loss from small active-set score gaps.Source: §1 · pp. 2–3
  • Whether the K^2/ρ^2 overhead is achievable for γ > 1 and the optimal dependence on K remain open.Source: §1 · p. 3

2 PRELIMINARIES

The preliminaries specify the stochastic comparison model, the best-arm regret objective, and the paper’s trajectory-level agreement criterion. They also describe the varying-action linear setting.

2 PRELIMINARIES

  • At each round, the learner selects an ordered pair of arms and observes a binary comparison outcome drawn with fresh comparison randomness.The preference matrix is unknown, and self-comparisons are allowed.Source: §2 · p. 3
  • The unique best arm beats every other arm with probability greater than 1/2, and regret is measured by cumulative probability regret relative to that arm.The arm’s preference gaps define the gap-dependent complexity H_p.Source: §2 · p. 3
  • Trajectory replicability requires two runs on the same instance to select the same ordered comparison at every round with probability at least 1−ρ.The runs share the algorithm’s internal seed but use independent comparison randomness; δ denotes statistical failure probability.Source: §2 · p. 3
  • In the linear setting, the finite action set may vary over time, and each action has an observed feature vector in R^d.Source: §2 · pp. 3–4

3 REPLICABLE ALGORITHMS FOR FINITE DUELING BANDITS

The finite-arm algorithms adapt Beat-the-Mean to preference scores that change with the active set, while coordinating randomized estimates so executions can remain replicable. A single-arm elimination method handles relaxed transitivity; under stronger transitivity at the best arm, simultaneous elimination improves the dependence on K.

3.1 Relaxed Stochastic Transitivity

  • Because the best arm need not maximize its active-set score under relaxed transitivity, RepBTM removes an estimated minimum only when the estimated score range is sufficiently large.This elimination rule is designed to preserve the best arm even when active-set scores are estimated with error.Source: §3.1 · p. 4
  • RepBTM uses shared rounding and retained opponent records to estimate active-set scores while coordinating decisions across executions with independent comparison outcomes.Its record lists are maintained as filtered prefixes of fixed sampling sequences, supporting uniform concentration over candidate active sets.Source: §3.1 · pp. 4–5, §3 · p. 4
  • Under relaxed transitivity and the stochastic triangle inequality, RepBTM is ρ-replicable and its regret is bounded by O(γ^8ΛRHp) with high probability.The analysis controls estimation error uniformly over active sets, preserves the best arm, and charges comparisons to the eliminations that delete their records.Source: §3.1 · p. 5

3.2 Sharper Guarantees under Strong Stochastic Transitivity

  • Strong transitivity lets the best arm maximize the active-set score, enabling simultaneous elimination of multiple arms using fresh phase samples.Each phase fixes the active set, cycles through leaders, draws uniform opponents, rounds scores at a checkpoint, and discards phase records.Source: §3.2 · p. 6
  • Under strong transitivity at the best arm, RepBTM9SST is ρ-replicable and achieves regret at most CΛSHp with high probability.Its best-arm transitivity condition is implied by relaxed transitivity and the triangle inequality when γ = 1, but only constrains triples containing the best arm.Source: §3 · p. 4, §3.2 · p. 6
  • The strong-transitivity improvement reduces scalar estimates from O(K^2 log T) to O(K log T), yielding a K^2/ρ^2 rather than K^4/ρ^2 regret prefactor.The rank-sensitive survivor bound avoids an additional K^2 sample penalty; whether the K^2/ρ^2 factor extends to relaxed γ ≥ 1 remains open.Source: §3.2 · pp. 6–7

4 REPLICABLE LINEAR DUELING BANDITS

RepMaxInP combines replicable generalized linear estimation with the Maximum-Informative-Pair rule for linear dueling bandits. It uses batched estimates and determinant-triggered batch changes, and its guarantee is stated in utility regret without explicit dependence on the number of available actions.

4 REPLICABLE LINEAR DUELING BANDITS

  • RepMaxInP combines replicable generalized linear estimation with the Maximum-Informative-Pair rule for linear dueling bandits.Preferences follow a link model applied to the difference between the compared action features and an unknown parameter.Source: §4 · p. 7
  • At each batch start, the algorithm fits an estimate from past comparisons and applies a fresh shared randomly shifted grid in covariance-induced coordinates.Within a batch, the estimate, confidence radius, and selection matrix remain fixed.Source: §4 · p. 7
  • The learner selects the promising pair with maximum feature-difference norm under the inverse selection matrix and starts a new batch when the determinant threshold is exceeded.The pair-selection rule breaks ties by index.Source: §4 · p. 8
  • The theorem states that RepMaxInP is ρ-replicable, uses at most B batches, and retains every utility-maximizing index as promising with high probability.The performance measure is utility regret, and the stated guarantees have no explicit dependence on the number of available actions Kt.Source: §4 · pp. 7–8

5 EXPERIMENTS

The experiments examine trajectory agreement and regret on synthetic and language-model preferences. Both replicable methods meet the empirical agreement target in the synthetic study, while the language-model study reports higher full-trajectory agreement for RepBTM9SST than for the unrounded batched baseline and BTM.

5.1 Synthetic Preferences

  • The synthetic study compares RepBTM and RepBTM9SST with non-replicable BTM on Bradley–Terry–Luce instances with K = 5 arms.Paired executions share internal randomness but receive independent comparison outcomes; the study reports the fraction of pairs whose trajectories coincide up to each round.Source: §5.1 · p. 8
  • At ρ = 0.2, both replicable methods exceed the target in aggregate empirical trajectory agreement, while no paired BTM executions agree throughout the horizon.On the synthetic preferences, cumulative regret plateaus after commitment to the best arm, with RepBTM9SST achieving lower regret than RepBTM.Source: §5.1 · p. 8

5.2 Preferences from a Language Model

  • The language-model study uses six summary-generation arms and measures regret against the long humanities-student summary, which is also the Condorcet winner.A version without randomized rounding provides non-trivial but insufficient replicability.Source: §5.2 · p. 8
  • At ρ = 0.25, full-trajectory agreement is 0.795 for RepBTM9SST, 0.290 for the unrounded batched baseline, and 0 for BTM.In the language-model study, regret curves plateau after commitment to the winner, and final regret decreases as ρ increases.Source: §5.2 · p. 8

6 CONCLUSION

The conclusion reports regret guarantees for replicable algorithms in finite-arm and linear dueling bandits, while identifying open questions about the cost of replicability. It also outlines extensions and discloses the paper’s use of generative AI tools.

6 CONCLUSION

  • RepBTM has regret ˜O(γ^8 K^4 H_p/ρ^2), while RepBTM9SST has regret ˜O(K^2 H_p/ρ^2); RepMaxInP has a utility-regret guarantee for linear dueling bandits.The conclusion also states that agreement on the entire comparison sequence is compatible with sublinear regret.Source: §6 · p. 8
  • The conclusion leaves open whether the K^2/ρ^2 overhead is necessary, whether it can be separated from log T, and how to improve or lower-bound linear regret’s d^3 dependence.For relaxed transitivity, reaching K^2/ρ^2 dependence with γ > 1 is also open.Source: §6 · p. 8
  • Figure 1 plots cumulative regret, trajectory agreement against the 1−ρ requirement, and final regret against ρ; its regret curves stay flat up to T.The caption reports 95% intervals across 30 test instances.Source: Figure 1 · p. 9
  • The paper identifies constrained, conservative, and safe settings, weaker winner notions, and active preference-data collection as directions for extending replicable dueling algorithms.The constrained, conservative, and safe settings require comparisons to meet cost or safety requirements.Source: §6 · p. 9

AI use statement

  • The authors report using generative AI for implementation, data analysis and preparation, interpretation, synthetic data, hypothesis refinement, figure work, and writing polish, but not for theoretical modeling or proof-critical work.They state that AI-assisted work was reviewed and that generated code was verified and tested by an author.Source: AI use statement · p. 9

A PROOFS FOR FINITE DUELING BANDITS

The appendix establishes rounding and agreement facts used in the finite-bandit proofs, then states the relaxed-transitivity setting for the subsequent analysis.

A PROOFS FOR FINITE DUELING BANDITS

  • The appendix defines ε_i as P_1,i−1/2, sets ε_1=0, and uses the strict preference order to give 0<ε_i<1/2 for i>1.It also identifies H_p as the sum over non-best arms of inverse preference gaps.Source: A PROOFS FOR FINITE DUELING BANDITS · p. 14

A.1 Shared-grid estimation

  • The rounding map Q_h has error less than h, and two rounded values can differ only when a grid boundary falls between their inputs.For a fixed interval of length a, the probability of containing a grid boundary is at most min{1, a/h}.Source: §A.1 · p. 14
  • Lemma A.2 establishes accuracy and two forms of agreement for rounded empirical estimates under its sampling conditions, using concentration bounds and a random grid offset.The proof conditions on opponent labels for one agreement result and averages over uniform labels for another.Source: §A.1 · pp. 14–15

A.2 Relaxed stochastic transitivity

  • The relaxed-transitivity analysis assumes the stated conditions for every ordered triple, with parameter γ≥1.The subsection also uses the worst-ranked arm in an active set to define η(W).Source: §A.2 · p. 16

A.2.1 Preference inequalities and preservation of the best arm

The proof derives preference inequalities that protect the best arm during elimination, then uses retained-prefix sampling to obtain uniform estimate accuracy across active sets.

A.2.1 Preference inequalities and preservation of the best arm

  • Under relaxed transitivity and the triangle inequality, the best arm’s active-set score exceeds the worst arm’s by at least η(W), while pairwise gaps and score differences admit γ-dependent upper bounds.The proof also notes that η(W) need not decrease as arms are removed.Source: §A.2.1 · p. 16
  • Removing an estimated minimum only when the estimated range exceeds 6γ^2τ preserves arm 1 whenever all active-set score estimates have error at most τ.When τ is large relative to η(W), the estimated range remains below the elimination threshold if arm 1 were estimated as a minimum.Source: §A.2.1 · p. 16

A.2.2 Retained prefixes and simultaneous accuracy

  • For a fixed active set W, each leader’s retained record list is a prefix of its label stream filtered to opponents in W.Nested active sets and the deletion rule preserve the order of records whose opponents remain in W.Source: §A.2.2 · p. 17
  • With probability at least 1−1/(4T), every realized rounded estimate is within τ(q) of its active-set score and arm 1 is never eliminated.The proof combines concentration over fixed sets and levels with the retained-prefix representation, rounding control, and repeated application of the preservation lemma.Source: §A.2.2 · p. 17

A.2.3 Retained counts and deleted records

The analysis bounds how many records leaders retain and how many records an elimination deletes, including records inherited from earlier active sets.

A.2.3 Retained counts and deleted records

  • On ER, a check at level q ≥ B(W) eliminates an arm, with B(W) ≤ q⋆(W) < 2B(W).The count analysis uses the threshold B(W) and bounds d(W) by 301A(W), with d(W) ≥ ΛR.Source: §A.2.3 · p. 18
  • On ER, every active leader retains at most d(W) records whenever the active set is W, including immediately before elimination and at the horizon.The bound follows by limiting counts within a stage and accounting for inherited records from earlier sets.Source: §A.2.3 · p. 18
  • With probability at least 1 − 1/(4T), every elimination from W deletes at most D(W) records, and at most |W|d(W) records remain if the terminal set has size at least two.The bound controls inherited and newly collected records using a uniform event over fixed set-arm pairs.Source: §A.2.3 · pp. 18–19

A.2.4 Regret under relaxed transitivity

The analysis bounds regret under relaxed transitivity and establishes trajectory agreement through a reference execution. It also gives phase-schedule and preference-gap properties under strong stochastic transitivity.

A.2.4 Regret under relaxed transitivity

  • Each relaxed-transitivity elimination from a set of size m costs at most 1204γ^8ΛR/ϵm; distinct pre-elimination sizes let the charges sum over m = 2,…,K.The terminal-set charge is at most 602γ^8ΛRHp, and the resulting expected regret bound follows after accounting for the exceptional event and ΛR.Source: §A.2.4 · pp. 19–20

A.2.5 Reference execution and trajectory agreement

  • With probability at least 1 − ρ, both executions agree on every ordered pair through time T, even though their comparison outcomes may differ.The induction matches decisions and sampling times to a reference execution; trajectory agreement does not require the regret-accuracy event.Source: §A.2.5 · pp. 20–21

A.3 Strong stochastic transitivity

  • Under Assumptions 3.1 and 3.4, Algorithm 2 uses H = 1 + ⌊log2 T⌋ phases and at most KH rounded-estimator calls.Only completed phases produce estimates, with at most K estimates per phase.Source: §A.3 · p. 21, §A.3.1 · p. 22

A.3.1 Deterministic properties of the phase schedule

  • On every sample path, active sets are nested and nonempty, each completed phase gives every arm exactly ng leads, and at most H phases are started.A truncated phase is a prefix of the cyclic schedule, so no arm leads more than ng times.Source: §A.3.1 · p. 21

A.3.2 Consequences of the preference assumptions

  • Under the strong-stochastic-transitivity consequences, the worst arm in any active set has the largest best-arm gap, and di(W) is bounded in terms of rank and ϵi.The proof bounds each opponent contribution between zero and 2ϵi, while at least ri(W) opponents contribute at least ϵi.Source: §A.3.2 · p. 22

A.3.3 Uniform accuracy and elimination

This section analyzes RepMaxInP’s accuracy, best-arm preservation, exploration cost, regret, and replicability guarantees, and states its fixed-context linear preference model and assumptions. It also describes how batchwise estimates, a promising set, shared randomness, and determinant growth govern comparison-pair choices and batch transitions.

A.3.3 Uniform accuracy and elimination

  • With probability at least 1−δ, all completed-phase estimates are uniformly accurate; on that event the best arm survives and elimination obeys the stated gap thresholds.An arm with d_i(W_g) ≥ 6τ_g is eliminated; survivors have d_i(W_g) < 6τ_g and ε_i < 6m_gτ_g/r_i(W_g).Source: §A.3.3 · pp. 22–23

A.3.4 The survivor barrier

  • On the uniform-accuracy event, consecutive started phases satisfy the survivor barrier η_{g+1} < 6a_gτ_g.Here η_g is the largest arm gap in the active set, and a_g=m_g/m_{g+1}.Source: §A.3.4 · p. 23

A.3.5 The nominal exploration cost

  • On the uniform-accuracy event, nominal exploration cost satisfies S ≤ 3698B.The bound charges each started phase its full nominal per-arm budget, including a truncated final phase.Source: §A.3.5 · pp. 24–25

A.3.6 High-probability regret

  • Deterministically, exploration leader-gap contribution is at most S; with probability at least 1−δ, opponent-gap contribution is at most 2S + log(1/δ).On the joint accuracy and opponent-gap event, exploitation compares the best arm with itself and incurs zero regret.Source: §A.3.6 · pp. 26–27

A.3.7 Trajectory replicability

  • The algorithm’s two executions select identical ordered pairs in every round with probability at least 1−ρ.The analysis bounds first disagreement at completed checkpoints, then uses agreement of checkpoint score vectors to establish trajectory agreement.Source: §A.3.7 · pp. 28–29

B COMPLETE ANALYSIS OF REPMAXINP

  • Appendix B specifies RepMaxInP and proves Theorem 4.1, using a feature sequence fixed before interaction and shared across executions.Probabilities include shared random shifts and comparison-noise variables.Source: B COMPLETE ANALYSIS OF REPMAXINP · p. 29

B.1 Model and algorithm specification

  • RepMaxInP assumes online-revealed contexts with K_t ≥ 1, bounded features and parameter space, and binary preference feedback governed by a known increasing symmetric link.The parameter set is compact and convex, contains 0 and θ⋆, and features have norm at most L; the supplied passage also states a noise moment bound and that R=1/2 is valid for binary feedback.Source: §B.1 · p. 29
  • At each batch start, RepMaxInP refits its estimate from past data, draws a fresh shared uniform shift, and forms the batch’s promising set.The algorithm also computes batch-specific quantities before selecting a pair.Source: §B.1 · pp. 29–30
  • The curvature assumption applies along the segment from the constrained estimate to the true parameter; the rounded estimate need not belong to the parameter set.Rounding error is controlled directly in the batch-start V-norm.Source: §B.1 · pp. 29–30
  • The algorithm selects the widest pair in the nonempty promising set and starts a new batch when the determinant exceeds e times its value at the batch start.Maximizations over indexed actions use lexicographic tie-breaking, including for identical features.Source: §B.1 · p. 30

B.2 Explicit regret guarantee

This section develops RepMaxInP’s finite-time regret guarantee and ρ-replicability, including the analysis linking trajectory replicability and regret measures. It also reports the synthetic BTL evaluation’s horizon and curve behavior.

B.2 Explicit regret guarantee

  • RepMaxInP is ρ-replicable, uses at most B batches, and achieves regret rate eO(d^3√T/(c_μρ)) under the stated assumptions.With probability at least 1 − δ, every utility-maximizing index belongs to Ct at every round; the same simplified rate holds in expectation for δ = min{ρ/4, 1/T}.Source: §B.2 · p. 30

B.3 Notation and probability model

  • At each batch start, RepMaxInP estimates from prior observations, reveals a shared shift, and freezes its estimate and selection quantities for that batch.Source: §B.3 · p. 31

B.7 Trajectory replicability

  • The trajectory analysis uses two independent comparison-noise streams and common shifts, with each batch’s shift independent of the joint histories formed before it is revealed.Source: §B.3 · p. 31, §B.7 · p. 36

B.4 Anytime confidence and randomized rounding

  • A simultaneous confidence event bounds the GLM estimates at every time and batch start, while shared uniform-grid shifts bound the probability that rounded estimates disagree.The shared-grid disagreement probability is bounded by the estimates’ L1 distance divided by the grid width.Source: §B.4 · pp. 32–33

B.5 Determinant growth and batch geometry

  • The determinant-growth bound limits the total number of batches to B and provides control used to bound accumulated uncertainty.Source: §B.5 · pp. 33–34

B.6 Optimism and high-probability regret

  • For δ = min{ρ/4, 1/T}, with 0 < δ < ρ/3, ρ − 2δ ≥ ρ/2, and Tδ ≤ 1, the additional term is at most 2LS, and Theorem 4.1’s expected rate follows.For the simplified rate, treating L, S, R, and λ as constants gives Dmax = eO(d) and B = eO(d).Source: §B.6 · p. 36

B.7 Trajectory replicability

  • Outside the batch disagreement events, shared features and frozen estimates, deterministic pair selection, and synchronized batch tests yield identical ordered trajectories, establishing trajectory replicability.Comparison outcomes may differ, but they do not affect pair selection within the current batch.Source: §B.7 · p. 38

B.8 Initialization and the relation between regret measures

  • With initial estimate 0 and V1 = λId, probability regret is bounded by kµ times the utility-regret bound (38) on the event in Theorem 4.1.Source: §B.8 · p. 39

C.1 Synthetic Preferences

  • At T = 2.5 × 10^8, all selected test runs had committed to the best arm by 6 × 10^6 rounds, after which regret and agreement curves stayed flat.The curves use ρ = 0.2; final regret is also reported for ρ ∈ {0.1, 0.2, 0.3, 0.4, 0.5}.Source: §C.1 · p. 40

C.2 Preferences from a Language Model

The language-model study tests preference comparisons among six summary arms using paired executions with independent comparison outcomes. It evaluates selected algorithm settings for regret and agreement, with reported evaluation agreement rates below their targets.

C.2 Preferences from a Language Model

  • The six arms were summaries for three audiences at two lengths, and the student-audience, long-summary arm was the Condorcet winner.The preference design treated this arm as the ground-truth winner.Source: §C.2 · p. 40
  • Paired executions shared the article sequence, opponent draws, and rounding offsets but used independent comparison outcomes; F(t) measures agreement through round t.Each round used an article-specific bank of 30 comparisons for the selected pair.Source: §C.2 · p. 40
  • RepBTM-SST rounded phase means to shared grid-cell midpoints and eliminated arms using a threshold; baselines were unrounded batched and sequential algorithms.The sequential baseline updated after each comparison and removed arms whose upper confidence bound fell below the largest lower confidence bound.Source: §C.2 · p. 40
  • Qwen3.8-27B generated summaries and judgments with 8-bit weights on one RTX 5090 GPU, using temperatures 0.2 and 0.7, respectively.Both used top-p 0.8 and top-k 20.Source: §C.2 · p. 40
  • At T=3 × 10^4 and ρ=0.25, the selected parameters were (c, a) = (16, 0.5); Figure 2 reports evaluation agreement below targets.Configurations had to eliminate the Condorcet arm in at most 5% of executions and finish with at most 1.05 active arms on average; among those meeting F(T) ≥ 1 − ρ, the lowest-regret configuration was selected.Source: §C.2 · p. 40

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