Original: Human-AI Collaborative Bayesian Optimisation
Revised: Human-AI Collaborative Bayesian Optimisation

Original: Abstract
Revised: Abstract

Original: Human-AI collaboration looks at harnessing the complementary strengths of both humans and AI.
Revised: Human-AI collaboration looks at harnessing the complementary strengths of both humans and AI.

Original: We propose a new method for human-AI collaboration in
Revised: We propose a new method for human-AI collaboration in

Original: Bayesian optimisation where the optimum is mainly pursued by the Bayesian optimisation algorithm following complex computation, whilst getting occasional help from the accompanying expert having a deeper knowledge of the underlying physical phenomenon.
Revised: Bayesian optimisation where the optimum is mainly pursued by the Bayesian optimisation algorithm following complex computation, whilst getting occasional help from the accompanying expert having a deeper knowledge of the underlying physical phenomenon.

Original: We expect experts to have some understanding of the correlation structures of the experimental system, but not the location of the optimum.
Revised: We expect experts to have some understanding of the correlation structures of the experimental system, but not the location of the optimum.

Original: The expert provides feedback by either changing the current recommendation or providing her belief on the good and bad regions of the search space based on the current observations.
Revised: The expert provides feedback by either changing the current recommendation or providing her belief on the good and bad regions of the search space based on the current observations.

Original: Our proposed method takes such feedback to build a model that aligns with the expert’s model and then uses it for optimisation.
Revised: Our proposed method takes such feedback to build a model that aligns with the expert’s model and then uses it for optimisation.

Original: We provide theoretical underpinning on why such an approach may be more efﬁcient than the one without expert’s feedback.
Revised: We provide theoretical underpinning on why such an approach may be more efﬁcient than the one without expert’s feedback.

Original: The empirical results show the robustness and superiority of our method with promising efﬁciency gains.
Revised: The empirical results show the robustness and superiority of our method with promising efﬁciency gains.

Original:  Introduction
Revised:  Introduction

Original: The deep penetration of AI systems into wider society demonstrates its ability to deal with complex real-world tasks.
Revised: The deep penetration of AI systems into wider society demonstrates its ability to deal with complex real-world tasks.

Original: However, there still exists a plethora of complicated real-world problems [Roccetti et al., 2020, Hechler et al., 2020] that AI systems are unable to properly address.
Revised: However, there still exists a plethora of complicated real-world problems [Roccetti et al., 2020, Hechler et al., 2020] that AI systems are unable to properly address.

Original: In such complex real-world tasks, human-AI collaborative systems can provide a viable alternative.
Revised: In such complex real-world tasks, human-AI collaborative systems can provide a viable alternative.

Original: Through appropriate collaboration strategy we can supplement the speed, scalability and quantitative abilities of
Revised: Through appropriate collaboration strategy we can supplement the speed, scalability and quantitative abilities of

Original: AI systems with the reasoning and abstraction ability of human users and make the combination exceedingly capable [Carroll et al., 2019, Maadi et al., 2021].
Revised: AI systems with the reasoning and abstraction ability of human users and make the combination exceedingly capable [Carroll et al., 2019, Maadi et al., 2021].

Original: Recent studies, mostly proposed in supervised learning setting, used either AI requesting help of human experts if its prediction lacks conﬁdence [Madras et al., 2017] or as guided by a meta-policy that focuses on the joint performance [Wilder et al., 2020].
Revised: Recent studies, mostly proposed in supervised learning setting, used either AI requesting help of human experts if its prediction lacks conﬁdence [Madras et al., 2017] or as guided by a meta-policy that focuses on the joint performance [Wilder et al., 2020].

Original: Mozannar and Sontag [2020], De et al.
Revised: Mozannar and Sontag [2020], De et al.

Original: [2020] considered also the cost associated with human decision making while learning a joint policy.
Revised: [2020] considered also the cost associated with human decision making while learning a joint policy.

Original: Bayesian optimisation (BO) [Brochu et al., 2010], an efﬁcient tool for global optimisation that can potentially play a much bigger role in scientiﬁc experimentation and product design, provides a rich playground for human-AI collaboration paradigm because the nature of applications ensures that there would be an expert running the optimisation process.
Revised: Bayesian optimisation (BO) [Brochu et al., 2010], an efﬁcient tool for global optimisation that can potentially play a much bigger role in scientiﬁc experimentation and product design, provides a rich playground for human-AI collaboration paradigm because the nature of applications ensures that there would be an expert running the optimisation process.

Original: There are some works on incorporating expert’s knowledge in Bayesian optimisation procedure.
Revised: There are some works on incorporating expert’s knowledge in Bayesian optimisation procedure.

Original: However, nearly all of the work relies on expert articulating a static type of knowledge in the beginning of the optimisation process either in terms of trends [Li et al., 2018, Riihimäki and Vehtari, 2010], shapes of functions [Andersen et al., 2017, Venkatesh et al., 2019] or suggesting a similar experimental system for transfer learning [Joy et al., 2016, Shilton et al., 2017].
Revised: However, nearly all of the work relies on expert articulating a static type of knowledge in the beginning of the optimisation process either in terms of trends [Li et al., 2018, Riihimäki and Vehtari, 2010], shapes of functions [Andersen et al., 2017, Venkatesh et al., 2019] or suggesting a similar experimental system for transfer learning [Joy et al., 2016, Shilton et al., 2017].

Original: In reality, most of the time experts may not be able to reveal knowledge in terms of such explicit speciﬁcations.
Revised: In reality, most of the time experts may not be able to reveal knowledge in terms of such explicit speciﬁcations.

Original: Expert’s knowledge will also more likely evolve over the course of optimisation process through assimilation of the new data points.
Revised: Expert’s knowledge will also more likely evolve over the course of optimisation process through assimilation of the new data points.

Original: Unfortunately,
Revised: Unfortunately,

Original: y 1.
Revised: y 1.

Original: (a) 0.
Revised: (a) 0.

Original: P r o b a b ili t y D e n s i t y
Revised: P r o b a b ili t y D e n s i t y

Original: HAT STD-BO
Revised: HAT STD-BO

Original: Ground truth
Revised: Ground truth

Original: Lengthscale for (b) a true collaborative system that takes into account such a ﬂuid knowledge from human expert via an intuitive communication interface is so far lacking.
Revised: Lengthscale for (b) a true collaborative system that takes into account such a ﬂuid knowledge from human expert via an intuitive communication interface is so far lacking.

Original: To address this problem, we propose a new H
Revised: To address this problem, we propose a new H

Original: uman- A I T eaming (HAT)
Revised: uman- A I T eaming (HAT)

Original: based Bayesian optimisation framework that takes expert input during the course of optimisation through one of the two ways: (i) correction of the current recommendation, or (ii) speciﬁcation of a good region and bad region based on the current observation set.
Revised: based Bayesian optimisation framework that takes expert input during the course of optimisation through one of the two ways: (i) correction of the current recommendation, or (ii) speciﬁcation of a good region and bad region based on the current observation set.

Original: We believe these are easier to specify, than say, a global trend or global shape of a function because an expert may be able to provide reasoning surrounding a small neighbourhood of a sample point.
Revised: We believe these are easier to specify, than say, a global trend or global shape of a function because an expert may be able to provide reasoning surrounding a small neighbourhood of a sample point.

Original: Once such feedback is given, we incorporate it into Bayesian optimisation by changing the model selection process that ensures that such relative knowledge that the corrected recommendation is a more promising one compared to the current recommendation or the good region is more promising than the bad region are maintained.
Revised: Once such feedback is given, we incorporate it into Bayesian optimisation by changing the model selection process that ensures that such relative knowledge that the corrected recommendation is a more promising one compared to the current recommendation or the good region is more promising than the bad region are maintained.

Original: We use the acquisition function as the means to compute the promise and use the aforesaid feedback during model selection via maximum likelihood.
Revised: We use the acquisition function as the means to compute the promise and use the aforesaid feedback during model selection via maximum likelihood.

Original: The model derived from these constraints is subsequently used for the next recommendation.
Revised: The model derived from these constraints is subsequently used for the next recommendation.

Original: Our framework allows an expert to intervene only when they are willing to intervene and also allows them to evolve their knowledge over the course of optimisation.
Revised: Our framework allows an expert to intervene only when they are willing to intervene and also allows them to evolve their knowledge over the course of optimisation.

Original: Theoretically, we show that when Gaussian process is used to model the function, this extra channel of information narrows the space of functions.
Revised: Theoretically, we show that when Gaussian process is used to model the function, this extra channel of information narrows the space of functions.

Original: We characterise such narrower distributions in terms of Sobolev spaces [Tartar, 2007] and show how it admits tighter upper bound of the total regret when an appropriate trade-off factor is used in the acquisition function.
Revised: We characterise such narrower distributions in terms of Sobolev spaces [Tartar, 2007] and show how it admits tighter upper bound of the total regret when an appropriate trade-off factor is used in the acquisition function.

Original: Further, we show in
Revised: Further, we show in

Original: Figure 1a and 1b, a case of HAT based Bayesian optimisation vs standard likelihood based model selection of length-scales of a Squared Exponential (SE) kernel.
Revised: Figure 1a and 1b, a case of HAT based Bayesian optimisation vs standard likelihood based model selection of length-scales of a Squared Exponential (SE) kernel.

Original: For a particular dimension ( x 2 ) we observe that the HAT framework with an expert having ground truth knowledge infers models that are distributed more narrowly than the standard one, inducing a smaller function space.
Revised: For a particular dimension ( x 2 ) we observe that the HAT framework with an expert having ground truth knowledge infers models that are distributed more narrowly than the standard one, inducing a smaller function space.

Original: We evaluate the performance of our proposed methods on both synthetic benchmark functions and real-world datasets.
Revised: We evaluate the performance of our proposed methods on both synthetic benchmark functions and real-world datasets.

Original: We tune the hyperparameters of the Support Vector Machine (SVM) classiﬁer [Burges, 1998, Christmann and Steinwart, 2008] operating on publicly available real-world datasets.
Revised: We tune the hyperparameters of the Support Vector Machine (SVM) classiﬁer [Burges, 1998, Christmann and Steinwart, 2008] operating on publicly available real-world datasets.

Original: We compare the performance of our proposed framework against different variations of the Bayesian optimisation algorithm.
Revised: We compare the performance of our proposed framework against different variations of the Bayesian optimisation algorithm.

Original: The empirical results prove the efﬁcacy of our proposed Human-AI Teaming (HAT) framework.
Revised: The empirical results prove the efﬁcacy of our proposed Human-AI Teaming (HAT) framework.

Original: We also provide a small-scale real-user study to demonstrate the superiority of our Human-AI Teaming approach over AI only approach.
Revised: We also provide a small-scale real-user study to demonstrate the superiority of our Human-AI Teaming approach over AI only approach.

Original:  Background
Revised:  Background

Original: Notations.
Revised: Notations.

Original: R for Reals.
Revised: R for Reals.

Original: X is an index set and x ∈ X .
Revised: X is an index set and x ∈ X .

Original: N n = { 1 , 2 , · · · , n } .
Revised: N n = { 1 , 2 , · · · , n } .

Original: ∪ for the set union.
Revised: ∪ for the set union.

Original: | · | is the determinant.
Revised: | · | is the determinant.

Original: We use lower case bold fonts v for vectors.
Revised: We use lower case bold fonts v for vectors.

Original: v i for i th element in a vector v .
Revised: v i for i th element in a vector v .

Original: v (cid:124) for the transpose of a vector v .
Revised: v (cid:124) for the transpose of a vector v .

Original: We use upper case bold fonts M (and bold Greek symbols) for matrices and M ij for the element in i th row and j th column of M .
Revised: We use upper case bold fonts M (and bold Greek symbols) for matrices and M ij for the element in i th row and j th column of M .

Original: abs ( · ) for the absolute value.
Revised: abs ( · ) for the absolute value.

Original: 2.1 Gaussian Process
Revised: 2.1 Gaussian Process

Original: Gaussian Process (GP)
Revised: Gaussian Process (GP)

Original: [Williams and Rasmussen, 2006] is a non-parametric model, that provides a ﬂexible framework for placing prior on functions.
Revised: [Williams and Rasmussen, 2006] is a non-parametric model, that provides a ﬂexible framework for placing prior on functions.

Original: Gaussian process deﬁnes a distribution over the set of possible functions given observations.
Revised: Gaussian process deﬁnes a distribution over the set of possible functions given observations.

Original: Though there are other popular surrogate models such as Wiener process [Kushner, 1964] and Student-t process [Shah et al., 2014], Gaussian process is still the preferred model because of its simplicity.
Revised: Though there are other popular surrogate models such as Wiener process [Kushner, 1964] and Student-t process [Shah et al., 2014], Gaussian process is still the preferred model because of its simplicity.

Original: A GP is completely speciﬁed by a mean function µ ( x ) and a kernel function k
Revised: A GP is completely speciﬁed by a mean function µ ( x ) and a kernel function k

Original: ( x , x 0 ) .
Revised: ( x , x 0 ) .

Original: The unknown objective function f ( x ) is modelled using Gaussian process as f ( x ) ∼ GP ( µ ( x ) , k
Revised: The unknown objective function f ( x ) is modelled using Gaussian process as f ( x ) ∼ GP ( µ ( x ) , k

Original: ( x , x 0 )) .
Revised: ( x , x 0 )) .

Original: If D 1: t = { x 1: t , y 1: t } denotes the set of t observations, then according to the properties of the Gaussian process, a new observation ( x t +1 , y t +1 ) and the set of previously observed data samples D 1: t are jointly Gaussian.
Revised: If D 1: t = { x 1: t , y 1: t } denotes the set of t observations, then according to the properties of the Gaussian process, a new observation ( x t +1 , y t +1 ) and the set of previously observed data samples D 1: t are jointly Gaussian.

Original: Therefore, the posterior distribution for the new observation y t +1 is computed as P ( y t +1 |D 1: t , x t +1 ) =
Revised: Therefore, the posterior distribution for the new observation y t +1 is computed as P ( y t +1 |D 1: t , x t +1 ) =

Original: N ( µ ( x t +1 ) , σ 2 ( x t +1 )) , where µ ( x t +1 )
Revised: N ( µ ( x t +1 ) , σ 2 ( x t +1 )) , where µ ( x t +1 )

Original: =
Revised: =

Original: k (cid:124)
Revised: k (cid:124)

Original: [ K + σ 2 GN I ] − 1 y , σ 2 ( x t +1 )
Revised: [ K + σ 2 GN I ] − 1 y , σ 2 ( x t +1 )

Original: =
Revised: =

Original: k ( x t +1 , x t +1 )
Revised: k ( x t +1 , x t +1 )

Original: − k (cid:124)
Revised: − k (cid:124)

Original: [ K + σ 2 GN I ] − 1 k , k =
Revised: [ K + σ 2 GN I ] − 1 k , k =

Original: [ k ( x 1 , x t +1 ) , · · · , k ( x t , x t +1 )] , K ij = k
Revised: [ k ( x 1 , x t +1 ) , · · · , k ( x t , x t +1 )] , K ij = k

Original: ( x i , x j ) ∀
Revised: ( x i , x j ) ∀

Original: i, ∀ j ∈ N t and σ 2 GN is the Gaussian noise.
Revised: i, ∀ j ∈ N t and σ 2 GN is the Gaussian noise.

Original: The kernel function k : X × X → R used in GP plays a crucial role in modelling the GP surrogates.
Revised: The kernel function k : X × X → R used in GP plays a crucial role in modelling the GP surrogates.

Original: The kernel function incorporates our prior belief about the unknown objective function.
Revised: The kernel function incorporates our prior belief about the unknown objective function.

Original: The kernel hyperparameters θ are usually estimated by maximising the marginal likelihood given as L = p ( y |
Revised: The kernel hyperparameters θ are usually estimated by maximising the marginal likelihood given as L = p ( y |

Original: X, θ ) =
Revised: X, θ ) =

Original: R p
Revised: R p

Original: ( y | f ) p ( f | X, θ ) d f .
Revised: ( y | f ) p ( f | X, θ ) d f .

Original: The closed-form formulation of GP log-likelihood is given as:
Revised: The closed-form formulation of GP log-likelihood is given as:

Original: [Equation]
Revised: [Equation]

Original: 2.2 Bayesian Optimisation
Revised: 2.2 Bayesian Optimisation

Original: Bayesian optimisation (BO) [Shahriari et al., 2015, Frazier, 2018] has proliferated into many domains, including sensor networks [Garnett et al., 2010, Zhang et al., 2018], robotics [MartinezCantin et al., 2007, Bossens and Tarapore, 2021], intelligent environmental monitoring [Marchant and Ramos, 2012], and information retrieval [Wang et al., 2014].
Revised: Bayesian optimisation (BO) [Shahriari et al., 2015, Frazier, 2018] has proliferated into many domains, including sensor networks [Garnett et al., 2010, Zhang et al., 2018], robotics [MartinezCantin et al., 2007, Bossens and Tarapore, 2021], intelligent environmental monitoring [Marchant and Ramos, 2012], and information retrieval [Wang et al., 2014].

Original: BO offers an efﬁcient framework for the global optimisation of an expensive and noisy black-box function f ( x ) , represented as:
Revised: BO offers an efﬁcient framework for the global optimisation of an expensive and noisy black-box function f ( x ) , represented as:

Original: [Equation]
Revised: [Equation]

Original: where, X corresponds to a compact search space.
Revised: where, X corresponds to a compact search space.

Original: The observations of the objective function f ( x ) are assumed to be corrupted with a white Gaussian noise i.e., y t
Revised: The observations of the objective function f ( x ) are assumed to be corrupted with a white Gaussian noise i.e., y t

Original: = f ( x t ) + (cid:15) t , where
Revised: = f ( x t ) + (cid:15) t , where

Original: (cid:15) t ∼ N (0 , σ 2 GN ) .
Revised: (cid:15) t ∼ N (0 , σ 2 GN ) .

Original: Bayesian optimisation models the objective function f by placing a prior distribution over the space of unknown objective functions and then use it to determine the next best locations to sample.
Revised: Bayesian optimisation models the objective function f by placing a prior distribution over the space of unknown objective functions and then use it to determine the next best locations to sample.

Original: Speciﬁcally it consists of two main components: (i) a Gaussian process model [Williams and Rasmussen, 2006], and (ii) an acquisition function [Kushner, 1964, Wilson et al., 2018].
Revised: Speciﬁcally it consists of two main components: (i) a Gaussian process model [Williams and Rasmussen, 2006], and (ii) an acquisition function [Kushner, 1964, Wilson et al., 2018].

Original: The GP predictive distribution obtained is used to select the next sampling location with the high promise of ﬁnding the optima.
Revised: The GP predictive distribution obtained is used to select the next sampling location with the high promise of ﬁnding the optima.

Original: The acquisition function guides the search for optima by suitably balancing the exploration-exploitation trade-off.
Revised: The acquisition function guides the search for optima by suitably balancing the exploration-exploitation trade-off.

Original: Srinivas et al.
Revised: Srinivas et al.

Original: [2012] have discussed the Gaussian Process-Upper
Revised: [2012] have discussed the Gaussian Process-Upper

Original: Conﬁdence Bound (GP-UCB) acquisition function using the upper conﬁdence bound selection criterion.
Revised: Conﬁdence Bound (GP-UCB) acquisition function using the upper conﬁdence bound selection criterion.

Original: A GP-UCB acquisition function at t th iteration is given by:
Revised: A GP-UCB acquisition function at t th iteration is given by:

Original: [Equation]
Revised: [Equation]

Original: where β t is a trade-off parameter that balances between the exploitation and exploration.
Revised: where β t is a trade-off parameter that balances between the exploitation and exploration.

Original: Chowdhury and Gopalan [2017] have discussed the possible choices for β t and their implications on the overall regret.
Revised: Chowdhury and Gopalan [2017] have discussed the possible choices for β t and their implications on the overall regret.

Original: In general, to achieve a faster convergence rate β t is increased with O (log t ) .
Revised: In general, to achieve a faster convergence rate β t is increased with O (log t ) .

Original: An iterative algorithm for the standard Bayesian optimisation is provided in the supplementary material.
Revised: An iterative algorithm for the standard Bayesian optimisation is provided in the supplementary material.

Original: 2.3 Reproducing Kernel Hilbert Spaces (RKHS)
Revised: 2.3 Reproducing Kernel Hilbert Spaces (RKHS)

Original: The kernel function used in GP surrogate modelling is associated with a unique Reproducing Kernel
Revised: The kernel function used in GP surrogate modelling is associated with a unique Reproducing Kernel

Original: Hilbert Spaces (RKHS).
Revised: Hilbert Spaces (RKHS).

Original: Let k be a symmetric positive deﬁnite kernel on R d and set k x ( t )
Revised: Let k be a symmetric positive deﬁnite kernel on R d and set k x ( t )

Original: =
Revised: =

Original: k ( t − x ) .
Revised: k ( t − x ) .

Original: For a given compact space X ⊆ R d , let φ ( X ) be the space of functions with the mapping X
Revised: For a given compact space X ⊆ R d , let φ ( X ) be the space of functions with the mapping X

Original: → R , spanned by the kernel k x .
Revised: → R , spanned by the kernel k x .

Original: If φ ( X ) is provided with the inner product represented as h k x , k x 0 i = k
Revised: If φ ( X ) is provided with the inner product represented as h k x , k x 0 i = k

Original: ( x − x 0 ) , then the completion of φ ( x ) is the reproducing kernel Hilbert space H k
Revised: ( x − x 0 ) , then the completion of φ ( x ) is the reproducing kernel Hilbert space H k

Original: ( X ) of k on X .
Revised: ( X ) of k on X .

Original:  Framework
Revised:  Framework

Original: We propose a Human-AI Teaming (HAT) based BO framework to involve the human experts in the optimisation loop to accelerate the optimisation by utilising their knowledge in the surrogate modelling.
Revised: We propose a Human-AI Teaming (HAT) based BO framework to involve the human experts in the optimisation loop to accelerate the optimisation by utilising their knowledge in the surrogate modelling.

Original: First, we discuss the additional knowledge accessible to human experts and then we provide two variants of the HAT framework differing mainly on how the expert knowledge is communicated.
Revised: First, we discuss the additional knowledge accessible to human experts and then we provide two variants of the HAT framework differing mainly on how the expert knowledge is communicated.

Original: The optimisation problem is same as that of the standard BO described in Background section.
Revised: The optimisation problem is same as that of the standard BO described in Background section.

Original: 3.1 Human Experimenter Knowledge
Revised: 3.1 Human Experimenter Knowledge

Original: In many real-world applications, we can expect the human experts to have knowledge about the correlations structure in the underlying function but not about the shape of the function or about the location of the optima.
Revised: In many real-world applications, we can expect the human experts to have knowledge about the correlations structure in the underlying function but not about the shape of the function or about the location of the optima.

Original: With such knowledge, expert may be able to provide a better recommendation, at least in the beginning.
Revised: With such knowledge, expert may be able to provide a better recommendation, at least in the beginning.

Original: We use those expert suggestions at different stages of the optimisation process to assess the deviations of the current AI model and correct them accordingly.
Revised: We use those expert suggestions at different stages of the optimisation process to assess the deviations of the current AI model and correct them accordingly.

Original: The inherent knowledge about the correlations structure of the objective function can take the form of full or partial knowledge of the kernel function, e.g., function is smoother along one dimension than the other.
Revised: The inherent knowledge about the correlations structure of the objective function can take the form of full or partial knowledge of the kernel function, e.g., function is smoother along one dimension than the other.

Original: In either case, the expert can provide input regarding the relative difference between two locations in the search space in terms of their utility as the next sample point based on her cognitive model and the selection criteria.
Revised: In either case, the expert can provide input regarding the relative difference between two locations in the search space in terms of their utility as the next sample point based on her cognitive model and the selection criteria.

Original: For example, if the expert knowledge of the kernel says that the objective function along one dimension is smoothly varying then an expert may be able to adjust the current recommendation by “pushing” that variable further in the direction that takes it away from the existing observations.
Revised: For example, if the expert knowledge of the kernel says that the objective function along one dimension is smoothly varying then an expert may be able to adjust the current recommendation by “pushing” that variable further in the direction that takes it away from the existing observations.

Original: If the expert has full knowledge of the kernel then she may be able to perform such rectiﬁcations along all the input variables.
Revised: If the expert has full knowledge of the kernel then she may be able to perform such rectiﬁcations along all the input variables.

Original: We also assume that the expert is running a form of Bayesian optimisation strategy as it has been demonstrated that human active search typically mimics Bayesian optimisation [Borji and Itti, 2013].
Revised: We also assume that the expert is running a form of Bayesian optimisation strategy as it has been demonstrated that human active search typically mimics Bayesian optimisation [Borji and Itti, 2013].

Original: Next, we discuss two ways of giving feedback.
Revised: Next, we discuss two ways of giving feedback.

Original: 3.2 Rectifying Current Recommendation
Revised: 3.2 Rectifying Current Recommendation

Original: In this approach, we propose to correct the previously suggested AI model’s observations X A in the light of human expert knowledge.
Revised: In this approach, we propose to correct the previously suggested AI model’s observations X A in the light of human expert knowledge.

Original: The human expert knowledge is represented using a set of observations X E generated using her cognitive model reﬂecting full or partial knowledge of the kernel function.
Revised: The human expert knowledge is represented using a set of observations X E generated using her cognitive model reﬂecting full or partial knowledge of the kernel function.

Original: At each iteration, the kernel hyperparameters Θ ∗ ∈ R d of the AI model ( i.e., the Bayesian optimisation process) are tuned by maximising the log-likelihood mentioned in Eq.
Revised: At each iteration, the kernel hyperparameters Θ ∗ ∈ R d of the AI model ( i.e., the Bayesian optimisation process) are tuned by maximising the log-likelihood mentioned in Eq.

Original: Using this tuned Gaussian process surrogate, the AI model suggests the next potential candidate ( x At +1 ) for the function evaluation.
Revised: Using this tuned Gaussian process surrogate, the AI model suggests the next potential candidate ( x At +1 ) for the function evaluation.

Original: At iterations that involve expert rectiﬁcation of the recommendations, we add constraints on the model selection via acquisition functions ( u GP-UCB ) such that the next potential candidate suggested by the AI model has to take into account the corrections enforced by the expert observations x
Revised: At iterations that involve expert rectiﬁcation of the recommendations, we add constraints on the model selection via acquisition functions ( u GP-UCB ) such that the next potential candidate suggested by the AI model has to take into account the corrections enforced by the expert observations x

Original: Ei ∀ i ∈ H collected in X E .
Revised: Ei ∀ i ∈ H collected in X E .

Original: Here we assume that a human expert is also using a form of Bayesian optimisation strategy but with a better model.
Revised: Here we assume that a human expert is also using a form of Bayesian optimisation strategy but with a better model.

Original: We should note that the expert does not necessarily need to provide the optimal point according the correct formulation.
Revised: We should note that the expert does not necessarily need to provide the optimal point according the correct formulation.

Original: The expert needs to only provide a more promising location than what the machine recommended.
Revised: The expert needs to only provide a more promising location than what the machine recommended.

Original: Hence, the constraints take the form as acquisition function ( u GP-UCB ) at x E is better than at x A .
Revised: Hence, the constraints take the form as acquisition function ( u GP-UCB ) at x E is better than at x A .

Original: The following equation shows the constrained model selection problem:
Revised: The following equation shows the constrained model selection problem:

Original: [Equation]
Revised: [Equation]

Original: where D corresponds to the set of all observations { ( x , y = f ( x )) } accumulated by the AI model, x
Revised: where D corresponds to the set of all observations { ( x , y = f ( x )) } accumulated by the AI model, x

Original: Ai corresponds to the i th observation suggested by the AI model collected in X A .
Revised: Ai corresponds to the i th observation suggested by the AI model collected in X A .

Original: These constraints (collected in a set C ) are enforced in the model selection process in the next round.
Revised: These constraints (collected in a set C ) are enforced in the model selection process in the next round.

Original: The intuition behind this constraint set is to restrict ourselves to the hyperparameter distribution that make the AI acquisition function more closely resemble the human acquisition function.
Revised: The intuition behind this constraint set is to restrict ourselves to the hyperparameter distribution that make the AI acquisition function more closely resemble the human acquisition function.

Original: One may wonder (i) why a human expert would behave like a BO algorithm?, and (ii) why humans themselves cannot perform the optimisation?.
Revised: One may wonder (i) why a human expert would behave like a BO algorithm?, and (ii) why humans themselves cannot perform the optimisation?.

Original: To answer the former we refer to the work of Borji and Itti [2013], where they showed that when humans are presented with a search problem they often tend to behave like a BO strategy, i.e., we tend to mix a bit of exploitation and exploration when sampling for the next point.
Revised: To answer the former we refer to the work of Borji and Itti [2013], where they showed that when humans are presented with a search problem they often tend to behave like a BO strategy, i.e., we tend to mix a bit of exploitation and exploration when sampling for the next point.

Original: Here we assume that the strategy is like the GP-UCB strategy.
Revised: Here we assume that the strategy is like the GP-UCB strategy.

Original: However, as we will see in the experiment, even if a human expert uses a noisy version of the GPUCB strategy, our proposed method can still be robust to that.
Revised: However, as we will see in the experiment, even if a human expert uses a noisy version of the GPUCB strategy, our proposed method can still be robust to that.

Original: To answer the latter, we say that while human may possibly get to the optima, it may be extremely taxing for her to do so at every iteration, and also when the number of observations are high.
Revised: To answer the latter, we say that while human may possibly get to the optima, it may be extremely taxing for her to do so at every iteration, and also when the number of observations are high.

Original: Here, we do not need the expert to specify the optima of their acquisition function, but to suggest a better location than x A .
Revised: Here, we do not need the expert to specify the optima of their acquisition function, but to suggest a better location than x A .

Original: We believe as humans we may be able to do that more easily than say running the actual BO which involves solving the global optimisation of the acquisition function.
Revised: We believe as humans we may be able to do that more easily than say running the actual BO which involves solving the global optimisation of the acquisition function.

Original: The complete procedure for Human-AI
Revised: The complete procedure for Human-AI

Original: Teaming (HAT) with experts rectifying recommendations (HAT-RR) is given in Algorithm 1.
Revised: Teaming (HAT) with experts rectifying recommendations (HAT-RR) is given in Algorithm 1.

Original: 3.3 Good Regions vs Bad Regions
Revised: 3.3 Good Regions vs Bad Regions

Original: In contrast to the aforesaid approach, the expert can suggest a good region and a bad region to sample next.
Revised: In contrast to the aforesaid approach, the expert can suggest a good region and a bad region to sample next.

Original: A good region is deﬁned by a set of candidate points that promise high probability of ﬁnding the optima i.e., a set of points that have similar utilities to be the next sampling location.
Revised: A good region is deﬁned by a set of candidate points that promise high probability of ﬁnding the optima i.e., a set of points that have similar utilities to be the next sampling location.

Original: Further, such good candidate points can be seen as the locations suggested by maximising expert’s acquisition functions.
Revised: Further, such good candidate points can be seen as the locations suggested by maximising expert’s acquisition functions.

Original: On the other hand, a bad region corresponds to the portion of the input space that has very low probability to further improve the current solution.
Revised: On the other hand, a bad region corresponds to the portion of the input space that has very low probability to further improve the current solution.

Original: A bad region can be visualised as a set of points that have a very low utility to be the next sampling location as indicated by the low values of the acquisition function.
Revised: A bad region can be visualised as a set of points that have a very low utility to be the next sampling location as indicated by the low values of the acquisition function.

Original: To keep the presentation simple, we formulate our problem based on a good point and a bad point.
Revised: To keep the presentation simple, we formulate our problem based on a good point and a bad point.

Original: The algorithm can be generalised to the case of region by adding constraint per pair of samples from those regions.
Revised: The algorithm can be generalised to the case of region by adding constraint per pair of samples from those regions.

Original: The expert suggestions about the good point ( x g ) and the bad point ( x b ) is used to ﬁne-tune the AI model by replacing its current optimal hyperparameters ( Θ ∗ ) with a hyperparameter set ( Θ + ) that is more aligned with the expert’s cognitive model, in addition to achieving good likelihood.
Revised: The expert suggestions about the good point ( x g ) and the bad point ( x b ) is used to ﬁne-tune the AI model by replacing its current optimal hyperparameters ( Θ ∗ ) with a hyperparameter set ( Θ + ) that is more aligned with the expert’s cognitive model, in addition to achieving good likelihood.

Original: To accomplish this we construct a co-objective that maximises the value difference between the acquisition function of the good point and the bad point.
Revised: To accomplish this we construct a co-objective that maximises the value difference between the acquisition function of the good point and the bad point.

Original: The co-objective is mathematically represented as:
Revised: The co-objective is mathematically represented as:

Original: [Equation]
Revised: [Equation]

Original: where D corresponds to the set of all observations accumulated by the AI model.
Revised: where D corresponds to the set of all observations accumulated by the AI model.

Original: The bi-objective problem is solved in two steps: (i) ﬁrst, the pure maximum likelihood (Eq.
Revised: The bi-objective problem is solved in two steps: (i) ﬁrst, the pure maximum likelihood (Eq.

Original: based hyperpa- rameter estimation (
Revised: based hyperpa- rameter estimation (

Original: ∗ argmax Θ log L ) is performed, and then (ii) the co-objective (Eq.
Revised: ∗ argmax Θ log L ) is performed, and then (ii) the co-objective (Eq.

Original: (5)) is solved by constraining the likelihood to be ∆ -close (threshold) to the likelihood obtained in the previous step.
Revised: (5)) is solved by constraining the likelihood to be ∆ -close (threshold) to the likelihood obtained in the previous step.

Original: The overall algorithm for the HAT framework with the aforementioned difference maximisation strategy (HAT-DM) is provided in Algorithm 2.
Revised: The overall algorithm for the HAT framework with the aforementioned difference maximisation strategy (HAT-DM) is provided in Algorithm 2.

Original:  Theoretical Analysis
Revised:  Theoretical Analysis

Original: Our theoretical analysis relies on the notion of Sobolev spaces [Tartar, 2007] and its extension to
Revised: Our theoretical analysis relies on the notion of Sobolev spaces [Tartar, 2007] and its extension to

Original: Hilbert spaces.
Revised: Hilbert spaces.

Original: We start with the preliminary results required for the theoretical analysis of our proposed method.
Revised: We start with the preliminary results required for the theoretical analysis of our proposed method.

Original: Chowdhury and Gopalan [2017] have discussed the regret bounds of a variant of the GP-UCB algorithm in terms of the norm bounds deﬁned on the objective function f .
Revised: Chowdhury and Gopalan [2017] have discussed the regret bounds of a variant of the GP-UCB algorithm in terms of the norm bounds deﬁned on the objective function f .

Original: Lemma 1 : Let f : X → R , where X ⊂ R d and d is the number of dimensions.
Revised: Lemma 1 : Let f : X → R , where X ⊂ R d and d is the number of dimensions.

Original: Let H k be the Reproducing Kernel Hilbert Space (RKHS) associated with the kernel function k ( · , · ) such that the RKHS norm is bounded in the ball of radius R i.e., k f k k ≤ B R .
Revised: Let H k be the Reproducing Kernel Hilbert Space (RKHS) associated with the kernel function k ( · , · ) such that the RKHS norm is bounded in the ball of radius R i.e., k f k k ≤ B R .

Original: Then there exists a sequence of trade-off factors β t ( R , δ ) such that it holds with probability at least 1 − δ that | f ( x ) − µ t ( x ) | ≤ √ β t σ t ( x ) , ∀ x ∈ X .
Revised: Then there exists a sequence of trade-off factors β t ( R , δ ) such that it holds with probability at least 1 − δ that | f ( x ) − µ t ( x ) | ≤ √ β t σ t ( x ) , ∀ x ∈ X .

Original: Lemma 1 states that f is highly probable to be contained in the conﬁdence intervals induced by the
Revised: Lemma 1 states that f is highly probable to be contained in the conﬁdence intervals induced by the

Original: GP predictions ( µ t ( · ) and σ t ( · ) ) using kernel k : X × X → R with the appropriate scaling induced by β t as discussed in Chowdhury and Gopalan [2017].
Revised: GP predictions ( µ t ( · ) and σ t ( · ) ) using kernel k : X × X → R with the appropriate scaling induced by β t as discussed in Chowdhury and Gopalan [2017].

Original: As the kernel k deﬁnes the size of the function space, the RKHS norm
Revised: As the kernel k deﬁnes the size of the function space, the RKHS norm

Original: k f k k = p
Revised: k f k k = p

Original: h
Revised: h

Original: f, f i k measures the complexity of functions f ∈ H k .
Revised: f, f i k measures the complexity of functions f ∈ H k .

Original: By considering larger norm-balls, we account for more complex functions in the given input space.
Revised: By considering larger norm-balls, we account for more complex functions in the given input space.

Original: In the Human-AI Teaming (HAT) framework, the kernel hyperparameters Θ HAT are tuned in the light of additional channel of information provided by the human experts.
Revised: In the Human-AI Teaming (HAT) framework, the kernel hyperparameters Θ HAT are tuned in the light of additional channel of information provided by the human experts.

Original: The expert suggestions incorporated in the modelling process have a signiﬁcant impact on the resultant hyperparameter distribution (see Figure 1a and 1b) and thus, the latent function space.
Revised: The expert suggestions incorporated in the modelling process have a signiﬁcant impact on the resultant hyperparameter distribution (see Figure 1a and 1b) and thus, the latent function space.

Original: Our HAT framework disregards some portions of the hyperparameter distribution that violates the constraints added in the modelling process and restricts itself to the (narrower) feasible regions.
Revised: Our HAT framework disregards some portions of the hyperparameter distribution that violates the constraints added in the modelling process and restricts itself to the (narrower) feasible regions.

Original: Further, the search for a better hyperparameter set that is more aligned with the expert’s cognitive model in the vicinity of the optimal hyperparameter set obtained by maximising the log-likelihood also results in a narrower distribution (governed by the threshold ∆ ).
Revised: Further, the search for a better hyperparameter set that is more aligned with the expert’s cognitive model in the vicinity of the optimal hyperparameter set obtained by maximising the log-likelihood also results in a narrower distribution (governed by the threshold ∆ ).

Original: Thus, our proposed framework deals with a narrower distribution of functions controlled by the modelling constraints enforced in the optimisation process.
Revised: Thus, our proposed framework deals with a narrower distribution of functions controlled by the modelling constraints enforced in the optimisation process.

Original: Such narrower distributions are spanned by the RKHS H k Θ HAT .
Revised: Such narrower distributions are spanned by the RKHS H k Θ HAT .

Original: We characterise the RKHS
Revised: We characterise the RKHS

Original: H k Θ HAT of our framework using the Sobolev spaces [Tartar, 2007].
Revised: H k Θ HAT of our framework using the Sobolev spaces [Tartar, 2007].

Original: Sobolev Hilbert space H s is a special case of Sobolev spaces obtained by restricting the RKHS associated with the kernel function k Θ .
Revised: Sobolev Hilbert space H s is a special case of Sobolev spaces obtained by restricting the RKHS associated with the kernel function k Θ .

Original: With mild regularity assumptions on the boundary conditions and smoothness of Sobolev
Revised: With mild regularity assumptions on the boundary conditions and smoothness of Sobolev

Original: Hilbert spaces, we draw similarities between the underlying constraints in the Sobolev Hilbert spaces and the constraints enforced to construct RKHS H k Θ HAT in the HAT framework.
Revised: Hilbert spaces, we draw similarities between the underlying constraints in the Sobolev Hilbert spaces and the constraints enforced to construct RKHS H k Θ HAT in the HAT framework.

Original: We use the mathematical formulations of the RKHS norm in Sobolev Hilbert spaces to characterise the
Revised: We use the mathematical formulations of the RKHS norm in Sobolev Hilbert spaces to characterise the

Original: RKHS norm of the HAT framework.
Revised: RKHS norm of the HAT framework.

Original: We refer to the supplementary material for the deﬁnition of
Revised: We refer to the supplementary material for the deﬁnition of

Original: Sobolev spaces.
Revised: Sobolev spaces.

Original: The following theorem establishes the relation between the RKHS norms of the
Revised: The following theorem establishes the relation between the RKHS norms of the

Original: HAT framework and the standard Bayesian optimisation framework.
Revised: HAT framework and the standard Bayesian optimisation framework.

Original: Theorem 1: Let H ( R d ) be the space of real continuous functions f k ∈ L 2 ( R d ) .
Revised: Theorem 1: Let H ( R d ) be the space of real continuous functions f k ∈ L 2 ( R d ) .

Original: Let k Θ HAT be the kernel function with hyperparameters Θ HAT used in the HAT framework .
Revised: Let k Θ HAT be the kernel function with hyperparameters Θ HAT used in the HAT framework .

Original: Let B R HAT be the norm-ball of radius R HAT induced in HAT based BO and let B R STD be the norm-ball for the standard
Revised: Let B R HAT be the norm-ball of radius R HAT induced in HAT based BO and let B R STD be the norm-ball for the standard

Original: BO.
Revised: BO.

Original: If H k Θ HAT ( X ) is the reproducing Hilbert space of functions f k = g | X associated with the kernel k Θ HAT , then with high certainty it holds k f k k Θ HAT ≤ B R HAT and B R HAT ≤ B R STD .
Revised: If H k Θ HAT ( X ) is the reproducing Hilbert space of functions f k = g | X associated with the kernel k Θ HAT , then with high certainty it holds k f k k Θ HAT ≤ B R HAT and B R HAT ≤ B R STD .

Original: The proof of Theorem 1 is provided in the supplementary material.
Revised: The proof of Theorem 1 is provided in the supplementary material.

Original: With the results established in
Revised: With the results established in

Original: Theorem 1, we conclude that the additional channel of human experimenter knowledge enforced as constraints results in a narrower distribution for f and thus signiﬁcantly reducing the radius ( R HAT ) of the norm ball i.e., B R HAT ≤ B R STD .
Revised: Theorem 1, we conclude that the additional channel of human experimenter knowledge enforced as constraints results in a narrower distribution for f and thus signiﬁcantly reducing the radius ( R HAT ) of the norm ball i.e., B R HAT ≤ B R STD .

Original: Now, we discuss the implications of the induced norm bounds on the overall regret of the algorithm.
Revised: Now, we discuss the implications of the induced norm bounds on the overall regret of the algorithm.

Original: Corollary 1: Pick δ ∈ (0 , 1) .
Revised: Corollary 1: Pick δ ∈ (0 , 1) .

Original: Let β t
Revised: Let β t

Original: ( R , δ ) =
Revised: ( R , δ ) =

Original: B R + (cid:15) q 2( γ t − 1 + 

Revised: B R + (cid:15) q 2( γ t − 1 +

Original: +
Revised: +

Original: ln
Revised: ln

Original: δ ) where γ t − 1 is the associated information gain [Chowdhury and Gopalan, 2017] after t − 1 rounds and (cid:15) is the
Revised: δ ) where γ t − 1 is the associated information gain [Chowdhury and Gopalan, 2017] after t − 1 rounds and (cid:15) is the

Original: Gaussian noise.
Revised: Gaussian noise.

Original: Let k f k k ≤ B R .
Revised: Let k f k k ≤ B R .

Original: If B R HAT ≤ B R STD , then the following holds with the probability at least 1 − δ ,
Revised: If B R HAT ≤ B R STD , then the following holds with the probability at least 1 − δ ,

Original: [Equation]
Revised: [Equation]

Original: where R T is the cumulative regret given by R T
Revised: where R T is the cumulative regret given by R T

Original: = P
Revised: = P

Original: Tt =1 r t and r t is the instantaneous regret given by r t = f
Revised: Tt =1 r t and r t is the instantaneous regret given by r t = f

Original: ( x ∗ ) − f ( x t ) .
Revised: ( x ∗ ) − f ( x t ) .

Original: Proof: As discussed in Lemma 1, the RKHS norm associated with a kernel function k ( · , · ) is bounded within the ball of radius R i.e., k f k k ≤ B R .
Revised: Proof: As discussed in Lemma 1, the RKHS norm associated with a kernel function k ( · , · ) is bounded within the ball of radius R i.e., k f k k ≤ B R .

Original: Further, it states that with high probability | f ( x ) − µ t ( x ) | ≤ √ β t σ t ( x ) is satisﬁed if β t ’s are appropriately chosen.
Revised: Further, it states that with high probability | f ( x ) − µ t ( x ) | ≤ √ β t σ t ( x ) is satisﬁed if β t ’s are appropriately chosen.

Original: If | f ( x ) − µ t ( x ) | ≤ √ β t σ t ( x ) holds, then the instantaneous regret is upper bounded by 2 √ β t σ t ( x ) (see Lemma 5 in [Srinivas et al., 2012]).
Revised: If | f ( x ) − µ t ( x ) | ≤ √ β t σ t ( x ) holds, then the instantaneous regret is upper bounded by 2 √ β t σ t ( x ) (see Lemma 5 in [Srinivas et al., 2012]).

Original: Therefore, the instantaneous regret r t is greatly inﬂuenced by the choice of β t .
Revised: Therefore, the instantaneous regret r t is greatly inﬂuenced by the choice of β t .

Original: By setting the value of β t as β t =
Revised: By setting the value of β t as β t =

Original: B R + (cid:15) q 2( γ t − 1 + 

Revised: B R + (cid:15) q 2( γ t − 1 +

Original: +
Revised: +

Original: ln
Revised: ln

Original: δ )
Revised: δ )

Original: [Chowdhury and Gopalan, 2017], we can impose a tighter bounds on the cumulative regret
Revised: [Chowdhury and Gopalan, 2017], we can impose a tighter bounds on the cumulative regret

Original: R T .
Revised: R T .

Original: The results of Theorem 1 state that, our proposed method places a tighter bound on the
Revised: The results of Theorem 1 state that, our proposed method places a tighter bound on the

Original: RKHS norm ball such that B R HAT ≤ B R STD .
Revised: RKHS norm ball such that B R HAT ≤ B R STD .

Original: Such restrictions forces smaller values for β t in our proposed method, thereby reducing the instantaneous regret r t signiﬁcantly.
Revised: Such restrictions forces smaller values for β t in our proposed method, thereby reducing the instantaneous regret r t signiﬁcantly.

Original: As a consequence, the overall cumulative regret R HATT =
Revised: As a consequence, the overall cumulative regret R HATT =

Original: P Tt =1 r t is also tightly bounded and grows at most as
Revised: P Tt =1 r t is also tightly bounded and grows at most as

Original: O (cid:0) B R HAT
Revised: O (cid:0) B R HAT

Original: [Equation]
Revised: [Equation]

Original: √
Revised: √

Original: [Equation]
Revised: [Equation]

Original: +
Revised: +

Original: [Equation]
Revised: [Equation]

Original: T ( γ T + ln 1 δ ) (cid:1) , where γ T is the kernel associated maximum information gain [Srinivas et al., 2012] after T iterations.
Revised: T ( γ T + ln 1 δ ) (cid:1) , where γ T is the kernel associated maximum information gain [Srinivas et al., 2012] after T iterations.

Original: Thus the result stated above follows.
Revised: Thus the result stated above follows.

Original:  Experiments
Revised:  Experiments

Original: We evaluate the optimisation performance of our proposed methods on various synthetic functions and several real-world datasets.
Revised: We evaluate the optimisation performance of our proposed methods on various synthetic functions and several real-world datasets.

Original: First, we validate our methods by demonstrating its sample efﬁciency in the global optimisation of synthetic benchmark functions.
Revised: First, we validate our methods by demonstrating its sample efﬁciency in the global optimisation of synthetic benchmark functions.

Original: Next, we tune the hyperparameters of the Support Vector Machine (SVM) classiﬁer operating on publicly available real-world datasets.
Revised: Next, we tune the hyperparameters of the Support Vector Machine (SVM) classiﬁer operating on publicly available real-world datasets.

Original: We compare the performance against different variations of the BO algorithm.
Revised: We compare the performance against different variations of the BO algorithm.

Original: The empirical results demonstrate the superiority of our proposed framework.
Revised: The empirical results demonstrate the superiority of our proposed framework.

Original: 5.1 Emulating Human Experts
Revised: 5.1 Emulating Human Experts

Original: The human expert is expected to know the underlying structures of the given input space.
Revised: The human expert is expected to know the underlying structures of the given input space.

Original: Here we assume that the expert is aware of the optimal ground truth kernel hyperparameters ( Θ GT ).
Revised: Here we assume that the expert is aware of the optimal ground truth kernel hyperparameters ( Θ GT ).

Original: We are motivated from the ﬁndings in Borji and Itti [2013] that the intelligent search strategy used by humans closely resembles Bayesian optimisation.
Revised: We are motivated from the ﬁndings in Borji and Itti [2013] that the intelligent search strategy used by humans closely resembles Bayesian optimisation.

Original: Therefore, we run a separate GP-UCB based
Revised: Therefore, we run a separate GP-UCB based

Original: BO algorithm with the ground truth kernel ( k GT ) for emulating human experts behaviour in our proposed framework.
Revised: BO algorithm with the ground truth kernel ( k GT ) for emulating human experts behaviour in our proposed framework.

Original: The knowledge about the (ground truth) kernel function is obtained apriori from a large number of random samples of the objective function under consideration.
Revised: The knowledge about the (ground truth) kernel function is obtained apriori from a large number of random samples of the objective function under consideration.

Original: Further, the human experts are given an option to intervene at any stage.
Revised: Further, the human experts are given an option to intervene at any stage.

Original: In this work, we have assumed that a human expert intervenes at every third iteration of the optimisation process.
Revised: In this work, we have assumed that a human expert intervenes at every third iteration of the optimisation process.

Original: We consider the following variants of standard BO and HAT based BO in our experiments.
Revised: We consider the following variants of standard BO and HAT based BO in our experiments.

Original: • HAT-RR: Human-AI Teaming framework with recommendation correction as mentioned in Algorithm 2.
Revised: • HAT-RR: Human-AI Teaming framework with recommendation correction as mentioned in Algorithm 2.

Original: The SE kernel is used as the ground truth kernel ( k GT ) for this baseline.
Revised: The SE kernel is used as the ground truth kernel ( k GT ) for this baseline.

Original: • HAT-RR (KFO) : Similar to HAT-RR, except that the ground truth kernel is chosen to be the kernel tuned from a Kernel Functional Optimisation (KFO) procedure [Venkatesh et al., 2021], a more expressive form of kernel than the Squared Exponential (SE) kernel.
Revised: • HAT-RR (KFO) : Similar to HAT-RR, except that the ground truth kernel is chosen to be the kernel tuned from a Kernel Functional Optimisation (KFO) procedure [Venkatesh et al., 2021], a more expressive form of kernel than the Squared Exponential (SE) kernel.

Original: • HAT-DM (KFO) : HAT framework with the expert feedback about good and bad regions (mentioned in Algorithm 2) and the kernel being tuned with KFO procedure.
Revised: • HAT-DM (KFO) : HAT framework with the expert feedback about good and bad regions (mentioned in Algorithm 2) and the kernel being tuned with KFO procedure.

Original: • STD-BO : A standard Bayesian optimisation algorithm mentioned in the supplementary material.
Revised: • STD-BO : A standard Bayesian optimisation algorithm mentioned in the supplementary material.

Original: • STD-BO
Revised: • STD-BO

Original: (MOD) :
Revised: (MOD) :

Original: Another variant of standard Bayesian optimisation procedure but naively considers the suggestions provided by human experts in its observation model.
Revised: Another variant of standard Bayesian optimisation procedure but naively considers the suggestions provided by human experts in its observation model.

Original: In the aforesaid methods, we have used SE kernel ( k SE ) for ﬁtting GP surrogate models.
Revised: In the aforesaid methods, we have used SE kernel ( k SE ) for ﬁtting GP surrogate models.

Original: The lengthscale parameter of k SE is tuned in the interval [0 . 1 , 1] .
Revised: The lengthscale parameter of k SE is tuned in the interval [0 . 1 , 1] .

Original: We standardise the function output, and thus use unit signal variance for k SE .
Revised: We standardise the function output, and thus use unit signal variance for k SE .

Original: We follow the guidelines mentioned in KFO framework [Venkatesh et al., 2021] for the hyperparameter selection of both the SE kernel ( k SE ) and hyperkernel ( κ ).
Revised: We follow the guidelines mentioned in KFO framework [Venkatesh et al., 2021] for the hyperparameter selection of both the SE kernel ( k SE ) and hyperkernel ( κ ).

Original: In the
Revised: In the

Original: KFO framework, the hyperparameters of k SE i.e., the lengthscale ( l ) and the signal variance ( σ 2 f ) are tuned in the interval [0 . 1 , 1] , as the bounds are always normalised.
Revised: KFO framework, the hyperparameters of k SE i.e., the lengthscale ( l ) and the signal variance ( σ 2 f ) are tuned in the interval [0 . 1 , 1] , as the bounds are always normalised.

Original: The hyperparameters ˜ λ
Revised: The hyperparameters ˜ λ

Original: h and l of the hyperkernel κ are tuned in the interval (0 , 1] and (0 , 1] , respectively.
Revised: h and l of the hyperkernel κ are tuned in the interval (0 , 1] and (0 , 1] , respectively.

Original: We use t 

Revised: We use t

Original: =
Revised: =

Original: d + 2 initial observations for a d − dimensional problem.
Revised: d + 2 initial observations for a d − dimensional problem.

Original: The threshold ( ∆ ) used in Algorithm 2 is set at 95% .
Revised: The threshold ( ∆ ) used in Algorithm 2 is set at 95% .

Original: 5.2 Synthetic Experiments
Revised: 5.2 Synthetic Experiments

Original: First, we evaluate our approach by ﬁnding the global optima of a benchmark function (Oscillator 2 D, see Figure 1a).
Revised: First, we evaluate our approach by ﬁnding the global optima of a benchmark function (Oscillator 2 D, see Figure 1a).

Original: The objective function considered has varying smoothness across its input dimensions.
Revised: The objective function considered has varying smoothness across its input dimensions.

Original: The expert is assumed to be aware of this smoothness information (lengthscale) via a ground truth kernel i.e., a kernel with the optimal hyperparameter set tuned in the light of numerous data points.
Revised: The expert is assumed to be aware of this smoothness information (lengthscale) via a ground truth kernel i.e., a kernel with the optimal hyperparameter set tuned in the light of numerous data points.

Original: The initial observations are chosen to be in the smoother regions of the objective function such that the methods have not much prior information about the input space.
Revised: The initial observations are chosen to be in the smoother regions of the objective function such that the methods have not much prior information about the input space.

Original: Comparing the kernel learning performances revealed that our approach encouraged larger lengthscale for x 2 and shorter lengthscale for x 1 .
Revised: Comparing the kernel learning performances revealed that our approach encouraged larger lengthscale for x 2 and shorter lengthscale for x 1 .

Original: The hyperparameter distribution along x 2 is depicted in Figure 1b.
Revised: The hyperparameter distribution along x 2 is depicted in Figure 1b.

Original: Furthermore, we evaluate our proposed method with the following multi-dimensional benchmark functions [Surjanovic and Bingham, 2017] with multiple local optima: (i) Ackley 1 D, (ii) Gramacy & Lee 1 D, (iii) Branin 2 D, (iv) Oscillator 2 D, (v) Hartmann 3 D, and (vi) Hartmann 6 D. We compare the performance of our method against the baselines by plotting the simple regret ( ˆ r + t ) given by:
Revised: Furthermore, we evaluate our proposed method with the following multi-dimensional benchmark functions [Surjanovic and Bingham, 2017] with multiple local optima: (i) Ackley 1 D, (ii) Gramacy & Lee 1 D, (iii) Branin 2 D, (iv) Oscillator 2 D, (v) Hartmann 3 D, and (vi) Hartmann 6 D. We compare the performance of our method against the baselines by plotting the simple regret ( ˆ r + t ) given by:

Original: [Equation]
Revised: [Equation]

Original: (6) where f ( x ∗ ) is the true optima of the objective function.
Revised: (6) where f ( x ∗ ) is the true optima of the objective function.

Original: We plot the simple regret for 10 × d +iterations, for a given d − dimensional problem.
Revised: We plot the simple regret for 10 × d +iterations, for a given d − dimensional problem.

Original: The convergence results obtained for the synthetic experiments after 10 random repeated runs are shown in Figure 2.
Revised: The convergence results obtained for the synthetic experiments after 10 random repeated runs are shown in Figure 2.

Original: It is evident from the results that our proposed method has outperformed the baselines with KFO being better than the SE kernel.
Revised: It is evident from the results that our proposed method has outperformed the baselines with KFO being better than the SE kernel.

Original: [Figure]
Revised: [Figure]

Original: 5.3 Real-world Experiments
Revised: 5.3 Real-world Experiments

Original: We evaluate our proposed method in tuning the hyperparameters of Support Vector Machine (SVM) classiﬁer operating on real-world datasets.
Revised: We evaluate our proposed method in tuning the hyperparameters of Support Vector Machine (SVM) classiﬁer operating on real-world datasets.

Original: In our experiments, we consider multi-dimensional realworld datasets publicly available in the UCI repository [Dua and Graff, 2017].
Revised: In our experiments, we consider multi-dimensional realworld datasets publicly available in the UCI repository [Dua and Graff, 2017].

Original: The real-world datasets used are randomly split into a training set containing 80% of the total instances and a test set consisting of the remaining 20% of the total instances.
Revised: The real-world datasets used are randomly split into a training set containing 80% of the total instances and a test set consisting of the remaining 20% of the total instances.

Original: We use C-SVM with Radial Basis Function (RBF) kernel to minimise the test classiﬁcation error ( E r ).
Revised: We use C-SVM with Radial Basis Function (RBF) kernel to minimise the test classiﬁcation error ( E r ).

Original: We tune the SVM hyperparameters i.e., the cost parameter and the RBF kernel width in the exponent space of [ − 3 , 3] and [ − 5 , 1] , respectively.
Revised: We tune the SVM hyperparameters i.e., the cost parameter and the RBF kernel width in the exponent space of [ − 3 , 3] and [ − 5 , 1] , respectively.

Original: The minimum classiﬁcation error (in % ) obtained for the test set averaged over 10 random repeated runs are shown in Table 1.
Revised: The minimum classiﬁcation error (in % ) obtained for the test set averaged over 10 random repeated runs are shown in Table 1.

Original: We italicise the best results of our HAT-RR framework to demonstrate that our proposed method outperforms the standard
Revised: We italicise the best results of our HAT-RR framework to demonstrate that our proposed method outperforms the standard

Original: Bayesian optimisation algorithm even if the ground truth kernel used (squared exponential kernel) is not very expressive of the given input space.
Revised: Bayesian optimisation algorithm even if the ground truth kernel used (squared exponential kernel) is not very expressive of the given input space.

Original: It is even better when more expressive kernel (KFO tuned kernel) is used.
Revised: It is even better when more expressive kernel (KFO tuned kernel) is used.

Original: Further, we believe that the superior performance of HAT-RR framework against the HAT-DM framework is because HAT-RR directly incorporates the suggestions from the human expert for function evaluation, whereas HAT-DM assesses and corrects the deviations in the current model by maximising the distance between good and bad regions.
Revised: Further, we believe that the superior performance of HAT-RR framework against the HAT-DM framework is because HAT-RR directly incorporates the suggestions from the human expert for function evaluation, whereas HAT-DM assesses and corrects the deviations in the current model by maximising the distance between good and bad regions.

Original: 5.4 Robustness to Imprecision in Human Knowledge
Revised: 5.4 Robustness to Imprecision in Human Knowledge

Original: In addition to the experiments discussed above, we further evaluate our proposed method to show its robustness in model learning against the noisy human expert inputs.
Revised: In addition to the experiments discussed above, we further evaluate our proposed method to show its robustness in model learning against the noisy human expert inputs.

Original: In this experiment, we emulate human errors by corrupting the human inputs by adding suitable noise to the (i) kernel parameters, and (ii) the trade-off factor β n .
Revised: In this experiment, we emulate human errors by corrupting the human inputs by adding suitable noise to the (i) kernel parameters, and (ii) the trade-off factor β n .

Original: In order to add noise in the kernel parameters, we use Multi
Revised: In order to add noise in the kernel parameters, we use Multi

Original: Kernel Learning (MKL) [Aiolli and Donini, 2015] - a weighted combination of SE kernel ( k SE ), linear kernel ( k LIN ) and polynomial kernel ( k POL ) i.e., k ( x , x 0 ) =
Revised: Kernel Learning (MKL) [Aiolli and Donini, 2015] - a weighted combination of SE kernel ( k SE ), linear kernel ( k LIN ) and polynomial kernel ( k POL ) i.e., k ( x , x 0 ) =

Original: w 1 k SE ( x , x 0 ) + w 2 k LIN ( x , x 0 ) + w 3 k POL ( x , x 0 ) for the ground truth kernel.
Revised: w 1 k SE ( x , x 0 ) + w 2 k LIN ( x , x 0 ) + w 3 k POL ( x , x 0 ) for the ground truth kernel.

Original: We add a white Gaussian noise to the weights w i.e., w i ∼ N ( w i , ˆ σ 2 GN ) to emulate the human expert error in understanding the ground truth knowledge.
Revised: We add a white Gaussian noise to the weights w i.e., w i ∼ N ( w i , ˆ σ 2 GN ) to emulate the human expert error in understanding the ground truth knowledge.

Original: To account for the human error in providing suggestions, we add noise to the trade-off factor β n using a Gamma distribution ( Γ ) i.e., β n = Γ( β 2 n ˆ σ 2 GN , ˆ σ 2 GN β n ) .
Revised: To account for the human error in providing suggestions, we add noise to the trade-off factor β n using a Gamma distribution ( Γ ) i.e., β n = Γ( β 2 n ˆ σ 2 GN , ˆ σ 2 GN β n ) .

Original: In all our experiments, we have set the noise variance ˆ σ 2 GN = 0 .
Revised: In all our experiments, we have set the noise variance ˆ σ 2 GN = 0 .

Original: We evaluate our proposed method supplemented with the imprecise human knowledge using the following synthetic functions: (i) Levy 2 D, (ii) Shubert 2 D, and (iii)
Revised: We evaluate our proposed method supplemented with the imprecise human knowledge using the following synthetic functions: (i) Levy 2 D, (ii) Shubert 2 D, and (iii)

Original: Egg 2 D.
Revised: Egg 2 D.

Original: We compare the convergence results with the standard Bayesian optimisation algorithm.
Revised: We compare the convergence results with the standard Bayesian optimisation algorithm.

Original: The results obtained for the aforesaid synthetic functions are shown in Figure 3.
Revised: The results obtained for the aforesaid synthetic functions are shown in Figure 3.

Original: It is evident from the results
Revised: It is evident from the results

Original: [Table]
Revised: [Table]

Original: best results of our HAT-RR framework to emphasise its superiority against the standard BO even when a standard unexpressive (SE kernel) is used.
Revised: best results of our HAT-RR framework to emphasise its superiority against the standard BO even when a standard unexpressive (SE kernel) is used.

Original: that our method performs on par with standard BO even when the expert knowledge is imprecise, as we try to incorporate expert knowledge to further improve the existing log-likelihood estimates.
Revised: that our method performs on par with standard BO even when the expert knowledge is imprecise, as we try to incorporate expert knowledge to further improve the existing log-likelihood estimates.

Original: [Figure]
Revised: [Figure]

Original: To further validate our proposed Human-AI collaborative Bayesian optimisation framework, we have conducted a study with real human experts.
Revised: To further validate our proposed Human-AI collaborative Bayesian optimisation framework, we have conducted a study with real human experts.

Original: We have provided these experimental results in the supplementary material.
Revised: We have provided the additional experimental results in the supplementary material (A.3).

Original: The code base used for the experiments mentioned above is available at https://github.com/mailtoarunkumarav/HumanAITeaming .
Revised: The code base used for the experiments mentioned above is available at https://github.com/mailtoarunkumarav/HumanAITeaming .

Original:  Conclusion
Revised:  Conclusion

Original: We propose a novel Bayesian optimisation framework to accelerate the optimisation process by incorporating extra information available from human experts.
Revised: We propose a novel Bayesian optimisation framework to accelerate the optimisation process by incorporating extra information available from human experts.

Original: We present two different expert intervention strategies.
Revised: We present two different expert intervention strategies.

Original: In the ﬁrst strategy, we let the expert to correct the current recommendation.
Revised: In the ﬁrst strategy, we let the expert to correct the current recommendation.

Original: In the second, we seek expert’s hunch on good region vs bad region for the selection of next sample based on the current observation set.
Revised: In the second, we seek expert’s hunch on good region vs bad region for the selection of next sample based on the current observation set.

Original: We then incorporate such feedback as constraints in the model selection.
Revised: We then incorporate such feedback as constraints in the model selection.

Original: We theoretically analyse our framework to show that expert’s knowledge can improve sample efﬁciency.
Revised: We theoretically analyse our framework to show that expert’s knowledge can improve sample efﬁciency.

Original: The experimental results show that our method outperforms other baselines.
Revised: The experimental results show that our method outperforms other baselines.

Original: Acknowledgements
Revised: Acknowledgements

Original: This research was partially funded by the Australian Government through Australian Research
Revised: This research was partially funded by the Australian Government through Australian Research

Original: Council (ARC).
Revised: Council (ARC).

Original: Professor Svetha Venkatesh is the recipient of an ARC Australian Laureate Fellowship (FL170100006).
Revised: Professor Svetha Venkatesh is the recipient of an ARC Australian Laureate Fellowship (FL170100006).