Original: Batch Bayesian Optimization on Permutations using the Acquisition Weighted Kernel
Revised: Batch Bayesian Optimization on Permutations using the Acquisition Weighted Kernel

Original: Abstract
Revised: Abstract

Original: In this work we propose a batch Bayesian optimization method for combinatorial problems on permutations, which is well suited for expensive-to-evaluate objectives.
Revised: In this work we propose a batch Bayesian optimization method for combinatorial problems on permutations, which is well suited for expensive-to-evaluate objectives.

Original: We first introduce LAW, an efficient batch acquisition method based on determinantal point processes using the acquisition weighted kernel.
Revised: We first introduce LAW, an efficient batch acquisition method based on determinantal point processes using the acquisition weighted kernel.

Original: Relying on multiple parallel evaluations, LAW enables accelerated search on combinatorial spaces.
Revised: Relying on multiple parallel evaluations, LAW enables accelerated search on combinatorial spaces.

Original: We then apply the framework to permutation problems, which have so far received little attention in the Bayesian Optimization literature, despite their practical importance.
Revised: We then apply the framework to permutation problems, which have so far received little attention in the Bayesian Optimization literature, despite their practical importance.

Original: We call this method LAW2ORDER.
Revised: We call this method LAW2ORDER.

Original: On the theoretical front, we prove that LAW2ORDER has vanishing simple regret by showing that the batch cumulative regret is sublinear.
Revised: On the theoretical front, we prove that LAW2ORDER has vanishing simple regret by showing that the batch cumulative regret is sublinear.

Original: Empirically, we assess the method on several standard combinatorial problems involving permutations such as quadratic assignment, flowshop scheduling and the traveling salesman, as well as on a structure learning task.
Revised: Empirically, we assess the method on several standard combinatorial problems involving permutations such as quadratic assignment, flowshop scheduling and the traveling salesman, as well as on a structure learning task.

Original:  Introduction
Revised:  Introduction

Original: From the celebrated traveling salesman problem[GP06] to flowshop and jobshop scheduling problems[GJS76], permutations are ubiquitous representations in combinatorial optimization.
Revised: From the celebrated traveling salesman problem[GP06] to flowshop and jobshop scheduling problems[GJS76], permutations are ubiquitous representations in combinatorial optimization.

Original: Such combinatorial problems on permutations arise in highly impactful application areas.
Revised: Such combinatorial problems on permutations arise in highly impactful application areas.

Original: For example, in chip design, permutations specify relative placements of memories and logical gates on a chip[AMS08].
Revised: For example, in chip design, permutations specify relative placements of memories and logical gates on a chip[AMS08].

Original: As another example, in 3D printing, scheduling is an important factor to determine the production time[CHHV18, GSE + 19, SFZ + 20].
Revised: As another example, in 3D printing, scheduling is an important factor to determine the production time[CHHV18, GSE + 19, SFZ + 20].

Original: In both cases, as well as in many others, evaluating the cost associated to a given permutation is expensive.
Revised: In both cases, as well as in many others, evaluating the cost associated to a given permutation is expensive.

Original: In situations where the evaluation is expensive, Bayesian optimization (BO) has shown good performance in many problems[SLA12, SRS + 15].
Revised: In situations where the evaluation is expensive, Bayesian optimization (BO) has shown good performance in many problems[SLA12, SRS + 15].

Original: Recently, BO on combinatorial spaces has made significant progress for categorical variables [BP18, DBDF20, OTGW19, SRDM20, DSR + 20].
Revised: Recently, BO on combinatorial spaces has made significant progress for categorical variables [BP18, DBDF20, OTGW19, SRDM20, DSR + 20].

Original: However,
Revised: However,

Original: BO on permutations is yet under-explored with a few exceptions [ZSBB14, ZYLW19, BBG + 20].
Revised: BO on permutations is yet under-explored with a few exceptions [ZSBB14, ZYLW19, BBG + 20].

Original: In this work we present a framework to deal with BO on permutations where the evaluation of the objective is expensive.
Revised: In this work we present a framework to deal with BO on permutations where the evaluation of the objective is expensive.

Original: We extend batch Bayesian optimization, which allows one to speed up the optimization by acquiring a batch of multiple points and evaluating the batch in parallel[AFF10,
Revised: We extend batch Bayesian optimization, which allows one to speed up the optimization by acquiring a batch of multiple points and evaluating the batch in parallel[AFF10,

Original: GDHL16], to the case of permutations.
Revised: GDHL16], to the case of permutations.

Original: Then, motivated by the observation that both the diversity of the points in the batches and the informativeness of the individual points in the batch improve the performance[GPL19], we propose a new batch acquisition method which is applicable to the search space of permutations and takes into account both the diversity of the batch and the informativeness of each point.
Revised: Then, motivated by the observation that both the diversity of the points in the batches and the informativeness of the individual points in the batch improve the performance[GPL19], we propose a new batch acquisition method which is applicable to the search space of permutations and takes into account both the diversity of the batch and the informativeness of each point.

Original: This method is based on determinantal point processes (DPPs), which have been widely used to model sets of diverse points[KT12], and can be conveniently incorporated into the Gaussian Process framework since DPPs are specified by a kernel.
Revised: This method is based on determinantal point processes (DPPs), which have been widely used to model sets of diverse points[KT12], and can be conveniently incorporated into the Gaussian Process framework since DPPs are specified by a kernel.

Original: To overcome the lack of informativeness of DDPs[KDK16] (more specifically, the selection of points in batches relies solely on the predictive variance of the surrogate model), we enhance DPPs by using a kernel weighted by acquisition values.
Revised: To overcome the lack of informativeness of DDPs[KDK16] (more specifically, the selection of points in batches relies solely on the predictive variance of the surrogate model), we enhance DPPs by using a kernel weighted by acquisition values.

Original: Therefore, we propose a new batch acquisition method using the so-called DPP Lensemble[BR05, KT12] augmented with the A cquisition W eight, dubbed LAW.
Revised: Therefore, we propose a new batch acquisition method using the so-called DPP Lensemble[BR05, KT12] augmented with the A cquisition W eight, dubbed LAW.

Original: The whole procedure to find the optimal ordering (permutation) through LAW is thus dubbed LAW2ORDER.
Revised: The whole procedure to find the optimal ordering (permutation) through LAW is thus dubbed LAW2ORDER.

Original: We compare
Revised: We compare

Original: LAW2ORDER and other competitors, firstly, on three combinatorial optimization benchmarks on permutations such as quadratic assignment problem, the flowshop scheduling problem, and the traveling salesman problem.
Revised: LAW2ORDER and other competitors, firstly, on three combinatorial optimization benchmarks on permutations such as quadratic assignment problem, the flowshop scheduling problem, and the traveling salesman problem.

Original: We also make comparisons on the structure learning problem.
Revised: We also make comparisons on the structure learning problem.

Original: In the structure learing problem, LAW2ORDER performs the best and the performance gap is more significant for larger permutation spaces.
Revised: In the structure learing problem, LAW2ORDER performs the best and the performance gap is more significant for larger permutation spaces.

Original: Moreover, LAW2ORDER still outperforms significantly genetic algorithms which use twice as many evaluations.
Revised: Moreover, LAW2ORDER still outperforms significantly genetic algorithms which use twice as many evaluations.

Original:  Preliminaries
Revised:  Preliminaries

Original: In this section, we briefly discuss some prerequisites for our proposed method and introduce notation.
Revised: In this section, we briefly discuss some prerequisites for our proposed method and introduce notation.

Original: Below we will denote a function f with one input as f ( · ) , and function K with two inputs as K ( · , · ) .
Revised: Below we will denote a function f with one input as f ( · ) , and function K with two inputs as K ( · , · ) .

Original: For B ∈ N , [ B ] = { 1 , · · · , B } while for a set X , | X | is the number of elements in X .
Revised: For B ∈ N , [ B ] = { 1 , · · · , B } while for a set X , | X | is the number of elements in X .

Original: 2.1 Batch Bayesian Optimization
Revised: 2.1 Batch Bayesian Optimization

Original: Bayesian Optimization (BO) aims at finding the global optimum of a black-box function f
Revised: Bayesian Optimization (BO) aims at finding the global optimum of a black-box function f

Original: over a search space X , namely, x opt = argmin x ∈X f ( x ) .
Revised: over a search space X , namely, x opt = argmin x ∈X f ( x ) .

Original: Two main components are the probabilistic modeling of the objective f ( x ) and the acquisition of new points to evaluate.
Revised: Two main components are the probabilistic modeling of the objective f ( x ) and the acquisition of new points to evaluate.

Original: Probabilistic modeling is performed by the surrogate model.
Revised: Probabilistic modeling is performed by the surrogate model.

Original: At the t -th round, the surrogate model attempts to approximate f ( x ) based on the evaluation data D t − 1 , producing the predictive mean µ t − 1 ( x ) = µ ( x | D t − 1 ) and the predictive variance σ 2 t − 1 ( x ) =
Revised: At the t -th round, the surrogate model attempts to approximate f ( x ) based on the evaluation data D t − 1 , producing the predictive mean µ t − 1 ( x ) = µ ( x | D t − 1 ) and the predictive variance σ 2 t − 1 ( x ) =

Original: σ 2 ( x | D t − 1 ) .
Revised: σ 2 ( x | D t − 1 ) .

Original: In the acquisition of a new point, the acquisition function a t ( x ) = a seq ( x | µ t − 1 ( · ) , σ 2 t − 1 ( · )) is specified, which is based on the predictive mean µ t − 1 ( · ) and the predictive variance σ 2 t − 1 ( · ) to score how informative points are for the optimization.
Revised: In the acquisition of a new point, the acquisition function a t ( x ) = a seq ( x | µ t − 1 ( · ) , σ 2 t − 1 ( · )) is specified, which is based on the predictive mean µ t − 1 ( · ) and the predictive variance σ 2 t − 1 ( · ) to score how informative points are for the optimization.

Original: Next, the point that maximizes the acquisition function is obtained, x t = argmax x a t ( x ) , and the objective evaluated, y t
Revised: Next, the point that maximizes the acquisition function is obtained, x t = argmax x a t ( x ) , and the objective evaluated, y t

Original: = f ( x t ) .
Revised: = f ( x t ) .

Original: Then, the new evaluation point is appended to the old dataset,
Revised: Then, the new evaluation point is appended to the old dataset,

Original: D t =
Revised: D t =

Original: D t − 1 ∪{ ( x t , y t ) } and the process repeats by fitting the surrogate model with D t .
Revised: D t − 1 ∪{ ( x t , y t ) } and the process repeats by fitting the surrogate model with D t .

Original: The process continues until the evaluation budget is depleted.
Revised: The process continues until the evaluation budget is depleted.

Original: To contrast with the proposed method, we call this basic BO as sequential BO.
Revised: To contrast with the proposed method, we call this basic BO as sequential BO.

Original: For a more extensive overview of Bayesian optimization, please refer to [SSW + 15, Fra18].
Revised: For a more extensive overview of Bayesian optimization, please refer to [SSW + 15, Fra18].

Original: With more computational resources, such as more GPUs and CPUs, we can speed up Bayesian optimization by allowing multiple evaluations in parallel.
Revised: With more computational resources, such as more GPUs and CPUs, we can speed up Bayesian optimization by allowing multiple evaluations in parallel.

Original: For this, we acquire a batch of multiple points, a method known as Batch Bayesian Optimization (BBO)[AFF10, GDHL16].
Revised: For this, we acquire a batch of multiple points, a method known as Batch Bayesian Optimization (BBO)[AFF10, GDHL16].

Original: In BBO, we need an acquisition function a batch scoring the quality of batches of B points { x b } b ∈ [ B ] instead of individual points.
Revised: In BBO, we need an acquisition function a batch scoring the quality of batches of B points { x b } b ∈ [ B ] instead of individual points.

Original: At time t , a batch of B points is acquired { x t,b } b ∈
Revised: At time t , a batch of B points is acquired { x t,b } b ∈

Original: [ B ] = argmax { x b } b ∈
Revised: [ B ] = argmax { x b } b ∈

Original: [ B ] a t ( { x b } b ∈ [ B ] ) , where a t ( { x b } b ∈ [ B ] ) =
Revised: [ B ] a t ( { x b } b ∈ [ B ] ) , where a t ( { x b } b ∈ [ B ] ) =

Original: a batch ( { x b } b ∈ [ B ] | µ t − 1 ( · ) , σ 2 t − 1 ( · )) .
Revised: a batch ( { x b } b ∈ [ B ] | µ t − 1 ( · ) , σ 2 t − 1 ( · )) .

Original: Then the points in the acquired batch are evaluated in parallel and the evaluation data is updated by D t
Revised: Then the points in the acquired batch are evaluated in parallel and the evaluation data is updated by D t

Original: = D t − 1 ∪{ ( x t,b , y t,b ) } b ∈ [ B ] .
Revised: = D t − 1 ∪{ ( x t,b , y t,b ) } b ∈ [ B ] .

Original: 2.2 Determinantal Point Processes
Revised: 2.2 Determinantal Point Processes

Original: Determinantal point processes (DPPs) are stochastic point processes well-suited to model sets of diverse points[KT12].
Revised: Determinantal point processes (DPPs) are stochastic point processes well-suited to model sets of diverse points[KT12].

Original: Let us assume that we want to sample a set of diverse points from a finite set
Revised: Let us assume that we want to sample a set of diverse points from a finite set

Original: X .
Revised: X .

Original: One way to define DPP is to use the so-called L-ensemble[KT12].
Revised: One way to define DPP is to use the so-called L-ensemble[KT12].

Original: For a given kernel L ( · , · ) on X , the L-ensemble is defined as the random point process with density P DPPL ( X ) = det([ L ( x , y )]
Revised: For a given kernel L ( · , · ) on X , the L-ensemble is defined as the random point process with density P DPPL ( X ) = det([ L ( x , y )]

Original: x , y ∈ X ) det( I + L ) where X ⊂ X
Revised: x , y ∈ X ) det( I + L ) where X ⊂ X

Original: and [ L ( x , y )]
Revised: and [ L ( x , y )]

Original: x , y ∈
Revised: x , y ∈

Original: X is a submatrix of L restricted to X [BR05].
Revised: X is a submatrix of L restricted to X [BR05].

Original: For a batch of just two points, X = { x , y } it is easy to observe that DPP encourages diversity — P DPPL ( { x , y } )
Revised: For a batch of just two points, X = { x , y } it is easy to observe that DPP encourages diversity — P DPPL ( { x , y } )

Original: ∝ L ( x , x ) L ( y , y ) − L ( x , y ) 2 .
Revised: ∝ L ( x , x ) L ( y , y ) − L ( x , y ) 2 .

Original: Indeed, for more similar points the value of
Revised: Indeed, for more similar points the value of

Original: L ( x , y ) is higher, resulting in a lower density.
Revised: L ( x , y ) is higher, resulting in a lower density.

Original: In DPPs there is no cardinality constraint on X .
Revised: In DPPs there is no cardinality constraint on X .

Original: We, therefore, define k -DPP, which is a DPP with the restriction that sampled sets have precisely k points.
Revised: We, therefore, define k -DPP, which is a DPP with the restriction that sampled sets have precisely k points.

Original: Denoting the set of subsets of X with k points by X k , the k -DPP density is defined for X ∈ X k by P k - DPP L ( X ) = det([ L ( x , y )]
Revised: Denoting the set of subsets of X with k points by X k , the k -DPP density is defined for X ∈ X k by P k - DPP L ( X ) = det([ L ( x , y )]

Original: x , y ∈ X ) (cid:80)
Revised: x , y ∈ X ) (cid:80)

Original: X ′∈X k det([ L ( x , y )] x , y ∈ X ′ ) .
Revised: X ′∈X k det([ L ( x , y )] x , y ∈ X ′ ) .

Original: Therefore, X ∗ = argmax X ∈X k P k - DPP L ( X ) is the most diverse set of k points with respect to the similarity encoded by the kernel L ( · , · ) .
Revised: Therefore, X ∗ = argmax X ∈X k P k - DPP L ( X ) is the most diverse set of k points with respect to the similarity encoded by the kernel L ( · , · ) .

Original: In our algorithm, we use that log of k-DPP density is submodular[SKKS09, KT12] and can be greedily maximized with approximation guarantees[NWF78, Sak20] (See Supp.
Revised: In our algorithm, we use that log of k-DPP density is submodular[SKKS09, KT12] and can be greedily maximized with approximation guarantees[NWF78, Sak20] (See Supp.

Original: Subsec.
Revised: Subsec.

Original: F.1 for a brief discussion).
Revised: F.1 for a brief discussion).

Original:  Method
Revised:  Method

Original: Batch acquisition on a combinatorial space poses two difficulties.
Revised: Batch acquisition on a combinatorial space poses two difficulties.

Original: First, the batch acquisition objectives of existing batch Bayesian optimization are designed based on the properties and intuition applicable to continuous spaces[GDHL16, GPL19].
Revised: First, the batch acquisition objectives of existing batch Bayesian optimization are designed based on the properties and intuition applicable to continuous spaces[GDHL16, GPL19].

Original: This may not always be suitable for discrete spaces.
Revised: This may not always be suitable for discrete spaces.

Original: For instance, the method in [GDHL16] is defined by using Euclidean distance.
Revised: For instance, the method in [GDHL16] is defined by using Euclidean distance.

Original: Also, the difficulties of combinatorial optimization are exacerbated when optimizing a batch jointly.
Revised: Also, the difficulties of combinatorial optimization are exacerbated when optimizing a batch jointly.

Original: This is in stark contrast to the continuous case where gradient based optimization is easily extended to batch optimization of multiple points in parallel[WCLF20].
Revised: This is in stark contrast to the continuous case where gradient based optimization is easily extended to batch optimization of multiple points in parallel[WCLF20].

Original: To cope with these challenges, we introduce a new batch acquisition method for Bayesian optimization, the maximization of the determinantal point process (DPP) density defined by an L -ensemble with
Revised: To cope with these challenges, we introduce a new batch acquisition method for Bayesian optimization, the maximization of the determinantal point process (DPP) density defined by an L -ensemble with

Original: A cquisition W eights, dubbed LAW .
Revised: A cquisition W eights, dubbed LAW .

Original: We describe LAW in Subsec.
Revised: We describe LAW in Subsec.

Original: 3.1 and its regret analysis is provided in Subsec.
Revised: 3.1 and its regret analysis is provided in Subsec.

Original: 3.2 and 3.3.
Revised: 3.2 and 3.3.

Original: 3.1 Batch Acquisition using LAW
Revised: 3.1 Batch Acquisition using LAW

Original: We start to define the main components of LAW.
Revised: We start to define the main components of LAW.

Original: Definition 3.1 (Weight function) .
Revised: Definition 3.1 (Weight function) .

Original: We call a function w : R → R a weight function if it is positive ( r ∈ R , w ( r ) > 0 ), increasing ( r 1 ≤ r 2 = ⇒ w ( r 1 ) ≤ w ( r 2 ) ), and bounded below and above by a positive number ( w − = inf r ∈ R w ( r ) >
Revised: We call a function w : R → R a weight function if it is positive ( r ∈ R , w ( r ) > 0 ), increasing ( r 1 ≤ r 2 = ⇒ w ( r 1 ) ≤ w ( r 2 ) ), and bounded below and above by a positive number ( w − = inf r ∈ R w ( r ) >

Original: 0 and w + = sup r ∈ R w ( r ) < ∞ ).
Revised: 0 and w + = sup r ∈ R w ( r ) < ∞ ).

Original: Definition 3.2 (Posterior covariance function) .
Revised: Definition 3.2 (Posterior covariance function) .

Original: Given a (prior) kernel K ( x 1 , x 2 ) , data D and noise variance σ 2 , the posterior (predictive) covariance function K post ( x 1 , x 2 | D , σ 2 ) is defined as
Revised: Given a (prior) kernel K ( x 1 , x 2 ) , data D and noise variance σ 2 , the posterior (predictive) covariance function K post ( x 1 , x 2 | D , σ 2 ) is defined as

Original: K ( x 1 , x 2 ) − K ( x 1 , D )( K ( D , D ) + σ 2 I ) − 1 K ( D , x 2 ) .
Revised: K ( x 1 , x 2 ) − K ( x 1 , D )( K ( D , D ) + σ 2 I ) − 1 K ( D , x 2 ) .

Original: Let us assume that we are running batch Bayesian optimization with Gaussian process surrogate model using the kernel K ( · , · ) and the acquisition function a ( · ) , and that we acquire a batch of B points in each round.
Revised: Let us assume that we are running batch Bayesian optimization with Gaussian process surrogate model using the kernel K ( · , · ) and the acquisition function a ( · ) , and that we acquire a batch of B points in each round.

Original: At the t -th round, we have the evaluation data D t − 1 , the posterior covariance function K t ( · , · ) =
Revised: At the t -th round, we have the evaluation data D t − 1 , the posterior covariance function K t ( · , · ) =

Original: K post ( · , ·| K, D t − 1 , σ 2 obs ) and the acquisition function a t ( { x b } b ∈ [ B ] ) =
Revised: K post ( · , ·| K, D t − 1 , σ 2 obs ) and the acquisition function a t ( { x b } b ∈ [ B ] ) =

Original: a ( { x b } b ∈ [ B ] | µ t − 1 ( · ) , σ 2 t − 1 ( · )) where µ t − 1 ( · ) and σ 2 t − 1 ( · ) are the predicitve mean and the predictive variance conditioned on D t − 1 .
Revised: a ( { x b } b ∈ [ B ] | µ t − 1 ( · ) , σ 2 t − 1 ( · )) where µ t − 1 ( · ) and σ 2 t − 1 ( · ) are the predicitve mean and the predictive variance conditioned on D t − 1 .

Original: In the existing work on batch Bayesian optimization using DPP[KDK16], the posterior covariance function is used as the kernel defining DPP.
Revised: In the existing work on batch Bayesian optimization using DPP[KDK16], the posterior covariance function is used as the kernel defining DPP.

Original: Even though the use of DPP in[KDK16] encourages diversity among points in batches, it essentially chooses points of high predictive variance.
Revised: Even though the use of DPP in[KDK16] encourages diversity among points in batches, it essentially chooses points of high predictive variance.

Original: However, the predictive mean also provides valuable information in Bayesian optimization.
Revised: However, the predictive mean also provides valuable information in Bayesian optimization.

Original: It is the acquisition function which harmonizes the predictive mean and the predictive variance to quantify how useful each point is.
Revised: It is the acquisition function which harmonizes the predictive mean and the predictive variance to quantify how useful each point is.

Original: Therefore, we propose a new batch acquisition method which actively uses the acquisition function while retaining the strength of DPP encouraging diversity in each batch.
Revised: Therefore, we propose a new batch acquisition method which actively uses the acquisition function while retaining the strength of DPP encouraging diversity in each batch.

Original: We define the Acquisition Weighted kernel L AW as follows
Revised: We define the Acquisition Weighted kernel L AW as follows

Original: [Equation]
Revised: [Equation]

Original: Here w is a positive weight function.
Revised: Here w is a positive weight function.

Original: We call the kernel L in L AW the diversity gauge of L AW .
Revised: We call the kernel L in L AW the diversity gauge of L AW .

Original: With the posterior covariance function as the diversity gauge L = K t and the acquisition function a = a t , the acquisition weighted kernel becomes
Revised: With the posterior covariance function as the diversity gauge L = K t and the acquisition function a = a t , the acquisition weighted kernel becomes

Original: [Equation]
Revised: [Equation]

Original: Due to the dependency of L = K t and a t to the round index t , we subscript L AW and L with t .
Revised: Due to the dependency of L = K t and a t to the round index t , we subscript L AW and L with t .

Original: We optimize the density of DPP using the acquisition weighted kernel to choose the points in the batch.
Revised: We optimize the density of DPP using the acquisition weighted kernel to choose the points in the batch.

Original: Before introducing how this optimization is actually performed, we provide an interpretation by rewriting the numerator of the density of the k -DPP defined by L AWt as
Revised: Before introducing how this optimization is actually performed, we provide an interpretation by rewriting the numerator of the density of the k -DPP defined by L AWt as

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: This shows that the maximization of eq.
Revised: This shows that the maximization of eq.

Original: (2) can be obtained by increasing both det([ L t ( x i , x j )] i,j ∈ [ B ] ) and (cid:81)
Revised: (2) can be obtained by increasing both det([ L t ( x i , x j )] i,j ∈ [ B ] ) and (cid:81)

Original: Bi =1 w ( a t ( x i )) 2 in a balanced way.
Revised: Bi =1 w ( a t ( x i )) 2 in a balanced way.

Original: Increasing the determinant term and increasing the product term promote diversity and acquisition values, respectively.
Revised: Increasing the determinant term and increasing the product term promote diversity and acquisition values, respectively.

Original: Now, we provide details on how the optimization is implemented.
Revised: Now, we provide details on how the optimization is implemented.

Original: In combinatorial spaces where the optimization of a single point is challenging, the joint optimization of multiple points is daunting.
Revised: In combinatorial spaces where the optimization of a single point is challenging, the joint optimization of multiple points is daunting.

Original: Thanks to the submodularity of the log of the determinant[KT12], the joint optimization of multiple points can be approximated by a sequence of single point optimizations with an approximation guarantee (See Supp.
Revised: Thanks to the submodularity of the log of the determinant[KT12], the joint optimization of multiple points can be approximated by a sequence of single point optimizations with an approximation guarantee (See Supp.

Original: Subsec.
Revised: Subsec.

Original: F.1 for submodularity).
Revised: F.1 for submodularity).

Original: The first point is obtained as in sequential Bayesian optimization by optimizing an acquisition function (line 3 in Alg. 1).
Revised: The first point is obtained as in sequential Bayesian optimization by optimizing an acquisition function (line 3 in Alg. 1).

Original: The rest of the B − 1 points are obtained by maximizing the k -DPP density defined by the acquisition weighted kernel, which we approximately perform with a greedy method (line 4 in Alg. 1).
Revised: The rest of the B − 1 points are obtained by maximizing the k -DPP density defined by the acquisition weighted kernel, which we approximately perform with a greedy method (line 4 in Alg. 1).

Original: Having chosen b − 1 points { x t,i } i ∈ [ b − 1] , the greedy maximization selects x b as follows:
Revised: Having chosen b − 1 points { x t,i } i ∈ [ b − 1] , the greedy maximization selects x b as follows:

Original: [Equation]
Revised: [Equation]

Original: where L t ( x , x |{ x t,i } i ∈ [ b − 1] ) is the posterior variance of the kernel L t conditioned on { x t,i }
Revised: where L t ( x , x |{ x t,i } i ∈ [ b − 1] ) is the posterior variance of the kernel L t conditioned on { x t,i }

Original: i ∈
Revised: i ∈

Original: [ b − 1] .
Revised: [ b − 1] .

Original: 3.2 Regret Analysis
Revised: 3.2 Regret Analysis

Original: In this subsection, we provide a theoretical analysis on the performance of LAW with two acquisition functions, GP-UBC[SKKS09] and EST[WZJ16].
Revised: In this subsection, we provide a theoretical analysis on the performance of LAW with two acquisition functions, GP-UBC[SKKS09] and EST[WZJ16].

Original: We begin with definitions needed in the analysis.
Revised: We begin with definitions needed in the analysis.

Original: Definition 3.3.
Revised: Definition 3.3.

Original: In the minimization of f using batch acquisition, where x ∗ = argmin x f ( x ) , r t,b = f ( x t,b ) − f ( x ∗ ) is called instantaneous regret and r ( B ) t
Revised: In the minimization of f using batch acquisition, where x ∗ = argmin x f ( x ) , r t,b = f ( x t,b ) − f ( x ∗ ) is called instantaneous regret and r ( B ) t

Original: [Equation]
Revised: [Equation]

Original: x
Revised: x

Original: [Equation]
Revised: [Equation]

Original: )) is called batch instantaneous regret.
Revised: )) is called batch instantaneous regret.

Original: Simple regret is defined as the minimum of batch instantaneous regrets R ( B ) T .
Revised: Simple regret is defined as the minimum of batch instantaneous regrets R ( B ) T .

Original: [Equation]
Revised: [Equation]

Original: Batch cumulative regret R ( B ) T is defined as the sum of batch instantaneous regrets
Revised: Batch cumulative regret R ( B ) T is defined as the sum of batch instantaneous regrets

Original: [Equation]
Revised: [Equation]

Original: Remark 3.4 .
Revised: Remark 3.4 .

Original: Note that S ( B ) T ≤ 1 T R ( B ) T .
Revised: Note that S ( B ) T ≤ 1 T R ( B ) T .

Original: Vanishing simple regret is proved by showing 1 T R ( B ) T → 0 .
Revised: Vanishing simple regret is proved by showing 1 T R ( B ) T → 0 .

Original: Definition 3.5.
Revised: Definition 3.5.

Original: For Gaussian processes with the kernel K and the variance of observation noise σ 2 , the maximum information gain γ T is defined as
Revised: For Gaussian processes with the kernel K and the variance of observation noise σ 2 , the maximum information gain γ T is defined as

Original: [Equation]
Revised: [Equation]

Original: For UCB and EST, we have the following regret bound.
Revised: For UCB and EST, we have the following regret bound.

Original: Theorem 3.6.
Revised: Theorem 3.6.

Original: Assume a kernel such that K ( · , · ) ≤ 1 , | X | < ∞ and f : X → R is sampled from
Revised: Assume a kernel such that K ( · , · ) ≤ 1 , | X | < ∞ and f : X → R is sampled from

Original: GP ( 0 , K ) .
Revised: GP ( 0 , K ) .

Original: In each round t ∈ [ T ] of batch Bayesian optimization, LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )
Revised: In each round t ∈ [ T ] of batch Bayesian optimization, LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )

Original: =
Revised: =

Original: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (Def. 3.1).
Revised: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (Def. 3.1).

Original: Let C 1 = 36 log(1+ σ − 2 ) where σ 2 is the variance of the observation noise and δ ∈ (0 , 1) .
Revised: Let C 1 = 36 log(1+ σ − 2 ) where σ 2 is the variance of the observation noise and δ ∈ (0 , 1) .

Original: At round t , define β ( B ) UCB t,
Revised: At round t , define β ( B ) UCB t,=

Original: [Equation]
Revised: [Equation]

Original: log (cid:16) |
Revised: log (cid:16) |

Original: [Equation]
Revised: [Equation]

Revised: |

Revised: [Equation]

Original: and ν t = min
Revised: and ν t = min

Original: [Equation]
Revised: [Equation]

Original:  where ˆ m t is the estimate of the optimum [WZJ16], ζ t = (cid:16) 2 log (cid:0) π 2 t 2 δ (cid:1)(cid:17) 1 / 2 , π t > 0 such that (cid:80) ∞ t =1 π − 1 t ≤ 1 .
Revised: (cid:19) where ˆ m t is the estimate of the optimum [WZJ16], ζ t = (cid:16) 2 log (cid:0) π 2 t 2 δ (cid:1)(cid:17) 1 / 2 , π t > 0 such that (cid:80) ∞ t =1 π − 1 t ≤ 1 .

Original: Then batch cumulative regret satisfies the following bound
Revised: Then batch cumulative regret satisfies the following bound

Original: [Equation]
Revised: [Equation]

Original: where for EST, η ( B ) t = ν
Revised: where for EST, η ( B ) t = ν

Original: [Equation]
Revised: [Equation]

Original: + ζ t and for UBC, η ( B
Revised: + ζ t and for UBC, η ( B

Original: [Equation]
Revised: [Equation]

Original: t
Revised: t

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: argmax
Revised: argmax

Original: [Equation]
Revised: [Equation]

Original: s .
Revised: s .

Original: Proof.
Revised: Proof.

Original: See.
Revised: See.

Original: Supp.
Revised: Supp.

Original: Sec. A.
Revised: Sec. A.

Original: Remark 3.7 .
Revised: Remark 3.7 .

Original: This theorem shows that, for the same kernel, the regret bound of LAW also enjoys the same asymptotic behavior as the regret bound of existing works[CBRV13, DKB14, KDK16].
Revised: This theorem shows that, for the same kernel, the regret bound of LAW also enjoys the same asymptotic behavior as the regret bound of existing works[CBRV13, DKB14, KDK16].

Original: Remark 3.8 .
Revised: Remark 3.8 .

Original: Note that Thm.
Revised: Note that Thm.

Original: 3.6 is about a bound on 1 T R ( B ) T while the analysis in [DKB14, KKSP18] is to bound 1 TB R T,B where R T,B
Revised: 3.6 is about a bound on 1 T R ( B ) T while the analysis in [DKB14, KKSP18] is to bound 1 TB R T,B where R T,B

Original: = (cid:80) t,b r t,b .
Revised: = (cid:80) t,b r t,b .

Original: Since 1 T R ( B ) T ≤ 1 TB R T,B , bounding 1 TB R T,B implies bounding 1 T R ( B ) T .
Revised: Since 1 T R ( B ) T ≤ 1 TB R T,B , bounding 1 TB R T,B implies bounding 1 T R ( B ) T .

Original: For the purpose of showing vanishing simple regret, both approaches are viable.
Revised: For the purpose of showing vanishing simple regret, both approaches are viable.

Original: Technically, two approaches require different treatments.
Revised: Technically, two approaches require different treatments.

Original: See Supp.
Revised: See Supp.

Original: Subsec.
Revised: Subsec.

Original: A.2 for the discussion on the differences between two approaches.
Revised: A.2 for the discussion on the differences between two approaches.

Original: Remark 3.9 .
Revised: Remark 3.9 .

Original: The ratio w + w − in Thm.
Revised: The ratio w + w − in Thm.

Original: 3.6 determines how LAW balances between the quality and the diversity.
Revised: 3.6 determines how LAW balances between the quality and the diversity.

Original: If the ratio is large, then the acquisition value is more influential in Eq. 2.
Revised: If the ratio is large, then the acquisition value is more influential in Eq. 2.

Original: Otherwise,
Revised: Otherwise,

Original: Eq. 2 is dominated by the determinant of the diversity gauge, and the diversity of the batch is more emphasized.
Revised: Eq. 2 is dominated by the determinant of the diversity gauge, and the diversity of the batch is more emphasized.

Original: The bound in Eq.
Revised: The bound in Eq.

Original: 6 reveals the necessity of the upper bound of w + w − .
Revised: 6 reveals the necessity of the upper bound of w + w − .

Original: Without the upper bound, i.e. virtually considering the acquisition value only, the batch acquisition may result in non-vanishing regret.
Revised: Without the upper bound, i.e. virtually considering the acquisition value only, the batch acquisition may result in non-vanishing regret.

Original: However, the bound is not tight enough considering the extreme case w + w − = 1 .
Revised: However, the bound is not tight enough considering the extreme case w + w − = 1 .

Original: Nonetheless, the necessity of the upper bound of w + w − guides how the weight function w ( · ) is set (See for details).
Revised: Nonetheless, the necessity of the upper bound of w + w − guides how the weight function w ( · ) is set (See for details).

Original: Moreover, the benefit of considering the acquisition weight is supported by the experimental results (Sec. 5)
Revised: Moreover, the benefit of considering the acquisition weight is supported by the experimental results (Sec. 5)

Original: Note that η ( B ) T
Revised: Note that η ( B ) T

Original: = O
Revised: = O

Original: ( (cid:112) log( TB ))
Revised: ( (cid:112) log( TB ))

Original: (See Supp.
Revised: (See Supp.

Original: Subsec.
Revised: Subsec.

Original: A.3 for details).
Revised: A.3 for details).

Original: In Thm.
Revised: In Thm.

Original: 3.6, we need η ( B ) T · (cid:112) γ TB TB → 0 to prove vanishing simple regret.
Revised: 3.6, we need η ( B ) T · (cid:112) γ TB TB → 0 to prove vanishing simple regret.

Original: We provide a bound for the maximum information gain γ T of a kernel on a finite space, which we use later to show the vanishing simple regret.
Revised: We provide a bound for the maximum information gain γ T of a kernel on a finite space, which we use later to show the vanishing simple regret.

Original: Theorem 3.10.
Revised: Theorem 3.10.

Original: K is a kernel on a finite set X ( | X | < ∞ ), σ 2 is the variance of the observation noise and Λ = { λ n } 1 , ··· , |X | ( λ n ≥ λ n +1 ≥ 0) is the set of eigenvalues of the gram matrix K ( X , X ) .
Revised: K is a kernel on a finite set X ( | X | < ∞ ), σ 2 is the variance of the observation noise and Λ = { λ n } 1 , ··· , |X | ( λ n ≥ λ n +1 ≥ 0) is the set of eigenvalues of the gram matrix K ( X , X ) .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: (7) where λ max is the largest eigenvalue of K ( X , X ) .
Revised: (7) where λ max is the largest eigenvalue of K ( X , X ) .

Original: Proof.
Revised: Proof.

Original: See.
Revised: See.

Original: Supp.
Revised: Supp.

Original: Subsec.
Revised: Subsec.

Original: B.
Revised: B.

Original: 3.3 Position Kernel
Revised: 3.3 Position Kernel

Original: Based on the comparative experiments in [ZSBB14] showing that the Position kernel outperforms others consistently, 2 , we use the position kernel in our BBO on permutations
Revised: Based on the comparative experiments in [ZSBB14] showing that the Position kernel outperforms others consistently, 2 , we use the position kernel in our BBO on permutations

Original: [Equation]
Revised: [Equation]

Original: The positive definiteness of the position kernel was empirically tested via simulation[ZSBB14] and has not been shown rigorously.
Revised: The positive definiteness of the position kernel was empirically tested via simulation[ZSBB14] and has not been shown rigorously.

Original: Therefore, we show the positive definiteness of the position kernel and bound its eigenvalues.
Revised: Therefore, we show the positive definiteness of the position kernel and bound its eigenvalues.

Original: Theorem 3.11.
Revised: Theorem 3.11.

Original: The position kernel K ( · , ·| τ ) defined on S N is positive definite and the eigenvalues of K ( X, X ) where X ⊂ X lie between (cid:16) 1 − ρ 

Revised: The position kernel K ( · , ·| τ ) defined on S N is positive definite and the eigenvalues of K ( X, X ) where X ⊂ X lie between (cid:16) 1 − ρ

Original: + ρ (cid:17) N and (cid:16) 1+ ρ 1 − ρ (cid:17)
Revised: + ρ (cid:17) N and (cid:16) 1+ ρ 1 − ρ (cid:17)

Original: N where ρ = exp( − τ ) .
Revised: N where ρ = exp( − τ ) .

Original: Proof.
Revised: Proof.

Original: See Supp.
Revised: See Supp.

Original: Subsec.
Revised: Subsec.

Original: B.
Revised: B.

Original: By utilizing the property of the position kernel, we provide a bound on the maximal information gain which is tighter than the one obtained in Thm.
Revised: By utilizing the property of the position kernel, we provide a bound on the maximal information gain which is tighter than the one obtained in Thm.

Original: Theorem 3.12.
Revised: Theorem 3.12.

Original: K ( · , ·| τ ) is the position kernel defined on S N , σ 2 is the variance of the observation noise, ρ = exp( − τ ) and, D max =
Revised: K ( · , ·| τ ) is the position kernel defined on S N , σ 2 is the variance of the observation noise, ρ = exp( − τ ) and, D max =

Original: ( N 2 − ( N mod 2)) / 2 .
Revised: ( N 2 − ( N mod 2)) / 2 .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: where λ max is the largest eigenvalue of K ( X , X ) and
Revised: where λ max is the largest eigenvalue of K ( X , X ) and

Original: [Equation]
Revised: [Equation]

Original: which is smaller than T · log(1 + σ − 2 max x ∈X K ( x, x )) .
Revised: which is smaller than T · log(1 + σ − 2 max x ∈X K ( x, x )) .

Original: Proof.
Revised: Proof.

Original: See Supp.
Revised: See Supp.

Original: Subsec.
Revised: Subsec.

Original: B.
Revised: B.

Original: Remark 3.13 .
Revised: Remark 3.13 .

Original: When ρ ∈ (0 , 1) is close to one, i.e. log(1 + σ − 2 (1 − ρ D max ))
Revised: When ρ ∈ (0 , 1) is close to one, i.e. log(1 + σ − 2 (1 − ρ D max ))

Original: ≈ 0 , we can observe that even in the finite-time regime, the regret is almost sublinear since it is dominated by log(1 + σ − 2 (1 + ( T − 1) ρ D max )) .
Revised: ≈ 0 , we can observe that even in the finite-time regime, the regret is almost sublinear since it is dominated by log(1 + σ − 2 (1 + ( T − 1) ρ D max )) .

Original: In this case, the theorem provides a bound which is significantly tighter than the bound in Thm.
Revised: In this case, the theorem provides a bound which is significantly tighter than the bound in Thm.

Original: 3.10 even in the finite-time regime.
Revised: 3.10 even in the finite-time regime.

Original: Remark 3.14 .
Revised: Remark 3.14 .

Original: If λ max is bounded, Thm.
Revised: If λ max is bounded, Thm.

Original: 3.10 can show the vanishing simple regret.
Revised: 3.10 can show the vanishing simple regret.

Original: For a kernel
Revised: For a kernel

Original: K on a finite space X , λ max ≤ trace ( K ( X , X )) <
Revised: K on a finite space X , λ max ≤ trace ( K ( X , X )) <

Original: ∞ .
Revised: ∞ .

Original: Therefore, γ T =
Revised: Therefore, γ T =

Original: O ( log ( T )) for any kernel.
Revised: O ( log ( T )) for any kernel.

Original: However, considering the magnitude of | X | and λ max for large spaces, Eq.
Revised: However, considering the magnitude of | X | and λ max for large spaces, Eq.

Original: 7is quite loose.
Revised: 7is quite loose.

Original: λ max in
Revised: λ max in

Original: Eq.
Revised: Eq.

Original: 7reflects kernel-dependent behavior of γ T .
Revised: 7reflects kernel-dependent behavior of γ T .

Original: Therefore, in Thm.
Revised: Therefore, in Thm.

Original: 3.12 we bound λ max for a specific kernel and analyze further kernel-dependent non-asymptotic behavior.
Revised: 3.12 we bound λ max for a specific kernel and analyze further kernel-dependent non-asymptotic behavior.

Original: The regret bounds of LAW are most informative in the asymptotic regime of large T .
Revised: The regret bounds of LAW are most informative in the asymptotic regime of large T .

Original: However, in
Revised: However, in

Original: Bayesian optimization where, typically, only a small number of evaluations can be afforded, the asymptotic bound may not be informative in terms of practical performance.
Revised: Bayesian optimization where, typically, only a small number of evaluations can be afforded, the asymptotic bound may not be informative in terms of practical performance.

Original: In Sec. 5, we show that, in practice, LAW significantly outperforms other methods.
Revised: In Sec. 5, we show that, in practice, LAW significantly outperforms other methods.

Original:  Related work
Revised:  Related work

Original: Most existing batch Bayesian optimization methods using Gaussian process surrogate models focus on continuous search spaces.
Revised: Most existing batch Bayesian optimization methods using Gaussian process surrogate models focus on continuous search spaces.

Original: Many of them are not applicable to combinatoral spaces because the algorithms use specific properties of Euclidean spaces, e.g, Euclidean distance [AFF10,
Revised: Many of them are not applicable to combinatoral spaces because the algorithms use specific properties of Euclidean spaces, e.g, Euclidean distance [AFF10,

Original: GDHL16, WF16, WZJ16, KDK16, LYY + 18], grid partitioning [WLJK17, WGKJ18], projection using Euclidean geometry [WCLF20].
Revised: GDHL16, WF16, WZJ16, KDK16, LYY + 18], grid partitioning [WLJK17, WGKJ18], projection using Euclidean geometry [WCLF20].

Original: The methods [SG15, KKSP18, GPL19] using Thompson sampling (TS) [Tho33, WBT + 20], random feature [RR + 07] or entropy search [HS12, HLHG14] require either closed-form expression of eigenfunctions or Choleksy decomposition of the gram matrix on all points in the search space.
Revised: The methods [SG15, KKSP18, GPL19] using Thompson sampling (TS) [Tho33, WBT + 20], random feature [RR + 07] or entropy search [HS12, HLHG14] require either closed-form expression of eigenfunctions or Choleksy decomposition of the gram matrix on all points in the search space.

Original: In general, a closed-form expression of eigenfunctions (RBF) is not available.
Revised: In general, a closed-form expression of eigenfunctions (RBF) is not available.

Original: For large combinatorial spaces, Choleksy decomposition is infeasible.
Revised: For large combinatorial spaces, Choleksy decomposition is infeasible.

Original: LAW is a batch acquisition method applicable to general spaces including permutation spaces.
Revised: LAW is a batch acquisition method applicable to general spaces including permutation spaces.

Original: Recently, BO on combinatorial spaces has made significant progress for categorical variables [BP18,
Revised: Recently, BO on combinatorial spaces has made significant progress for categorical variables [BP18,

Original: DBDF20, OTGW19, SRDM20, DSR + 20].
Revised: DBDF20, OTGW19, SRDM20, DSR + 20].

Original: However, relatively few works in Bayesian optimization have focused on permutations[ZSBB14, ZYLW19, BBG + 20].
Revised: However, relatively few works in Bayesian optimization have focused on permutations[ZSBB14, ZYLW19, BBG + 20].

Original: While existing works focus on the effect of the kernel on performance, our focus is to scale up Bayesian optimization on permutations via batch acquisition, which has not been studied in previous works.
Revised: While existing works focus on the effect of the kernel on performance, our focus is to scale up Bayesian optimization on permutations via batch acquisition, which has not been studied in previous works.

Original: The application of determinantal point processes (DPPs) to Bayesian optimization is not new.
Revised: The application of determinantal point processes (DPPs) to Bayesian optimization is not new.

Original: The use of DPP and the regret analysis on continuous search spaces[KDK16] is closely related to our work.
Revised: The use of DPP and the regret analysis on continuous search spaces[KDK16] is closely related to our work.

Original: We focus on optimization problems on permutations rather than continuous spaces and use acquisition weighted kernels in our DPP.
Revised: We focus on optimization problems on permutations rather than continuous spaces and use acquisition weighted kernels in our DPP.

Original: We provide a regret bound, which includes the unweighted case as a special case.
Revised: We provide a regret bound, which includes the unweighted case as a special case.

Original: Moreover, we show the behavior of the information gain of the position kernel, which, in turn, helps to understand the behavior of BO on permutations.
Revised: Moreover, we show the behavior of the information gain of the position kernel, which, in turn, helps to understand the behavior of BO on permutations.

Original: The idea of using weighted kernels was investigated in DPP [KT10, KT12], also recently in the context of active learning[BWAS19] and more recently in architecture search [NLYAO21].
Revised: The idea of using weighted kernels was investigated in DPP [KT10, KT12], also recently in the context of active learning[BWAS19] and more recently in architecture search [NLYAO21].

Original: In addition to the use of the acquisition weights, we provide a regret analysis and the bound on the information gain of the position kernel for BO on permutations.
Revised: In addition to the use of the acquisition weights, we provide a regret analysis and the bound on the information gain of the position kernel for BO on permutations.

Original: In existing works on regret analysis of batch Bayesian optimization, the cumulative regret is analyzed as an end goal[DKB14] and as a medium to show vanishing simple regret[KKSP18].
Revised: In existing works on regret analysis of batch Bayesian optimization, the cumulative regret is analyzed as an end goal[DKB14] and as a medium to show vanishing simple regret[KKSP18].

Original: On the other hand, we analyze the batch cumulative regret not the cumulative regret (see Def. 3.3 and remarks thereafter).
Revised: On the other hand, we analyze the batch cumulative regret not the cumulative regret (see Def. 3.3 and remarks thereafter).

Original: The batch cumulative regret is analyzed in [CBRV13] but without the acquisition weight.
Revised: The batch cumulative regret is analyzed in [CBRV13] but without the acquisition weight.

Original: More detailed discussion on the difference among all these analyses is provided in Supp.
Revised: More detailed discussion on the difference among all these analyses is provided in Supp.

Original: Subsec.
Revised: Subsec.

Original: A.2.
Revised: A.2.

Original:  Experiments
Revised:  Experiments

Original: We empirically demonstrate the benefit of LAW on many optimization problems on permutations.
Revised: We empirically demonstrate the benefit of LAW on many optimization problems on permutations.

Original: In all Gaussian process (GP) based BO including baselines, we use the position kernel (see.
Revised: In all Gaussian process (GP) based BO including baselines, we use the position kernel (see.

Original: Subsec. 3.3).
Revised: Subsec. 3.3).

Original: At each round, evaluation outputs are normalized.
Revised: At each round, evaluation outputs are normalized.

Original: GP surrogate models are trained with output normalized evaluation data by optimizing the marginal likelihood until convergence withdifferent random initializations.
Revised: GP surrogate models are trained with output normalized evaluation data by optimizing the marginal likelihood until convergence withdifferent random initializations.

Original: We use the Adam optimizer[KB14] with default PyTorch[PGC + 17] settings except for the learning rate of 0 . 1 .
Revised: We use the Adam optimizer[KB14] with default PyTorch[PGC + 17] settings except for the learning rate of 0 . 1 .

Original: When the optimization is performed on a single permutation variable, for example in greedy optimization, hill climbing is used until convergence and the neighbors are defined as the set of permutations obtainable by swapping two locations.
Revised: When the optimization is performed on a single permutation variable, for example in greedy optimization, hill climbing is used until convergence and the neighbors are defined as the set of permutations obtainable by swapping two locations.

Original: 5.1 Weight function
Revised: 5.1 Weight function

Original: The motivation of the acquisition weight is to promote the quality of the queries in the batch by using acquisition weights.
Revised: The motivation of the acquisition weight is to promote the quality of the queries in the batch by using acquisition weights.

Original: In order to reflect this motivation, the weight function should be monotonically increasing.
Revised: In order to reflect this motivation, the weight function should be monotonically increasing.

Original: The better the quality (acquisition value) is, the larger the batch acquisition objective is.
Revised: The better the quality (acquisition value) is, the larger the batch acquisition objective is.

Original: In Eq. 2, the batch acquisition objective is factorized into the DPP with the diversity gauge and the product of weights, thus the weight function should be positive to prevent the product of an even number of large negative values becomes a large positive value.
Revised: In Eq. 2, the batch acquisition objective is factorized into the DPP with the diversity gauge and the product of weights, thus the weight function should be positive to prevent the product of an even number of large negative values becomes a large positive value.

Original: Also, in Eq. 2, zero weight value nullifies the diversity component, thus the weights function is required to be nonzero.
Revised: Also, in Eq. 2, zero weight value nullifies the diversity component, thus the weights function is required to be nonzero.

Original: In Thm.
Revised: In Thm.

Original: 3.6, for vanishing regret, the ratio w + w − should be upper-bounded.
Revised: 3.6, for vanishing regret, the ratio w + w − should be upper-bounded.

Original: Not only it facilitates the proof, but the upper bound is also intuitively appealing because we do not want to overly emphasize the quality of the queries.
Revised: Not only it facilitates the proof, but the upper bound is also intuitively appealing because we do not want to overly emphasize the quality of the queries.

Original: We want to balance quality and diversity.
Revised: We want to balance quality and diversity.

Original: Weight functions with an unbounded ratio may erase the diversity consideration.
Revised: Weight functions with an unbounded ratio may erase the diversity consideration.

Original: Combining the rationale behind LAW and its regret analysis, we set the weight function to be, monotonically increasing, positive valued, bounded below, bounded above.
Revised: Combining the rationale behind LAW and its regret analysis, we set the weight function to be, monotonically increasing, positive valued, bounded below, bounded above.

Original: 5.2 Combinatorial Optimization Benchmarks
Revised: 5.2 Combinatorial Optimization Benchmarks

Original: We consider LAW with two acquisition functions 4 , EST[WZJ16] and EI[JSW98], LAW-EST and
Revised: We consider LAW with two acquisition functions 4 , EST[WZJ16] and EI[JSW98], LAW-EST and

Original: LAW-EI.
Revised: LAW-EI.

Original: Even though the regret bound of LAW-EI is not provided in Thm.
Revised: Even though the regret bound of LAW-EI is not provided in Thm.

Original: 3.6, we include LAW-EI because EI is the most popular acquisition function and this reveals the effect of the acquisition weights with different acquisition functions.
Revised: 3.6, we include LAW-EI because EI is the most popular acquisition function and this reveals the effect of the acquisition weights with different acquisition functions.

Original: We use the sigmoid w ( a ) = 0 . 01 + 0 .
Revised: We use the sigmoid w ( a ) = 0 . 01 + 0 .

Original: 99(1 + exp( − 0 .
Revised: 99(1 + exp( − 0 .

Original: 2 · a )) −for LAW-EST and w ( a ) = 0 .
Revised: 2 · a )) −for LAW-EST and w ( a ) = 0 .

Original: + a for LAW-EI.
Revised: + a for LAW-EI.

Original: The baselines are q-EI, q-EST[GLRC08], BUCB[DKB14], DPP-MAX-EST, DPP-SAMPLEEST[KDK16] 6 and MACE-UCB, MACE-EST[LYY + 18] 7 .
Revised: The baselines are q-EI, q-EST[GLRC08], BUCB[DKB14], DPP-MAX-EST, DPP-SAMPLEEST[KDK16] 6 and MACE-UCB, MACE-EST[LYY + 18] 7 .

Original: Even though the original names of the baselines are used to emphasize their batch acquisition strategy, all baselines use the position kernel.
Revised: Even though the original names of the baselines are used to emphasize their batch acquisition strategy, all baselines use the position kernel.

Original: Hence, the batch acquisition strategy is the only differentiating factor among baselines and
Revised: Hence, the batch acquisition strategy is the only differentiating factor among baselines and

Original: LAW(ours).
Revised: LAW(ours).

Original: Note that DPP-MAX-EST [KDK16] corresponds to LAW-EST with w ( · ) ≡ const.
Revised: Note that DPP-MAX-EST [KDK16] corresponds to LAW-EST with w ( · ) ≡ const.

Original: , i.e., no acquisition weight.
Revised: , i.e., no acquisition weight.

Original: The β t in UCB balancing between exploitation and exploration increases as the size of the search space increases in the finite search space case[SKKS09].
Revised: The β t in UCB balancing between exploitation and exploration increases as the size of the search space increases in the finite search space case[SKKS09].

Original: In the experiments, due to the size of the search space,
Revised: In the experiments, due to the size of the search space,

Original: GP-UCB virtually becomes the predictive variance.
Revised: GP-UCB virtually becomes the predictive variance.

Original: Thus we exclude LAW-UCB.
Revised: Thus we exclude LAW-UCB.

Original: LAW-EI is included to check the influence of different acquisition functions despite the lack of regret analysis.
Revised: LAW-EI is included to check the influence of different acquisition functions despite the lack of regret analysis.

Original: Therefore, the weight function is chosen to prevent zero values from numerical truncation.
Revised: Therefore, the weight function is chosen to prevent zero values from numerical truncation.

Original: 6 PE[CBRV13] is equivalent to DPP-MAX-UCB[KDK16].
Revised: 6 PE[CBRV13] is equivalent to DPP-MAX-UCB[KDK16].

Original: Since on continuous problems DPP-MAX-EST outperforms DPP-MAX-UCB[KDK
Revised: Since on continuous problems DPP-MAX-EST outperforms DPP-MAX-UCB[KDK

Original: [Table]
Revised: [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: Note that, due to the reasons discussed in Sec. 4, existing works based on Thompson sampling or the properties of Euclidean space are excluded from the baselines.
Revised: Note that, due to the reasons discussed in Sec. 4, existing works based on Thompson sampling or the properties of Euclidean space are excluded from the baselines.

Original: We consider three types of combinatorial optimization on permutations, Quadratic Assignment
Revised: We consider three types of combinatorial optimization on permutations, Quadratic Assignment

Original: Problems(QAP), Flowshop Scheduling Problems(FSP) and Traveling Salesman Problems(TSP)
Revised: Problems(QAP), Flowshop Scheduling Problems(FSP) and Traveling Salesman Problems(TSP)

Original: (See
Revised: (See

Original: Supp.
Revised: Supp.

Original: Subsec.
Revised: Subsec.

Original: F.3 for data source).
Revised: F.3 for data source).

Original: For each benchmark, all methods share 5 randomly generated initial evaluation data sets of 20 points and for each initial evaluation data set, each method is run three times — 15 runs in total.
Revised: For each benchmark, all methods share 5 randomly generated initial evaluation data sets of 20 points and for each initial evaluation data set, each method is run three times — 15 runs in total.

Original: DPP-MAX-EST uses the position kernel as LAW-EST, this is equivalent to LAW-EST without the acquisition weight, i.e. w ( a ) = 1 .
Revised: DPP-MAX-EST uses the position kernel as LAW-EST, this is equivalent to LAW-EST without the acquisition weight, i.e. w ( a ) = 1 .

Original: By comparing LAW-EST with DPP-MAX-EST, we can directly evaluate the benefits of using the acquisition weight.
Revised: By comparing LAW-EST with DPP-MAX-EST, we can directly evaluate the benefits of using the acquisition weight.

Original: As shown in Tab. 5.2, LAW-EI, LAW-EST, q-EI and q-EST are in top four except for FSP-car and TSP-att48.
Revised: As shown in Tab. 5.2, LAW-EI, LAW-EST, q-EI and q-EST are in top four except for FSP-car and TSP-att48.

Original: LAW-EST performs the best on FSP and TSP while q-EI or q-EST perform the best on QAP.
Revised: LAW-EST performs the best on FSP and TSP while q-EI or q-EST perform the best on QAP.

Original: Along with the experiment on the structure learning (Subsec. 5.3), we conjecture that
Revised: Along with the experiment on the structure learning (Subsec. 5.3), we conjecture that

Original: QAP has a certain structure more friendly to q-EI and q-EST.
Revised: QAP has a certain structure more friendly to q-EI and q-EST.

Original: Also LAW-EI exhibits comparable performance with q-EI and q-EST while outperforming other baselines, which supports the benefit of the acquisition weight.
Revised: Also LAW-EI exhibits comparable performance with q-EI and q-EST while outperforming other baselines, which supports the benefit of the acquisition weight.

Original: In terms of the average rank over all benchmarks, LAW-EST performs the best with the average rank of 1 . 89 against q-EST ( 2 . 44 ) and q-EI ( 2 . 78 ).
Revised: In terms of the average rank over all benchmarks, LAW-EST performs the best with the average rank of 1 . 89 against q-EST ( 2 . 44 ) and q-EI ( 2 . 78 ).

Original: Overall, among the baselines, LAW-EST exhibits stable and competitive performance across different benchmarks.
Revised: Overall, among the baselines, LAW-EST exhibits stable and competitive performance across different benchmarks.

Original: 5.2.1 Comparison to the local penalization (LP)
Revised: 5.2.1 Comparison to the local penalization (LP)

Original: Two additional variants of LAW, LAW-PRIOR-EST and LAW-PRIOR-EI, are also compared (Supp.
Revised: Two additional variants of LAW, LAW-PRIOR-EST and LAW-PRIOR-EI, are also compared (Supp.

Original: Sec.
Revised: Sec.

Original: G), which use the prior covariance function as the diversity gauge, L = K , of L AWt .
Revised: G), which use the prior covariance function as the diversity gauge, L = K , of L AWt .

Original: These variants do not use evaluation data in the diversity gauge.
Revised: These variants do not use evaluation data in the diversity gauge.

Original: [Table]
Revised: [Table]

Original: Interestingly, LAW-PRIOR-EST and LAW-PRIOR-EI resemble LP[GDHL16] (Supp.
Revised: Interestingly, LAW-PRIOR-EST and LAW-PRIOR-EI resemble LP[GDHL16] (Supp.

Original: Sec. C), and thus this allows an indirect comparison to LP which is not applicable to combinatorial spaces.
Revised: Sec. C), and thus this allows an indirect comparison to LP which is not applicable to combinatorial spaces.

Original: These variants using the prior covariance function performs worse than LAW using the posterior covariance function, which is natural since using more data for the diversity gauge enhances the performance.
Revised: These variants using the prior covariance function performs worse than LAW using the posterior covariance function, which is natural since using more data for the diversity gauge enhances the performance.

Original: More importantly, LAW-PRIOR-EST and LAW-PRIOR-EI outperform DPP-MAXEST which uses the posterior covariance function without the acquisition weight, which supports that the acquisition weight is key in the performance improvement
Revised: More importantly, LAW-PRIOR-EST and LAW-PRIOR-EI outperform DPP-MAXEST which uses the posterior covariance function without the acquisition weight, which supports that the acquisition weight is key in the performance improvement

Original: [Table]
Revised: [Table]

Original: Based on the empirical analysis above, we choose LAW-EST as our final recommendation, which we call LAW2ORDER.
Revised: Based on the empirical analysis above, we choose LAW-EST as our final recommendation, which we call LAW2ORDER.

Original: 5.3 Structure Learning
Revised: 5.3 Structure Learning

Original: We apply LAW2ORDER to the score-based structure learning problem[DM17].
Revised: We apply LAW2ORDER to the score-based structure learning problem[DM17].

Original: Existing score-based methods assume a computationally amenable structure of the score to be optimized (decomposability)[KF09, SGG19].
Revised: Existing score-based methods assume a computationally amenable structure of the score to be optimized (decomposability)[KF09, SGG19].

Original: Distinctively our approach does not necessitate the decomposability of the score to be optimized.
Revised: Distinctively our approach does not necessitate the decomposability of the score to be optimized.

Original: We consider the NML score as below
Revised: We consider the NML score as below

Original: [Equation]
Revised: [Equation]

Original: where p BN ( ·| G , θ ) is the density of Bayesian Network (BN) with DAG G and the parameter θ , and
Revised: where p BN ( ·| G , θ ) is the density of Bayesian Network (BN) with DAG G and the parameter θ , and

Original: REG NML is the normalized marginal likelihood (NML) which is a complexity measure from the minimum description length principle[GG07].
Revised: REG NML is the normalized marginal likelihood (NML) which is a complexity measure from the minimum description length principle[GG07].

Original: NML is not decomposable, and thus the methods assuming a decomposable score are not applicable.
Revised: NML is not decomposable, and thus the methods assuming a decomposable score are not applicable.

Original: Since it is infeasible to compute NML exactly and we resort on MC estimate, the NML score evaluation is noisy.
Revised: Since it is infeasible to compute NML exactly and we resort on MC estimate, the NML score evaluation is noisy.

Original: In addition to not being decomposable, noisy evaluation also makes existing methods inapplicable to the NML score objective.
Revised: In addition to not being decomposable, noisy evaluation also makes existing methods inapplicable to the NML score objective.

Original: For more details of NML and its MC estimate, see Supp.
Revised: For more details of NML and its MC estimate, see Supp.

Original: Sec.
Revised: Sec.

Original: E.
Revised: E.

Original: Similarly to [SWU17, RU18] we search over permutations specifying topological order of DAGs and the existence of edges is determined by the conditional independence test.
Revised: Similarly to [SWU17, RU18] we search over permutations specifying topological order of DAGs and the existence of edges is determined by the conditional independence test.

Original: In addition to qEI and q-EST performing well on the benchmarks, we compare LAW2ORDER with the genetic algorithm (GA), which is one of the most popular choices for optimization problems on permutations including TSP[Pot96].
Revised: In addition to qEI and q-EST performing well on the benchmarks, we compare LAW2ORDER with the genetic algorithm (GA), which is one of the most popular choices for optimization problems on permutations including TSP[Pot96].

Original: We generated 5 sets of 20 random initial points.
Revised: We generated 5 sets of 20 random initial points.

Original: LAW2ORDER, q-EI and q-EST is run on each of these 5 sets using a batch size 20.
Revised: LAW2ORDER, q-EI and q-EST is run on each of these 5 sets using a batch size 20.

Original: Assuming the same resource constraint (at most 20 evaluations in parallel), GA generates 20 off-springs in each generation.
Revised: Assuming the same resource constraint (at most 20 evaluations in parallel), GA generates 20 off-springs in each generation.

Original: GA is also run 5 times with a population size of 100 points using Pymoo[BD20].
Revised: GA is also run 5 times with a population size of 100 points using Pymoo[BD20].

Original: The first 20 points of each initial population in the 5 runs are equal to the 20 initial points used in LAW2ORDER, q-EI and q-EST.
Revised: The first 20 points of each initial population in the 5 runs are equal to the 20 initial points used in LAW2ORDER, q-EI and q-EST.

Original: Even though the real deployment of Bayesian optimization assumes that the cost of evaluation is expensive enough to render the time to acquire new batches negligible, as a stress test, we allowed twice the evaluation budget for GA.
Revised: Even though the real deployment of Bayesian optimization assumes that the cost of evaluation is expensive enough to render the time to acquire new batches negligible, as a stress test, we allowed twice the evaluation budget for GA.

Original: On data generated from 4 real-world BNs[Scu10, SGG19], the results are reported in Tab. 2.
Revised: On data generated from 4 real-world BNs[Scu10, SGG19], the results are reported in Tab. 2.

Original: LAW2ORDER outperforms q-EI and q-EST with a significant margin except for Child where all three find the putative optimum quickly.
Revised: LAW2ORDER outperforms q-EI and q-EST with a significant margin except for Child where all three find the putative optimum quickly.

Original: Still, in Child, LAW2ORDER finds a point of negligible differences with the putative optimum the most quickly.
Revised: Still, in Child, LAW2ORDER finds a point of negligible differences with the putative optimum the most quickly.

Original: Also, except for Sachs, we observe that the performance gap increases as the permutation size (the size of the search space) increases.
Revised: Also, except for Sachs, we observe that the performance gap increases as the permutation size (the size of the search space) increases.

Original: On this realistic problem, our argument that LAW2ORDER is stable and efficient batch acquisition method on permutations is reinforced.
Revised: On this realistic problem, our argument that LAW2ORDER is stable and efficient batch acquisition method on permutations is reinforced.

Original: In comparison with GA, we consider GA(620) with the same evaluation budget and GA(1240) with twice large evaluation budget.
Revised: In comparison with GA, we consider GA(620) with the same evaluation budget and GA(1240) with twice large evaluation budget.

Original: LAW2ORDER dominates GA(620) in all problems with a significant margin.
Revised: LAW2ORDER dominates GA(620) in all problems with a significant margin.

Original: Even compared with GA(1240), LAW2ORDER significantly outperforms except for Sachs which has one of the smallest search spaces (See Tab. 2 and Supp.
Revised: Even compared with GA(1240), LAW2ORDER significantly outperforms except for Sachs which has one of the smallest search spaces (See Tab. 2 and Supp.

Original: Sec.
Revised: Sec.

Original: G).
Revised: G).

Original: Contrary to our expectation that BO has the sample efficiency higher than GA, GA(620) outperforms q-EI and q-EST on Sachs.
Revised: Contrary to our expectation that BO has the sample efficiency higher than GA, GA(620) outperforms q-EI and q-EST on Sachs.

Original: LAW2ORDER shows robust performance even in the problem where the performance of q-EI and q-EST is degraded.
Revised: LAW2ORDER shows robust performance even in the problem where the performance of q-EI and q-EST is degraded.

Original: Together with the experiments on the benchmarks, LAW2ORDER is shown to be a robust batch acquisition method on permutations.
Revised: Together with the experiments on the benchmarks, LAW2ORDER is shown to be a robust batch acquisition method on permutations.

Original: Moreover, promoting the diversity in batches while taking into account the acquisition weight appears more beneficial with larger batch size (20) as shown in the structure learning experiment.
Revised: Moreover, promoting the diversity in batches while taking into account the acquisition weight appears more beneficial with larger batch size (20) as shown in the structure learning experiment.

Original:  Conclusion
Revised:  Conclusion

Original: In this paper we have focused on combinatorial optimization problems over permutations where each evaluation is assumed to be expensive.
Revised: In this paper we have focused on combinatorial optimization problems over permutations where each evaluation is assumed to be expensive.

Original: This class of problems has many interesting applications, ranging from chip design (where we wish to place cells while minimizing area and wire-length), warehouse optimization (where we need to order the retrieval of items from a warehouse using a robot), neural architecture search and so on.
Revised: This class of problems has many interesting applications, ranging from chip design (where we wish to place cells while minimizing area and wire-length), warehouse optimization (where we need to order the retrieval of items from a warehouse using a robot), neural architecture search and so on.

Original: In spite of its practical significance, BO on permutations is under-explored in contrast to the recent progress on combinatorial BO with categorical variables.
Revised: In spite of its practical significance, BO on permutations is under-explored in contrast to the recent progress on combinatorial BO with categorical variables.

Original: In response to this, we have proposed a batch Bayesian optimization algorithm on permutations,
Revised: In response to this, we have proposed a batch Bayesian optimization algorithm on permutations,

Original: LAW2ORDER, which uses an extension of the determinantal point processes with the acquisition weighted kernel.
Revised: LAW2ORDER, which uses an extension of the determinantal point processes with the acquisition weighted kernel.

Original: This allows the search process over the surrogate function to make optimal use of all parallel available computational resources and be guided by both the expected objective value and its posterior uncertainty.
Revised: This allows the search process over the surrogate function to make optimal use of all parallel available computational resources and be guided by both the expected objective value and its posterior uncertainty.

Original: On the theory side we offer a regret analysis, which shows that the regret bound of LAW enjoys the same asymptotic behavior as existing methods.
Revised: On the theory side we offer a regret analysis, which shows that the regret bound of LAW enjoys the same asymptotic behavior as existing methods.

Original: On the empirical side, we show that LAW variants consistently exhibit competitive performance on a wide range of combinatorial optimization tasks, including a challenging structure learning problem.
Revised: On the empirical side, we show that LAW variants consistently exhibit competitive performance on a wide range of combinatorial optimization tasks, including a challenging structure learning problem.

Original: From these we conclude that the acquisition weights are indeed a key factor in the success of the proposed method, and that the performance gains increase for large batch sizes.
Revised: From these we conclude that the acquisition weights are indeed a key factor in the success of the proposed method, and that the performance gains increase for large batch sizes.

Revised: LAW is applicable to general search spaces for which a kernel can be defined.

Revised: We leave the exploration of our method to applications outside searching over permutations for future work.

Original: 6.1 Limitations
Revised: 6.1 Limitations

Original: LAW achieves improved sample efficiency in the sense that the quality of batches from LAW is maintained for large batch sizes.
Revised: LAW achieves improved sample efficiency in the sense that the quality of batches from LAW is maintained for large batch sizes.

Original: However, due to the sequential nature of the greedy maximization of LAW objective, its computational complexity is linear with respect to the batch size.
Revised: However, due to the sequential nature of the greedy maximization of LAW objective, its computational complexity is linear with respect to the batch size.

Original: Especially for large permutation spaces, this may be a nonnegligible cost.
Revised: Especially for large permutation spaces, this may be a nonnegligible cost.

Original: We hope that the sample efficiency of LAW is complemented by the computational efficiency allowing massive parallelization.
Revised: We hope that the sample efficiency of LAW is complemented by the computational efficiency allowing massive parallelization.

Original: Even though the regret bound in Thm.
Revised: Even though the regret bound in Thm.

Original: 3.6 describes the effect of the acquisition weight, as mentioned in Rmk.
Revised: 3.6 describes the effect of the acquisition weight, as mentioned in Rmk.

Original: 3.9, it only sheds light on the demerit of excessive emphasis on the acquisition weights but it does not detect the demerit of considering the diversity only.
Revised: 3.9, it only sheds light on the demerit of excessive emphasis on the acquisition weights but it does not detect the demerit of considering the diversity only.

Original: We hope our work inspires the regret bound for acquisition methods taking into account properties other than diversity.
Revised: We hope our work inspires the regret bound for acquisition methods taking into account properties other than diversity.

Original: A Regret Analysis
Revised: A Regret Analysis

Original: In this section, we show that LAW with GP-UCB or EST has the vanishing simple regret with high probability.
Revised: In this section, we show that LAW with GP-UCB or EST has the vanishing simple regret with high probability.

Original: In Bayesian optimization (BO), the goal is to find a minimum for a given objective f
Revised: In Bayesian optimization (BO), the goal is to find a minimum for a given objective f

Original: [Equation]
Revised: [Equation]

Original: First, we introduce different types of regret.
Revised: First, we introduce different types of regret.

Original: Our analysis on the vanishing simple regret of LAW only requires batch version of all regrets below.
Revised: Our analysis on the vanishing simple regret of LAW only requires batch version of all regrets below.

Original: Therefore, the proof for the vanishing simple regret can be read without referring to sequential version of regrets below.
Revised: Therefore, the proof for the vanishing simple regret can be read without referring to sequential version of regrets below.

Original: The sequential version definitions are used when we contrast our regret analysis with the regret analysis in existing works [DKB14,
Revised: The sequential version definitions are used when we contrast our regret analysis with the regret analysis in existing works [DKB14,

Original: KKSP18].
Revised: KKSP18].

Original: We begin with two equivalent round indexing in the batch setting , the sequential indexing, an 1-tuple and the batch indexing, an ordered 2-tuple which are related via following mappings.
Revised: We begin with two equivalent round indexing in the batch setting , the sequential indexing, an 1-tuple and the batch indexing, an ordered 2-tuple which are related via following mappings.

Original: [Equation]
Revised: [Equation]

Original: The batch indexing is primarily used and the sequential indexing is expressed via T ( B ) seq .
Revised: The batch indexing is primarily used and the sequential indexing is expressed via T ( B ) seq .

Original: With two indexing, we have batch and sequential versions of regret definitions with the instantaneous regret r t,b = f ( x ⋆ ) − f ( x t,b ) at a query point x t,b .
Revised: With two indexing, we have batch and sequential versions of regret definitions with the instantaneous regret r t,b = f ( x ⋆ ) − f ( x t,b ) at a query point x t,b .

Original: In the case of a noisy objective, y t,b = f ( x t,b ) + ϵ
Revised: In the case of a noisy objective, y t,b = f ( x t,b ) + ϵ

Original: Note that we use the definition of sequential simple/cumulative regret in the context of batch BO.
Revised: Note that we use the definition of sequential simple/cumulative regret in the context of batch BO.

Original: Since sequential simple regret is equal to batch simple regret, we call both simple regret without prefixes.
Revised: Since sequential simple regret is equal to batch simple regret, we call both simple regret without prefixes.

Original: In [CBRV13], sequential cumulative regret is termed full cumulative regret to contrast with batch cumulative regret.
Revised: In [CBRV13], sequential cumulative regret is termed full cumulative regret to contrast with batch cumulative regret.

Original: The simple regret is in accord with the goal of BO[KKSP18] while the cumulative regret is prevalent in bandit[LS20].
Revised: The simple regret is in accord with the goal of BO[KKSP18] while the cumulative regret is prevalent in bandit[LS20].

Original: [Table]
Revised: [Table]

Original: Between Simple
Revised: Between Simple

Original: [Table]
Revised: [Table]

Original: Between Cumulative
Revised: Between Cumulative

Original: [Table]
Revised: [Table]

Original: When an algorithm exhibits that cumulative regret averaged over rounds converges to zero, then the algorithm is called no regret.
Revised: When an algorithm exhibits that cumulative regret averaged over rounds converges to zero, then the algorithm is called no regret.

Original: As stated in Table 3, simple regret is bounded above by cumulative regret averaged over rounds, therefore, vanishing simple regret is often proved by showing that the algorithm is no regret[KKSP18].
Revised: As stated in Table 3, simple regret is bounded above by cumulative regret averaged over rounds, therefore, vanishing simple regret is often proved by showing that the algorithm is no regret[KKSP18].

Original: In batch BO, there are two types of query depending on the accessible information.
Revised: In batch BO, there are two types of query depending on the accessible information.

Original: Non-delayed query point uses all previous query points with all corresponding evaluations, e.g. { x t, 1 } t ∈
Revised: Non-delayed query point uses all previous query points with all corresponding evaluations, e.g. { x t, 1 } t ∈

Original: [ T ] in
Revised: [ T ] in

Original: LAW while delayed query point uses all previous query points but some of evaluations for previous query points are not used, e.g. { x t,b } t ∈ [ T ] ,b =2 , ··· ,B in LAW.
Revised: LAW while delayed query point uses all previous query points but some of evaluations for previous query points are not used, e.g. { x t,b } t ∈ [ T ] ,b =2 , ··· ,B in LAW.

Original: For evaluation, instantaneous regret and posterior variance, we can say non-delayed and delayed according to the query point with which it is defined.
Revised: For evaluation, instantaneous regret and posterior variance, we can say non-delayed and delayed according to the query point with which it is defined.

Original: The our analysis consists of steps below
Revised: The our analysis consists of steps below

Original: Bound batch cumulative regret with the sum of non-delayed instantaneous regrets, i.e, the regrets from the first points in each batch
Revised: Bound batch cumulative regret with the sum of non-delayed instantaneous regrets, i.e, the regrets from the first points in each batch

Original: [Equation]
Revised: [Equation]

Original: Bound non-delayed instantaneous regrets with non-delayed posterior variance,
Revised: Bound non-delayed instantaneous regrets with non-delayed posterior variance,

Original: i
Revised: i

Original: e, posterior variance conditioned on all previous query points with their evaluations.
Revised: e, posterior variance conditioned on all previous query points with their evaluations.

Original: [Equation]
Revised: [Equation]

Original: η t depends on the acquisition function and the details are given in Theorem A.18.
Revised: η t depends on the acquisition function and the details are given in Theorem A.18.

Original: Bound non-delayed posterior variance with all posterior variance (Lemma A.4)
Revised: Bound non-delayed posterior variance with all posterior variance (Lemma A.4)

Original: [Equation]
Revised: [Equation]

Original: While the regret analysis on sequential cumulative regret [DKB14, KKSP18] 10 requires high probability confidence interval for r t,b for all t ∈
Revised: While the regret analysis on sequential cumulative regret [DKB14, KKSP18] 10 requires high probability confidence interval for r t,b for all t ∈

Original: [ T ] and b ∈ [ B ] , our analysis on batch cumulative regret requires high probability confidence interval for r t, 1 for all t ∈
Revised: [ T ] and b ∈ [ B ] , our analysis on batch cumulative regret requires high probability confidence interval for r t, 1 for all t ∈

Original: [ T ] .
Revised: [ T ] .

Original: More detailed discussion on the differences between two approaches is given after the proof (see Subsection A.2).
Revised: More detailed discussion on the differences between two approaches is given after the proof (see Subsection A.2).

Original: A.1 Vanishing simple regret of LAW
Revised: A.1 Vanishing simple regret of LAW

Original: In Bayesian optimization using LAW, the surrogate model is Gaussian processes with a kernel K .
Revised: In Bayesian optimization using LAW, the surrogate model is Gaussian processes with a kernel K .

Original: At t -th round of batch Bayesian optimization with the batch size of B , we have L AWt which defines
Revised: At t -th round of batch Bayesian optimization with the batch size of B , we have L AWt which defines

Original: L-ensembles of k-DPP.
Revised: L-ensembles of k-DPP.

Original: L AWt is obtained using the product of the predictive covariance function of K conditioned on D t − 1 =
Revised: L AWt is obtained using the product of the predictive covariance function of K conditioned on D t − 1 =

Original: ( x s,b , y s,b ) } s ∈ [ t − 1] ,b ∈
Revised: ( x s,b , y s,b ) } s ∈ [ t − 1] ,b ∈

Original: [ B ] and the acquisition function a t using the evaluation data D t − 1 as follows
Revised: [ B ] and the acquisition function a t using the evaluation data D t − 1 as follows

Original: [Equation]
Revised: [Equation]

Original: 13) where L t ( x , x ′ )
Revised: ) where L t ( x , x ′ )

Original: =
Revised: =

Original: K ( x , x ′ | D t − 1 ) is the diversity gauge and w : R → R is positive increasing, w − =
Revised: K ( x , x ′ | D t − 1 ) is the diversity gauge and w : R → R is positive increasing, w − =

Original: inf x ∈ R w ( x ) > 0 and w + = sup x ∈ R w ( x ) < ∞ , which we call the weight function .
Revised: inf x ∈ R w ( x ) > 0 and w + = sup x ∈ R w ( x ) < ∞ , which we call the weight function .

Original: The batch with B points x t, 1 , · · · , x t,B are acquired by
Revised: The batch with B points x t, 1 , · · · , x t,B are acquired by

Original: [Equation]
Revised: [Equation]

Original: 14) where L t ( x , x |{ x t,i } i ∈ [ b − 1] ) is the posterior variance of the kernel L t conditioned on { x t,i } i ∈
Revised: [Equation] where L t ( x , x |{ x t,i } i ∈ [ b − 1] ) is the posterior variance of the kernel L t conditioned on { x t,i } i ∈

Original: [ b − 1] .
Revised: [ b − 1] .

Original: Note that the posterior variance of the posterior covariance function K t conditioned on { x t,i } i ∈ [ b − 1] is equal to the posterior variance of the prior covariance function K conditioned on D t − 1 ∪{ x t,i }
Revised: Note that the posterior variance of the posterior covariance function K t conditioned on { x t,i } i ∈ [ b − 1] is equal to the posterior variance of the prior covariance function K conditioned on D t − 1 ∪{ x t,i }

Original: i ∈
Revised: i ∈

Original: [ b − 1] .
Revised: [ b − 1] .

Original: In the rest of the section, we use below simpler notation
Revised: In the rest of the section, we use below simpler notation

Original: [Equation]
Revised: [Equation]

Original: µ t ( x ) is the predictive mean conditioned on D t − 1 .
Revised: µ t ( x ) is the predictive mean conditioned on D t − 1 .

Original: [Equation]
Revised: [Equation]

Original: Note that σ 2 t − 1 ,b is well defined for b = 2 , · · · , B since the posterior variance does not depend on output values while the predictive mean is defined only when b = 1 where evaluated output y s,b exists for each x s,b in conditioning data.
Revised: Note that σ 2 t − 1 ,b is well defined for b = 2 , · · · , B since the posterior variance does not depend on output values while the predictive mean is defined only when b = 1 where evaluated output y s,b exists for each x s,b in conditioning data.

Original: We start with lemmas used in the regret bound analysis.
Revised: We start with lemmas used in the regret bound analysis.

Original: Lemma A.1.
Revised: Lemma A.1.

Original: Assume a kernel such that K ( · , · ) ≤ 1 .
Revised: Assume a kernel such that K ( · , · ) ≤ 1 .

Original: For each t ∈ [ T ] , LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )
Revised: For each t ∈ [ T ] , LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )

Original: =
Revised: =

Original: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (as defined below Eq. 13).
Revised: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (as defined below Eq. 13).

Original: The posterior variance defined as Eq. 15.
Revised: The posterior variance defined as Eq. 15.

Original: has the following relation
Revised: has the following relation

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Proof.
Revised: Proof.

Original: By the definition of x t,b x
Revised: By the definition of x t,b x

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: = argmax
Revised: = argmax

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: ) thus
Revised: ) thus

Original: [Equation]
Revised: [Equation]

Original: By the "Information never hurts" principle[KSG08], i.e. the posterior variance decreases as the conditioning set increases, we have
Revised: By the "Information never hurts" principle[KSG08], i.e. the posterior variance decreases as the conditioning set increases, we have

Original: [Equation]
Revised: [Equation]

Original: 21) since σ t is conditioned by D t
Revised: 21) since σ t is conditioned by D t

Original: =
Revised: =

Original: D t − 1 ∪{ x t,i } i ∈ [ B ] while σ t,b is conditioned by D t − 1 ∪{ x t,i } i ∈ [ b − 1] .
Revised: D t − 1 ∪{ x t,i } i ∈ [ B ] while σ t,b is conditioned by D t − 1 ∪{ x t,i } i ∈ [ b − 1] .

Original: Combining these two inequalities, we have
Revised: Combining these two inequalities, we have

Original: [Equation]
Revised: [Equation]

Original: (22) which also applies when x = x t +1 , 1 .
Revised: (22) which also applies when x = x t +1 , 1 .

Original: Remark A.2 .
Revised: Remark A.2 .

Original: LAW does not use the heuristic called the relevant region[CBRV13, KDK16], which makes the proof simpler compared with the Lemma 6.5 in [KDK16].
Revised: LAW does not use the heuristic called the relevant region[CBRV13, KDK16], which makes the proof simpler compared with the Lemma 6.5 in [KDK16].

Original: Remark A.3 .
Revised: Remark A.3 .

Original: The Lemma 6.5 in [KDK16] claims that the inequality similar to Eq.
Revised: The Lemma 6.5 in [KDK16] claims that the inequality similar to Eq.

Original: 19 and Eq.holds for sampling(DPP-SAMPLE).
Revised: 19 and Eq.holds for sampling(DPP-SAMPLE).

Original: However, such inequality relies on fact that x t,b is the maximum of an objective which is not guaranteed to hold for sampling(DPP-SAMPLE).
Revised: However, such inequality relies on fact that x t,b is the maximum of an objective which is not guaranteed to hold for sampling(DPP-SAMPLE).

Original: The regret analysis of
Revised: The regret analysis of

Original: DPP-SAMPLE in [KDK16] appears to need a revision.
Revised: DPP-SAMPLE in [KDK16] appears to need a revision.

Original: In our version, we do not make any claim in the case of sampling.
Revised: In our version, we do not make any claim in the case of sampling.

Original: Lemma A.4.
Revised: Lemma A.4.

Original: Assume a kernel such that K ( · , · ) ≤ 1 .
Revised: Assume a kernel such that K ( · , · ) ≤ 1 .

Original: For each t ∈ [ T ] , LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )
Revised: For each t ∈ [ T ] , LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )

Original: =
Revised: =

Original: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (as defined below Eq. 13).
Revised: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (as defined below Eq. 13).

Original: The posterior variance defined as Eq. 15.
Revised: The posterior variance defined as Eq. 15.

Original: has the following relation
Revised: has the following relation

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: From Lemma A.1, for b = 2 , · · · , B , we have
Revised: From Lemma A.1, for b = 2 , · · · , B , we have

Original: [Equation]
Revised: [Equation]

Original: Summing these for b = 2 , · · · , B and σ t − 1 , 1 ( x t,b )
Revised: Summing these for b = 2 , · · · , B and σ t − 1 , 1 ( x t,b )

Original: [Equation]
Revised: [Equation]

Original: since w − ≤ w + .
Revised: since w − ≤ w + .

Original: Summing this with respect to t , we have
Revised: Summing this with respect to t , we have

Original: T
Revised: T

Original: [Equation]
Revised: [Equation]

Original: The term on the left hand side can be rewritten
Revised: The term on the left hand side can be rewritten

Original: [Equation]
Revised: [Equation]

Original: Since ( B − 1)( σ 0 , 1 ( x 1 , 1 ) − σ T, 1 ( x T +1 , 1 )) ≤ Bσ 0 , 1 ( x 1 , 1 )
Revised: Since ( B − 1)( σ 0 , 1 ( x 1 , 1 ) − σ T, 1 ( x T +1 , 1 )) ≤ Bσ 0 , 1 ( x 1 , 1 )

Original: T (cid:88) t = σ
Revised: T (cid:88) t = σ

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: ( x t
Revised: t

Original: [Equation]
Revised: [Equation]

Original: T (cid:88)t
Revised: T (cid:88)

Original: [Equation]
Revised: [Equation]

Original: b
Revised: b

Original: [Equation]
Revised: [Equation]

Original: Q.E.D.
Revised: Q.E.D.

Original: Remark A.5 .
Revised: Remark A.5 .

Original: In Lemma 3 in [CBRV13] and Lemma 6.5 in [KDK16], the second term in Eq.is ignored.
Revised: In Lemma 3 in [CBRV13] and Lemma 6.5 in [KDK16], the second term in Eq.is ignored.

Original: However, σ T, 1 ( x T +1 , 1 ) − σ 0 , 1 ( x 1 , 1 ) can be negative, which should not be ignored.
Revised: However, σ T, 1 ( x T +1 , 1 ) − σ 0 , 1 ( x 1 , 1 ) can be negative, which should not be ignored.

Original: Nevertheless, this error does not change the regret analysis in [CBRV13] because constant terms divided by T vanishes.
Revised: Nevertheless, this error does not change the regret analysis in [CBRV13] because constant terms divided by T vanishes.

Original: Our version has the additional constant 1 on the right hand side of Eq. 23.
Revised: Our version has the additional constant 1 on the right hand side of Eq. 23.

Original: Definition A.6.
Revised: Definition A.6.

Original: The maximum information gain γ T is defined as below
Revised: The maximum information gain γ T is defined as below

Original: [Equation]
Revised: [Equation]

Original: where Y is the observation at X and H is the differential entropy.
Revised: where Y is the observation at X and H is the differential entropy.

Original: For Gaussian processes with the kernel K and the variance of observation noise σ
Revised: For Gaussian processes with the kernel K and the variance of observation noise σ

Original: [Equation]
Revised: [Equation]

Original: We rephrase lemmas from previous works with the batch indexing for notational ease and discuss the noteworthy points in the rephrased version compared with the original ones.
Revised: We rephrase lemmas from previous works with the batch indexing for notational ease and discuss the noteworthy points in the rephrased version compared with the original ones.

Original: Lemma A.7 (Lemma 3 [SKKS09], Lemma 4 [CBRV13], Theorem 3.1 [WZJ16]) .
Revised: Lemma A.7 (Lemma 3 [SKKS09], Lemma 4 [CBRV13], Theorem 3.1 [WZJ16]) .

Original: Assume a kernel such that K ( · , · ) ≤ 1 .
Revised: Assume a kernel such that K ( · , · ) ≤ 1 .

Original: For each t ∈ [ T ] , LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )
Revised: For each t ∈ [ T ] , LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )

Original: =
Revised: =

Original: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (as defined below Eq. 13).
Revised: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (as defined below Eq. 13).

Original: The posterior variance defined as Eq. 15.
Revised: The posterior variance defined as Eq. 15.

Original: has the following relation
Revised: has the following relation

Original: [Equation]
Revised: [Equation]

Original: where C 1 =
Revised: where C 1 =

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: − 2 ) and γ TB is the maximum information gain at TB
Revised: − 2 ) and γ TB is the maximum information gain at TB

Original: Proof.
Revised: Proof.

Original: Following the trick used in the proof of Lemma 5.4 in [SKKS09],
Revised: Following the trick used in the proof of Lemma 5.4 in [SKKS09],

Original: [Equation]
Revised: [Equation]

Original: In LAW, x t, 1 and x t,b deterministic conditioned respectively on D t − 

Revised: In LAW, x t, 1 and x t,b deterministic conditioned respectively on D t −

Original: = { ( x s,b , y s,b ) } s ∈ [ t − 1] ,b ∈
Revised: = { ( x s,b , y s,b ) } s ∈ [ t − 1] ,b ∈

Original: [ B ] and D t − 1 ∪{ x t,c } c =2 , ··· ,b − 1 for b = 2 , · · · , B .
Revised: [ B ] and D t − 1 ∪{ x t,c } c =2 , ··· ,b − 1 for b = 2 , · · · , B .

Original: Also, x t,b does not depend on { y s,c } T ( B ) seq ( s,c ) < T ( B ) seq ( t,b ) as long as { x s,c } T ( B ) seq ( s,c ) ≤ T ( B ) seq ( t,b ) .
Revised: Also, x t,b does not depend on { y s,c } T ( B ) seq ( s,c ) < T ( B ) seq ( t,b ) as long as { x s,c } T ( B ) seq ( s,c ) ≤ T ( B ) seq ( t,b ) .

Original: Therefore, the proof of Lemma 5.
Revised: Therefore, the proof of Lemma 5.

Original: in [SKKS09] can be applied
Revised: in [SKKS09] can be applied

Original: [Equation]
Revised: [Equation]

Original: Remark A.8 .
Revised: Remark A.8 .

Original: In contrast to Lemma 5.4 in [SKKS09] which bounds the sum of square of regrets,
Revised: In contrast to Lemma 5.4 in [SKKS09] which bounds the sum of square of regrets,

Original: Lemma A.7 bounds the sum of the posterior variances.
Revised: Lemma A.7 bounds the sum of the posterior variances.

Original: The delayed evaluation { y t,b } t ∈
Revised: The delayed evaluation { y t,b } t ∈

Original: [ T ] ,b ∈
Revised: [ T ] ,b ∈

Original: [ B ] does not cause any impediment in the proof.
Revised: [ B ] does not cause any impediment in the proof.

Original: Lemma A.9 (Lemma 6.1 [KDK16], Lemma 3.2 [WZJ16]) .
Revised: Lemma A.9 (Lemma 6.1 [KDK16], Lemma 3.2 [WZJ16]) .

Original: For ζ t = (cid:16) 2 log (cid:0) π 2 t 2 δ (cid:1)(cid:17) 1 / 2 with δ ∈ (0 , 1) and π t > 0 such that (cid:80) ∞ t =1 π t ≤ 1 , an arbitrary sequence of actions x 1 , 1 , · · · , x T, 1 ∈ X
Revised: For ζ t = (cid:16) 2 log (cid:0) π 2 t 2 δ (cid:1)(cid:17) 1 / 2 with δ ∈ (0 , 1) and π t > 0 such that (cid:80) ∞ t =1 π t ≤ 1 , an arbitrary sequence of actions x 1 , 1 , · · · , x T, 1 ∈ X

Original: P (cid:18) (cid:92)
Revised: P (cid:18) (cid:92)

Original: [Equation]
Revised: [Equation]

Original: ∈ [ T
Revised: ∈ [ T

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Remark A.10 .
Revised: Remark A.10 .

Original: ζ t only depends on the number of batch round t and is independent with the batch size
Revised: ζ t only depends on the number of batch round t and is independent with the batch size

Original: B .
Revised: B .

Original: Therefore, ζ t in batch BO is the same as one in the sequential BO.
Revised: Therefore, ζ t in batch BO is the same as one in the sequential BO.

Original: Lemma A.11 (Lemma 3.3 [WZJ16]) .
Revised: Lemma A.11 (Lemma 3.3 [WZJ16]) .

Original: If | f ( x t, 1 ) − µ t − 1 ( x t, 1 ) | ≤ ζ t σ t − 1 , 1 ( x t, 1 ) r t, 1 = f ( x t, 1 ) − f ( x ⋆ ) ≤ ( ν t + ζ t ) σ t − 1 , 1 ( x t, 1 ) (34) where ν t = (cid:16) min x ∈X µ t − 1 ( x ) − ˆ m σ t − 1 , 1 ( x ) (cid:17) , ˆ m is the estimate of the optimum[WZJ16] and ζ t = (cid:16) 2 log (cid:0)
Revised: If | f ( x t, 1 ) − µ t − 1 ( x t, 1 ) | ≤ ζ t σ t − 1 , 1 ( x t, 1 ) r t, 1 = f ( x t, 1 ) − f ( x ⋆ ) ≤ ( ν t + ζ t ) σ t − 1 , 1 ( x t, 1 ) (34) where ν t = (cid:16) min x ∈X µ t − 1 ( x ) − ˆ m σ t − 1 , 1 ( x ) (cid:17) , ˆ m is the estimate of the optimum[WZJ16] and ζ t = (cid:16) 2 log (cid:0)

Original: π 2 t 2 δ (cid:1)(cid:17) 1 /with δ ∈ (0 , 1) and π t > 0 such that (cid:80) ∞ t =1 π − 1 t ≤ 1 .
Revised: π 2 t 2 δ (cid:1)(cid:17) 1 /with δ ∈ (0 , 1) and π t > 0 such that (cid:80) ∞ t =1 π − 1 t ≤ 1 .

Original: Remark A.12 .
Revised: Remark A.12 .

Original: In Lemma A.11, we only bound regrets in ( t, 1) -th round where there is no delayed evaluation.
Revised: In Lemma A.11, we only bound regrets in ( t, 1) -th round where there is no delayed evaluation.

Original: Remark A.13 .
Revised: Remark A.13 .

Original: In contrast to the original condition (cid:80) Tt =
Revised: In contrast to the original condition (cid:80) Tt =

Original: 1 π t ≤ 1 , we use (cid:80) ∞ t =1 π t ≤ 1 so that π t s are T independent as the recommendation of the choice π t = 

Revised: 1 π t ≤ 1 , we use (cid:80) ∞ t =1 π t ≤ 1 so that π t s are T independent as the recommendation of the choice π t =

Original: π 2 t 2 in [WZJ16].
Revised: π 2 t 2 in [WZJ16].

Original: When the number of rounds T is known in advance, T dependent π t is possible, e.g, π t = T [WZJ16].
Revised: When the number of rounds T is known in advance, T dependent π t is possible, e.g, π t = T [WZJ16].

Original: By making π t independent with T , EST becomes anytime, i.e. not requiring that the number of rounds is known in advance.
Revised: By making π t independent with T , EST becomes anytime, i.e. not requiring that the number of rounds is known in advance.

Original: Lemma A.14 (Lemma 5.1 [SKKS09]) .
Revised: Lemma A.14 (Lemma 5.1 [SKKS09]) .

Original: For β ( B ) UCB t,= 2 log (cid:16) |X | π 2 ( T ( B ) seq ( t, 1)) 2 

Revised: For β ( B ) UCB t,= 2 log (cid:16) |X | π 2 ( T ( B ) seq ( t, 1)) 2

Original: δ (cid:17) with δ ∈ (0 , 1) ,
Revised: δ (cid:17) with δ ∈ (0 , 1) ,

Original: P (cid:18) (cid:92) x ∈X (cid:110) f
Revised: P (cid:18) (cid:92) x ∈X (cid:110) f

Original: (cid:12)(cid:12)(cid:12) | f ( x ) − µ t − 1 ( x ) |
Revised: (cid:12)(cid:12)(cid:12) | f ( x ) − µ t − 1 ( x ) |

Original: ≤ ( β ( B ) UCB t,) 1 / 2 · σ t − 1 , 1 ( x ) (cid:111)(cid:19) ≥ 1 − δ. 
Revised: ≤ ( β ( B ) UCB t,) 1 / 2 · σ t − 1 , 1 ( x ) (cid:111)(cid:19) ≥ 1 − δ. [Equation]

Original: Remark A.15 .
Revised: Remark A.15 .

Original: Note that β ( B ) UCB t,= β [Equation] T ( B ) seq ( t, 1) .
Revised: Note that β ( B ) UCB t,= β UCB T ( B ) seq ( t, 1) .

Original: in batch BO with the batch size of B , β is set as if there is B times more rounds.
Revised: in batch BO with the batch size of B , β is set as if there is B times more rounds.

Original: Lemma A.16 (Lemma 5.2 [SKKS09], Lemma 1 [CBRV13]) .
Revised: Lemma A.16 (Lemma 5.2 [SKKS09], Lemma 1 [CBRV13]) .

Original: If | f ( x ) − µ t − 1 ( x ) | ≤ ( β ( B ) UCB t,) 1 / 2 σ t − 1 , 1 ( x ) for all x ∈ X , then
Revised: If | f ( x ) − µ t − 1 ( x ) | ≤ ( β ( B ) UCB t,) 1 / 2 σ t − 1 , 1 ( x ) for all x ∈ X , then

Original: [Equation]
Revised: [Equation]

Original: Remark A.17 .
Revised: Remark A.17 .

Original: In Lemma A.16, we only bound regrets in ( t, 1) -th round where there is no delayed evaluation.
Revised: In Lemma A.16, we only bound regrets in ( t, 1) -th round where there is no delayed evaluation.

Original: Theorem A.18.
Revised: Theorem A.18.

Original: Assume a kernel such that K ( · , · ) ≤ 1 , | X | < ∞ and f : X → R is sampled from
Revised: Assume a kernel such that K ( · , · ) ≤ 1 , | X | < ∞ and f : X → R is sampled from

Original: GP ( 0 , K ) .
Revised: GP ( 0 , K ) .

Original: In each round t ∈ [ T ] of batch Bayesian optimization, LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )
Revised: In each round t ∈ [ T ] of batch Bayesian optimization, LAW acquires a batch using the evaluation data D t − 1 , the diversity measure L t ( · , · )

Original: =
Revised: =

Original: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (as defined below Eq. 13).
Revised: K ( · , ·| D t − 1 ) , an acquisition function a t ( · ) and a weight function w ( · ) (as defined below Eq. 13).

Original: Let C 1 = 36 log(1+ σ − 2 ) where σ 2 is the variance of the observation noise and δ ∈ (0 , 1) .
Revised: Let C 1 = 36 log(1+ σ − 2 ) where σ 2 is the variance of the observation noise and δ ∈ (0 , 1) .

Original: For GP-UCB, define β ( B ) UCB t,
Revised: For GP-UCB, define β ( B ) UCB t,

Original: [Equation]
Revised: [Equation]

Original: (cid:17) and let
Revised: (cid:17) and let

Original: [Equation]
Revised: [Equation]

Original: For EST, define ν t = min x
Revised: For EST, define ν t = min x

Original: [Equation]
Revised: [Equation]

Original: where ˆ m t is the estimate of the optimum [WZJ16], ζ t = (cid:16) 2 log (cid:0) π 2 t 2 δ (cid:1)(cid:17) 1 / 2 , π t > 0 such that (cid:80) ∞ t =1 π − 1 t ≤ 1 and let η ( B ) t = ν t ∗ + ζ t (38) where t ∗ = argmax s ∈
Revised: where ˆ m t is the estimate of the optimum [WZJ16], ζ t = (cid:16) 2 log (cid:0) π 2 t 2 δ (cid:1)(cid:17) 1 / 2 , π t > 0 such that (cid:80) ∞ t =1 π − 1 t ≤ 1 and let η ( B ) t = ν t ∗ + ζ t (38) where t ∗ = argmax s ∈

Original: [ t ] ν s .
Revised: [ t ] ν s .

Original: Then batch cumulative regret satisfies the following bound
Revised: Then batch cumulative regret satisfies the following bound

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Let η ( B ) t =
Revised: Let η ( B ) t =

Original: [Equation]
Revised: [Equation]

Original: EST
Revised: EST

Original: [Equation]
Revised: [Equation]

Original: UCB.
Revised: UCB.

Original: For the batch cumulative regret case, we use Lemma A.1.
Revised: For the batch cumulative regret case, we use Lemma A.1.

Original: [Equation]
Revised: [Equation]

Original: 40) ≤
Revised: 40) ≤

Original: [Equation]
Revised: [Equation]

Original: (41) ≤ η ( B ) T ·
Revised: ≤ η ( [Equation] T ·

Original: T (cid:88) t
Revised: T (cid:88) t

Original: = σ t − 1 , 1 ( x t, 1 ) by (cid:26) Lemma A.11 EST
Revised: = σ t − 1 , 1 ( x t, 1 ) by (cid:26) Lemma A.11 EST

Original: Lemma A.16 UCB ≤ η ( B ) T · (cid:16) 1 + w + w −
Revised: Lemma A.16 UCB ≤ η ( B ) T · (cid:16) 1 + w + w −

Original: B
Revised: B

Original: T (cid:88) t =
Revised: T (cid:88) t =

Original: B (cid:88) b
Revised: B (cid:88) b

Original: = σ t − 1 ,b ( x t,b ) (cid:17) by Lemma A.
Revised: = σ t − 1 ,b ( x t,b ) (cid:17) by Lemma A.

Original: ≤ η ( B ) T · (cid:16) 1 + w + w −
Revised: ≤ η ( B ) T · (cid:16) 1 + w + w −

Original: (cid:118) (cid:117) (cid:117) (cid:116) T B
Revised: (cid:118) (cid:117) (cid:117) (cid:116) T B

Original: T (cid:88) t =
Revised: T (cid:88) t =

Original: B (cid:88)
Revised: B (cid:88)

Original: b
Revised: b

Original: [Equation]
Revised: [Equation]

Original: by Cauchy-Schwarz ≤ η ( B ) T ·
Revised: by Cauchy-Schwarz ≤ η ( B ) T ·

Original: [Equation]
Revised: [Equation]

Original: by Lemma A.
Revised: by Lemma A.

Original: By Lemma A.9 for EST and Lemma A.14 for UCB, above two inequalities hold with the probability at least 1 − δ .
Revised: By Lemma A.9 for EST and Lemma A.14 for UCB, above two inequalities hold with the probability at least 1 − δ .

Original: Remark A.19 .
Revised: Remark A.19 .

Original: Due to the difference of the statements in Lemma A.1, the batch cumulative regret bound additionally has the term w + w − .
Revised: Due to the difference of the statements in Lemma A.1, the batch cumulative regret bound additionally has the term w + w − .

Original: Even with this additional term, it shows that the bound of the batch cumulative regret of LAW enjoys the same asymptotic behavior as existing methods [CBRV13,
Revised: Even with this additional term, it shows that the bound of the batch cumulative regret of LAW enjoys the same asymptotic behavior as existing methods [CBRV13,

Original: DKB14, KDK16].
Revised: DKB14, KDK16].

Original: Remark A.20 .
Revised: Remark A.20 .

Original: This theorem provides a rough guideline how to choose a weight function, that is, bounded below by a positive value and bounded above, which is the condition we specify for the weight function.
Revised: This theorem provides a rough guideline how to choose a weight function, that is, bounded below by a positive value and bounded above, which is the condition we specify for the weight function.

Original: Even though this shows that simple regret vanishes, this regret bound for LAW is loose because not much specific structure of the weight function other than the bound is used.
Revised: Even though this shows that simple regret vanishes, this regret bound for LAW is loose because not much specific structure of the weight function other than the bound is used.

Original: We expect that, using other properties of the weight function along with the boundedness, the bound can be improved.
Revised: We expect that, using other properties of the weight function along with the boundedness, the bound can be improved.

Original: A.2 Difference to analysis of sequential cumulative regret
Revised: A.2 Difference to analysis of sequential cumulative regret

Original: In our regret analysis, we analyze batch cumulative regret.
Revised: In our regret analysis, we analyze batch cumulative regret.

Original: In existing works, sequential cumulative regret is analyzed as an end goal [DKB14] and as a medium to show vanishing simple regret in [KKSP18].
Revised: In existing works, sequential cumulative regret is analyzed as an end goal [DKB14] and as a medium to show vanishing simple regret in [KKSP18].

Original: In [CBRV13], 11 both batch cumulative regret and sequential cumulative regret are analyzed.
Revised: In [CBRV13], 11 both batch cumulative regret and sequential cumulative regret are analyzed.

Original: 12 We discuss the differences between these two approaches and the technical details in their proofs.
Revised: 12 We discuss the differences between these two approaches and the technical details in their proofs.

Original: By definition, the analysis of sequential cumulative regret takes into account all instantaneous regrets incurred while batch cumulative regret considers minimum instantaneous regrets in each batch.
Revised: By definition, the analysis of sequential cumulative regret takes into account all instantaneous regrets incurred while batch cumulative regret considers minimum instantaneous regrets in each batch.

Original: Therefore, bounds on sequential cumulative regret are stronger than ones on batch cumulative regret in this sense (as shown in Table 3 Relation between two Cumulative).
Revised: Therefore, bounds on sequential cumulative regret are stronger than ones on batch cumulative regret in this sense (as shown in Table 3 Relation between two Cumulative).

Original: However, each has its own more appropriate scenario to use.
Revised: However, each has its own more appropriate scenario to use.

Original: The sequential cumulative regret is often appropriate in the situation where the optimization objective represents the cost of evaluations.
Revised: The sequential cumulative regret is often appropriate in the situation where the optimization objective represents the cost of evaluations.

Original: For example, in multi-armed bandit, each instantaneous regret represents the cost of evaluation (playing arm-pulling) and the goal is to minimize the incurred cost in finding the best bandit machine.
Revised: For example, in multi-armed bandit, each instantaneous regret represents the cost of evaluation (playing arm-pulling) and the goal is to minimize the incurred cost in finding the best bandit machine.

Original: On the other hand, batch cumulative regret is often reasonable when the optimization objective is different to the cost of evaluations.
Revised: On the other hand, batch cumulative regret is often reasonable when the optimization objective is different to the cost of evaluations.

Original: For example, in hyperparameter optimization, the cost of evaluations can be wall-clock time and the objective is the cross-validation error.
Revised: For example, in hyperparameter optimization, the cost of evaluations can be wall-clock time and the objective is the cross-validation error.

Original: In this case, we want to find a good hyperparameter no matter how bad hyperparameters are evaluated, which possibly acts as exploratory query points.
Revised: In this case, we want to find a good hyperparameter no matter how bad hyperparameters are evaluated, which possibly acts as exploratory query points.

Original: In proofs, each analysis takes a slightly different route.
Revised: In proofs, each analysis takes a slightly different route.

Original: As argued in [DKB14], to bound all instantaneous regret, a wider confidence bound is needed to bound instantaneous regret with the corresponding posterior variance.
Revised: As argued in [DKB14], to bound all instantaneous regret, a wider confidence bound is needed to bound instantaneous regret with the corresponding posterior variance.

Original: While the posterior mean is not update in the batch acquisition until all query points are evaluated, the posterior variance is updated whenever a new query point is given no matter whether it is evaluated or not.
Revised: While the posterior mean is not update in the batch acquisition until all query points are evaluated, the posterior variance is updated whenever a new query point is given no matter whether it is evaluated or not.

Original: To guarantee high probability bound for all instantaneous regrets, an additional kernel-dependent constant is introduced and the constant is controlled with an initialization scheme[DKB14].
Revised: To guarantee high probability bound for all instantaneous regrets, an additional kernel-dependent constant is introduced and the constant is controlled with an initialization scheme[DKB14].

Original: In [KKSP18], the analysis relies on such kernel-dependent constant and the initialization scheme but it is empirically shown that the algorithm performs well without the initialization scheme.
Revised: In [KKSP18], the analysis relies on such kernel-dependent constant and the initialization scheme but it is empirically shown that the algorithm performs well without the initialization scheme.

Original: The necessity of the kernel-dependent constant suggests that the analysis of sequential cumulative regret in [CBRV13] requires a revision.
Revised: The necessity of the kernel-dependent constant suggests that the analysis of sequential cumulative regret in [CBRV13] requires a revision.

Original: For the purpose to show vanishing simple regret, batch cumulative regret can be used circumventing the additional constant and the initialization scheme proposed in [DKB14].
Revised: For the purpose to show vanishing simple regret, batch cumulative regret can be used circumventing the additional constant and the initialization scheme proposed in [DKB14].

Original: In the analysis using batch cumulative regret, only non-delayed regret is considered and bounded by non-delayed posterior variance (Eq. 41).
Revised: In the analysis using batch cumulative regret, only non-delayed regret is considered and bounded by non-delayed posterior variance (Eq. 41).

Original: Then non-delayed posterior variance is bounded by the average of non-delayed posterior variance and delayed posterior variances in the same batch (Lemma A.4).
Revised: Then non-delayed posterior variance is bounded by the average of non-delayed posterior variance and delayed posterior variances in the same batch (Lemma A.4).

Original: Therefore, the effect of the batch size influences the bound in this posterior variance bounding step.
Revised: Therefore, the effect of the batch size influences the bound in this posterior variance bounding step.

Original: However, in the analysis using sequential cumulative regret[DKB14, KKSP18], both non-delayed and delayed instantaneous regrets need to be bounded.
Revised: However, in the analysis using sequential cumulative regret[DKB14, KKSP18], both non-delayed and delayed instantaneous regrets need to be bounded.

Original: The bound is the corresponding posterior variance multiplied by a specially design number to handle delayed cases.
Revised: The bound is the corresponding posterior variance multiplied by a specially design number to handle delayed cases.

Original: In response to this, the additional kernel-dependent constant and the initialization scheme are introduced in [DKB14].
Revised: In response to this, the additional kernel-dependent constant and the initialization scheme are introduced in [DKB14].

Original: Batch cumulative regret is enough in showing vanishing simple regret.
Revised: Batch cumulative regret is enough in showing vanishing simple regret.

Original: The proof only considers non-delayed instantaneous regrets in batches.
Revised: The proof only considers non-delayed instantaneous regrets in batches.

Original: Therefore, the analysis of batch cumulative regret reveals how delayed query points in a batch explore effectively and help to reduce future non-delayed instantaneous regrets.
Revised: Therefore, the analysis of batch cumulative regret reveals how delayed query points in a batch explore effectively and help to reduce future non-delayed instantaneous regrets.

Original: We admit that some may argue that a tighter bound is possible by taking into account delayed evaluations with smaller instantaneous regrets.
Revised: We admit that some may argue that a tighter bound is possible by taking into account delayed evaluations with smaller instantaneous regrets.

Original: Still, this is aligned with the intuition of many batch acquisition methods promoting diversity in batches.
Revised: Still, this is aligned with the intuition of many batch acquisition methods promoting diversity in batches.

Original: In practice, it is not unlikely to observe a delayed evaluation is better than the non-delayed evaluation in the same batch.
Revised: In practice, it is not unlikely to observe a delayed evaluation is better than the non-delayed evaluation in the same batch.

Original: [Equation]
Revised: [Equation]

Original: where β ( B ) UCB t,
Revised: where β ( B ) UCB t,

Original: [Equation]
Revised: [Equation]

Original: log
Revised: log

Original: [Equation]
Revised: [Equation]

Original: |
Revised: |

Original: [Equation]
Revised: [Equation]

Original: (cid:17) , t ∗ = argmax s ∈ [ t ] ν s , ν t = min x (cid:18) µ t − 1 ( x ) − ˆ m t σ t − 1 , 1 ( x ) (cid:19) where ˆ m t is the estimate of the optimum [WZJ16], ζ t = (cid:16) 2 log (cid:0) π 2 t 2 δ (cid:1)(cid:17) 1 / 2 , π t > 0 such that (cid:80) ∞ t =1 π − 1 t ≤ 1 .
Revised: (cid:17) , t ∗ = argmax s ∈ [ t ] ν s , ν t = min x (cid:18) µ t − 1 ( x ) − ˆ m t σ t − 1 , 1 ( x ) (cid:19) where ˆ m t is the estimate of the optimum [WZJ16], ζ t = (cid:16) 2 log (cid:0) π 2 t 2 δ (cid:1)(cid:17) 1 / 2 , π t > 0 such that (cid:80) ∞ t =1 π − 1 t ≤ 1 .

Original: For UCB, it is clear that 2( β ( B ) UCB t,) 1 / 

Revised: For UCB, it is clear that 2( β ( B ) UCB t,) 1 /

Original: =
Revised: =

Original: O ((log( tB )) 1 / 2 ) .
Revised: O ((log( tB )) 1 / 2 ) .

Original: For EST, we first look into ζ t .
Revised: For EST, we first look into ζ t .

Original: If we choose π t = π 2 t 2as suggested in [WZJ16], then ζ t =
Revised: If we choose π t = π 2 t 2as suggested in [WZJ16], then ζ t =

Original: O ((log( tB )) 1 / 2 ) .
Revised: O ((log( tB )) 1 / 2 ) .

Original: Since ˆ m t =
Revised: Since ˆ m t =

Original: E f ∼ GP ( µ t − 1 ( · ) ,σ 2 t − 1 ( · ))
Revised: E f ∼ GP ( µ t − 1 ( · ) ,σ 2 t − 1 ( · ))

Original: [inf x f ( x )]
Revised: [inf x f ( x )]

Original: [WZJ16], from Lemma 5.
Revised: [WZJ16], from Lemma 5.

Original: in [SKKS09], we have
Revised: in [SKKS09], we have

Original: [Equation]
Revised: [Equation]

Original: where τ 1 / 2 t = 2 log (cid:0) |X | π 2 t 2 6 δ
Revised: where τ 1 / 2 t = 2 log (cid:0) |X | π 2 t 2 6 δ

Original: [Equation]
Revised: [Equation]

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: with x lb = argmin x
Revised: with x lb = argmin x

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Since, τ t is increasing, we have ν t ∗ ≤ τ t and ν t ∗ =
Revised: Since, τ t is increasing, we have ν t ∗ ≤ τ t and ν t ∗ =

Original: O ((log( tB )) 1 / 2 ) .
Revised: O ((log( tB )) 1 / 2 ) .

Original: Therefore, for EST, η ( B ) t
Revised: Therefore, for EST, η ( B ) t

Original: [Equation]
Revised: [Equation]

Original: B Information Gain
Revised: B Information Gain

Original: In this section, we present results of our analysis on the information gain and the position kernel.
Revised: In this section, we present results of our analysis on the information gain and the position kernel.

Original: In Theorem.
Revised: In Theorem.

Original: B.1 in Subsection B.1, we show that for a kernel on a finite space, the information gain grows O (log( T )) .
Revised: B.1 in Subsection B.1, we show that for a kernel on a finite space, the information gain grows O (log( T )) .

Original: In combination with Section A, this shows that an arbitrary kernel on a finite space including the position kernel achieves sublinear regret.
Revised: In combination with Section A, this shows that an arbitrary kernel on a finite space including the position kernel achieves sublinear regret.

Original: To our knowledge, the positive definiteness of the position kernel has not been shown rigorously, [ZSBB14] used randomly generated data to empirically check that whether the position kernel is positive definite and [ZYLW19] argued that the exponential of a metric is positive definite, which is not true in general.
Revised: To our knowledge, the positive definiteness of the position kernel has not been shown rigorously, [ZSBB14] used randomly generated data to empirically check that whether the position kernel is positive definite and [ZYLW19] argued that the exponential of a metric is positive definite, which is not true in general.

Original: Therefore, we show that the position kernel is positive definite and further provide a lower and upper bound of the eigenvalues of the position kernel in Subsection B.2.
Revised: Therefore, we show that the position kernel is positive definite and further provide a lower and upper bound of the eigenvalues of the position kernel in Subsection B.2.

Original: In Theorem B.4 in Subsection B.3, we show that by using the properties of the position kernel, a tighter bound on the maximum information gain is achievable.
Revised: In Theorem B.4 in Subsection B.3, we show that by using the properties of the position kernel, a tighter bound on the maximum information gain is achievable.

Original: B.1 Information gain of kernels on a finite space
Revised: B.1 Information gain of kernels on a finite space

Original: In this subsection, we show a bound of the information gain of kernels defined on a finite set.
Revised: In this subsection, we show a bound of the information gain of kernels defined on a finite set.

Original: Theorem B.1.
Revised: Theorem B.1.

Original: K is a kernel on a finite set X ( | X | < ∞ ), σ 2 is the variance of the observation noise and Λ = { λ n } 1 , ··· , |X | ( λ n ≥ λ n +1 ≥ 0) is the set of eigenvalues of the gram matrix K ( X , X ) .
Revised: K is a kernel on a finite set X ( | X | < ∞ ), σ 2 is the variance of the observation noise and Λ = { λ n } 1 , ··· , |X | ( λ n ≥ λ n +1 ≥ 0) is the set of eigenvalues of the gram matrix K ( X , X ) .

Original: The number of elements in a set A is denoted by N A , so N X is the number of elements of X and is equal to the number of eigenvalues of K ( X , X ) .
Revised: The number of elements in a set A is denoted by N A , so N X is the number of elements of X and is equal to the number of eigenvalues of K ( X , X ) .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Let us consider the eigenvalues and the eigenvectors of the gram matrix K ( X , X ) .
Revised: Let us consider the eigenvalues and the eigenvectors of the gram matrix K ( X , X ) .

Original: [Equation]
Revised: [Equation]

Original: with Λ = diag ( λ 1 , · · · , λ N X ) ,
Revised: with Λ = diag ( λ 1 , · · · , λ N X ) ,

Original: U =
Revised: U =

Original: [ u 1 , · · · , u N X ] ∈ R N X × N X where λ i is an eigenvalue and
Revised: [ u 1 , · · · , u N X ] ∈ R N X × N X where λ i is an eigenvalue and

Original: u i is the corresponding eigenvector.
Revised: u i is the corresponding eigenvector.

Original: Since
Revised: Since

Original: [Equation]
Revised: [Equation]

Original: the map
Revised: the map

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: is a N X dimensional feature map
Revised: is a N X dimensional feature map

Original: [Equation]
Revised: [Equation]

Original: For a sequence A = { a 1 , · · · , a N A } of a i ∈ X
Revised: For a sequence A = { a 1 , · · · , a N A } of a i ∈ X

Original: = x
Revised: = x

Original: iN X i =1 , the gram matrix K ( A, A ) can be expressed with the projection matrix P X
Revised: iN X i =1 , the gram matrix K ( A, A ) can be expressed with the projection matrix P X

Original: A ∈ { 0 , 1 } N A × N X from X to A such that [ P X A ] ij = 

Revised: A ∈ { 0 , 1 } N A × N X from X to A such that [ P X A ] ij =

Original: if a i = x j
Revised: if a i = x j

Original: [Equation]
Revised: [Equation]

Original: Remark B.2 .
Revised: Remark B.2 .

Original: Note that ( P X A ) T · P X A is N X × N X diagonal matrix and [( P X A ) T · P X A ] ii is how many times x i appears in the sequence A .
Revised: Note that ( P X A ) T · P X A is N X × N X diagonal matrix and [( P X A ) T · P X A ] ii is how many times x i appears in the sequence A .

Original: We obtain two bounds.
Revised: We obtain two bounds.

Original: The first one is
Revised: The first one is

Original: [Equation]
Revised: [Equation]

Original: 52) using Hadamard’s inequality.
Revised: 52) using Hadamard’s inequality.

Original: Adopting the proof for the information gain of the linear kernel from [SKKS09], the second one is log det
Revised: Adopting the proof for the information gain of the linear kernel from [SKKS09], the second one is log det

Original: [Equation]
Revised: [Equation]

Original: 53) =
Revised: 53) =

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: by Eq.=
Revised: by Eq.=

Original: [Equation]
Revised: [Equation]

Original: by Weinstein-Aronszajn identity ≤
Revised: by Weinstein-Aronszajn identity ≤

Original: [Equation]
Revised: [Equation]

Original: by Hadamard’s inequality ≤
Revised: by Hadamard’s inequality ≤

Original: [Equation]
Revised: [Equation]

Original: by Eq.(54) using Weinstein-Aronszajn identity 14 , Hadamard’s inequality 15 and below
Revised: by Eq.(54) using Weinstein-Aronszajn identity 14 , Hadamard’s inequality 15 and below

Original: [Equation]
Revised: [Equation]

Original: Rmk
Revised: Rmk

Original: [Equation]
Revised: [Equation]

Original: B.
Revised: B

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: i
Revised: i

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where the second equality comes from the fact that ( P X A ) T · P X A is N X × N X diagonal matrix and the last inequality is possible because every numbers are non-negative since [( P X A ) T · P X A ] ii is how many times x i appears in the sequence A .
Revised: where the second equality comes from the fact that ( P X A ) T · P X A is N X × N X diagonal matrix and the last inequality is possible because every numbers are non-negative since [( P X A ) T · P X A ] ii is how many times x i appears in the sequence A .

Original: Putting Eq.
Revised: Putting Eq.

Original: 52 and Eq.
Revised: 52 and Eq.

Original: 53 together,
Revised: 53 together,

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Q.E.D.
Revised: Q.E.D.

Original: B.2 Positive definiteness of the position kernel
Revised: B.2 Positive definiteness of the position kernel

Original: In this subsection, we show that the positive definiteness of the position kernel and the bound of its eigenvalues.
Revised: In this subsection, we show that the positive definiteness of the position kernel and the bound of its eigenvalues.

Original: Theorem B.3.
Revised: Theorem B.3.

Original: The position kernel K ( · , ·| τ ) defined on S N is positive definite and the eigenvalues of the K ( A, A ) where A ⊂ X lie between (cid:16) 1 − ρ 

Revised: The position kernel K ( · , ·| τ ) defined on S N is positive definite and the eigenvalues of the K ( A, A ) where A ⊂ X lie between (cid:16) 1 − ρ

Original: + ρ (cid:17) N and (cid:16) 1+ ρ 1 − ρ (cid:17)
Revised: + ρ (cid:17) N and (cid:16) 1+ ρ 1 − ρ (cid:17)

Original: N where ρ = exp( − τ ) .
Revised: N where ρ = exp( − τ ) .

Original: Proof.
Revised: Proof.

Original: We show that the kernel is positive definite on a larger set
Revised: We show that the kernel is positive definite on a larger set

Original: [Equation]
Revised: [Equation]

Original: Since S N ⊂ X , K ( S N , S N ) is a principal submatrix of K ( X , X ) With Poincaré seperation theorem (or Cauchy interlacing theorem), we show that the position kernel is positive definite and that the eigenvalues of K ( S N , S N ) lie between the smallest eigenvalue and the largest eigenvalue of
Revised: Since S N ⊂ X , K ( S N , S N ) is a principal submatrix of K ( X , X ) With Poincaré seperation theorem (or Cauchy interlacing theorem), we show that the position kernel is positive definite and that the eigenvalues of K ( S N , S N ) lie between the smallest eigenvalue and the largest eigenvalue of

Original: K ( X , X ) .
Revised: K ( X , X ) .

Original: On X , the position kernel is a product kernel of N kernels defined { 1 , · · · , N } as below
Revised: On X , the position kernel is a product kernel of N kernels defined { 1 , · · · , N } as below

Original: [Equation]
Revised: [Equation]

Original: and its gram matrix on each component has following form
Revised: and its gram matrix on each component has following form

Original: [Equation]
Revised: [Equation]

Original: 58) where ρ = exp( − τ ) .
Revised: where ρ = exp( − τ ) .

Original: This form of matrix is known as Kac-Murdock-Szegö ( KMS ) matrix [GS58, Tre99], which we denote by KMS ( ρ ) ( 0 < ρ < 1 ).
Revised: This form of matrix is known as Kac-Murdock-Szegö ( KMS ) matrix [GS58, Tre99], which we denote by KMS ( ρ ) ( 0 < ρ < 1 ).

Original: Their eigenvalues λ n are bounded as below [GS58, Tre99]
Revised: Their eigenvalues λ n are bounded as below [GS58, Tre99]

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: We observe that the each component kernel is positive definite with above bounds on the eigenvalues.
Revised: We observe that the each component kernel is positive definite with above bounds on the eigenvalues.

Original: Since
Revised: Since

Original: [Equation]
Revised: [Equation]

Original: where (cid:78) is the Kronecker product, the lower bound and the upper bound of the eigenvalues of
Revised: where (cid:78) is the Kronecker product, the lower bound and the upper bound of the eigenvalues of

Original: K ( X , X ) are (cid:16) 1 − ρ 1+ ρ (cid:17) N and (cid:16) 1+ ρ 1 − ρ (cid:17) N , respectively.
Revised: K ( X , X ) are (cid:16) 1 − ρ 1+ ρ (cid:17) N and (cid:16) 1+ ρ 1 − ρ (cid:17) N , respectively.

Original: For A ∈ S N ∈ X , these bounds also apply to the eigenvalues of K ( A, A ) by Poincaré seperation theorem (or Cauchy interlacing theorem).
Revised: For A ∈ S N ∈ X , these bounds also apply to the eigenvalues of K ( A, A ) by Poincaré seperation theorem (or Cauchy interlacing theorem).

Original: Q.E.D. B.3 Information gain of the position kernel
Revised: Q.E.D. B.3 Information gain of the position kernel

Original: Theorem B.4.
Revised: Theorem B.4.

Original: K ( · , ·| τ ) is the position kernel defined on S N , σ 2 obs is the variance of the observation noise, ρ = exp( − τ ) and
Revised: K ( · , ·| τ ) is the position kernel defined on S N , σ 2 obs is the variance of the observation noise, ρ = exp( − τ ) and

Original: [Equation]
Revised: [Equation]

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: which is smaller than T · log det(1 + σ − 2 obs max x ∈X K ( x, x )) .
Revised: which is smaller than T · log det(1 + σ − 2 obs max x ∈X K ( x, x )) .

Original: Proof.
Revised: Proof.

Original: γ T is defined as
Revised: γ T is defined as

Original: [Equation]
Revised: [Equation]

Original: By Lem.
Revised: By Lem.

Original: B.6 in Supp.
Revised: B.6 in Supp.

Original: Subsec.
Revised: Subsec.

Original: B.3, for i, , j = 1 , · · · , T , ρ
Revised: B.3, for i, , j = 1 , · · · , T , ρ

Original: [Equation]
Revised: [Equation]

Original: max ≤ [ K ( A, A )] ij ≤ 1 .
Revised: max ≤ [ K ( A, A )] ij ≤ 1 .

Original: By Perron-Frobenius theorem, the largest eigenvalue of K ( A, A ) is bounded below by
Revised: By Perron-Frobenius theorem, the largest eigenvalue of K ( A, A ) is bounded below by

Original: [Equation]
Revised: [Equation]

Original: When λ ( A ) i is the i-th eigenvalue of K ( A, A ) , with the constraint λ
Revised: When λ ( A ) i is the i-th eigenvalue of K ( A, A ) , with the constraint λ

Original: ( A )
Revised: ( A )

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: is bounded above by
Revised: is bounded above by

Original: [Equation]
Revised: [Equation]

Original: Here, we use the fact that for (cid:80) i x i =
Revised: Here, we use the fact that for (cid:80) i x i =

Original: C , x i > 0 if there are p and q such that x p < x q , then for x ′ i defined as x ′ i
Revised: C , x i > 0 if there are p and q such that x p < x q , then for x ′ i defined as x ′ i

Original: = x i for i ̸
Revised: = x i for i ̸

Original: = p, 1 and x ′ p =
Revised: = p, 1 and x ′ p =

Original: x p + d , x ′ q = x q − d where d ≤ ( x q − x p ) /
Revised: x p + d , x ′ q = x q − d where d ≤ ( x q − x p ) /

Original: [Equation]
Revised: [Equation]

Original: Note that without the constraint on the lower bound of the largest eigenvalue,
Revised: Note that without the constraint on the lower bound of the largest eigenvalue,

Original: [Equation]
Revised: [Equation]

Original: where trace ( K ( A, A ))
Revised: where trace ( K ( A, A ))

Original: =
Revised: =

Original: T for kernels such that K ( x, x ) is a constant independent of x ∈ X as is for the position kernel.
Revised: T for kernels such that K ( x, x ) is a constant independent of x ∈ X as is for the position kernel.

Original: This shows that the bound in this theorem is tighter than that of Thm.
Revised: This shows that the bound in this theorem is tighter than that of Thm.

Original: B.1.
Revised: B.1.

Original: Q.E.D.
Revised: Q.E.D.

Original: Remark B.5 .
Revised: Remark B.5 .

Original: specially when σ 2 obs and/or ρ is large, i.e. log(1 + σ − 2 obs (1 − ρ D max ))
Revised: specially when σ 2 obs and/or ρ is large, i.e. log(1 + σ − 2 obs (1 − ρ D max ))

Original: ≈ 0 , we can observe that even in the finite-time regime, the regret is almost sublinear since it is dominated by log(1 + σ − 2 obs (1 + ( T − 1) ρ D max )) .
Revised: ≈ 0 , we can observe that even in the finite-time regime, the regret is almost sublinear since it is dominated by log(1 + σ − 2 obs (1 + ( T − 1) ρ D max )) .

Original: In this case, the theorem provide a bound which is significantly tighter than the bound in Thm.
Revised: In this case, the theorem provide a bound which is significantly tighter than the bound in Thm.

Original: 3.10 even in the finite-time regime.
Revised: 3.10 even in the finite-time regime.

Original: Even though both are the same in the asymptotic regime, they may differ significantly in the finite-time regime.
Revised: Even though both are the same in the asymptotic regime, they may differ significantly in the finite-time regime.

Original: Lemma B.6.
Revised: Lemma B.6.

Original: For π 1 , π 2 in S N ,
Revised: For π 1 , π 2 in S N ,

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Note that d pos is left-invariant, that is,
Revised: Note that d pos is left-invariant, that is,

Original: [Equation]
Revised: [Equation]

Original: for π ∈ S N , and thus
Revised: for π ∈ S N , and thus

Original: [Equation]
Revised: [Equation]

Original: where π id = (1 , · · · , N ) .
Revised: where π id = (1 , · · · , N ) .

Original: By induction on N , we show that
Revised: By induction on N , we show that

Original: [Equation]
Revised: [Equation]

Original: Base case( N = 2 )
Revised: Base case( N = 2 )

Original: This is trivial.
Revised: This is trivial.

Original: Induction Step As the induction hypothesis, assume that above is true for N = k .
Revised: Induction Step As the induction hypothesis, assume that above is true for N = k .

Original: When N = k +1 , let us consider π = ( − , − , ·
Revised: When N = k +1 , let us consider π = ( − , − , ·

Original: · · , , a ) ∈ S
Revised: · · , , a ) ∈ S

Original: k +1 an arbitrary permutation whose last element is a ̸ = 1 , d pos ( [Equation] id , [Equation]
Revised: k +1 an arbitrary permutation whose last element is a ̸ = 1 , d pos ( π id , π ) = (cid:88)

Original: = (cid:88)

Original: [Equation]
Revised: [Equation]

Original: where a ̸ = N .
Revised: where a ̸ = N .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: where (cid:88)
Revised: where (cid:88)

Original: [Equation]
Revised: [Equation]

Original: is from the induction hypothesis.
Revised: is from the induction hypothesis.

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: Q.E.D.
Revised: Q.E.D.

Original: C Resemblance to the Local Penalization
Revised: C Resemblance to the Local Penalization

Original: Taken from [GDHL16], the local penalization strategy selects b -th point in a batch as follows
Revised: Taken from [GDHL16], the local penalization strategy selects b -th point in a batch as follows

Original: [Equation]
Revised: [Equation]

Original: where ϕ ( x , x t,i ) is a local penalizer which is non-decreasing function of Euclidean distance ∥ x − x t,i ∥ 2 and g ( · ) is a positive increasing function similar to our weight function.
Revised: where ϕ ( x , x t,i ) is a local penalizer which is non-decreasing function of Euclidean distance ∥ x − x t,i ∥ 2 and g ( · ) is a positive increasing function similar to our weight function.

Original: If we use the prior covariance function K ( · , · ) , which is the kernel of the GP surrogate model in place of the posterior covariance function K t ( · , · ) as the diversity gauge of L AWt , the greedy maximization objective becomes
Revised: If we use the prior covariance function K ( · , · ) , which is the kernel of the GP surrogate model in place of the posterior covariance function K t ( · , · ) as the diversity gauge of L AWt , the greedy maximization objective becomes

Original: [Equation]
Revised: [Equation]

Original: We call this LAW variant as LAW-prior-EST and LAW-prior-EI according to the acquisition function each uses.
Revised: We call this LAW variant as LAW-prior-EST and LAW-prior-EI according to the acquisition function each uses.

Original: Since the closer to the conditioning data it is, the smaller the predictive variance is, the predictive variance behaves exactly as what the local penalizer aims at.
Revised: Since the closer to the conditioning data it is, the smaller the predictive variance is, the predictive variance behaves exactly as what the local penalizer aims at.

Original: Another key difference is that, while the local penalizer Eq.
Revised: Another key difference is that, while the local penalizer Eq.

Original: 76 is heuristically designed, LAW-prior-EST\EI use the kernel whose hyperparameters are fitted in the surrogate model fitting step.
Revised: 76 is heuristically designed, LAW-prior-EST\EI use the kernel whose hyperparameters are fitted in the surrogate model fitting step.

Original: Therefore, the diversity measured in
Revised: Therefore, the diversity measured in

Original: LAW-prior-EST\EI is more guided by the collected evaluation data.
Revised: LAW-prior-EST\EI is more guided by the collected evaluation data.

Original: Additional comparison to these variants (Supp.
Revised: Additional comparison to these variants (Supp.

Original: Sec.
Revised: Sec.

Original: G) reveals the contribution of the acquisition weights and thus further confirms the benefit of using acquisition weights in the optimization performance
Revised: G) reveals the contribution of the acquisition weights and thus further confirms the benefit of using acquisition weights in the optimization performance

Original: D Score-based Structure Learning
Revised: D Score-based Structure Learning

Original: In score-based structure learning, general-purpose optimization methods are utilized to optimize a score S ( G , D ) of DAG G
Revised: In score-based structure learning, general-purpose optimization methods are utilized to optimize a score S ( G , D ) of DAG G

Original: [SGG19] for a given data D .
Revised: [SGG19] for a given data D .

Original: Typically, S ( · , · ) is a penalized likelihood score or an information theoretic criterion[DM17].
Revised: Typically, S ( · , · ) is a penalized likelihood score or an information theoretic criterion[DM17].

Original: The prevalent choice of the optimization method is a local search which relies on the efficient computation of the DAG score to afford many score evaluations.[Chi02, KF09, SGG19].
Revised: The prevalent choice of the optimization method is a local search which relies on the efficient computation of the DAG score to afford many score evaluations.[Chi02, KF09, SGG19].

Original: For the efficient computation, it is critical for a score to be decomposable[SSS19], defined as below.
Revised: For the efficient computation, it is critical for a score to be decomposable[SSS19], defined as below.

Original: [Equation]
Revised: [Equation]

Original: where s ( · ) is a score defined for a node v ∈ V .
Revised: where s ( · ) is a score defined for a node v ∈ V .

Original: In local search with a decomposable score, changes made by local modification of the DAG G can be reflected to the network score S ( G , D ) by updating corresponding components without calculating the network score from scratch.
Revised: In local search with a decomposable score, changes made by local modification of the DAG G can be reflected to the network score S ( G , D ) by updating corresponding components without calculating the network score from scratch.

Original: Despite of the computational benefit, the constraint on the decomposability of the score restricts the use of more suitable scores, such as, scores with non-factorized priors[CCD15] or sophisticated information criteria[GR19].
Revised: Despite of the computational benefit, the constraint on the decomposability of the score restricts the use of more suitable scores, such as, scores with non-factorized priors[CCD15] or sophisticated information criteria[GR19].

Original: E Normalized Maximum Likelihood E.1 Model Selection with Minimum Description Length
Revised: E Normalized Maximum Likelihood E.1 Model Selection with Minimum Description Length

Original: In minimum description length (MDL) principle[GG07], a distribution called a universal distribution is associated with each model class, for example, ¯ p G ( · ) is associated with the M G , BNs with a given DAG G .
Revised: In minimum description length (MDL) principle[GG07], a distribution called a universal distribution is associated with each model class, for example, ¯ p G ( · ) is associated with the M G , BNs with a given DAG G .

Original: For a given data D , model selection can be performed by comparing the universal distribution relative to a model class
Revised: For a given data D , model selection can be performed by comparing the universal distribution relative to a model class

Original: [Equation]
Revised: [Equation]

Original: E.2 Normalized Maximum Likelihood
Revised: E.2 Normalized Maximum Likelihood

Original: Normalized Maximum Likelihood (NML) is regarded as the most fundamental universal distribution[GR19].
Revised: Normalized Maximum Likelihood (NML) is regarded as the most fundamental universal distribution[GR19].

Original: For the discrete BNs with a DAG G with the data D , NML is defined as
Revised: For the discrete BNs with a DAG G with the data D , NML is defined as

Original: [Equation]
Revised: [Equation]

Original: where ˆ θ ML ( G , D ) is the maximum likelihood estimator of the parameters of the BN with the DAG G on the data D .
Revised: where ˆ θ ML ( G , D ) is the maximum likelihood estimator of the parameters of the BN with the DAG G on the data D .

Original: The summation over all possible data with the same cardinality is the computational bottleneck.
Revised: The summation over all possible data with the same cardinality is the computational bottleneck.

Original: The log of the denominator REG NML ( G , N ) = log( (cid:80) |D ′ |
Revised: The log of the denominator REG NML ( G , N ) = log( (cid:80) |D ′ |

Original: = N p BN ( D ′ | G , ˆ θ ML
Revised: = N p BN ( D ′ | G , ˆ θ ML

Original: ( G , D ′ ))) is called NML regret.
Revised: ( G , D ′ ))) is called NML regret.

Original: E.3 NML regret estimation
Revised: E.3 NML regret estimation

Original: Even though it is strongly principled, NML computation is restricted to certain classes of models, e.g, multinomial distribution[KM07], naive Bayes[MM07], which prevents its use in score-based structure learning.
Revised: Even though it is strongly principled, NML computation is restricted to certain classes of models, e.g, multinomial distribution[KM07], naive Bayes[MM07], which prevents its use in score-based structure learning.

Original: In Bayesian networks, efficient approximations were proposed and shown to perform better in model selection[RSKM08, SLAJR18].
Revised: In Bayesian networks, efficient approximations were proposed and shown to perform better in model selection[RSKM08, SLAJR18].

Original: Even though REG NML ( G , N ) cannot be exactly computed, the summation can be estimated using
Revised: Even though REG NML ( G , N ) cannot be exactly computed, the summation can be estimated using

Original: Monte carlo with proper scaling when BN is discrete.
Revised: Monte carlo with proper scaling when BN is discrete.

Original: [Equation]
Revised: [Equation]

Original: (81) where LSE is the logsumexp whose implementation increases the numerical stability significantly 17 .
Revised: (81) where LSE is the logsumexp whose implementation increases the numerical stability significantly 17 .

Original: In our scaled MC estimate of REG NML ( · , · ) , we observed that smaller samples tend to marginally underestimate the value.
Revised: In our scaled MC estimate of REG NML ( · , · ) , we observed that smaller samples tend to marginally underestimate the value.

Original: However, the estimation seems quickly saturated with respect to the sample size.
Revised: However, the estimation seems quickly saturated with respect to the sample size.

Original: We observed that a MC-estimate of NML regret using | S | = 10 , 000 is a good compromise between the stability of the estimation and the time needed for the evaluation.
Revised: We observed that a MC-estimate of NML regret using | S | = 10 , 000 is a good compromise between the stability of the estimation and the time needed for the evaluation.

Original: With 10,000 samples, the estimation is stable and the difference made by using more samples is marginal to the difference made by the choice of different DAGs.
Revised: With 10,000 samples, the estimation is stable and the difference made by using more samples is marginal to the difference made by the choice of different DAGs.

Original: On machines with Intel(R) Xeon(R) CPU E5-2630 v 2.40GHz, the evaluation time of the objectives (Tab. 2) ranges from one minute to four minutes.
Revised: On machines with Intel(R) Xeon(R) CPU E5-2630 v 2.40GHz, the evaluation time of the objectives (Tab. 2) ranges from one minute to four minutes.

Original: F Additional Information on Experiments F.1 Submodular Maximization
Revised: F Additional Information on Experiments F.1 Submodular Maximization

Original: A set function g : 2 Ω → R , where 2 Ω is the power set of Ω , is submodular when it has the diminishing returns property, that is, for all P ⊂ Q ⊂ Ω and
Revised: A set function g : 2 Ω → R , where 2 Ω is the power set of Ω , is submodular when it has the diminishing returns property, that is, for all P ⊂ Q ⊂ Ω and

Original: p ∈
Revised: p ∈

Original: Ω \ Q
Revised: Ω \ Q

Original: [Equation]
Revised: [Equation]

Original: As a combinatorial version of convexity [Lov83], submodularity has been playing a critical role in combinatorial optimization [Fuj05].
Revised: As a combinatorial version of convexity [Lov83], submodularity has been playing a critical role in combinatorial optimization [Fuj05].

Original: One important property of the submodular function is that when it is positive ( g ( · ) ≥ 0 ) and monotone ( P ⊂ Q = ⇒ g
Revised: One important property of the submodular function is that when it is positive ( g ( · ) ≥ 0 ) and monotone ( P ⊂ Q = ⇒ g

Original: ( P ) ≤ g ( Q ) ), its maximization can be performed greedily with an approximation guarantee [NWF78] as given below.
Revised: ( P ) ≤ g ( Q ) ), its maximization can be performed greedily with an approximation guarantee [NWF78] as given below.

Original: In the maximization of a positive monotone submodular function with the cardinality constraints, g ( P ∗ ) = max | P | =
Revised: In the maximization of a positive monotone submodular function with the cardinality constraints, g ( P ∗ ) = max | P | =

Original: M g ( P ) , the solution P ∗ greedy =
Revised: M g ( P ) , the solution P ∗ greedy =

Original: { p ∗ 1 , · · · , p ∗ M } from the greedy strategy which sequentially solves p ∗ m = argmax p ∈ Ω g ( { p ∗ 1 , · · · , p ∗ m − 1 , p } ) has the following approximation guarantee
Revised: { p ∗ 1 , · · · , p ∗ M } from the greedy strategy which sequentially solves p ∗ m = argmax p ∈ Ω g ( { p ∗ 1 , · · · , p ∗ m − 1 , p } ) has the following approximation guarantee

Original: [Equation]
Revised: [Equation]

Original: In practice, this greedy method often provides almost optimum solutions [SKD15].
Revised: In practice, this greedy method often provides almost optimum solutions [SKD15].

Original: Moreover, it is possible to relax the conditions (positivity, monotonicity, and even submodularity)
Revised: Moreover, it is possible to relax the conditions (positivity, monotonicity, and even submodularity)

Original: [FMV11,
Revised: [FMV11,

Original: BBKT17, Sak20].
Revised: BBKT17, Sak20].

Original: F.2 DPP-SAMPLE-EST implementation
Revised: F.2 DPP-SAMPLE-EST implementation

Original: In [KDK16], DPP-MAX-EST and DPP-SAMPLE-EST are compared on continuous spaces using the median of multiple runs, and the median of DPP-SAMPLE-EST outperforms the median of
Revised: In [KDK16], DPP-MAX-EST and DPP-SAMPLE-EST are compared on continuous spaces using the median of multiple runs, and the median of DPP-SAMPLE-EST outperforms the median of

Original: DPP-MAX-EST.
Revised: DPP-MAX-EST.

Original: On the permutation spaces of our experiments, DPP-SAMPLE-EST performs worse than DPP-MAX-EST in terms of the mean of multiple runs.
Revised: On the permutation spaces of our experiments, DPP-SAMPLE-EST performs worse than DPP-MAX-EST in terms of the mean of multiple runs.

Original: We attribute this to the sample size used in sampling, rather than the use of a different performance measure.
Revised: We attribute this to the sample size used in sampling, rather than the use of a different performance measure.

Original: Due to the large size of permutation spaces, it is infeasible to collect many samples as suggested in [AGR16], which is O ( S log( S/ϵ )) where S is the size of the permutation space and ϵ is the desired approximation level.
Revised: Due to the large size of permutation spaces, it is infeasible to collect many samples as suggested in [AGR16], which is O ( S log( S/ϵ )) where S is the size of the permutation space and ϵ is the desired approximation level.

Original: As suggested in [AGR16], we first use DPP-MAX-EST to pick the initial point of sampling and then we perform 100 MCMC sampling.
Revised: As suggested in [AGR16], we first use DPP-MAX-EST to pick the initial point of sampling and then we perform 100 MCMC sampling.

Original: Therefore, MCMC includes the maximization routine in its initialization.
Revised: Therefore, MCMC includes the maximization routine in its initialization.

Original: And we conjectured that not many sampling steps are necessary because it already starts from the most likely point expecting that being perturbed from the most likely point, it is still likely but retains reasonable diversity.
Revised: And we conjectured that not many sampling steps are necessary because it already starts from the most likely point expecting that being perturbed from the most likely point, it is still likely but retains reasonable diversity.

Original: However, from the results we have, it appears that more sampling steps are necessary.
Revised: However, from the results we have, it appears that more sampling steps are necessary.

Original: Since the sufficient condition given in [AGR16] requires an infeasibly huge number on permutation space and such huge number can make the batch acquisition time not negligible compared with evaluation time, LAW focuses on the maximization of the k -DPP density.
Revised: Since the sufficient condition given in [AGR16] requires an infeasibly huge number on permutation space and such huge number can make the batch acquisition time not negligible compared with evaluation time, LAW focuses on the maximization of the k -DPP density.

Original: F.3 Combinatorial optimization problems on permutations
Revised: F.3 Combinatorial optimization problems on permutations

Original: We consider three types of combinatorial optimization on permutations
Revised: We consider three types of combinatorial optimization on permutations

Original: Quadratic Assignment Problems [KB57] Given N facilities P and N locations L , a distance d ( · , · ) is given for each pair of locations and a weight k ( · , · ) is given for each pair of facilities, for example, the cost of delivery between facilities.
Revised: Quadratic Assignment Problems [KB57] Given N facilities P and N locations L , a distance d ( · , · ) is given for each pair of locations and a weight k ( · , · ) is given for each pair of facilities, for example, the cost of delivery between facilities.

Original: Then the goal is to find an assignment represented by a permutation π ∗ minimizing f ( π )
Revised: Then the goal is to find an assignment represented by a permutation π ∗ minimizing f ( π )

Original: = (cid:80) a,b ∈P k
Revised: = (cid:80) a,b ∈P k

Original: ( a, b ) · d ( π ( a ) , π ( b )) .
Revised: ( a, b ) · d ( π ( a ) , π ( b )) .

Original: Data source ( https://www.opt.math.tugraz.at/qaplib/inst.html ): char12a[CB89], nug22[NVR68], esc32a[EW90]
Revised: Data source ( https://www.opt.math.tugraz.at/qaplib/inst.html ): char12a[CB89], nug22[NVR68], esc32a[EW90]

Original: Flowshop Scheduling Problems [Wik20] There are N machines and M jobs.
Revised: Flowshop Scheduling Problems [Wik20] There are N machines and M jobs.

Original: Each job requires
Revised: Each job requires

Original: N operations to complete.
Revised: N operations to complete.

Original: The n -th operation of the job must be executed on the m -th machine.
Revised: The n -th operation of the job must be executed on the m -th machine.

Original: Each machine can process at most one operation at a time.
Revised: Each machine can process at most one operation at a time.

Original: Each operation in each job has its own execution specified.
Revised: Each operation in each job has its own execution specified.

Original: Even though jobs can be executed in any order, operations in each job should obey the given order.
Revised: Even though jobs can be executed in any order, operations in each job should obey the given order.

Original: The problem is to find an optimal order of jobs to minimize execution time.
Revised: The problem is to find an optimal order of jobs to minimize execution time.

Original: For a formal description, please refer to [Ree95].
Revised: For a formal description, please refer to [Ree95].

Original: Data source ( http://people.brunel.ac.uk/~mastjjb/jeb/orlib/flowshopinfo.
Revised: Data source ( http://people.brunel.ac.uk/~mastjjb/jeb/orlib/flowshopinfo.

Original: html ):
Revised: html ):

Original: car5[Car78], hel2[Hel60], reC19[Ree95]
Revised: car5[Car78], hel2[Hel60], reC19[Ree95]

Original: Traveling Salesman Problems For given cities, a salesman visits each city exactly once while minimizing a given cost incurred in travelling.
Revised: Traveling Salesman Problems For given cities, a salesman visits each city exactly once while minimizing a given cost incurred in travelling.

Original: TSP is the most widely known example of combinatorial optimization on permutations.
Revised: TSP is the most widely known example of combinatorial optimization on permutations.

Original: Data source ( http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/ )
Revised: Data source ( http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/ )

Original: G Experiment Results
Revised: G Experiment Results

Original: In this section, we provide the additional experimental results which we cannot present in the main text due to the page limit.
Revised: In this section, we provide the additional experimental results which we cannot present in the main text due to the page limit.

Original: Following results are presented from the next page
Revised: Following results are presented from the next page

Original: • Comparison with other LAW variants as combinatorial versions of the local penalization[GDHL16] (Subsec.
Revised: • Comparison with other LAW variants as combinatorial versions of the local penalization[GDHL16] (Subsec.

Original: 5.2.1) • Figures of the structure learning experiments (Sec. 5.3)
Revised: 5.2.1) • Figures of the structure learning experiments (Sec. 5.3)

Original: Method
Revised: Method

Original: [Table]
Revised: [Table]

Original: #Eval
Revised: #Eval

Original: [Table]
Revised: [Table]

Original: +18104 .
Revised: +18104 .

Original: 80 ± 955 .
Revised: 80 ± 955 .

Original: [Table]
Revised: [Table]

Original: EST +14731 .
Revised: EST +14731 .

Original: 60 ± 633 .
Revised: 60 ± 633 .

Original: [Table]
Revised: [Table]

Original: SAMPLE
Revised: SAMPLE

Original: [Table]
Revised: [Table]

Original: EST +19969 .
Revised: EST +19969 .

Original: 60 ± 718 .
Revised: 60 ± 718 .

Original: MACE
Revised: MACE

Original: [Table]
Revised: [Table]

Original: EST +14126 .
Revised: EST +14126 .

Original: 13 ± 596 .
Revised: 13 ± 596 .

Original: MACE
Revised: MACE

Original: [Table]
Revised: [Table]

Original: UCB +13440 .
Revised: UCB +13440 .

Original: 13 ± 347 .
Revised: 13 ± 347 .

Original: q
Revised: q

Original: [Table]
Revised: [Table]

Original: EI +12769 .
Revised: EI +12769 .

Original: 20 ± 457 .
Revised: 20 ± 457 .

Original: q
Revised: q

Original: [Table]
Revised: [Table]

Original: EST +11790 .
Revised: EST +11790 .

Original: 13 ± 497 .
Revised: 13 ± 497 .

Original: LAW
Revised: LAW

Original: [Table]
Revised: [Table]

Original: EI +11914 .
Revised: EI +11914 .

Original: 40 ± 345 .
Revised: 40 ± 345 .

Original: LAW
Revised: LAW

Original: [Table]
Revised: [Table]

Original: EST +12067 .
Revised: EST +12067 .

Original: 07 ± 237 .
Revised: 07 ± 237 .

Original: LAW
Revised: LAW

Original: [Table]
Revised: [Table]

Original: prior
Revised: prior

Original: [Table]
Revised: [Table]

Original: EI +11875 .
Revised: EI +11875 .

Original: 67 ± 771 .
Revised: 67 ± 771 .

Original: LAW
Revised: LAW

Original: [Table]
Revised: [Table]

Original: EST +11842 .
Revised: EST +11842 .

Original: 53 ± 301 .
Revised: 53 ± 301 .

Original: Method
Revised: Method

Original: Mean ±
Revised: Mean ±

Original: [Table] [Table]
Revised: [Table] [Table]

Original: #Eval
Revised: #Eval

Original: DPP
Revised: DPP

Original: [Table]
Revised: [Table]

Original: EST +3899 .
Revised: EST +3899 .

Original: 60 ± 23 .
Revised: 60 ± 23 .

Original: DPP
Revised: DPP

Original: [Table]
Revised: [Table]

Original: SAMPLE
Revised: SAMPLE

Original: [Table]
Revised: [Table]

Original: EST +4446 .
Revised: EST +4446 .

Original: 13 ± 22 .
Revised: 13 ± 22 .

Original: MACE
Revised: MACE

Original: [Table]
Revised: [Table]

Original: EST +4085 .
Revised: EST +4085 .

Original: 87 ± 19 .
Revised: 87 ± 19 .

Original: MACE
Revised: MACE

Original: [Table]
Revised: [Table]

Original: UCB +4030 .
Revised: UCB +4030 .

Original: 80 ± 26 .
Revised: 80 ± 26 .

Original: q
Revised: q

Original: [Table]
Revised: [Table]

Original: +3653 .
Revised: +3653 .

Original: 07 ± 10 .
Revised: 07 ± 10 .

Original: q
Revised: q

Original: [Table]
Revised: [Table]

Original: EST +3690 . 

Revised: EST +3690 .

Original: ± 15 .
Revised: ± 15 .

Original: LAW
Revised: LAW

Original: [Table]
Revised: [Table]

Original: +3724 . 

Revised: +3724 .

Original: ± 12 .
Revised: ± 12 .

Original: LAW
Revised: LAW

Original: [Table]
Revised: [Table]

Original: EST +3730 .
Revised: EST +3730 .

Original: 93 ± 9 .
Revised: 93 ± 9 .

Original: LAW
Revised: LAW

Original: [Table]
Revised: [Table]

Original: prior
Revised: prior

Original: [Table]
Revised: [Table]

Original: EI +3777 .
Revised: EI +3777 .

Original: 47 ± 7 .
Revised: 47 ± 7 .

Original: LAW
Revised: LAW

Original: [Table]
Revised: [Table]

Original: EST +3695 .
Revised: EST +3695 .

Original: 47 ± 11 .
Revised: 47 ± 11 .

Original: Method
Revised: Method

Original: Mean ±
Revised: Mean ±

Original: [Table]
Revised: [Table]

Original: .Err.
Revised: .Err.

Original: #Eval
Revised: #Eval

Original: DPP
Revised: DPP

Original: [Table]
Revised: [Table]

Original: MAX-EST +276 .
Revised: MAX-EST +276 .

Original: 53 ± 3 .
Revised: 53 ± 3 .

Original: DPP
Revised: DPP

Original: [Table]
Revised: [Table]

Original: SAMPLE
Revised: SAMPLE

Original: [Table]
Revised: [Table]

Original: EST +319 .
Revised: EST +319 .

Original: 60 ± 3 .
Revised: 60 ± 3 .

Original: MACE
Revised: MACE

Original: [Table]
Revised: [Table]

Original: EST +285 .
Revised: EST +285 .

Original: 60 ± 3 .
Revised: 60 ± 3 .

Original: MACE
Revised: MACE

Original: [Table]
Revised: [Table]

Original: UCB +250 .
Revised: UCB +250 .

Original: 27 ± 3 .
Revised: 27 ± 3 .

Original: q
Revised: q

Original: [Table]
Revised: [Table]

Original: EI +172 .
Revised: EI +172 .

Original: 67 ± 3 .
Revised: 67 ± 3 .

Original: [Table]
Revised: [Table]

Revised: [Table]

Original: [Table] [Table] 
Revised: [Table] [Table] [Table]

Original: [Table] [Table][Table]
Revised: [Table] [Table]

Original: [Table] [Table][Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: +
Revised: +

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table]
Revised: [Table]

Original: +3575 .
Revised: +3575 .

Original: 53 ± 25 .
Revised: 53 ± 25 .

Original: [Table]
Revised: [Table]

Original: +3582 .
Revised: +3582 .

Original: [Table] [Table]
Revised: [Table] [Table]

Original: +3426 .
Revised: +3426 .

Original: 53 ± 39 .
Revised: 53 ± 39 .

Original: [Table]
Revised: [Table]

Original: +3526 .
Revised: +3526 .

Original: [Table] [Table]
Revised: [Table] [Table]

Original: +3465 .
Revised: +3465 .

Original: 87 ± 25 .
Revised: 87 ± 25 .

Original: [Table]
Revised: [Table]

Original: +3369 .
Revised: +3369 .

Original: 27 ± 7 .
Revised: 27 ± 7 .

Original: [Table]
Revised: [Table]

Original: +3445 .
Revised: +3445 .

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: Method
Revised: Method

Original: [Table]
Revised: [Table]

Original: +
Revised: +

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table] [Table]
Revised: [Table] [Table]

Original: +3652 .
Revised: +3652 .

Original: 87 ± 29 .
Revised: 87 ± 29 .

Original: [Table]
Revised: [Table]

Original: +2939 .
Revised: +2939 .

Original: 67 ± 49 .
Revised: 67 ± 49 .

Original: [Table]
Revised: [Table]

Original: +2697 .
Revised: +2697 .

Original: 93 ± 50 .
Revised: 93 ± 50 .

Original: [Table]
Revised: [Table]

Original: +2065 .
Revised: +2065 .

Original: 13 ± 36 .
Revised: 13 ± 36 .

Original: [Table]
Revised: [Table]

Original: EST +2059 .
Revised: EST +2059 .

Original: [Table] [Table]
Revised: [Table] [Table]

Original: +2486 .
Revised: +2486 .

Original: 87 ± 47 .
Revised: 87 ± 47 .

Original: [Table]
Revised: [Table]

Original: +2038 .
Revised: +2038 .

Original: 40 ± 36 .
Revised: 40 ± 36 .

Original: [Table]
Revised: [Table]

Original: +2491 .
Revised: +2491 .

Original: 27 ± 46 .
Revised: 27 ± 46 .

Original: [Table]
Revised: [Table]

Original: +2250 . 

Revised: +2250 .

Original: ± 56 .
Revised: ± 56 .

Original: Method
Revised: Method

Original: [Table]
Revised: [Table]

Original: EST +39539 .
Revised: EST +39539 .

Original: 47 ± 486 .
Revised: 47 ± 486 .

Original: [Table]
Revised: [Table]

Original: +40893 .
Revised: +40893 .

Original: 30 ± 265 .
Revised: 30 ± 265 .

Original: MACE
Revised: MACE

Original: [Table]
Revised: [Table]

Original: EST +32710 .
Revised: EST +32710 .

Original: 55 ± 212 .
Revised: 55 ± 212 .

Original: MACE
Revised: MACE

Original: [Table]
Revised: [Table]

Original: +25772 .
Revised: +25772 .

Original: 51 ± 370 .
Revised: 51 ± 370 .

Original: q
Revised: q

Original: [Table]
Revised: [Table]

Original: +20472 .
Revised: +20472 .

Original: 44 ± 502 .
Revised: 44 ± 502 .

Original: [Table] [Table]
Revised: [Table] [Table]