Title: Improving the Knowledge Gradient Algorithm
Venue: NeurIPS 2023
Accepted: yes
Keywords: best arm identification

knowledge gradient

asymptotic optimality

convergence rate
Abstract: The knowledge gradient (KG) algorithm is a popular policy for the best arm identification (BAI) problem. It is built on the simple idea of always choosing the measurement that yields the greatest expected one-step improvement in the estimate of the best mean of the arms. In this research, we show that this policy has limitations, causing the algorithm not asymptotically optimal. We next provide a remedy for it, by following the manner of one-step look ahead of KG, but instead choosing the measurement that yields the greatest one-step improvement in the probability of selecting the best arm. The new policy is called improved knowledge gradient (iKG). iKG can be shown to be asymptotically optimal. In addition, we show that compared to KG, it is easier to extend iKG to variant problems of BAI, with the $\epsilon$-good arm identification and feasible arm identification as two examples. The superior performances of iKG on these problems are further demonstrated using numerical examples.

Improving the Knowledge Gradient Algorithm

Abstract

The knowledge gradient (KG) algorithm is a popular policy for the best arm identification (BAI) problem. It is built on the simple idea of always choosing the measurement that yields the greatest expected one-step improvement in the estimate of the best mean of the arms. In this research, we show that this policy has limitations, causing the algorithm not asymptotically optimal. We next provide a remedy for it, by following the manner of one-step look ahead of KG, but instead choosing the measurement that yields the greatest one-step improvement in the probability of selecting the best arm. The new policy is called improved knowledge gradient (iKG). iKG can be shown to be asymptotically optimal. In addition, we show that compared to KG, it is easier to extend iKG to variant problems of BAI, with the ϵ -good arm identification and feasible arm identification as two examples. The superior performances of iKG on these problems are further demonstrated using numerical examples.

1 Introduction

The best arm identification (BAI) is a sequential decision problem where in each stage, the agent pulls one out of k given arms and observes a noisy sample of the chosen arm. At the end of the sampling stage, the agent needs to select the arm that is believed to be the best according to the samples. In this research, we let the best arm be the one with the largest mean. BAI is a useful abstraction of issues faced in many practical settings [1, 2] and has been widely studied in the machine learning community [3, 4]. Since in practical problems, the target arm(s) (to be identified) is not necessarily the best arm, some variant models of BAI have also been proposed in the literature, e.g., topm arm identification [5, 6], Pareto front identification [7], ϵ -good arm identification [8], feasible arm identification [9, 10], etc.

In this research, we focus on the fixed-budget BAI, in which the total number of samples (budget) is fixed and known by the agent. The goal is to correctly identify the best arm when the budget is used up. To solve this problem, many methods have been proposed, e.g., successive rejects (SR) [4], expected improvements (EI) [11], top-two sampling [12, 13], knowledge gradient (KG) [14, 15], optimal computing budget allocation (OCBA)[16, 17, 18], etc. Among these methods, KG has been prevailing. It was first proposed in [19] and further analyzed in [20, 21]. It is built on the simple idea of always pulling the arm that yields the greatest expected one-step improvement in the estimate of the best mean of the arms. This improvement measure is analytical, making the algorithm easily implementable. KG often offers reasonable empirical performances and has been successfully applied in a number of real applications [22, 23].

However, we observe that this definition of KG has limitations, causing the algorithm not asymptotically optimal. Here by not being asymptotically optimal, we mean that the KG algorithm is not rate optimal, in the sense that the probability of the best arm being falsely selected based on the posterior means of the k arms does not converge to zero at the fastest possible rate. This is resulted from KG allocating too few samples to the best arm and excessive samples to the remaining arms. Note that Frazier et al. [20] claimed that KG is 'asymptotically optimal', but in their context, 'asymptotically optimal' is consistent, i.e., all the arms will be infinitely sampled as the round n →∞ , so that the best arm will be correctly selected eventually. This is a relatively weak result for BAI algorithms (the simple equal allocation is also consistent). In this paper, asymptotically optimal refers to rate optimal.

Contributions. We propose a new policy that can overcome this limitation of KG. The new policy follows the manner of one-step look ahead of KG, but pulls the arm that yields the greatest one-step improvement in the probability of selecting the best arm. We call it improved knowledge gradient (iKG) and show that it is asymptotically optimal. This policy is originated from the thought of looking at whether the best arm has been selected at the end of sampling, instead of looking at the extent that the mean of the selected arm has been maximized. Although both ways can identify the best arm, it turns out that the algorithms developed from them are significantly different in the rates of posterior convergence. Another advantage of iKG over KG is that iKG is more general and can be more easily extended to variant problems of BAI. We use ϵ -good arm identification and feasible arm identification as examples, develop algorithms for them using the idea of iKG and establish asymptotic optimality for the algorithms.

This paper is conceptually similar to [12] which improves the EI algorithm for BAI. However, for EI, sampling ratios of any two arms in the non-best set are already asymptotically optimal. One only needs to introduce a parameter β to balance the probabilities of sampling the best arm and the non-best set without changing the sampling policy within the non-best set to further improve EI. For KG, sampling ratios are not asymptotically optimal for any two out of the k arms. It requires a fundamental change on the sampling policy that influences the sampling rates of all the arms to improve KG. Moreover, the improved rate of posterior convergence of EI in [12] still depends on β which is not necessarily optimal, while we can show that this rate of iKG is optimal.

2 Knowledge Gradient and its Limitations

In this section, we review KG and discuss its limitations. Suppose there are k arms in BAI. In each round t , the agent chooses any arm i to pull and obtains a noisy sample X t +1 ,i . After n rounds, the agent needs to select an arm that he/she believes to be the best. Under the framework of the KG algorithm, X t +1 ,i 's are assumed to be independent across different rounds t and arms i and following the normal distribution N ( µ i , σ 2 i ) with unknown means µ i and known variances σ 2 i . The best arm is assumed to be unique. Without loss of generality, let µ ⟨ 1 ⟩ > µ ⟨ 2 ⟩ ≥ . . . ≥ µ ⟨ k ⟩ , where ⟨ i ⟩ indicates the arm with i -th largest mean.

The KG algorithm can be derived from a dynamic programming (DP) formulation of BAI. The state space S consists of all the possible posterior means and variances of the arms, denoted as S ≜ R k × (0 , ∞ ) k . State S t in round t can be written as S t = ( µ t, 1 , µ t, 2 , . . . , µ t,k , σ 2 t, 1 , σ 2 t, 2 , . . . , σ 2 t,k ) ⊤ . In the Bayesian model, the unknown mean µ i is treated as random and let θ i be the random variable following its posterior distribution. We adopt normal distribution priors N ( µ 0 ,i , σ 2 0 ,i ) . With samples of the arms, we can compute their posterior distributions, which are still normal N ( µ t,i , σ 2 t,i ) in round t by conjugacy. The posterior mean and variance of arm i are

̸

̸

In this paper, we adopt a non-informative prior for each arm i ∈ A , i.e., µ 0 ,i = 0 and σ 0 ,i = ∞ . Denote the action space as A ≜ { 1 , 2 , . . . , k } and transition function as T ≜ S × A × S → S . Suppose θ t,i is a random variable following the posterior distribution N ( µ t,i , σ 2 i ) of arm i . Then, the state transition can be written as S t +1 = T ( S t , i, θ t,i ) . Let π be the sampling policy that guides the agent to pull arm I t in round t and Π be the set of sampling policies π = ( I 0 , I 1 , . . . , I n -1 ) adapted to the filtration I 0 , X 1 ,I 0 , . . . , I t -1 , X t,I t -1 . After n rounds, the estimated best arm I ∗ n is selected

and a terminal reward v n ( S n ) is received. We can write our objective as

The DP principle implies that the value function in round 0 ≤ t < n can be computed recursively by

and the DP principle tells us that any policy satisfying

is optimal. However, the optimal policy is basically intractable unless for problems with very small scales, known as the 'curse of dimensionality'.

On the other hand, note that except the terminal reward v n ( S n ) , this problem has no rewards in the other rounds, so we can restructure v n ( S n ) as a telescoping sequence

Thus, v n ( S n ) can be treated as the cumulation of multiple one-step improvements v n ( S l ) -v n ( S l -1 ) , l = t +1 , . . . , n . A class of one-step look ahead algorithms iteratively pull the arm that maximizes the expectation of the one-step improvement on the value function

These algorithms are not optimal in general unless there is only one round left, i.e., n = t +1 .

The KG algorithm falls in this class. It sets the terminal reward as v n ( S n ) = µ I ∗ n . With this reward, the one-step improvement in (3) becomes

̸

and in each round, the KG algorithm pulls the arm I t ( S t ) ∈ argmax i ∈ A KG t,i .

Algorithm 1: KG Algorithm

Input: k ≥ 2 , n 1 Collect n 0 samples for each arm i ; 2 while t < n do 3 Compute KG t,i and set I t = argmax i ∈ A KG t,i ; 4 Play I t ; 5 Update µ t +1 ,i and σ t +1 ,i ; Output: I ∗ n

We next characterize for the KG algorithm the rate of posterior convergence of 1 -P { I ∗ n = I ∗ } , the probability that the best arm is falsely selected.

Proposition 1. Let c ⟨ i ⟩ = ( µ ⟨ 1 ⟩ -µ ⟨ i ⟩ ) /σ ⟨ i ⟩ ( µ ⟨ 1 ⟩ -µ ⟨ 2 ⟩ ) /σ ⟨ 2 ⟩ , i = 2 , ..., k . For the KG algorithm,

̸

where

̸

̸

We observe that Γ KG is not optimal. To make this point, Proposition 2 gives an example that Γ KG is no better than this rate of the TTEI algorithm [12] when the parameter β (probability of sampling the best arm) of TTEI is set to some suboptimal value.

We define the Q-factors as

̸

Proposition 2. For the TTEI algorithm [12], the rate of posterior convergence of 1 -P { I ∗ n = I ∗ } exists and is denoted as Γ TTEI . Let its probability of sampling the best arm β = ( σ ⟨ 2 ⟩ /σ ⟨ 1 ⟩ ∑ i =1 1 /c ⟨ i ⟩ + 1) -1 . We have Γ KG ≤ Γ TTEI .

̸

̸

According to the proof of Proposition 2, there are configurations of the BAI problem leading to Γ KG < Γ TTEI , i.e., Γ KG is not optimal. In fact, with β = ( σ ⟨ 2 ⟩ /σ ⟨ 1 ⟩ ∑ i =1 1 /c ⟨ i ⟩ +1) -1 , Γ KG = Γ TTEI is achieved only in some special cases, e.g., when k = 2 .

3 Improved Knowledge Gradient

In this section, we propose an improved knowledge gradient (iKG) algorithm. We still follow the manner of one-step look ahead of KG, but set the terminal reward of problem (2) as v n ( S n ) = 1 { I ∗ n = I ∗ } . That is, for the goal of identifying the best arm, we reward the selected arm by a 0-1 quantity showing whether this arm is the best arm, instead of the mean of this arm (as in KG).

̸

̸

However, the probability P { ⋃ i = I ∗ n ( θ i > θ I ∗ n ) } in (4) does not have an analytical expression. To facilitate the algorithm implementation and analysis, we adopt an approximation to it using the Bonferroni inequality [24]:

̸

and E [ v n ( S n )] can be approximately computed as

̸

̸

Note that the Bonferroni inequality has been adopted as an approximation of the probability of correct selection in the literature for development of BAI algorithms [16]. For our purpose, we can show that the use of this approximation still makes the resulting algorithm asymptotically optimal and empirically superior.

Algorithm 2: iKG Algorithm

Input: k ≥ 2 , n 1 Collect n 0 samples for each arm i ; 2 while t < n do 3 Compute iKG t,i and set I t = argmax i ∈ A iKG t,i ; 4 Play I t ; 5 Update µ t +1 ,i , σ t +1 ,i and I ∗ t +1 ; Output: I ∗ n

Let iKG t,i be the one-step improvement in (3) with I ∗ t treated as unchanged after one more sample and E [ v n ( S n )] approximated by (5). We have the following proposition to compute iKG t,i . The iKG algorithm pulls the arm with the largest iKG t,i in each round.

Proposition 3. With the definition of iKG t,i above, we have

̸

̸

̸

̸

Both KG and iKG are greedy algorithms that look at the improvement only one-step ahead. The essential difference between them is on the reward they use for the event of best arm identification. For KG, it is the mean of the arm selected, while for iKG, it is a 0-1 quantity showing whether the best arm is selected. It is interesting to note that the choice between these two rewards has been discussed in the control community for optimization of complex systems, known as cardinal optimization (similar to KG) vs. ordinal optimization (similar to iKG) [25], with the discussion result in line with this research, indicating that ordinal optimization has advantages over cardinal optimization in the convergence rates of the optimization algorithms [26].

and w i is the sampling rate of arm i satisfying

In addition, for any BAI algorithms,

Theorem 1 shows that the rate of posterior convergence Γ iKG of the iKG algorithm is the fastest possible. We still use TTEI as an example. This theorem indicates that Γ TTEI ≤ Γ iKG for any β ∈ (0 , 1) and the equality holds only when β is set to β ∗ , where β ∗ is the optimal value of β and is typically unknown.

4 Variant Problems of BAI

Another advantage of iKG over KG is that iKG is more general, in the sense that it can be easily extended to solve variant problems of BAI. In the variants, the target arms to be identified are not the single best arm, but no matter how the target arms are defined, one can always look at the event that whether these arms are correctly identified at the end of sampling and investigate the probability of this event to develop iKG and the algorithm. In contrast, it is difficult to extend KG to identify arms that cannot be found through optimizing means of these (and/or other) arms. In this section, we extend iKG to two BAI variants: ϵ -good arm identification [8] and feasible arm identification [10]. We develop algorithms for them and establish their asymptotic optimality. Note that in these two variant problems, the target arms need to be found by comparing their means with some fixed values. In such cases, the idea of KG is not straightforward.

4.1 ϵ -Good Arm Identification

We follow the notation in Sections 2 and 3. For the k arms, suppose µ ⟨ 1 ⟩ ≥ µ ⟨ 2 ⟩ ≥ . . . ≥ µ ⟨ k ⟩ . Given ϵ > 0 , the ϵ -good arm identification problem aims to find all the arms i with µ ⟨ i ⟩ > µ ⟨ 1 ⟩ -ϵ , i.e., all the arms whose means are close enough to the best ( ϵ -good). Assume that no arms have means lying on µ ⟨ 1 ⟩ -ϵ . Denote the set of ϵ -good arms as G ϵ and the estimated set of ϵ -good arms after n rounds as G ϵ n . We set the terminal reward v n ( S n ) = 1 { G ϵ n = G ϵ } , i.e., whether the set G ϵ is correctly selected. Then, E [ v n ( S n )] = P { G ϵ n = G ϵ } , where

Again, applying the Bonferroni inequality,

̸

̸

Let iKG ϵ t,i be the one-step improvement in (3) with I ∗ t treated as unchanged after one more sample and E [ v n ( S n )] approximated by the right-hand side of (9). We have the following proposition to compute iKG ϵ t,i .

Proposition 4. With the definition of iKG ϵ t,i above, we have

̸

̸

Algorithm 3: iKGϵ Algorithm ( ϵ -good Arm Identification)

To identify the ϵ -good arms, the iKGϵ algorithm pulls the arm with the largest iKG ϵ t,i in each round. For this algorithm, we can show that the rate of posterior convergence of 1 -P { G ϵ n = G ϵ } is the fastest possible.

Theorem 2. For the iKGϵ algorithm, lim n →∞ -1 n log(1 -P { G ϵ n = G ϵ } ) = Γ ϵ , where

and w i is the sampling rate of arm i satisfying

In addition, for any ϵ -good arm identification algorithms,

4.2 Feasible Arm Identification

In the feasible arm identification, samples from pulling arms i are m -dimensional vectors X t +1 ,i = [ X t +1 ,i 1 , . . . , X t +1 ,im ] instead of scalars, where each dimension of the vector corresponds to some measure of the system performance and X t +1 ,ij is the observation associated with arm i and measure j . Suppose X t +1 ,ij 's follow the normal distribution with unknown means µ ij and known variances σ 2 ij . We impose constraints µ ij ≤ γ j on arms i = 1 , 2 , . . . , k and measures j = 1 , 2 , . . . , m . The goal of this problem is to find the set of feasible arms S 1 . Let the estimated set of feasible arms after n rounds be S 1 n and S 2 = A \ S 1 . We assume that X t +1 ,ij 's are independent across different rounds t and measures j , and µ ij 's do not lie on the constraint limits γ j . To facilitate the analysis, we also define for round t the set of measures E 1 t,i ≜ { j : µ t,ij ≤ γ j } satisfied by arm i and the set of measures E 2 t,i ≜ { j : µ t,ij > γ j } violated by arm i .

̸

̸

̸

Set the terminal reward v n ( S n ) = 1 {S 1 n = S 1 } , i.e., whether the set S 1 is correctly selected. Then, E [ v n ( S n )] = P {S 1 n = S 1 } , where

Applying the Bonferroni inequality,

The inequality holds because 0 < ∏ j ∈E 1 n,i P ( θ ij ≤ γ j ) ≤ 1 .

Let iKG F t,i be the one-step improvement in (3) with S 1 t , S 2 t and E 2 t,i treated as unchanged after one more sample and E [ v n ( S n )] approximated by the right-hand side of (13). We have the following proposition to compute iKG F t,i .

Proposition 5. With the definition of iKG F t,i above, we have

Algorithm 4: iKG-F Algorithm (Feasible Arm Identification)

Input: k ≥ 2 , n

1 Collect n 0 samples for each arm i ;

2 while t < n do

3 Compute iKG F t,i and set I t = argmax i ∈ A iKG F t,i ;

4 Play I t

;

5

Update

µ

t

+1

,i

,

σ

t

+1

,i

,

S

1

t

+1

,

S

2

t

+1

,

E

1

t

+1

,i

and

E

2

t

+1

,i

;

Output: S 1 n

To identify the feasible arms, the iKG-F algorithm pulls the arm with the largest iKG F t,i in each round. For this algorithm, we can show that the rate of posterior convergence of 1 -P {S 1 n = S 1 } is also the fastest possible.

Theorem 3. For the iKG-F algorithm, lim n →∞ -1 n log(1 -P {S 1 n = S 1 } ) = Γ F , where

and w i is the sampling rate of arm i satisfying

̸

In addition, for any feasible arm identification algorithms

5 Numerical Experiments

In this section, we show empirical performances of the iKG, iKGϵ and iKG-F algorithms on synthetic and real-world examples. For the best arm identification problem, we compare iKG with the following algorithms.

Expected Improvement (EI) [11]. This is another common strategy for BAI. In each round, it pulls the arm offering the maximal expected improvement over the current estimate of the best mean of the arms.

Top-Two Expected Improvement (TTEI) [12]. This is a modification of the EI algorithm by introducing a parameter β to control the probabilities of sampling the best arm and the non-best set. We set the parameter β in TTEI as its default value 1 / 2 .

Knowledge Gradient. This is the algorithm under study in this research.

For the ϵ -good arm identification problem, we compare iKGϵ with the following algorithms.

APT Algorithm [27]. It is a fixed-budget algorithm for identifying the arms whose means are above a given threshold. We set the input tolerance parameter as 0 . 0001 and the threshold as the posterior mean of the estimated best arm minus ϵ .

( ST ) 2 Algorithm [8]. It is a fixed-confidence algorithm for ϵ -good arm identification. It pulls three arms in each round, the estimated best arm, one arm above the threshold and one arm below the threshold. We set the input tolerance parameter as 0 . 0001 and γ = 0 .

For the feasible arm identification problem, we compare iKG-F with the following algorithms.

MD-UCBE Algorithm [10]. This is a fixed-budget algorithm for feasible arm identification based on the upper confidence bound. We set the input tolerance parameter as 0 . 0001 and hyperparameter a = 25 36 n -k H , where H is a constant that can be computed. Katz-Samuels and Scott [10] showed that with a = 25 36 n -k H , the performance of MD-UCBE is nearly optimal.

MD-SAR Algorithm [10]. This is a fixed-budget algorithm for feasible arm identification based on successive accepts and rejects. We set the input tolerance parameter as 0 . 0001 .

In addition, iKG, iKGϵ and iKG-F will be compared with the equal allocation, where each arm is simply played with the same number of rounds. It is a naive method and is often used as a benchmark against which improvements might be measured.

The examples for testing include three synthetic examples, called Examples 1-3, and three real examples, namely the Dose-Finding Problem, Drug Selection Problem, and Caption Selection Problem. For Example 1-3 and the Dose-Finding problem, samples of the arms are two-dimensional. We call the measures of them measures 1 and 2. When the examples are tested for the best arm identification and ϵ -good identification, only measure 1 will be used for identifying good/best arms. When the examples are tested for the feasible arm identification, both measures will be used for feasibility detection. For the Drug Selection and Caption Selection problems, samples of the arms are one-dimensional. They are tested for the best arm identification, ϵ -good identification and feasible arm identification.

Synthetic Datasets . We consider three examples, all containing ten arms.

Example 1. The means in measure 1 of the ten arms are 0 . 1927 , 0 . 6438 , 3 . 0594 , 3 . 0220 , 1 . 3753 , 1 . 4215 , 0 . 9108 , 1 . 0126 , 0 . 1119 and 1 . 8808 , and the means in measure 2 of the ten arms are 0 . 4350 , 0 . 7240 , 1 . 1566 , 0 . 8560 , 3 . 4712 , 0 . 8248 , 3 . 8797 , 1 . 9819 , 3 . 2431 and 1 . 4315 , all of which are uniformly generated in (0 , 4) . Samples of the arms are corrupted by normal noises N (0 , 1) . The best arm is arm 3 and 0 . 1 -good arms are arms 3 and 4. For the feasible arm identification, we choose arms with means in both measures less than 2 . Then the feasible arms are arms 1 , 2 , 6 , 8 and 10 .

Example 2. We keep the setting of Example 1. Distributions of the noises for arms 1-5 are changed to N (0 , 4) .

Example 3. Consider functions y 1 ( x ) = -0 . 05 x 2 , y 2 ( x ) = -0 . 06(7 -x ) and y 3 ( x ) = 0 . 06( x -6) . The means in measure 1 of the ten arms are y 1 ( x ) with x = 1 , 2 , . . . , 10 . The means in measure 2 of the ten arms are y 2 ( x ) with x = 1 , . . . , 6 and y 3 ( x ) with x = 7 , . . . , 10 . Noises follow the normal distribution N (0 , 1) . The best arm is arm 1 and 0 . 5 -good arms are arms 1-3. For the feasible arm identification, we choose arms with means in measure 1 greater than -0 . 5 and means in measure 2 less than 0 . The feasible arms are arms 1-3.

Dose-Finding Problem . We use the data in [28] (see ACR 50 in week 16 ) for treating rheumatoid arthritis by the drug secukinumab. There are four dosage levels, 25 mg, 75 mg, 150 mg, and 300 mg, and a placebo, which are treated as five arms. We develop a simulation model based on the dataset. Each arm is associated with two performance measures: the probability of the drug being effective and the probability of the drug causing infections. The means of the five arms are µ 1 = (0 . 151 , 0 . 259) , µ 2 = (0 . 184 , 0 . 184) , µ 3 = (0 . 209 , 0 . 209) , µ 4 = (0 . 171 , 0 . 293) and µ 5 = (0 . 06 , 0 . 16) . Samples of each arm are corrupted by normal noises N (0 , 0 . 25) . The best arm is arm 3 and the 0.03-good arms are arms 2 and 3. For the feasible arm identification, we find the arms whose probability of being effective is larger than 0 . 18 and the probability of causing infections is less than 0 . 25 . The feasible arms are arms 2 and 3.

Drug Selection Problem . We consider five contraceptive alternatives based on the Drug Review Dataset (https://doi.org/10.24432/C5SK5S): Ethinyl estradiol / levonorgest, Ethinyl estradiol / norethindro, Ethinyl estradiol / norgestimat, Etonogestrel and Nexplanon, which can be treated as five arms. The dataset provides user reviews on the five drugs along with related conditions and ratings reflecting overall user satisfaction. We set the means of the five arms as µ 1 = 5 . 8676 , µ 2 = 5 . 6469 , µ 3 = 5 . 8765 , µ 4 = 5 . 8298 and µ 5 = 5 . 6332 , and the variances of the five arms as σ 2 1 = 3 . 2756 , σ 2 2 = 3 . 4171 , σ 2 3 = 3 . 2727 , σ 2 4 = 3 . 3198 and σ 2 5 = 3 . 3251 , all calculated by the data. When this example is used for the best arm identification and ϵ -good arm identification, the best arm (with the highest user satisfaction) and 0.003-good arm are both arm 3 (Ethinyl estradiol / norgestimat). When this example is used for feasible arm identification, we will select the drugs whose ratings are over 5 . 6 , and the feasible arms are arm 1 (Ethinyl estradiol / levonorgest), arm 2 (Ethinyl estradiol / norethindro), arm 3 (Ethinyl estradiol / norgestimat), arm 4 (Etonogestrel) and arm 5 (Nexplanon).

Caption Selection Problem . We aim to select good captions based on the New Yorker Cartoon Caption Contest Dataset (https://nextml.github.io/caption-contest-data/). In the contests, each caption can be treated as an arm. The dataset provides the mean and variance of each arm, which can be used to set up our experiments. We will test contests 853 (Caption 853) and 854 (Caption 854).

In Caption 853, we randomly select ten captions as arms. We set the means of the ten arms as µ 1 = 1 . 1400 , µ 2 = 1 . 0779 , µ 3 = 1 . 4160 , µ 4 = 1 . 0779 , µ 5 = 1 . 1081 , µ 6 = 1 . 1467 , µ 7 = 1 . 1333 , µ 8 = 1 . 1075 , µ 9 = 1 . 1026 and µ 10 = 1 . 4900 , and the variances of the arms as σ 2 1 = 0 . 1418 , σ 2 2 = 0 . 0991 , σ 2 3 = 0 . 4871 , σ 2 4 = 0 . 0728 , σ 2 5 = 0 . 0977 , σ 2 6 = 0 . 1809 , σ 2 7 = 0 . 1843 , σ 2 8 = 0 . 0970 , σ 2 9 = 0 . 0932 and σ 2 10 = 0 . 4843 , which are all calculated by the data. When this example is used for the best arm identification, the best arm (with the highest funniness score) is arm 10. When this example is used for ϵ -good arm identification, the 0.1-good arms are arms 3 and 10. When this example is used for feasible arm identification, we will select the captions whose funniness scores are over 1.4, and the feasible arms are arms 3 and 10.

In Caption 854, we also randomly select ten captions as arms. We set the means of the ten arms as µ 1 = 1 . 1986 , µ 2 = 1 . 1890 , µ 3 = 1 . 1400 , µ 4 = 1 . 2621 , µ 5 = 1 . 1544 , µ 6 = 1 . 0339 , µ 7 = 1 . 1349 , µ 8 = 1 . 2786 , µ 9 = 1 . 1765 and µ 10 = 1 . 1367 , and the variances of the arms as σ 2 1 = 0 . 1879 ,

σ 2 2 = 0 . 2279 , σ 2 3 = 0 . 1346 , σ 2 4 = 0 . 3186 , σ 2 5 = 0 . 1314 , σ 2 6 = 0 . 0330 , σ 2 7 = 0 . 1337 , σ 2 8 = 0 . 3167 , σ 2 9 = 0 . 1858 and σ 2 10 = 0 . 1478 , all calculated by the data. When this example is used for the best arm identification, the best arm is arm 8. When this example is used for ϵ -good arm identification, the 0.05-good arms are arms 4 and 8. When this example is used for feasible arm identification, we will select the captions whose funniness scores are over 1.25, and the feasible arms are arms 4 and 8.

For the tested algorithms, probabilities of false selection (PFS) are obtained based on the average of 100 macro-replications. Tables 1-3 show the PFS of the algorithms under some fixed sample sizes (additional numerical results about the PFS and sampling rates of the tested algorithms are provided in the Supplement). The proposed iKG, iKGϵ and iKG-F perform the best. For the best arm identification, EI tends to allocate too many samples to the estimated best arm, leading to insufficient exploration in the remaining arms, while KG tends to allocate too few samples to the estimated best arm, leading to excessive exploration in the remaining arms. TTEI always allocates approximately one-half budget to the estimated best arm when β = 1 / 2 , leading to the budget not being the best utilized. For the ϵ -good identification, APT and ( ST ) 2 are inferior because the former insufficiently pulls the estimate best arm, leading to inaccurate estimates of the threshold, while the latter falls in the fixed-confidence regime that focuses on making guarantees on the probability of false selection instead of minimizing it. For the feasible arm identification, both MD-UCBE and MD-SAR allocate too many samples to the arms near the constraint limits. For the three problems, equal allocation performs the worst in general, because it does not have any efficient sampling mechanisms for identifying the target arms in these problems.

6 Conclusion

This paper studies the knowledge gradient (KG), a popular policy for the best arm identification (BAI). We observe that the KG algorithm is not asymptotically optimal, and then propose a remedy for it. The new policy follows KG's manner of one-step look ahead, but utilizes different evidence to identify the best arm. We call it improved knowledge gradient (iKG) and show that it is asymptotically optimal. Another advantage of iKG is that it can be easily extended to variant problems of BAI. We use ϵ -good arm identification and feasible arm identification as two examples for algorithm development and analysis. The superior performances of iKG on BAI and the two variants are further demonstrated using numerical examples.

References

D. A. Berry. Modified two-armed bandit strategies for certain clinical trials. Journal of the American Statistical Association , 73(362):339-345, 1978.

A. Gilotte, C. Calauzènes, T. Nedelec, A. Abraham, and S. Dollé. Offline a/b testing for recommender systems. In Proceedings of the 11th ACM International Conference on Web Search and Data Mining , pages 198-206, 2018.

E. Even-Dar, S. Mannor, Y. Mansour, and S. Mahadevan. Action elimination and stopping conditions for the multi-armed bandit and reinforcement learning problems. Journal of Machine Learning Research , 7(6), 2006.

J.-Y. Audibert, S. Bubeck, and R. Munos. Best arm identification in multi-armed bandits. In 23rd Conference on Learning Theory , pages 41-53, 2010.

S. Bubeck, T. Wang, and N. Viswanathan. Multiple identifications in multi-armed bandits. In International Conference on Machine Learning , pages 258-265. PMLR, 2013.

H. Xiao and S. Gao. Simulation budget allocation for selecting the top-m designs with input uncertainty. IEEE Transactions on Automatic Control , 63(9):3127-3134, 2018.

P. Auer, C. K. Chiang, R. Ortner, and M. Drugan. Pareto front identification from stochastic bandit feedback. In Artificial Intelligence and Statistics , pages 939-947. PMLR, 2016.

B. Mason, L. Jain, A. Tripathy, and R. Nowak. Finding all ϵ -good arms in stochastic bandits. Advances in Neural Information Processing Systems , 33:20707-20718, 2020.

S. Gao and W. Chen. Efficient feasibility determination with multiple performance measure constraints. IEEE Transactions on Automatic Control , 62(1):113-122, 2016.

J. Katz-Samuels and C. Scott. Feasible arm identification. In International Conference on Machine Learning , pages 2535-2543. PMLR, 2018.

S. E. Chick, J. Branke, and C. Schmidt. Sequential sampling to myopically maximize the expected value of information. INFORMS Journal on Computing , 22(1):71-80, 2010.

C. Qin, D. Klabjan, and D. Russo. Improving the expected improvement algorithm. In Proceedings of the 31st International Conference on Neural Information Processing Systems , pages 5387-5397, 2017.

D. Russo. Simple Bayesian algorithms for best arm identification. Operations Research , 68(6):1625-1647, 2020.

I. O. Ryzhov, P. I. Frazier, and W. B. Powell. On the robustness of a one-period look-ahead policy in multi-armed bandit problems. Procedia Computer Science , 1(1):1635-1644, 2010.

Y. Li and S. Gao. On the finite-time performance of the knowledge gradient algorithm. In International Conference on Machine Learning , pages 12741-12764. PMLR, 2022.

C. H. Chen, J. Lin, E. Yücesan, and S. E. Chick. Simulation budget allocation for further enhancing the efficiency of ordinal optimization. Discrete Event Dynamic Systems , 10:251-270, 2000.

S. Gao, W. Chen, and L. Shi. A new budget allocation framework for the expected opportunity cost. Operations Research , 65(3):787-803, 2017.

Y. Li and S. Gao. Convergence rate analysis for optimal computing budget allocation algorithms. Automatica , 153:111042, 2023.

S. S. Gupta and K. J. Miescke. Bayesian look ahead one stage sampling allocations for selecting the largest normal mean. Statistical Papers , 35(1):169-177, 1994.

P. I. Frazier, W. B. Powell, and S. Dayanik. A knowledge-gradient policy for sequential information collection. SIAM Journal on Control and Optimization , 47(5):2410-2439, 2008.

P. Frazier, W. Powell, and S. Dayanik. The knowledge-gradient policy for correlated normal beliefs. INFORMS Journal on Computing , 21(4):599-613, 2009.

C. Schoppe. Wind and pumped-hydro power storage: Determining optimal commitment policies with knowledge gradient non-parametric estimation. B.Sc. Thesis at Princeton University , 2010.

D. M. Negoescu, P. I. Frazier, and W. B. Powell. The knowledge-gradient algorithm for sequencing experiments in drug discovery. INFORMS Journal on Computing , 23(3):346-363, 2011.

J. Galambos. Bonferroni inequalities. The Annals of Probability , pages 577-581, 1977.

Y.-C. Ho, S. Sreenivas, and P. Vakili. Ordinal optimization of deds. Discrete Event Dynamic Systems , 2(1):61-88, 1992.

Y.-C. Ho. An explanation of ordinal optimization: Soft computing for hard problems. Information Sciences , 113(3-4):169-192, 1999.

A. Locatelli, M. Gutzeit, and A. Carpentier. An optimal algorithm for the thresholding bandit problem. In International Conference on Machine Learning , pages 1690-1698. PMLR, 2016.

M. C. Genovese, P. Durez, H. B. Richards, J. Supronik, E. Dokoupilova, V. Mazurov, J. A. Aelion, S. H. Lee, C. E. Codding, H. Kellner, et al. Efficacy and safety of secukinumab in patients with rheumatoid arthritis: a phase ii, dose-finding, double-blind, randomised, placebo controlled study. Annals of the Rheumatic Diseases , 72(6):863-869, 2013.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper tackles the problem of best arm identification (BAI). The paper introduces \"improved knowledge gradient (iKG)\" that can be shown to be asymptotically optimal. Some numerical results are also provided\n\nSTRENGTHS:\nThe contribution of the paper has an interesting scope of showing limitations of an existing algorithm for best-arm identification (BAI), providing a variant along with theoretical analysis and numerical experiments. The paper also discussed variant problems of BAI making the scope quite large within the multi-arm bandit setting.\nThe paper seem to me scientifically sound and is overall well-written.\n\nWEAKNESSES:\n(minor) Some elements in the formalization, some notations and minor typos could possibly be improved.\n\nDisclaimer: I can't fully assess the novelty as I'm not an expert in the field of the best arm identification problem and I didn't check in details all the proofs.\n\nQUESTIONS:\n- In the beginning of Section 2, $i$ refers to the arms and then it is mentioned that $\\sigma_i$ are known variances but then in Equation 1, the sigma is estimated. Can you clarify this?\n\nMinor additional comment:\n- There are few typos/english mistakes: e.g. \"this policy has limitations, causing the algorithm not asymptotically optimal\", \"These algorithms are not optimal in general except there is only one round left\".\n- I know that it is a usual notation to say something like \"$\\mu_1 > \\mu_2 \\ge . . . \\ge \\mu_k$\". But this feels strange as it would (naively seem to) imply that one already knows that the arm i=1 is the best. Is it possible to introduce another notation or do I miss something?\n- Line 70: $\\theta_i$ is defined as \"the random variable following *its* posterior distribution\". Is it the $\\mu$ only or is it for the whole state $S_t$.\n- Line 74: The transition function that depends on two states and one action and deterministically transition to a new state is not entirely clear to me from a formal perspective. $\\theta_{t,i}$ is defined as a random variable following some (posterior) distribution but it is then apparently also a \"state\"? Isn't it a probability distribution over the state space?\n\nLIMITATIONS:\nThe limitations are adequately discussed.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-12"}

comments: SUMMARY:
The paper focuses on the fixed-budget best arm identification (BAI) problem, in which an agent aims to correctly identify the best arm (one with the highest mean) among k given arms within a known fixed number of samples. An improved knowledge gradient (iKG) policy is proposed, which prioritizes the arm that maximizes the one-step improvement in the probability of selecting the best arm. The algorithm yields an asymptotically optimal solution, overcoming the limitations of the knowledge gradient (KG) policy that is not rate optimal. The paper also extends iKG to the problem of ε-good arm identification and feasible arm identification examples.

STRENGTHS:
The idea of considering probability instead of expectation is new in this context (even though it has already been used in other contexts). The analyses of extensions is also an added benefit.

WEAKNESSES:
The computational experiments are weak, in particular the choice (more precisely, creation) of datasets bags for more. In short, the experiments are mostly based on synthetic data and some of them come across as very far fetched. 

The concepts are very similar to TTEI which makes the contributions marginal.

QUESTIONS:
Why not comparing iKG with KG, for example in the 'one average' sense? I do understand that one is asymptotically optimal and the other not, but perhaps the cases when KG is not are very rare and thus the choice between iKG and KG would be a wash. 

The datasets are really contrived. Why not getting more real-world data and then tweaking it to fit the purpose?

LIMITATIONS:
NA
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2023-07-04

comments: SUMMARY:
The paper is concerned with the best arm identification (BAI) problem. The main contribution of the paper is the proposal of an improved version of the knowledge gradient algorithm which is also shown to be asymptotically optimal. Moreover, the proposed algorithm is extended to the variant of BAI problem and illustrated on certain applications.

STRENGTHS:
- The main message and contributions of the paper are clearly stated and demonstrated.  
- The advantage of the proposed algorithm is theoretically proven, by showing that it is asymptotically optimal. 
- By addressing several variants of the BAI problem, the versatility of the approach is shown, supported by theoretical results. 
- The impact of the proposed method on several applications are shown.

WEAKNESSES:
- The limitations of the proposed method is not discussed
- The range of the problems the proposed approach can solve might be narrow.

QUESTIONS:
- Can you describe the limitations of the proposed approach, and possibly directions for future research?

LIMITATIONS:
No
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2023-07-05

comments: SUMMARY:
The paper considers knowledge gradient (KG) algorithm and its limitations in the best arm identification (BAI) problem. The authors first demonstrate that the KG policy is not asymptotically optimal, highlighting its shortcomings, and then provide the improved knowledge gradient (iKG) policy. The key idea is to select measurements that maximize the one-step improvement in the probability of selecting the best arm, rather than the mean improvement. The paper establishes that iKG is asymptotically optimal, offering a remedy to the limitations of KG. The empirical evidence for the effectiveness of the proposed method is also provided.

STRENGTHS:
1. The theoretical analysis on the KG algorithm on its sub-optimality gap
2. This work proposed a novel improved KG algorithm with optimality guarantees
3. Discussion on the variant problems of BAI shows that the proposed iKG method is general and can be used in various scenarios.

WEAKNESSES:
1. Some notations are rather unclear, e.g., the definition of subscript $i$ in line 61 is not introduced until line 64, definition of $I_t$
2. What is the computational complexity comparing with KG? It is beneficial to discuss its relationship with the scale of the problem.

QUESTIONS:
1. How does the last inequality in Eqn. F.2 hold?

LIMITATIONS:
The author has addressed the limitations well.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-07

comments: SUMMARY:
The researchers tackle the problem of fixed-budget Best Arm Identification (BAI). Specifically, they observed that typical Knowledge Gradient approaches based on the policy of choosing the step with the best expected one step improvement are not asymptotically optimal. They define a new policy, improved Knowledge Gradient (iKG) that, instead, seeks to improve to create the best one-step improvement in the probability of selecting the best arm.

STRENGTHS:
iKG has more applicability than standard KG. They specifically show applications to epsilon-good arm identification and feasible arm identification, which KG cannot handle. They show the application of iKG on synthetic and real datasets and the iKG variants perform the best overall.

WEAKNESSES:
Although I can appreciate the complexity of the numerical examples and the breadth of comparison algorithms for iKG-e and iKG-F, I do not believe there are enough different datasets to talk about the performance generally. The authors should have chosen several datasets of varying natures to show general applicability. It is hard for me to gain a sense of whether these datasets are just lucky examples.

QUESTIONS:
Can you show comparison algorithms for several other datasets that can duplicate the type of performance seen here? Are there datasets where performance of iKG is worse or not as effective, and can you explain why?

LIMITATIONS:
The authors discuss the limitations of KG, but not of their techniques specifically. I think this could be accomplished by showing some contrary datasets or examples.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2023-07-07