Enhancing Preference-based Linear Bandits via Human Response Time

Authors: shen  li
yuyang  zhang
zhaolin  ren
claire  liang
na  li
julie a shah
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 01:49:31

Abstract: Interactive preference learning systems present humans with queries as pairs of options; humans then select their preferred choice, allowing the system to infer preferences from these binary choices. While binary choice feedback is simple and widely used, it offers limited information about preference strength. To address this, we leverage human response times, which inversely correlate with preference strength, as complementary information. We introduce a computationally efficient method based on the EZ-diffusion model, combining choices and response times to estimate the underlying human utility function. Theoretical and empirical comparisons with traditional choice-only estimators show that for queries where humans have strong preferences (i.e., "easy" queries), response times provide valuable complementary information and enhance utility estimates. We integrate this estimator into preference-based linear bandits for fixed-budget best-arm identification. Simulations on three real-world datasets demonstrate that incorporating response times significantly accelerates preference learning.

Main paper: 
 introduction: 
 Interactive preference learning from human binary choices is essential in systems like recommender systems [9, 21, 32, 56] , assistive robots [54, 65] , and fine-tuning of large language models [5, 43, 46, 47, 59 ]. This process is often framed as a preference-based bandit problem [7, 31] , where the system repeatedly presents pairs of options, the human selects their preferred option, and the system infers preferences based on these choices. Choices feedback is widely used for its simplicity and low cognitive burden on users [37, 72, 74] . However, while binary choices capture preferences, they offer limited insight into preference strength [77] . To address this, researchers have incorporated additional human explicit feedback, such as ratings [50, 58] , labels [74] , and slider bars [5, 72] , but these approaches tend to complicate the interface and increase users' cognitive load [36, 37] .In this paper, we propose leveraging human implicit feedback, specifically response times, to provide additional insights into preference strength. Unlike explicit feedback, response time is unobtrusive and effortless to measure [17] , offering valuable information that complements binary choices [2, 16] . For instance, consider an online retailer that repeatedly presents users with a binary query, whether to purchase or skip a recommended product [35] . Since most users skip products most of the time [33] , the probability of skipping becomes nearly 1 for most items. This lack of variation in choices makes it difficult to assess how much a user likes or dislikes any specific product, limiting the system's ability to accurately infer their preferences. Response time can help overcome this limitation. Psychological research shows an inverse relationship between response time and preference strength [17] : users who strongly prefer to skip a product tend to do so quickly, while longer response times can indicate weaker preferences. Thus, even when choices appear similar, response time can uncover subtle differences in preference strength, helping to accelerate preference learning.Leveraging response times for preference learning presents notable challenges. Psychological research has extensively studied the relationship between human choices and response times [17, 19] using complex models like Drift-Diffusion Models [51] and Race Models [12, 66] . While these models align with both behavioral and neurobiological evidence [70] , they rely on computationally intensive methods, such as hierarchical Bayesian inference [71] and maximum likelihood estimation (MLE) [52] , to estimate the underlying human utility functions from both human choices and response times, making them impractical for real-time interactive systems. Although faster estimators exist [8, 28, 30, 67, 68] , they typically estimate the utility functions for a single pair of options, without aggregating data across multiple pairs. This limits their ability to leverage structures like linear utility functions, which are critical both in preference learning with large option spaces [21, 24, 41, 54, 56] and in cognitive models for human multi-attribute decision-making [26, 64, 76] .To address these challenges, we propose a computationally efficient method for estimating linear human utility functions by incorporating both choices and response times, based on the differencebased EZ diffusion model [8, 67] . Our method leverages response times to transform binary choice signals into continuous signals, framing utility estimation as a linear regression that aggregates data across multiple pairs. We compare our estimator to traditional logistic regression methods that rely solely on choices [3, 31] . Our theoretical and empirical analyses show that for binary queries with strong preferences (i.e., "easy" queries), choices alone provide limited information, while response times offer valuable insights into preference strength, significantly enhancing utility estimates. Thus, incorporating response times makes easy queries more informative.Our linear-regression-based estimator integrates seamlessly into algorithms for preference-based bandits with linear human utility functions [3, 31] , enabling interactive learning systems to leverage response times for faster learning. We specifically integrated our estimator into the Generalized Successive Elimination algorithm [3] for fixed-budget best-arm identification [29, 34] . Simulations using three real-world datasets [16, 39, 57] consistently show that incorporating response times significantly reduces identification errors, compared to traditional methods that rely solely on choices. To the best of our knowledge, this is the first work to integrate response times into bandit (and RL).Section 2 introduces the preference-based linear bandit problem and the difference-based EZ diffusion model. Section 3 presents our utility estimator, incorporating both choices and response times, and offers a theoretical comparison to the choice-only estimator. Section 4 integrates both estimators into the Generalized Successive Elimination algorithm. Section 5 presents empirical results for estimation and bandit learning. Section 6 discusses the limitations of our approach. Appendix B reviews response time models, parameter estimation techniques, and their connection to preference-based RL.Nomenclature: We use [n] to denote the set {1, . . . , n}. For a scalar random variable x, E [x] and V [x] denote its expectation and variance, respectively. The function sgn(x) denotes the sign of x.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? Answer: [Yes] Justification: The main claim in the abstract and introduction is the benefit of incorporating response times in preference learning, which is the key contribution and scope. Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
problem setting and preliminaries: 
 Preference-based bandits with a linear utility function. The learner is given a finite set of options (or "arms"), each represented by a feature vector in Z ⊂ R d , and a finite set of binary queries, where each query is the difference between two arms, denoted by X ⊂ R d . For instance, if the learner can query any pair of arms, the query space is X = {zz ′ : z, z ′ ∈ Z}. In the online retailer example from section 1, the query space is X = {zz skip : z ∈ Z}, where z represents purchasing a product and z skip represents skipping (often set as 0). For each arm z ∈ Z, the human utility is assumed to be linear in the feature space, defined as u z := z ⊤ θ * , where θ * ∈ R d represents the human's preference parameters. For any query x ∈ X , the utility difference is then defined as u x := x ⊤ θ * . Given a query x := z 1z 2 ∈ X , we model human choices and response times using the differencebased EZ-Diffusion Model (dEZDM) [8, 67] , integrated with our linear utility structure. (See appendix B.1 for a comparison with other models.) This model interprets human decision-making as a stochastic process in which evidence accumulates over time to compare two options. As shown in fig. 1a , after receiving a query x, the human first spends a fixed amount of non-decision time, denoted by t nondec > 0, to perceive and encode the query. Then, evidence E x accumulates over time following a Brownian motion with drift x ⊤ θ * and two symmetric absorbing barriers, a > 0 and -a. Specifically, at time t nondec + τ where τ ≥ 0, the evidence is E x,τ = x ⊤ θ * • τ + B(τ ), where B(τ ) ∼ N (0, τ ) is standard Brownian motion. This process continues until the evidence reaches either the upper barrier a or lower barrier -a, at which point a decision is made. The random stopping time, t x := min {τ > 0 : E x,τ ∈ {a, -a}}, represents the decision time. If E x,tx = a, the human chooses z 1 ; if E x,tx = -a, they choose z 2 . The choice is represented by the random variable c x , where c x = 1 if z 1 is chosen, and -1 if z 2 is chosen. The total response time, t RT,x , is the sum of the non-decision time and the decision time: t RT,x = t nondec + t x . The choice probability, expected choice, choice variance, and expected decision time are given as follows [48, eq. (A.16 ) and (A.17)]:EQUATIONThis choice probability matches that of the Bradley and Terry [10] model. If the learner relies solely on choices, then our bandit problem reduces to the transductive linear logistic bandit problem [31] .Figures 1b and 1c illustrate the roles of the parameters x ⊤ θ * and a. First, the absolute drift (or the absolute utility difference), |x ⊤ θ * |, represents the query's easiness [40] . A higher |x ⊤ θ * | corresponds to an easier query, resulting in shorter decision times and more consistent choices. In contrast, when |x ⊤ θ * | is low (close to 0), the query becomes harder, leading to longer decision times and less consistent choices. Second, the barrier a represents the human's conservativeness in decisionmaking [40] . 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B W x 8 M P N 6 b Y W a e G 0 m h 0 b K + j c L W 9 s 7 u X n G / d H B 4 d H x i n p Y 7 O o w V h z Y P Z a h 6 L t M g R Q B t F C i h F y l g v i u h 6 0 7 v F 3 7 3 C Z Q W Y d D C e Q S O z 8 a B 8 A R n m E l D s z x A m G H S E j 7 Q q g Z + l Z a G Z s W q W U v Q T W L n p E J y N I f m 1 2 A U 8 t i H A L l k W v d t K 0 I n Y Q o F l 5 C W B r G G i P E p G 0 M / o w H z Q T v J 8 v a U X m b K i H q h y i p A u l R / T y T M 1 3 r u u 1 m n z 3 C i 1 7 2 F + J / X j 9 G 7 c x I R R D F C w F e L v FE [t x ] ± σ t,x E [c x ] ± σ c,x (b) a = 0.8 -4 -2 0 2 4 x θ * 0 2 4 6 E [t x ] ± σ t,x E [c x ] ± σ c,x(c) a = 1.8Figure 1 : (a) depicts the human decision-making process for a binary query x ∈ X , where the human selects between two arms. The human first spends a fixed non-decision time t nondec encoding the query. Then, the human's evidence accumulates according to a Brownian motion with drift x ⊤ θ * . When the evidence reaches the upper barrier a or lower barrier -a, the human makes a choice, denoted by c x = 1 or c x = -1, respectively. The random stopping time of the accumulation process is the decision time t x , and the total response time is t RT,x = t nondec + t x . (b) and (c) plot the expected choice E[c x ] and the expected decision time E[t x ], with shaded regions representing one standard deviation, plotted as functions of the utility difference x ⊤ θ * for two barrier values a.We adopt the common assumption that t nondec is constant across all queries for a given human [16, 76] and further assume that t nondec is known to the learner. This assumption enables the learner to perfectly recover t x from the observed t RT,x . In section 5.2, we empirically show that even when t nondec is unknown, its impact on the performance of our method that relies on decision times is negligible.Learning objective: Best-arm identification with a fixed budget. We focus on the fixed-budget best-arm identification problem [29, 34] . The learner is provided with a total interaction time budget B > 0, an arm space Z, a query space X , and a non-decision time t nondec . Both the human's preference vector θ * and the decision barrier a are unknown. In each episode s ∈ N, the learner selects a query x s ∈ X , receives human feedback (c xs,s , t xs,s ) generated by the dEZDM, and consumes t RT,xs,s time. When the cumulative interaction time exceeds the budget B at some episode S, i.e., S s=1 t RT,xs,s > B, the learner must stop and recommend an arm z ∈ Z. The goal is to recommend the unique best arm z * := arg max z∈Z z ⊤ θ * , minimizing the error probabilityP [ z ̸ = z * ].To solve this problem, we use the Generalized Successive Elimination (GSE) algorithm [1, 3, 75] . This algorithm divides the total budget B evenly into multiple phases. In each phase s, it strategically samples queries till this phase's allocated budget is exhausted, collecting both human choices and response times. It then computes an estimate θ s of the true preference vector θ * and eliminates arms with low estimated utilities based on θ s . The key to improving estimation is effectively leveraging decision times to gain additional insights into preference strength, complementing the information provided by choices. In section 3, we introduce this estimator and compare it theoretically to the traditional choice-only estimator. section 4 details the integration of both estimators into GSE.Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: section 6 contains discussions about limitations. Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
utility estimation: 
 This section addresses the problem of estimating human preference θ * from a fixed dataset, denoted byx, c x,sx,i , t x,sx,i x∈Xsample,i∈ [nx] . Here, X sample denotes the set of queries in the dataset, n x denotes the number of samples for each query x ∈ X sample , and s x,i denotes the episode when x is sampled for the i-th time. Samples from the same query x are i.i.d., while samples from different queries are independent. Section 3.1 introduces a new estimator, the "choice-decision-time estimator," which uses both choices and decision times, in contrast to the commonly used "choice-only estimator" that only uses choices [3, 31] . Sections 3.2 and 3.3 theoretically compares these estimators, analyzing both asymptotic and non-asymptotic performance and highlighting the advantages of incorporating decision times. Section 5.1 presents empirical results that validate our theoretical insights.The choice-decision-time estimator is based on the following relationship between human utilities, choices, and decision times, derived from eq. ( 1):EQUATIONIntuitively, when a human provides consistent choices (i.e., large |E[c x ]|) and makes decisions quickly (i.e., small E[t x ]), it suggests that the query is easy and the preference is strong (i.e., large |x ⊤ θ * |). This relationship reframes the estimation of θ * as a linear regression problem. Accordingly, the choice-decision-time estimator calculates the empirical means of both choices and decision times, aggregates the ratios across all sampled queries, and applies ordinary least squares (OLS) to estimate θ * /a. Since the ranking of arm utilities based on θ * /a is identical to that based on θ * , estimating θ * /a is sufficient for identifying the best arm. This estimate of θ * /a, denoted by θ CH,DT , is given by:EQUATIONIn contrast, the choice-only estimator is based on eq. ( 1), which shows that for each query x ∈ X , the random variable (c x + 1)/2 follows a Bernoulli distribution with mean1/[1 + exp(-x ⊤ • 2aθ * )].Similar to the choice-decision-time estimator, the parameter 2a does not impact the ranking of arms, so estimating 2aθ * is sufficient for best-arm identification. This estimation is formulated as a logistic regression problem [3, 31] , with MLE providing the following estimate of 2aθ * , denoted by θ CH :EQUATIONwhere µ(y) := 1/[1+exp(-y)] is the standard logistic function. While this MLE lacks a closed-form solution, it can be efficiently solved using optimization methods like Newton's algorithm [25, 44] .The choice-decision-time estimator from eq. (3) satisfies the following asymptotic normality result:Theorem 3.1 (Asymptotic normality of θ CH,DT ). Given an i.i.d. dataset x, c x,sx,i , t x,sx,i i∈[n] for each x ∈ X sample , where x∈Xsample xx ⊤ ≻ 0, and assuming that the datasets for different x ∈ X sample are independent, then, for any vector y ∈ R d , as n → ∞, the following holds:√ n y ⊤ θ CH,DT,n -θ * /a D -→ N (0, ζ 2 /a 2 ).Here, the asymptotic variance depends on a problem-specific constant, ζ 2 , with an upper bounded:ζ 2 ≤ ∥y∥ 2 x∈X sample min x ′ ∈X sample E[t x ′ ] •xx ⊤ -1 .The proof is provided in appendix C.2. The asymptotic variance upper bound shows that all sampled queries are weighted by a common factor min x ′ ∈Xsample E [t x ′ ], which is the smallest expected decision time among all the sampled queries. This weight represents the amount of information provided by each query's choices and decision times to the estimation of θ * . A larger weight indicates that every query provides more information, leading to lower variance and better estimates.In contrast, the choice-only estimator from eq. ( 4) has the following asymptotic normality result, as derived from Fahrmeir and Kaufmann [23, √ ny ⊤ θ CH,n -2aθ * D -→ N 0, 4a 2 ∥y∥ 2 x∈X sample [a 2 V[cx]]•xx ⊤ -1.This asymptotic variance shows that each sampled query x is weighted by its own factora 2 V [c x ],representing the amount of information the query's choices contribute to estimating θ * . A larger weight indicates that the query contributes more information, leading to better estimates.The weights in both theorems highlight the different contributions of choices and decision times to estimating θ * . In the choice-only estimator (theorem 3.2), each query is weighted bya 2 V [c x ],which is a function of the utility difference x ⊤ θ * for a fixed barrier a. As shown in the dashed or solid gray curve in fig. 2a , as queries become easier (i.e., as |x ⊤ θ * | increases), the weight quickly decays to zero, indicating that choices from easy queries provide much less information than harder queries. Intuitively, for easy queries, humans consistently choose the same option, making it difficult for the learner to gauge whether their preference for that option is moderate or strong. Thus, choices from easy queries provide limited information about preference strength, contributing minimally to estimating θ * . This intuition aligns with the online retailer example in section 1.In the choice-decision-time estimator (theorem 3.1), each query is weighted bymin x ′ ∈Xsample E [t x ′ ].The orange curves in fig. 2a show E [t x ], but not the 'min' operator for clarity. For each query x ∈ X sample , comparing the orange and gray curves reveals that E [t x ] is generally higher than the choice-only weight, a 2 V [c x ]. However, the choice-decision-time estimator's actual weight,min x ′ ∈Xsample E [t x ′ ],can vary with X sample , and may be either larger or smaller than a 2 V [c x ]. For example, when most queries are hard, the choice-decision-time estimator's weight may be smaller than some of the choice-only estimator's weights, suggesting that decision times do not always enhance estimation. However, when most queries are easy, the choice-only estimator's weights are close to zero, while the choice-decision-time estimator's weight remains large. This demonstrates that incorporating decision times makes easy queries more informative, enhancing estimation.Additionally, as the barrier a increases, all curves shift upward. Intuitively, a higher barrier, indicating greater conservativeness in human decision-making, leads to longer decision times and more consistent choices, as discussed in fig. 1 , providing more information for utility estimation.In this section, we focus on the simpler problem of estimating the utility difference for a single query, without aggregating data from multiple queries. Comparing the non-asymptotic concentration bounds of both estimators, in this case, provides insights similar to those discussed in section 3.2. Extending this non-asymptotic analysis to the full estimation of the preference vector θ * is left for future work.Given a query x ∈ X , the task is to estimate the utility difference ux := x ⊤ θ * using the i.i.d. dataset {(c x,sx,i , t x,sx,i )} i∈[nx]. Applying the choice-decision-time estimator from eq. ( 3), we get the following estimate (for details, see appendix C.3.1), which estimates u x /a rather than u x : EQUATIONE [t x ] , a =1.8 E [t x ] , a =0.8 a 2 V [c x ] , a =1.8 a 2 V [c x ] , a =0.8 (a) E [tx] and a 2 V [cx] in asymptotic variances -6 -4 -2 0 2 4 6x θ * E [t x ] and a 2 V [c x ]in the asymptotic variances for the choice-decision-time estimator (orange, theorem 3.1) and the choice-only estimator (gray, theorem 3.2), respectively. This comparison shows that incorporating decision times makes easy queries more informative. Additionally, higher barrier values a lead to more conservative decision-making, increasing informativeness for both choices and decision times. (b) compares the weights in the non-asymptotic concentration bounds (theorems 3.3 and 3.4), showing similar trends, though these terms may not be optimal due to proof techniques.In contrast, applying the choice-only estimator from eq. ( 4), we get the following estimate (for details, see appendix C.3.2), which estimates 2au x rather than u x :EQUATIONwhere (c x,sx,i + 1)/2 is the binary choice coded as 0 or 1, and µ -1 (p) := log (p/(1p)) is the logit function (inverse of µ introduced in eq. ( 4)).Notably, the choice-only estimator in eq. ( 6) aligns with the EZ-diffusion model's drift estimator [67, eq. ( 5)]. Moreover, the estimators in Xiang Chiong et al. [73, eq. ( 6)] and Berlinghieri et al. [8, eq. ( 7)] combine elements of both estimators from eqs. ( 5) and (6) . In section 5, we demonstrate that both estimators from Wagenmakers et al. [67, eq. ( 5)] and Xiang Chiong et al. [73, eq. ( 6)] are outperformed by our proposed estimator in eq. ( 3) for the full bandit problem.Assuming the utility difference u x ̸ = 0, the choice-decision-time estimator in eq. ( 5) satisfies the following non-asymptotic concentration bound, proven in appendix C.3.1: Theorem 3.3 (Non-asymptotic concentration of u x,CH,DT ). For each query x ∈ X withu x ̸ = 0, given an i.i.d. dataset c x,sx,i , t x,sx,i i∈[nx] , for any ϵ > 0 satisfying ϵ ≤ min |u x |/( √ 2a), 1 + √ 2 a|u x |/E [t x ], the following holds:P u x,CH,DT - u x a > ϵ ≤ 4 exp -m non-asym CH,DT x ⊤ θ * 2 n x [ϵ • a] 2 ,where m non-asym CH,DTx ⊤ θ * := E [t x ] / (2 + 2 √ 2) a .In contrast, the choice-only estimator in eq. ( 6) has the following non-asymptotic concentration result, adapted from Jun et al. [31, theorem 5]2 : Theorem 3.4 (Non-asymptotic concentration of u x,CH ). For each query x ∈ X , given an i.i.d. datasetc x,sx,i i∈[nx] , for any positive ϵ < 0.3, if n x ≥ 1/ μ(2au x ) • max{3 2 log(6e)/ϵ 2 , 64 log(3)/(1 - ϵ 2 /0.3 2 )}, the following holds:P (| u x,CH -2au x | > ϵ) ≤ 6 exp -m non-asym CH x ⊤ θ * 2 n x [ϵ/(2a)] 2 ,where m non-asymCH x ⊤ θ * := a V [c x ] / 2.4.The weights m non-asym CH,DT (•) and m non-asym CH (•) from theorems 3.3 and 3.4, respectively, are functions of the utility difference x ⊤ θ * for a fixed barrier a. These weights determine how quickly the estimation error decays as the dataset size n x grows, with larger weights indicating better estimation. While these weights may not be optimal due to our proof techniques, they still reveal the different contributions of choices and decision times to estimating u x , similar to our asymptotic analysis in section 3.2. Figure 2b compares the weights for the choice-decision-time estimator (orange, m non-asym CH,DT (•)) and the choice-only estimator (gray, m non-asym CH (•)). As shown, for hard queries, the choice-decisiontime estimator's weights may be smaller. However, for easy queries, the choice-only estimator's weights are close to zero, while the choice-decision-time estimator's weight remains relatively large, reinforcing that decision times enhance estimation by making easy queries more informative.In summary, both our asymptotic (section 3.2) and non-asymptotic (section 3.3) analyses show that the choice-decision-time estimator extracts more information from easy queries, while the choice-only estimator might perform better when most queries are hard. This aligns with the empirical findings of Clithero [16] and will be further supported by our empirical results in section 5.1. To illustrate, consider a teacher trying to identify the top student through two-choice questions. If the questions are easy and all students answer correctly, identifying the top performer is challenging. The teacher can either: (1) ask harder questions to gain more information from choices, or (2) keep the questions easy but use response times (i.e., how quickly students finish) to reveal additional insights.In fixed-budget best-arm identification, our choice-decision-time estimator's ability to extract more information from easy queries is crucial. A bandit learner like GSE [3] strategically samples queries, updates its estimate of θ * , and eliminates lower-utility arms. Unlike the choice-only estimator, our approach extracts more information from easy queries, which take less time (i.e., fewer resources) to answer, providing better 'bang per buck' (information per resource) [4] . Moreover, since the learner doesn't know θ * in advance, it cannot selectively sample only hard queries to exploit the strengths of the choice-only estimator. Next, we will integrate both estimators into a bandit learning algorithm.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [Yes]Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [No] Justification: We plan to release our code later after we receive approvals from the funding agency.Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
interactive learning algorithm: 
 We introduce the Generalized Successive Elimination (GSE) algorithm [1, 3, 75] for fixed-budget best-arm identification in preference-based linear bandits, and outline the key options for each GSE component, which we empirically compare in section 5.The pseudo-code for GSE is shown in algorithm 1. The algorithm uses a hyperparameter η to control the number of phases, the budget allocation per phase, and the number of arms eliminated in each phase. GSE divides the total budget B evenly into phases and reserves a buffer, sized by another hyperparameter B buff , to prevent over-consuming resources within each phase (line 4). In each phase, GSE computes an experimental design, a probability distribution λ over the query space to determine which queries to sample. We consider two designs: the transductive design [24] , λ trans (line 5), and the hard-query design [31] , λ hard (line 6). Both designs minimize the worst-case variance of utility differences between the surviving arms. The transductive design treats all queries equally to achieve this, while the hard-query design focuses more on sampling hard queries to leverage the choice-only estimator's advantage in extracting information from hard queries, as discussed in section 3. GSE then randomly samples queries according to the design λ s (line 7). After the phase's budget is exhausted, GSE estimates θ * using either the choice-decision-time estimator θ CH,DT (line 8) or the choice-only estimator θ CH (line 9). Arms with low estimated utilities are eliminated, and this process repeats phase-by-phase until only one arm remains in Z S+1 , which GSE then recommends.The key difference between algorithm 1 and previous GSE algorithms [1, 3, 75] is that, in our setting, queries consume random response times, which are unknown to the learner. Prior work assumes fixed resource consumption per query and uses deterministic rounding methods [3, 24] to pre-allocate queries to be sampled. This approach is unsuitable for our case with random resource usage. Instead, we adopt a random sampling procedure [13, 61] in line 7 to allocate queries based on the design. The random resource usage also necessitates careful tuning of both the elimination parameter η, to balance data collection and arm elimination, and the buffer size B buff , to prevent over-consuming resources during each phase. In our empirical study (section 5.2), we manually tune both parameters. Our results show that the choice of η impacts performance, unlike prior studies, which typically set η = 2 by default [3, section 3]. Further theoretical analysis is needed to fully understand the algorithm's behavior and optimize both the elimination parameter and the buffer size.Algorithm 1 Generalized Successive Elimination (GSE) [3] 1: Input: Arm space Z, query space X , non-decision time t nondec , and total budget B. Estimate 1. θ s := θ CH,DT,s ← apply eq. ( 3) to all the J s samples. 9:Estimate 2. θ s := θ CH,s ← apply eq. ( 4) to all the J s samples. Output: the single one z ∈ Z S+1 . 
empirical results: 
 This section empirically compares the GSE variations introduced in section 4: (1) (λ trans , θ CH,DT ):Transductive design with the choice-decision-time estimator. (2) (λ trans , θ CH ): Transductive design with the choice-only estimator. (3) (λ hard , θ CH ): Hard-query design with the choice-only estimator.We benchmark the estimation performance of these GSE variations using the "sphere" synthetic problem from the linear bandit literature [20, 42, 61] . In this problem, the arm space Z ⊂ {z ∈ R 5 : ∥z∥ 2 = 1} contains 10 randomly generated arms. To define the true preference vector θ * , we select the two arms z and z ′ that are closest in direction, i.e., (z, z ′ ) ∈ arg max z,z ′ ∈Z z ⊤ z ′ , and set θ * = z + 0.01(z ′z). In this way, z is the best arm. The query space is X := {zz ′ : z ∈ Z}.Estimation performance, as discussed in section 3, depends on the utility difference x ⊤ θ * and the barrier a. To adjust the utility differences, we scale each query by scaling each arm z to c Z • z. We vary a over a range of typical values from the psychology literature [16, 71] . For each (c Z , a) pair, the system generates 10 random problem instances and runs 200 repeated simulations per instance. In each simulation, the GSE variations sample 50 queries without considering the response time budget and then compute θ. Performance is measured by P[arg max z∈Z z ⊤ θ ̸ = z * ], reflecting the best-arm identification goal discussed in section 2. To focus purely on estimation, we allow λ hard access to the true θ * , enabling it to perfectly compute the weights μ(x ⊤ θ * ) used in line 6 of algorithm 1.As shown in fig. 3a , when fixing the barrier a and examining the vertical line, we observe that the choice-only estimator with the transductive design performs well for small c Z (hard queries). However, as c Z increases and queries become easier, performance declines, even though larger c Z generally makes best-arm identification easier. This decline, illustrated by the dark curved band, aligns with the insights from section 3, that choices from easy queries provide limited information.In Figure 3b , for moderate c Z , the choice-only estimator with the hard-query design outperforms the transductive design (fig. 3a ), showing that focusing on harder queries improves estimation. The lower dark band in fig. 3b compared to fig. 3a shows that focusing on hard queries improves estimation when most queries are easy. However, as c Z becomes too large, performance declines, likely because many weights μ(x ⊤ θ * ) approach zero, preventing informative queries from being sampled. This advantage of the hard-query design relies on perfect knowledge of θ * and the same resource consumption across all queries. In practice, where θ * is unknown and hard queries require longer response times, the hard-query design can be outperformed by the transductive design, as shown in the next section.In contrast, fig. 3c shows that the choice-decision-time estimator consistently outperforms the choiceonly estimators in both the transductive and hard-query designs. The choice-decision-time estimator's performance improves as c Z increases (i.e., queries become easier), confirming the theoretical insights suggesting that choices from easy queries provide limited information. In contrast, in (c), the choicedecision-time estimator θ CH,DT with transductive design (λ trans ) consistently achieves better estimation across all c Z values, indicating that decision times make easy queries more informative. from section 3 that incorporating decision times make easy queries more informative. Performance also improves with a higher barrier a, supporting the insights discussed at the end of section 3.2.This section compares the bandit performance of six GSE variations. The first three are as previously defined: (λ trans , θ CH,DT ), (λ trans , θ CH ), and (λ hard , θ CH ).The 4th GSE variation, (λ trans , θ CH,RT ), evaluates the performance of the choice-decision-time estimator when the non-decision time t nondec is unknown. The estimator, θ CH,RT , is identical to the original choice-decision-time estimator from Eq. ( 3), but with response times used in place of decision times.The 5th GSE variation, (λ trans , θ CH,logit ), is based on Wagenmakers et al. [67, eq. ( 5)], which states that x ⊤ • (2aθ * ) = µ -1 (P[c x = 1]), where µ -1 (p) := log (p/ (1p)). By incorporating our linear utility structure, we obtain the following choice-only estimator θ CH,logit :θ CH,logit :=   x∈Xsample n x xx ⊤   -1 x∈Xsample n x x • µ -1 C x ,whereEQUATION2 c x,sx,i + 1 is the empirical mean of the binary choices coded as 0 or 1.The 6th GSE variation, (λ trans , θ CH,DT,logit ), is based on Xiang Chiong et al. [73, eq. ( 6)], which states thatx ⊤ θ * = sgn (c x ) E [c x ] /E [t x ] • 0.5 µ -1 (P [c x = 1]). This identity forms the foundation of the estimator in Berlinghieri et al. [8, eq. ( 7)]. By incorporating our linear utility structure, we obtain the following choice-decision-time estimator θ CH,DT,logit :θ CH,DT,logit :=   x∈Xsample n x xx ⊤   -1 x∈Xsample n x x • sgn (c x ) E [c x ] E [t x ] • 1 2 µ -1 C x .We evaluate the six GSE variations by simulating their performance on bandit instances constructed from three real-world datasets of human choices and response times. The first dataset, food-risk with choices (-1 or 1) [57] , contains choices and response times from 42 participants answering queries comparing two sets of food items. For each participant, we identified the dEZDM parameters and built a bandit instance with Z ⊂ R (a) Food-risk, (-1 or 1) choices [57] . dataset, snack with choices (yes or no) [16] , contains choices and response times from 31 participants comparing one snack item to a fixed reference snack. For each participant, we built a bandit instance with Z ⊂ R 17 , where |Z| = 17, and X := {z -0 : z ∈ Z}. The third dataset, snack with choices (-1 or 1) [39] , contains choices and response times from 39 participants comparing two snack items. For each participant, we built a bandit instance with Z ⊂ R 21 , |Z| = 21, and X := {zz ′ : z ∈ Z}. Details on data processing, tuning the elimination parameter η and buffer size B buff as defined in algorithm 1, and experimental procedures are provided in appendix D. Key results for the three domains are shown in fig. 4 , with full results in appendix D. First, (λ trans , θ CH,DT ) consistently outperforms (λ trans , θ CH ), demonstrating the benefit of incorporating decision times. Second, both of these GSE variations outperform (λ hard , θ CH ), which, as discussed in section 5.1, suffers from (1) relying on θ for query selection, which is prone to estimation errors, and (2) favoring hard queries with longer response times, reducing the total number of queries sampled within a fixed budget. Third, (λ trans , θ CH,DT ) performs similarly to (λ trans , θ CH,RT ), indicating that not knowing the non-decision time has minimal impact. Finally, (λ trans , θ CH,logit ) [67] and (λ trans , θ CH,DT,logit ) [73] do not perform as consistently well as (λ trans , θ CH,DT ), underscoring the effectiveness of our proposed choice-decision-time estimator from eq. ( 3). 
conclusion and future work: 
 This work leverages human response times to enhance fixed-budget best-arm identification in preference-based linear bandits. We proposed a utility estimator that incorporates both choices and response times. Our theoretical and empirical analyses show that response times make easy queries more informative. When integrated into a bandit algorithm, incorporating response times consistently improved performance across simulations based on three real-world datasets.A limitation of this work is that reliable response time data requires participants to stay focused [45] , which can be challenging in crowdsourcing environments. Future work could incorporate eye movements into the DDM framework, as in attentional DDMs [26, 38, 39, 57, 76] , to detect attention lapses and filter unreliable data. Additionally, while response times are effective for easy queries, they may be less so for hard ones. Future research could develop algorithms that adaptively decide when to incorporate response times. Another direction is to remove the assumption of known non-decision times by estimating them from data, with approaches like those proposed by Wagenmakers et al. [67] .Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes]Justification: We state our full algorithm algorithm 1 in section 4 and the implementation details in appendix D. We also provide the detailed procedures that we have taken to convert several datasets to bandit instances in section 5 and appendix D.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
experiment statistical significance: 
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 Appendix: Incorporating human response times in human-interactive AI systems provides significant benefits, such as efficiently eliciting user preferences, reducing cognitive loads on users, and improving accessibility for users with disabilities and various cognitive abilities. These benefits can greatly improve recommendation systems, assistive robots, online shopping platforms, and fine-tuning for large language models. However, using human response times also raises concerns about privacy, manipulation, and bias against individuals with slower response times. Governments and law enforcement should work together to mitigate these negative consequences by establishing ethical standards and regulations. Businesses should always obtain user consent before recording response times.Bounded Accumulation Models (BAMs) describe human decision-making using an accumulator (or sampling rule) and a stopping rule [70] . In binary choice tasks, such as two-alternative forced choice tasks, a widely used BAM is the drift-diffusion model (DDM) [51] , which models decisions as Brownian motion with fixed boundaries. To capture differences in human response times for correct and incorrect answers, Ratcliff and McKoon [51] allows drift, starting point, and non-decision time to vary across trials. Wagenmakers et al. [67] later introduced the EZ-diffusion model (EZDM), a simplified version of DDM with closed-form solutions for choice and response time moments, making parameter estimation easier and more robust. EZDM assumes deterministic drift, starting point, and non-decision time, fixed across trials, with the starting point equidistant from the boundaries. Berlinghieri et al. [8] specialized EZDM to the difference-based EZDM (dEZDM), where the drift represents the utility difference between two options. For binary queries with arms z 1 and z 2 , the drift is modeled as u z1u z2 , where u z1 and u z2 are the utilities of z 1 and z 2 .As discussed in section 2, we impose a linear utility structure on the dEZDM, where each arm's utility is given by u z = z ⊤ θ * , with θ * denotes the human preference vector. This approach is supported by both bandit and psychology literature. In bandits, linear utility models scale efficiently with a large number of arms [15, 41] . In psychology, linear combinations of attributes are commonly used in multi-attribute decision-making models [26, 64, 76] . The standard dEZDM in [8, Definition 1] is a special case of our dEZDM with a linear utility structure, where arms correspond to the standard basis vectors in Euclidean space R d . This mirrors the relationship between multi-armed bandits and linear bandits.Similarly to our approach, Shvartsman et al. [55] parameterize the human utility function as a Gaussian process and propose a moment-matching Bayesian inference method that uses both choices and response times to estimate latent utilities. Unlike our work, their focus is solely on estimation and does not address bandit optimization. Integrating their estimation techniques into bandit optimization presents an interesting avenue for future research.Another widely used BAM is the race model [11, 66] , which naturally extends to queries with more than two options. In race models, each option has its own accumulator, and the decision ends when any accumulator reaches its barrier. BAMs can also model human attention during decision-making. For example, the attentional-DDM [38, 39, 76] jointly models choices, response times, and eye movements across different options or attributes. Similarly, Thomas et al. [62] introduce the gazeweighted linear accumulator model to study gaze bias at the trial level. To incorporate learning effects, Pedersen et al. [49] combine reinforcement learning (RL) with DDM, where the human adjusts the drift through RL. In contrast, our work uses RL for AI decision-making when interacting with humans. BAMs also connect to Bayesian RL models of human cognition. For example, Fudenberg et al. [27] propose a model where humans balance decision accuracy and time cost, showing it is equivalent to a DDM with time-decaying boundaries. Neurophysiological evidence supports BAMs. For instance, EEG recordings demonstrate that neurons exhibit accumulation processes and decision thresholds [70] . Additionally, diffusion processes have been used to model neural firing rates [53] .BAMs often lack closed-form density functions, so hierarchical Bayesian inference is commonly used for parameter estimation [71] . While flexible, these methods are computationally intensive, making them impractical for real-time applications in online learning systems. Faster estimators [8, 67, 73] usually estimate parameters for individual option pairs without leveraging data across pairs. To address this, we propose a computationally efficient method for estimating linear human utility functions, which we integrate into bandit learning. In section 5.2, we empirically show that our estimator outperforms those from prior work [67, 73] . In practice, using response time data requires pre-processing and model fitting, as outlined by Myers et al. [45] . Additionally, Alós-Ferrer et al. [2] , Baldassi et al. [6] , Fudenberg et al. [28] propose statistical tests to assess the suitability of various DDM extensions for a given dataset.Response times serve multiple purposes, as highlighted by Clithero [17] . A primary use is improving choice prediction. For instance, Clithero [16] showed that DDM predicts choice probabilities more accurately than the logit model, with parameters estimated through Bayesian Markov chain Monte Carlo. Similarly, Alós-Ferrer et al. [2] demonstrated that response times enhance the identifiability of human preferences compared to using choices alone.Response times also shed light on human decision-making processes. Castro et al. [14] applied DDM analysis to explore how cognitive workload, induced by secondary tasks, influences decision-making. Analyzing response times has been a long-standing method in cognitive testing to assess mental capabilities [19] . Additionally, Zhang et al. [78, 79] introduced a framework that uses human planning time to infer their intended goals.Response times can also enhance AI decision-making. In dueling bandits and preference-based RL [7] , human choice models are commonly used for preference elicitation. One such model, the random utility model, can be derived from certain BAMs [2] . For example, as discussed after eq. ( 1), both the Bradley-Terry model [10] and dEZDM [8, 67] yield logistic choice probabilities in the formP[z 1 ≻ z 2 ] = σ logistic (u z1 -u z2 ) = 1/ (1 + exp (-(u z1 -u z2 ))), where u z1 and u z2 denote the utilities of z 1 and z 2 [7, section 3.2]. Our work leverages this connection between random utility models and choice-response-time models to estimate human utilities using both choices and response times.We hypothesize that the insight that response times make easy queries more informative extends beyond the dEZDM and the specific link function σ logistic . Many psychological models capture both choices and response times but lack closed-form choice distributions. In these cases, choice probability is often expressed as P[z 1 ≻ z 2 ] = σ † (u z1 , u z2 ), where σ † is a potentially complex function of u z1 and u z2 without a closed form. If we fix u z2 and vary only u z1 , the function σ † (•, u z2 ) is known as a psychometric function, which typically exhibits an "S" shape [60, above fig. 1 .1]. As preference strength becomes extreme, σ † flattens, providing less information, similar to the green curves in figs. 1b and 1c. In such cases, we conjecture that response times can offer valuable complementary information.If we further assume the choice probability depends only on the utility difference, u z1u z2 , we obtain P[z 1 ≻ z 2 ] = σ ‡ (u z1u z2 ). This link function, σ ‡ , is commonly used in preference-based RL [7, section 3.2], where it is assumed to be strictly monotonic and bounded within [0, 1], again reflecting the "S"-shaped behavior. As the absolute value of the utility difference grows large, σ ‡ (u z1u z2 ) flattens, providing less information. We conjecture that response times can offer valuable complementary information.In summary, BAMs, like DDMs and race models, offer a strong theoretical framework for understanding human decision-making, supported by both behavioral and neurophysiological evidence. These models have been widely applied to choice prediction and the study of human cognitive processes. Our work connects BAMs with bandit algorithms by introducing a computationally efficient estimator for online preference learning. Future research could explore other BAM variants to further examine the benefits of incorporating response times.C.1 Parameters of the difference-based EZ-Diffusion Model (dEZDM) [8, 67] Given a human preference vector θ * , for each query x ∈ X , the utility difference is defined as u x := x ⊤ θ * . In the dEZDM model (introduced in section 2), with barrier a, according to Wagenmakers et al. [67, eq. ( 4), (6) , and ( 9)], the human choice c x has the following properties:P (c x = 1) = 1 1 + exp (-2au x ) , P (c x = -1) = exp (-2au x ) 1 + exp (-2au x ).Thus, the expected choice is E [c x ] = tanh(au x ), and the choice variance is V [c x ] = 1-tanh(au x ) 2 (restating eq. ( 1)).The human decision time t x has the following properties:E [t x ] = a ux tanh(au x ) if u x ̸ = 0 a 2 if u x = 0(restating eq. ( 1)),V [t x ] = a ux 3 exp(4aux)-1-4aux exp(2aux) (exp(2aux)+1) 2 if u x ̸ = 0 2a 4 /3 if u x = 0 .From this, we obtain the following key relationship:E [c x ] E [t x ] = u x a = x ⊤ 1 a θ *(restating eq. ( 2)).All these parameters depend solely on the utility difference u x := x ⊤ θ * and the barrier a.We now present the proof of the asymptotic normality result for the choice-decision-time estimator, θ CH,DT , as stated in theorem 3.1, which is restated as follows:Theorem 3.1 (Asymptotic normality of θ CH,DT ). Given an i.i.d. dataset x, c x,sx,i , t x,sx,i i∈[n] for each x ∈ X sample , where x∈Xsample xx ⊤ ≻ 0, and assuming that the datasets for different x ∈ X sample are independent, then, for any vector y ∈ R d , as n → ∞, the following holds:√ n y ⊤ θ CH,DT,n -θ * /a D -→ N (0, ζ 2 /a 2 ).Here, the asymptotic variance depends on a problem-specific constant, ζ 2 , with an upper bounded:ζ 2 ≤ ∥y∥ 2 x∈X sample min x ′ ∈X sample E[t x ′ ] •xx ⊤ -1 .Proof. To simplify notation, we define:EQUATIONFor brevity, we abbreviate X sample as X and θ CH,DT,n as θ. The estimator θ can be expressed as:θ = x ′ ∈X nx ′ x ′⊤ -1 x∈X nx C xT x (restating eq. ( 3)).We rewrite θ * /a as:EQUATIONTherefore, for any vector y ∈ R d , we have:EQUATIONwhere ξ x is defined as ξ x := y ⊤ x ′ ∈X nx ′ x ′⊤ -1 nx. In eq. ( 9), the only random variables are C x and T x . For simplicity, for any x i ∈ X := {x 1 , • • • , x |X | }, we slighly abuse the notation and useξ i , c i , t i , C i , T i , C i and T i denote ξ xi , c xi , t xi , C xi , T xi , C xi ,and T xi , respectively. By applying the multidimensional central limit theorem, we have:√ n        C 1 -C 1 T 1 -C 1 . . . C |X | -C |X | T |X | -C |X |        D -→ N        0,        V [c 1 ] cov [c 1 , t 1 ] cov [t 1 , c 1 ] V [t 1 ] . . . V c |X | cov c |X | , t |X | cov t |X | , c |X | V t |X |               = N 0, diag V [c 1 ] , V [t 1 ] , • • • , V c |X | , V t |X | .(10) In the first line of eq. ( 10), the block-diagonal structure of the covariance matrix emerges because( C i , T i ) i∈[|X |]are independent of each other. For any fixed x i , to derive the second line of eq. ( 10), we use the fact that:EQUATIONwhere(i) is because E [t i |c i = 1] = E [t i |c i = -1][48, eq. (A.7) and (A.9)]. Therefore, eq. ( 11) implies that cov(c i , t i ) = 03 , which justifies the second line of eq. ( 10). Now, let us define the function g(c1 , t 1 , • • • , c |X | , t |X | ) := i∈[|X |] ξ i c i /t i .The gradient of g is:EQUATION)Using the multivariate delta method, we obtain:EQUATIONBy applying the identities outlined in appendix C.1, we can establish the following identity:EQUATION)Substituting this identity into eq. ( 13), we obtain:EQUATION)Finally, the asymptotic variance can be upper bounded as follows:EQUATIONC.3 Non-asymptotic concentration of the two estimators for estimating the utility difference u x given a query xSection 3.3 focuses on the problem of estimating the utility difference for a single query. Given a query x ∈ X , the objective is to estimate the utility difference u x := x ⊤ θ * using an i.i.d. dataset, denoted by (c x,sx,i , t x,sx,i ) i∈ [nx] .We begin by applying the choice-decision-time estimator from eq. ( 3), which is derived by solving the following least squares problem:θ CH,DT = arg min θ∈R d x∈Xsample x ⊤ θ - i∈[nx] c x,sx,i i∈[nx] t x,sx,i 2 .Similarly, the utility difference for a single query is estimated as the solution to the following least squares problem, yielding the estimate:u x,CH,DT = arg min u∈R u - i∈[nx] c x,sx,i i∈[nx] t x,sx,i 2 = i∈[nx] c x,sx,i i∈[nx] t x,sx,i(restating eq. ( 5)).The resulting estimate, u x,CH,DT , approximates u x /a rather than u x . However, since the ranking of arm utilities is preserved between u x /a and u x , estimating u x /a is sufficient for the purpose of best-arm identification.For the case where the utility difference u x ̸ = 0, the non-asymptotic concentration inequality for this estimator is presented in theorem 3.3. To prove this, we first introduce lemma C.1, which demonstrates that for any given query x, the decision time is a sub-exponential random variable.To simplify notation, we define:EQUATION) Lemma C.1. If u x ̸ = 0, then (t x -T x ) is sub-exponential SE ν 2 x , α x , where ν x = √ 2a/|u x | and α x = 2/u 2 x .Proof. For simplicity, we will omit the subscript x throughout the proof and assume, without loss of generality, that u > 0.Our objective is to establish the following inequality, which holds for all s ∈ (-u 2 /2, u 2 /2):EQUATION)This implies that (t -T ) is sub-exponential SE ν 2 , α , as defined by Wainwright [69, Definition 2.7].Step 1: Transform eq. ( 18) into a more manageable inequality (eq. ( 24)).Using Cox [18, eq. ( 128)], with ∆ := u 2 -2s, θ 1 := -u -√ ∆ and θ 2 := -u + √ ∆, we have 4 :EQUATIONIn the last line, we define N = 2 cosh(au) and D(s) = 2 cosh(a √ ∆). Thus, we arrive at:EQUATIONTo prove the original inequality in eq. ( 18), it is now sufficient to show:EQUATIONFor s = 0, the inequality holds trivially, as:EQUATIONFor s ̸ = 0, taking the derivative of the left-hand side of eq. ( 21) yields:EQUATIONIn step 2, we will prove the following inequality:EQUATIONEquation (24) implies that D(s) • exp a u tanh(au)s + a 2 u 2 s 2 ≥ N , which finishes the proof.Step 2. Prove eq. (24) .For s ≥ 0, the following holds:EQUATIONHere, (i) follows from tanh(au) ≥ tanh(a√ ∆) = tanh(a √ u 2 -2s) and (ii) follows from tanh(x)/x ≤ 1.For s < 0, the following holds:EQUATIONHere, (i) follows from tanh(au) ≤ tanh(a √ ∆) = tanh(a √ u 2 -2s) and (ii) follows from tanh(x)/x ≤ 1.By combining both cases, we conclude that the inequality in eq. ( 24) holds, which completes Step 2 and proves the desired result.Next, we prove theorem 3.3, which provides the non-asymptotic concentration inequality for the estimator from eq. ( 5), restated as follows: Theorem 3.3 (Non-asymptotic concentration of u x,CH,DT ). For each query x ∈ X withu x ̸ = 0, given an i.i.d. dataset c x,sx,i , t x,sx,i i∈[nx] , for any ϵ > 0 satisfying ϵ ≤ min |u x |/( √ 2a), 1 + √ 2 a|u x |/E [t x ], the following holds:P u x,CH,DT - u x a > ϵ ≤ 4 exp -m non-asym CH,DT x ⊤ θ * 2 n x [ϵ • a] 2 ,where m non-asym CH,DTx ⊤ θ * := E [t x ] / (2 + 2 √ 2) a .Proof. For clarity, we will omit the subscripts x throughout this proof. Based on lemma C.1, we define the constants ν := √ 2a/|u| and α := 2/u 2 .We begin by introducingϵ C := T / √ 2 + √ 2ν|C|/T •ϵ and ϵ T := νϵ C . From the identities provided in appendix C.1, we know that ν|C|/T = √ 2a/|u| • |u|/a = √ 2.This allows us to simplify the constants ϵ C and ϵ T as:EQUATIONFor any ϵ satisfying the following condition:EQUATIONwe observe that ϵ T < min T (1 -1/ √ 2), ν 2 /α . We can now apply lemma C.2 to derive the following:EQUATIONThus, by combining the results, we conclude:EQUATIONHere, (i) follows from lemma C.3, (ii) uses lemma C.2 and eq. ( 29), and (iii) follows from eq. ( 27).Lemma C.2. For each query x with u x ̸ = 0, and constants ϵ C > 0 and ϵ T ∈ (0, ν 2 x /α x ], the following inequalities hold:EQUATIONHere, the constants areν x := √ 2a/|u x | and α x := 2/u 2 x .Proof. Since c x ∈ {-1, 1}, by applying Hoeffding's inequality [69, proposition 2.5], we obtain:EQUATIONFrom lemma C.1, we know that t x is sub-exponential SE(ν 2 x , α x ). By applying Wainwright [69, proposition 2.9 and eq. (2.18)], we obtain:EQUATIONEQUATIONProof. The maximum value of C/ T -C/T is attained at the extremum of C/ T . Since C/ T is linear in C, the extremum of C/ T is attained atC * ∈ {C -ϵ C , C + ϵ C } for any T ∈ [T -ϵ T , T + ϵ T ] > 0.Given that T > 0, the extremum of C * / T is attained at T * ∈ {Tϵ T , T + ϵ T }. Therefore, the extremum of C/ T lies in the set:EQUATIONFor any combination (s C , s T ) ∈ {±1} × {±1}, and using the function ϵ T ≤ (1 -1/ √ 2)T , we have:EQUATION)By combining these results, we conclude that:max C∈[C-ϵ C ,C+ϵ C ] T ∈[T -ϵ T ,T +ϵ T ] C T - C T = max (s C ,s T )∈{±1}×{±1} C + s C ϵ C T + s T ϵ T - C T ≤ √ 2 ϵ C + ϵ T |C|/T T .We now apply the logistic-regression-based choice-only estimator from eq. ( 4) to estimate the utility difference for a single query. Recall that for each query x ∈ X , the human choice c x ∈ {-1, 1}. We define the binary-encoded choice as e x := (c x + 1) /2 ∈ {0, 1}. We reformulate the MLE in eq. ( 4) into a utility difference estimation problem for a single query, leading to the following optimization problem:u x,CH = arg max u∈R i∈[nx] log µ(c x,sx,i u) = arg max u∈R i∈[nx] log (µ(u)) ex,s x,i • (µ(-u)) 1-ex,s x,i .The first-order optimality condition provides the optimal solution:u x,CH = µ -1   1 n x i∈[nx] e x,sx,i  (restating eq. ( 6)), where µ -1 (p) := log (p/(1p)) is the logit function (also known as the log-odds), defined as the inverse of the function µ(•) introduced in eq. ( 4).The resulting estimate, u x,CH , from eq. ( 6) gives an estimate of 2au x , not u x . However, since the ranking of arm utilities based on 2au x is the same as that based on the true u x , estimating 2au x suffices for identifying the best arm.The non-asymptotic concentration inequality for this estimator is stated in theorem 3.4. This result is directly adapted from Jun et al. [31, theorem 5] , by lettingx 1 = • • • = x t = 1 and t eff = d = 1.Our empirical experiments (Sec. 5) were conducted on a MacBook Pro (M3 Pro, Nov 2023) with 36 GB of memory.Our implementation of algorithm 1 is done in Julia, building on the implementation of Tirinzoni and Degenne [63] , where the transductive and hard-query designs are solved using the Frank-Wolfe algorithm [24] . Their implementation is available at https://github.com/AndreaTirinzoni/ bandit-elimination. The simulation and Bayesian inference for DDM are implemented using the Julia package SequentialSamplingModels.jl at https://itsdfish.github.io/ SequentialSamplingModels.jl/dev/#SequentialSamplingModels.jl.For a query x ∈ X , the estimators from Wagenmakers et al. [67] and Xiang Chiong et al. [73] , analyzed in section 3.3 and benchmarked in section 5.2, require calculating µ -1 (p) := log (p/ (1p)), where µ -1 (•) is the logit function and p := 1/n x • nx i=1 c x,sx,i + 1 /2 represents the empirical mean of the human binary choices coded as 0 or 1. Since p = 0 or p = 1 makes this calculation undefined, we follow Wagenmakers et al. [67, the discussion below fig. 6 ] and approximate p as 1 -1/(2n x ) when p = 1 and 1/(2n x ) when p = 0. D.2 Processing the snack dataset with choices (yes or no) [16] We accessed the snack dataset with choices (yes or no) [16] through the supplementary material provided by Alós-Ferrer et al. [2] at https://www.journals.uchicago.edu/doi/abs/10.1086/ 713732. This dataset consists of training and testing data. The training data was collected from a "YN" task, where 31 participants provided binary feedback ("Yes" or "No") and response times for queries comparing each of the 17 snack items to a fixed reference snack, with each query repeated 10 times. The reference snack, assigned a utility of 0, remained fixed throughout the experiment. The testing data was collected using a two-alternative forced-choice task, where participants provided binary choices and response times for queries comparing two snack items, with each query repeated once. Clithero [16] fit a difference-based EZ-diffusion model [8, 67] to the training data using Bayesian inference with non-informative priors, without imposing a linear utility structure, and tested the model using the testing data.In our work, we fit each participant's training data to a difference-based EZ-diffusion model with a linear utility structure, as described in section 2, and used the fitted model to simulate human feedback for bandit experiments. We preprocessed the data by removing outliers, following Clithero [16, footnote 22] , excluding trials with response times below 200 ms or greater than five standard deviations above the mean. After cleaning, the number of trials per participant ranged from 167 to 170. Since the dataset does not provide feature vectors for the 17 non-reference snack items, we used one-hot encoding to represent each snack item as a feature vector in R 17 . This allowed us to construct a bandit instance for each participant with a preference vector θ * ∈ R 17 , an arm space Z ⊂ R 17 with |Z| = 17, and a query space X := {z -0 : z ∈ Z} to represent comparisons with the reference snack. We applied Bayesian inference with non-informative priors [16] to estimate each participant's preference vector θ * , non-decision time t nondec , and barrier a. Across participants, the barrier a ranged from 0.759 to 1.399, with a mean of 1.1, and t nondec ranged from 0.139 to 0.485 seconds, with a mean of 0.367 seconds.For each of the six GSE variations (introduced in section 5.2): (λ trans , θ CH,DT ), (λ trans , θ CH,RT ), (λ trans , θ CH ), (λ hard , θ CH ), (λ trans , θ CH,logit ), and (λ trans , θ CH,DT,logit ), we tuned the elimination parameter η in algorithm 1 using the following procedure: We considered η ∈ {2, 3, 4, 5, 6, 7, 8, 9}, resulting in the number of phases := log η |Z| = log η (17) (line 4 of algorithm 1) being {5, 3, 3, 2, 2, 2, 2, 2}, respectively. We excluded η > ⌈17/2⌉ = 9, as those cases also result in 2 phases, the same as η ∈ {5, 6, 7, 8, 9}. Then, for each η, for each of the 31 bandit instances, and for each budget ∈ {50, 75, 100, 125, 150, 200, 250, 300} seconds, we ran 50 repeated simulations per GSE variation under different random seeds, sampling human feedback from the fitted dEZDM. We then aggregated the results into a single best-arm identification error probability for each GSE variation, η, bandit instance, and budget. These error probabilities were compiled into violin and box plots, as shown in fig. 6 .For each GSE variation, we selected the η that minimized the median error probability, as shown in the box plots in fig. 6 . If multiple η values yielded the same median, we used the third quartile, and if necessary, the first quartile, to break ties. Based on this approach, we selected: η = 6 for (λ trans , θ CH,DT ), η = 6 for (λ trans , θ CH,RT ), η = 9 for (λ trans , θ CH ), η = 9 for (λ hard , θ CH ), η = 9 for (λ trans , θ CH,logit ), and η = 5 for (λ trans , θ CH,DT,logit ).After tuning η, we manually set the buffer size B buff in algorithm 1 to 10 seconds based on empirical results, ensuring the budget was not exceeded in any phase. We then benchmarked each GSE variation on all 31 bandit instances using its own manually tuned η and B buff . Each variation was evaluated over 300 repeated simulations with different random seeds, where human choices and response times were sampled from the dEZDM with the identified parameters. The full results are shown in fig. 7 , with selected results presented in fig. 4b . , as a function of budget for each GSE variation, simulated using the snack dataset with choices (yes or no) [16] , as described in appendix D.2. The box plots follow the convention of the matplotlib Python package. For each GSE variation and budget, the horizontal line in the middle of the box represents the median of the error probabilities across all bandit instances. Each error probability is averaged over 300 repeated simulations under different random seeds. The box's upper and lower borders represent the third and first quartiles, respectively, with whiskers extending to the farthest points within 1.5× the interquartile range. Flier points indicate outliers beyond the whiskers. Figure 9 : A violin plot overlaid with a box plot showing the best-arm identification error probability, P [ z ̸ = z * ], as a function of budget for each GSE variation, simulated using the snack dataset with choices (-1 or 1) [39] , as described in appendix D.3. The box plots follow the convention of the matplotlib Python package. For each GSE variation and budget, the horizontal line in the middle of the box represents the median of the error probabilities across all bandit instances. Each error probability is averaged over 300 repeated simulations under different random seeds. The box's upper and lower borders represent the third and first quartiles, respectively, with whiskers extending to the farthest points within 1.5× the interquartile range. Flier points indicate outliers beyond the whiskers.In Jun et al. [31, theorem 5], we let x1 = • • • = xt = 1 and teff = d = 1.Equation(11) implies that for any query xi, the human choice ci and decision time ti are uncorrelated. Moreover, they are independent, as discussed by Drugowitsch [22, the discussion above eq. (7)] and Baldassi et al.[6, proposition 3].In Cox[18, eq. (128)], setting a = 2a and x0 = a leads to the desired result. 


Decision: The paper studies preference-based linear bandits in fixed-budget best-arm identification problem. The authors investigate the benefit of incorporating human response time which inversely correlates with preference strength. The authors propose to incorporate the drift-diffusion model (DDM) from psychology with an elimination-based solution.  The paper presents both theoretical and empirical results showing that incorporating response time makes easy queries more useful and thus improves overall performance.  The reviewers are unanimously positive, recognizing the motivation and novelty of introducing response time and appreciating the solid analysis and clear paper presentation. The reviewers' questions about real-world datasets and the non-asymptotic results are addressed by the rebuttal. I recommend acceptance of the paper. The authors are suggested incorporate the discussion and new results from the rebuttal into the final version.

Reviews: review: paper_summary: paper_summary: This paper studies whether leveraging human response times can lead to better performance in bandit learning from preference feedback. More specifically, the paper integrates the drift-diffusion model (DDM) from psychology into the best-arm identification problem in linear bandits. Given a fixed interaction time, the goal is to utilize response times alongside binary choices so as to maximize the probability of recommending the optimal arm.

The paper introduces an estimator of the preference/reward vector using both binary responses and response times via linear regression. This estimator can be incorporated into bandit algorithms. Asymptotic normality results and three simulations indicate that this new estimator leveraging response times can make easier queries more useful, in comparison with traditional estimators that only use binary responses.
strength_weakness: strength_weakness: - It would be helpful to see more discussion on related work, especially on DDMs and race models. For example, what evidence has been given in the psychology literature that DDMs can explain the human decision-making process? What are the typical objectives in papers that study DDMs, and how are they different from the best-arm identification problem in this work? Have DDMs been considered in bandits and reinforcement learning?

- On the theoretical side, there are no non-asymptotic results regarding the performance of the bandit algorithm, such as bounds on the error probability.

- On the empirical side, only one dataset contains the response times of the participants. The last two experiments simulate response times according to the DDM -- it is reasonable to expect that they are then useful to estimating $\theta^*$. The algorithm also has a hyperparameter that requires tuning.
questions: questions: See above for questions on related work.

I am also curious -- the estimator that uses only binary responses essentially assumes the Bradley-Terry model, if my understanding is correct. Looking at the logistic sigmoid function, I can see that the when $x^\top \theta^*$ is away from 0, the curve becomes flat and therefore not much information can be gained here. Have you considered other noise models that do not use a link function like the logistic sigmoid function? Would you expect similar behavior?
limitations: limitations: Limitations have been discussed.
paper_summary: paper_summary: The submission proposes to use response times to obtain additional information from participants in preference learning settings. They apply a variant of the Drift-Diffusion Model, a popular model of human decision making from psychology, and combine it with an algorithm applicable to linear bandits. In asymptotic analysis and application to real data, they demonstrate the benefits of taking advantage of response times, especially for queries with large value differences.
strength_weakness: strength_weakness: * I think the paper slightly misrepresents the actual model it's using. Unless I'm missing something, the DDM is usually defined as the first passage time of a two-boundary Wiener process, and the likelihood of a RT under that model involves solving an SDE which has no closed-form solution. However, moments of the DDM RT distribution are available, which is what enables approaches that use them to approximately solve for parameters (this includes the E-Z Diffusion model of Wagenmakers et al., the current contribution (as far as I can tell), as well as the related work of Shvartsman et al. 2023 (arxiv:2306.06296). I think using E-Z diffusion type approaches is great, I just think the paper shouldn't represent them as the full DDM. Also note that the E-Z diffusion line of work provides for closed-form estimation of nondecision time (by estimating drift (i.e. value difference) based on response time variance, and then backing into the response time mean from there -- this might help address the issue of unknown nondecision time identified by the authors. 
* Related to the above, moving away from the simplifying assumption of E-Z diffusion would introduce the concern of other unknown DDM parameters such as drift variability, nonsymmetrical initial conditions, etc.
questions: questions: * Why should choice-only would ever do better asymptotically. L186 makes this claim w.r.t. section 3.2 but I don't see it in the figure -- the orange lines (dashed or solid) seem always above the gray lines. 
* For figure 2, if possible it should be moved below section 3.2., so that it can be interpreted after the asymptotic min-weight of both estimators has been discussed. This is also the part of the submission which is most confusing -- mental calisthenics are needed to map from asymptotic min-weight to the variance of theta estimates and therefore to the influence of observations at various value differences. Another editing / clarification pass could be helpful. 

Additional notes: 
* All figure axis tick labels are very tiny, should ideally be larger. In addition, adding color legends in fig 4 (outside the caption) would help with readability.
limitations: limitations: Discussed and addressed, even with additional analyses (and as noted above, the limitation regarding nondecision time might be possible to work around, though I could be missing something there).
paper_summary: paper_summary: The paper studies linear bandit preference learning, where binary preference data have been augmented with response times. A joint model for choice and response time falls from setting the linear preference model as the drift parameter in a drift-diffusion model. Experiments and theoretical analysis show that including response time in the model increases the value of "easy" responses, and thus improves bandit performance.
strength_weakness: strength_weakness: The paper has one significant weakness: all of the experiments use simulated response times, simulated from the model developed in the paper. We thus don't get a sense for the "real-world" performance of the method, where response times will not adhere precisely to the DDM model. I agree with the paper that response time is easy to collect alongside binary preferences. Surely there are some datasets available that include actual human response times? Real human response times are the big missing piece of the experimental evaluation.
questions: questions: On the topic of real human response time data, should we expect the model to be robust to lapses, or would some sort of data pre-processing be necessary to remove those?
limitations: limitations: Addressed.
paper_summary: paper_summary: The paper explores the interactive preference learning problem.
Traditional binary choice feedback is limited in conveying preference strength. 
The authors propose leveraging human response time, which inversely correlates with preference strength, as additional information. 
They adopt the difference-based Drift-Diffusion Model with linear human utility functions and propose a computationally efficient linear utility estimator that incorporates human response times.
Discussions about the proposed estimator and traditional estimators relying on the logistic regression are provided.
The authors also integrate the proposed estimator into the Generalized Successive Elimination(GME) algorithm for the fixed-budget best arm identification problem.
The proposed method is evaluated on both synthetic and real-world datasets, demonstrating the effectiveness of incorporating human response times for essay queries.
strength_weakness: strength_weakness: - Although the authors provide the asymptotic analysis for their estimators, they do not provide a theoretical analysis for the proposed algorithm.
- As mentioned in the paper, when the barrier $a$ is small, incorporating response times may not improve estimation performance. Since the parameter $a$ is also unknown, like $\theta^{*}$, learners may not know whether to incorporate response times. It would be beneficial if the authors could have a discussion about the practical choice of the estimator under these conditions.
questions: questions: Please refer to the weaknesses mentioned above.
limitations: limitations: Limitations are adequately discussed in the paper.

Comments: comment: I appreciate the clarifications regarding EZ-DDM and the asymptotic behavior of the choice-only strategy. I recommend that the authors try to make room for both clarifications (the distinction between the paper's approach and EZ-DDM's estimators, and the limiting case where response times don't help) in the main text or at least the supplement, alongside the other minor reorganizations. 

I'm happy to see that the other reviewers agree that this is a good paper. I will add more and say that it's the most straightforwardly applicable approach to using response times in preference learning that I've seen (without requiring MCMC sampling, likelihood approximations, or funky numerics), and the only one with any sort of non-vacuous theoretical guarantees. As such, it's the most likely to have impact more broadly (e.g. on current "hot" areas like RLHF). I have raised my rating by a point accordingly, and hope to see it at the conference.
Thank you for the additional experiments and analysis. I will raise my score as a result and hope that the paper is accepted. There was some related work just published at UAI that may be worth mentioning: Shvartsman et al. "Response time improves Gaussian process models for perception and preferences", https://openreview.net/forum?id=oUZ5JweNRc . It's essentially hooking a DDM model up into a Gaussian process bandit. Similar in concept though the mechanics and application area are of course quite different.
Thank you for your detailed responses. I have updated my confidence score to 3 and intend to maintain my rating.
The reviewer thanks the authors for addressing their concerns. The reviewer will raise the score accordingly.