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Sharp Non-Asymptotic Analysis of the Penalized Challenger in $\beta$-EB-TCI for Bernoulli Bandits

Nam Nguyen, Tuan Quang Dam

arXiv:2610.01951v1cs.LGstat.ML

TL;DR

Sharp finite-confidence behavior for top-two Bernoulli best-arm identification remains incompletely understood, including how penalized challenger sampling and entry into a stable regime affect stopping. This paper analyzes β-EB-TCI and establishes a sharp fixed-β leading stopping bound in that regime, alongside guarantees that address reaching it. [p25, p24, p01, p08, p27]

  • Problem

    Finite-confidence analysis must control adaptive GLR accuracy, challenger sampling after the best arm becomes leader, and the time to reach that stable regime.

    Source: §1 · p. 2

  • Method

    The paper analyzes Bernoulli β-EB-TCI’s logarithmically penalized challenger using a population allocation benchmark and time-uniform empirical-to-population GLR control.

    Source: §1 · p. 2, §4.2 · p. 6

  • Results

    In the stabilized regime, every suboptimal arm is sampled linearly often and the stopping-time bound has leading term T⋆_β(µ) log(1/δ).

    Source: §1 · p. 2

  • Takeaways & Limitations

    A guard with O(√(Kt)) pulls by time t yields an expected sample-complexity bound for any number of arms under a unique-best-arm assumption.

    Source: §8 · p. 10

    The analysis is Bernoulli-specific with fixed β; finite-mean stabilization for tied suboptimal means and extensions beyond Bernoulli rewards or to online β adaptation remain open.

    Source: §8 · p. 10

Abstract

from arXiv · show

Top-two algorithms are simple and effective for fixed-confidence best-arm identification, but their sharp non-asymptotic behavior is still not well understood. We study this problem for Bernoulli bandits through $\beta$-EB-TCI, the empirical-best top-two rule of Jourdan et al., whose challenger is chosen using a Bernoulli transportation cost with a logarithmic count penalty. We prove that, after the empirical leader has become the true best arm and its sampling fraction stays close to $\beta$, the stopping time is $T_{\beta}^{\star}(\mu)\log(1/\delta)$ up to lower-order concentration terms. We also show that, in this regime, every challenger is sampled linearly often. Thus, for the original algorithm without forced exploration, the main remaining difficulty is to control when the empirical leader becomes permanently correct. These results imply a non-asymptotic high-probability bound for all Bernoulli instances with a unique best arm. If the algorithm satisfies a finite-mean sufficient-exploration condition, the bound further yields the sharp expected sample complexity. In particular, this gives the sharp expectation result for the unguarded Bernoulli rule when all arm means are pairwise distinct, using the sufficient-exploration result of Jourdan et al. Finally, if we add a mild forced-exploration rule that contributes only $O(\sqrt{Kt})$ pulls up to time $t$, we obtain a self-contained expected sample-complexity theorem for any number of arms under the unique-best-arm assumption. We also identify a limitation of proof strategies that try to handle equal suboptimal means through a single index-comparison argument.

1 Introduction

The paper analyzes finite-confidence behavior of Bernoulli β-EB-TCI, a top-two rule with a logarithmically penalized challenger, addressing gaps in sharp non-asymptotic guarantees. It proves sharp leading-order stopping behavior after stabilization and develops bounds that account for reaching that regime. [p23, p24, p25, p01, p03]

Method

  • β-EB-TCI samples the empirical leader with probability β and selects the challenger by minimizing its one-sided Bernoulli GLR plus log Nt,j.The logarithmic count penalty favors less-sampled challengers when evidence values are comparable.Source: §1 · p. 2

Motivation

  • The analysis targets three coupled issues: empirical-to-population GLR accuracy, challenger sampling after the best arm leads, and time to reach stabilization.The cited stopping-rule result controls wrong recommendations on runs that stop, but does not itself prove termination.Source: §1 · p. 2

Main result

  • In the stabilized regime, every suboptimal arm is sampled linearly often, and the stopping-time bound has leading term T⋆_β(µ) log(1/δ).Stabilization means the true best arm remains the empirical leader while its sampling fraction stays near β.Source: §1 · p. 2

Guarantees beyond stabilization

  • For every unique-best Bernoulli instance, the unguarded algorithm has a non-asymptotic high-probability stopping bound; a sharp expected bound additionally requires finite-mean sufficient exploration.Jourdan et al. [9] establish that exploration property for their covered tie-breaking variant when means are pairwise distinct; a guard yields an expected bound for any number of arms.Source: §1 · pp. 2–3

Scope

  • The paper’s sharpness claim concerns the exact fixed-β leading coefficient, not optimized lower-order constants or a matching non-asymptotic lower bound.The guarded rule contributes only O(√(Kt)) pulls by time t.Source: §1 · p. 2

2 Related Work

The paper places its analysis among allocation-tracking, top-two, and time-uniform concentration approaches. Its focus is the exact Bernoulli penalized challenger and finite-time stopping behavior.

Fixed-confidence identification and allocation tracking

  • Track-and-Stop tracks a globally optimal allocation, whereas this work fixes the best-arm fraction at β and uses local leader–challenger updates.Other fixed-confidence families include LUCB and KL-based refinements.Source: §2 · p. 3

Top-two methods

  • This work analyzes Bernoulli β-EB-TCI’s exact logarithmically penalized challenger, unlike unpenalized, Gaussian-approximation, or other top-two rules.The closest existing non-asymptotic analysis cited is Gaussian TTUCB, which uses a different rule and reward model.Source: §2 · p. 3

Time-uniform concentration and stopping-time tails

  • The analysis combines mixture-martingale confidence sequences with uniform control of pairwise Bernoulli GLRs, while treating correctness and sharp expected stopping time as separate questions.The error guarantee for the GLR stopping rule is taken from Jourdan et al. [9]; termination and finite-time challenger-count behavior are proved here.Source: §2 · p. 3

3 Setting and Algorithm

The paper specifies β-EB-TCI for Bernoulli bandits with a unique best arm, including its penalized challenger rule, GLR stopping criterion, and optional exploration guard.

Setting

  • The setting is a Bernoulli bandit with K ≥ 2 arms, a unique best arm, and a fixed deterministic tie-breaking rule.The experiments use the smallest-index tie-breaking rule; Corollary 6.5 uses the convention covered by Jourdan et al. [9].Source: §3 · p. 3

Evidence statistics

  • For a leader–challenger pair, the symmetric two-sample GLR uses empirical means and Bernoulli KL divergence; the one-sided transportation cost is zero when the first arm’s empirical mean is lower.When the first arm is the empirical leader, its transportation cost equals the symmetric GLR.Source: §3 · pp. 3–4

Sampling rule

  • On unguarded rounds, the algorithm chooses the challenger minimizing Wt(Bt+1,j) + log Nt,j, then samples the leader with probability β and the challenger otherwise.The optional guard instead pulls a least-sampled arm whenever any arm has fewer than t/K pulls.Source: §3 · p. 4

Stopping rule

  • The GLR stopping rule uses a time-dependent threshold, and its error guarantee controls wrong recommendations on runs that stop but does not establish termination.Almost-sure termination is proved separately in later stopping-time results.Source: §3 · p. 4

4 Population Allocation Benchmark and Uniform GLR Control

The section develops a discrete population benchmark for penalized challenger allocation and a time-uniform comparison between empirical and population pairwise GLR statistics.

4.1 A discrete penalized allocation benchmark

  • The continuous allocation assigns the remaining mass 1−α across suboptimal arms, while β remains fixed rather than being estimated or adapted online.The constrained characteristic time describes the fixed-best-arm-fraction allocation; the global characteristic time allows optimization over the best-arm coordinate.Source: §4.1 · p. 5
  • The discrete benchmark maximizes the minimum population penalized index over integer challenger counts with fixed total pulls, and approximates the continuous allocation value up to Oµ(1) when α stays in a compact subset of (0,1).It uses true means and is an analytical benchmark, not an implementable algorithm.Source: §4.1 · p. 5

4.2 Uniform comparison of empirical and population pairwise GLR statistics

  • A time-uniform empirical-to-population GLR comparison operates at the two-arm effective-sample-size fluctuation scale, enabling population allocation bounds to transfer to adaptive empirical indices.The population statistic grows on the effective sample-size scale, while deviations are of order √(Neff,t Ht(η)) + Ht(η).Source: §4.2 · pp. 5–6

5 Sharp Guarantees in the Stabilized Regime

Once the true best arm remains the empirical leader and its sampling fraction stays in U around β, the section establishes a non-asymptotic stopping bound and linear challenger growth. The remaining control is the time needed to enter that regime.

5 Sharp Guarantees in the Stabilized Regime

  • The analysis gives a uniform lower bound on evidence against all challengers after stabilization.The event supporting the bound has probability at least 1 − 2η and holds simultaneously over stabilization times and later rounds.Source: §5 · p. 6
  • The stabilized-regime stopping bound retains the fixed-β leading coefficient; lower-order terms include stabilization, concentration, confidence, and discretization effects.The bound assumes an almost-surely finite U-stabilization time.Source: §5 · p. 6
  • A high-probability bound on tU yields a high-probability stopping bound, while a summable upper-quantile bound yields an expected stopping-time bound.The quantile bound is summable when its dyadic weighted values have a finite sum.Source: §5 · pp. 6–7
  • Every suboptimal arm’s pull count grows linearly after a transient determined by the stabilization time and concentration error.This rules out persistent under-sampling by the penalized challenger rule once the regime is stabilized.Source: §5 · p. 7
  • The time until all arm counts support empirical separation is at most Cµ,β,U times tU + log(K/η) + log log(e/η) + 1.This bound holds on the event of Proposition 5.1.Source: §5 · p. 7

6 Reaching the Stabilized Regime

The section controls entry into the stabilized regime and derives stopping guarantees for the unguarded and guarded algorithms. Expected guarantees for the unguarded rule require sufficient exploration, while the guard supplies deterministic exploration under a unique-best-arm assumption.

6 Reaching the Stabilized Regime

  • The unguarded algorithm has a non-asymptotic high-probability stopping bound under the unique-best-arm assumption.The stabilization time has a high-probability upper bound polynomial in log(1/η).Source: §6 · pp. 7–8
  • Under finite-mean sufficient exploration, the unguarded algorithm obtains an expected stopping-time bound.The condition requires an almost-surely finite exploration time with finite expectation; it yields a summable upper-quantile bound for stabilization.Source: §6 · p. 8
  • For pairwise-distinct means, the unguarded variant covered by Jourdan et al. [9] has an expected bound with their specified tie-breaking convention.The corollary includes their uniform tie-breaking convention.Source: §6 · p. 8
  • With the exploration guard enabled, the algorithm has an expected stopping-time bound for any number of arms under a unique-best-arm assumption.The guard samples a least-sampled arm when its count falls below the stated threshold.Source: §6 · p. 8
  • The guard provides a deterministic lower bound on every arm count, whereas the unguarded expected bound remains unresolved for deterministic tie-breaking with equal-mean suboptimal arms.The guarded analysis also controls the time to enter the stabilized regime under a unique best arm.Source: §6 · pp. 8–9
  • M(η) ≥ η^-a for every a < C d(µu, λ), so the one-step index-comparison argument does not provide a sufficiently strong tail bound for finite-mean stabilization control.This limits that proof route, not the possibility of finite-mean exploration under a more global analysis.Source: §6 · p. 9

7 Experiments

Experiments test predictor agreement, sensitivity to β, guard overhead, and the effect of removing the logarithmic challenger penalty. On the tested Bernoulli instances, observed mean stopping times closely follow the threshold-crossing predictor, while removing the penalty substantially worsens the equal-mean stress test.

Experimental protocol

  • The experiments are diagnostic rather than a broad comparison of best-arm algorithms.The idealized reference knows the true means and tracks a globally optimal allocation; it provides scale, not an algorithm ranking.Source: §7 · p. 9
  • The evaluation compares empirical mean stopping time with a threshold-crossing predictor across three representative mean-vector instances, three β values, five δ values, and 200 independent replications per setting.The β values were {0.3, 0.5, 0.7}; the δ values were {10−4, 10−5, 10−6, 10−7, 10−8}.Source: §7 · p. 9

Predictor agreement and sensitivity to β

  • 1.53% mean and 3.86% maximum absolute relative error for guarded settings, versus 1.66% and 4.20% for unguarded settings, relative to tpred.Across the 45 guarded settings, none of the 18,000 penalized runs reached the cap or recommended a wrong arm.Source: §7 · p. 9
  • The raw mean gaps alone do not determine the best choice of β.Figure 2 reports β-sensitivity for CLUSTER-CLOSE5 and TIE-CLOSE4 at δ = 10^-6.Source: §7 · p. 9, Figure 2 · p. 10

Guard overhead, effect of removing the penalty, and an idealized reference

  • On TIE-CLOSE4 with β = 0.5 and δ = 10^-8, removing log Nt,j raised mean stopping time from 105,593 to 188,036, and 12.8% of runs reached the cap.With the guard enabled, the median changed from 104,918 to 103,743 and the empirical 95th percentile from 134,140 to 130,792; the largest mean guarded-pull fraction in tested equal-mean instances was 0.0069%.Source: §7 · p. 9

8 Conclusion

The analysis establishes sharp stopping-time results for Bernoulli β-EB-TCI, with conclusions depending on the exploration setting. The unguarded rule has a high-probability bound for unique-best-arm instances, while expected bounds require sufficient exploration; the guard yields an expected bound for any K.

8 Conclusion

  • In the stabilized regime, every suboptimal-arm pull count grows linearly and the stopping-time upper bound has leading term T⋆β(µ) log(1/δ).The regime requires the true best arm to remain the empirical leader while its sampling fraction stays near β.Source: §8 · p. 10
  • The unguarded rule has a non-asymptotic high-probability stopping bound for any Bernoulli instance with a unique best arm, but its expected bound requires finite-mean sufficient exploration.For pairwise-distinct means, Jourdan et al. [9] establish this condition for the tie-breaking variant covered by their analysis.Source: §8 · p. 10
  • The exploration guard gives an expected stopping-time bound for any number of arms under a unique-best-arm assumption, with only O(√(Kt)) guarded pulls by time t.The proof and fixed-β analysis are Bernoulli-specific; extensions beyond Bernoulli rewards, online β adaptation, and finite-mean stabilization with tied suboptimal means remain open.Source: §8 · p. 10

A Technical Ingredients from Prior Work

The paper separates stopping-error control, adaptive-sampling concentration, and sufficient exploration because each supports a different part of its guarantees. For Bernoulli rewards, the one-sided transportation cost equals the pairwise Bernoulli GLR for ordered arm pairs.

A Technical Ingredients from Prior Work

  • Stopping-rule error control on runs that stop does not establish termination or a stopping-time bound, so separate analysis is needed for those guarantees.The paper also notes that a high-probability bound on a random time need not imply finite expectation.Source: A Technical Ingredients from Prior Work · pp. 14–15
  • Armwise KL confidence sequences evaluated at adaptive pull counts provide the probabilistic starting point for uniform pairwise-GLR control, not stabilization or stopping by themselves.The paper's analysis separately establishes challenger growth, entry into the stabilized regime, and stopping.Source: A Technical Ingredients from Prior Work · p. 14
  • The all-distinct expected-bound corollary uses Jourdan et al. [9] sufficient exploration only for their uniform tie-breaking convention, not the paper's fixed deterministic convention.This prior result is not used for the unconditional high-probability analysis or the guarded expected result.Source: A Technical Ingredients from Prior Work · pp. 14–15

B.1 Bernoulli specialization of the transportation cost

  • For ordered Bernoulli pairs, the one-sided transportation cost equals the pairwise Bernoulli GLR for testing whether both arms can share a common mean.This identity is the Bernoulli specialization used in the algorithm's challenger score.Source: §B.1 · pp. 15–16

B.2 Uniform error control

The Bernoulli stopping rule has a time-uniform error guarantee with or without the exploration guard, though error control alone does not ensure termination. The appendix also establishes convexity and regularity properties for Bernoulli population allocations.

B.2 Uniform error control

  • At the displayed threshold, the Bernoulli stopping rule's probability of stopping with an incorrect recommendation is at most δ under any adaptive sampling rule.The guarantee applies to Algorithm 1 with or without the optional exploration guard.Source: §B.2 · p. 16
  • The error guarantee does not imply almost-sure termination for an arbitrary sampling rule; termination requires one of the stated regimes.Combining termination with the error bound gives δ-correctness in those regimes.Source: §B.2 · p. 16

C.1 Bernoulli two-arm calculus

  • The Bernoulli population-allocation objective is strictly convex, so its minimizer is unique.Source: §C.1 · p. 17

C.2 Boundary behavior and inverse maps

  • For each suboptimal arm j, the stated map is a strictly increasing C∞ bijection from (0, ∞) onto (0, α d(µ_i⋆, µ_j)).Source: §C.2 · p. 18
  • For each α in (0, 1), the fixed-α allocation optimizer and scalar are unique, and the optimizer map is C1 on compact subsets.Source: §C.2 · p. 18

C.3 Discrete rounding and proof of Proposition 4.1

The appendix bounds the best attainable penalized index through discrete allocation rounding, then supplies confidence-sequence and local regularity tools for the Bernoulli GLR.

C.3 Discrete rounding and proof of Proposition 4.1

  • Fixed-sum rounding produces integers m_j≥1 that sum to s and satisfy |m_j−a_j|≤1 for every j.The construction rounds down, then adds one to coordinates with the largest fractional parts as needed.Source: §C.3 · p. 19
  • For α=n/t in compact U, the rounded allocation gives n_s(µ) ≤ tT*_α(µ)+log t, while every challenger index is at least tT*_α(µ)+log t−L_U+log(c_U/2).Source: §C.3 · p. 20

D.1 Armwise KL confidence sequences

  • Armwise KL confidence sequences give a high-probability event on which the stated control holds simultaneously for every arm and all t≥K.Source: §D.1 · pp. 20–21

D.2 Local Lipschitz control of the Bernoulli GLR

  • The Bernoulli GLR V_m,ℓ has a Bregman representation and admits local Lipschitz control when its arguments lie in [ε,1−ε].Source: §D.2 · p. 21

D.3 Proof of Theorem 4.2

This section develops stabilization, concentration, and challenger-index bounds, including a high-probability bound that holds simultaneously over times and arm pairs, and uses them to derive stopping-time guarantees. It treats stabilization and expected-complexity bounds for the unguarded rule and shows how a guard yields a polylogarithmic upper-quantile bound; the sufficient-exploration corollary applies to the tie-breaking variant, not Algorithm 1’s fixed deterministic tie-breaking convention.

D.3 Proof of Theorem 4.2

  • On an event of probability at least 1−η, the displayed Theorem 4.2 bound holds simultaneously for all t≥K and all distinct arm pairs.The proof combines small- and large-count regimes using armwise confidence bounds.Source: §D.3 · p. 22

E.1 Continuous equalizers and slope bounds

  • γ_n,S = t T*_α(µ) + log t + O_µ(1) and r_j,n,S = t w_j,α + O_µ(1), with a unique equalizer once S exceeds a finite cutoff.The equalizer allocates challenger counts so each Φ_j(n,r_j,n,S) has the same value; the bounds hold uniformly for α in U.Source: §E.1 · pp. 23–24

E.2 Guard-count control

  • The least-sampled-arm guard admits a deterministic activation-count bound, proved by partitioning time into epochs in which at most K guard pulls are needed.Within an epoch, after every arm reaches count r+1, the guard condition cannot recur.Source: §E.2 · p. 24

E.3 The uniform challenger-index lower bound

  • After any U-stabilization time, every challenger index has a uniform lower bound on the event G_η.The proof compares challenger pulls with deficits from the continuous equalizer and controls those deficits over time.Source: §E.3 · pp. 25–26

E.4 Proof of Proposition 5.1

  • Once the empirical leader is the true best arm, its challenger transportation-cost index equals the corresponding evidence index for every challenger.The proof of Proposition 5.1 combines the stabilized-regime bound with leader-fraction control.Source: §E.4 · p. 27

E.5 Challenger growth and separation

  • The challenger-growth and separation arguments establish logarithmic separation thresholds and show that, after a sufficiently late stabilized time, arm counts reach the required thresholds.The threshold argument is uniform over η∈(0,1/2].Source: §E.5 · pp. 27–28

F.1 Uniform leader-fraction concentration

  • With probability at least 1−η, leader-fraction control holds uniformly over start times m and later times t whenever the leader remains the true best arm.The proposition accounts for guard activations in the interval from m+1 through t.Source: §F.1 · p. 28

F.2 Entrance into a compact neighborhood of β

  • Entry into a fixed compact neighborhood of β follows after the empirical leader is correct, with the guard contributing an additive K term that disappears when disabled.Lemma F.2 bounds the entrance time in terms of the leader-correctness time L; the displayed bound is incomplete in the supplied passages.Source: §F.2 · p. 29

F.3 Centered quantile integration; Define the monotone envelope

  • Any random time with a summable upper-quantile bound also has a nonincreasing summable upper-quantile bound; for nonnegative X with finite mean, dyadic quantiles are summable.Source: Define the monotone envelope · pp. 30–31

F.4 Threshold inversion

  • Threshold inversion supplies a bound for crossing the stopping threshold when the process incurs a linear entrance delay.The stated inversion constant depends only on T, C0, a, b, d, and K.Source: §F.4 · p. 31

F.5 Proof of Theorem 5.2

  • Theorem 5.2’s proof converts stabilized-regime bounds into a high-probability stopping-time bound and an expectation bound when the stabilization time has summable upper quantiles.The proof uses quantile monotonicity to obtain the expectation conclusion from the high-probability bound.Source: §F.5 · p. 34

G.1 Separation implies permanent leader correctness

  • After separation, the armwise confidence event ensures that the empirical leader is permanently the true best arm.Source: §G.1 · p. 34

G.2 High-probability stabilization bound for the unguarded rule

  • For the unguarded rule, separation-deficient nonleaders remain cheap to sample, yielding polynomial high-probability stabilization and a polynomial upper-quantile bound for entry into the stabilized regime.The resulting high-probability bound is unconditional, but need not be integrable.Source: §G.2 · pp. 35–36

G.3 Finite-mean sufficient exploration

  • Finite-mean sufficient exploration yields a summable upper-quantile bound for stabilization, allowing Theorem 5.2 to give the expected-complexity bound.Source: §G.3 · p. 37

G.4 All-distinct means instances

  • For all-distinct Bernoulli means, Jourdan et al. [9] establish the sufficient-exploration property for the EB-leader, TCI-challenger rule, with a finite-mean random time Tse.Source: §G.4 · p. 37
  • Theorem 6.4 applies with c_se = 1 to the tie-breaking variant, but not to Algorithm 1’s fixed deterministic tie-breaking convention.The corollary does not extend the cited sufficient-exploration result to that convention.Source: §G.4 · p. 37

G.5 Guarded expected bound

  • With epoch boundaries t_m = Km^2, the guard pulls a least-sampled arm until every arm has at least m+1 samples by t_{m+1}.This inductive argument establishes the minimum-count lower bound; the bound on t_sep(η) is immediate.Source: §G.5 · p. 38
  • The guard gives a polylogarithmic upper-quantile bound for U-stabilization, satisfying Theorem 5.2’s conditions and completing the proof of the expected bound.The proof combines Proposition 6.6 and Lemma F.2 to establish the probability requirement and dyadic summability condition.Source: §G.5 · p. 38

G.6 Limitation of one-step index comparisons

The appendix analyzes a frozen wrong leader in the equal-suboptimal-mean setting and also documents the synthetic experiment grid.

G.6 Limitation of one-step index comparisons

  • With an equal-mean challenger λ, the stated bound gives M(η)≥η^-a for sufficiently small η and any a<C d(µ_u,λ).Source: §G.6 · p. 38

H Experiment Details

  • The experiments use synthetic Bernoulli instances without external datasets, and Table 1 lists SEP5, CLUSTER-CLOSE5, TIE-CLOSE4, and confidence levels 10^-4 through 10^-8.Source: H Experiment Details · p. 38, Table 1 · p. 39

H.1 Algorithms and implementation

This section documents the simulation variants, benchmark calculations, and diagnostics, alongside reproducibility and compute-resource disclosures. It also summarizes the study’s synthetic Bernoulli simulations and checklist declarations concerning human subjects, artifacts, societal impact, assets, and limited LLM assistance.

H.1 Algorithms and implementation

  • The unpenalized challenger ablation hits the 700,000-round cap in 12.8% of stress-test runs, while no penalized evaluation setting hits the cap or yields a Monte Carlo misidentification.The ablation removes log N_t,j from the challenger index but leaves the stopping rule and other details unchanged.Source: Table 3 · p. 40, §H.1 · p. 40, §H.3 · p. 41
  • The empirical mean stopping time tracks the nonmonotone dependence of T⋆_β(µ) on the leader sampling probability β.Table 5 gives the numerical values underlying the right-hand panel of Figure 2.Source: §H.1 · p. 40

H.2 Idealized population-allocation reference

  • The population-allocation reference uses the true-mean Bernoulli characteristic-time allocation and the same GLR threshold, so its comparison with guarded EB-TCI is diagnostic, not an algorithm ranking.The comparison uses β = 0.4, the closest sweep value to the optimal best-arm fractions 0.357 and 0.367 for the two instances.Source: §H.2 · pp. 40–41

H.3 Supplementary figures

  • Figure 3 reports stress-instance stopping-time survival curves and normalized guard overhead across the equal-close family.The stress curves use TIE-CLOSE4 with β = 0.5 and δ = 10^-8; the overhead spans K ∈ {4, 8, 12, 16} at δ = 10^-6.Source: §H.3 · p. 41, Figure 3 · p. 42
  • Figure 4 shows the effect of removing the challenger penalty and compares guarded-rule final sampling fractions with optimal population weights.The allocation comparison uses β = 0.5 and δ = 10^-8.Source: §H.3 · pp. 41–42, Figure 4 · p. 42

NeurIPS Paper Checklist

  • The supplementary code package and specifications support reproducing the synthetic Bernoulli simulations, including grids, seeds, stopping thresholds, caps, uncertainty summaries, and reported metrics.The package includes scripts for the calibration grid, β-sweep, guard diagnostics, penalty ablation, tracking-reference comparison, and figure and table generation.Source: NeurIPS Paper Checklist · pp. 44–45
  • The experiments are CPU-only and require no specialized hardware, but exact hardware, memory, and wall-clock requirements are not reported.The supplementary package provides smoke, medium, and full reproduction modes.Source: NeurIPS Paper Checklist · p. 46
  • The authors answer Yes on ethics compliance, describing theoretical work based only on synthetic Bernoulli simulations and no human subjects, private or scraped data, or deployed systems.Source: NeurIPS Paper Checklist · pp. 46–47
  • Societal impacts are marked N/A because the work is foundational theory; any impacts would be indirect and depend on downstream bandit applications.Source: NeurIPS Paper Checklist · p. 47
  • The paper reports no third-party datasets, pretrained models, or code assets, and identifies its only new asset as anonymized supplementary code for reproducing synthetic experiments.The paper also says it releases no high-misuse-risk artifacts.Source: NeurIPS Paper Checklist · pp. 47–48
  • Crowdsourcing, human-subject research, participant data, and human annotation are absent, so participant-risk discussion and IRB review are not applicable.Source: NeurIPS Paper Checklist · p. 49
  • LLM assistance was limited to non-core support such as writing, editing, and experiment-script organization; it was not part of the method or scientific component.The authors state that they specified and verified the theoretical results, simulations, outputs, plots, and claims.Source: NeurIPS Paper Checklist · p. 49
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