1 Introduction
Bayesian optimization (BO) is a popular approach to optimizing blackbox functions that are expensive to evaluate. Because of expensive evaluations, BO aims to approximately locate the function maximizer without evaluating the function too many times. This requires a good strategy to adaptively choose where to evaluate based on the current observations.
BO adopts a Bayesian perspective and assumes that there is a prior on the function; typically, we use a Gaussian process (GP) prior. Then, the information collection strategy can rely on the prior to focus on good inputs, where the goodness is determined by an acquisition function derived from the GP prior and current observations. In past literature, it has been shown both theoretically and empirically that if the function is indeed drawn from the given prior, there are many acquisition functions that BO can use to locate the function maximizer quickly [51, 5, 53].
However, in reality, the prior we choose to use in BO often does not reflect the distribution from which the function is drawn. Hence, we sometimes have to estimate the hyperparameters of a chosen form of the prior on the fly as we collect more data [50]. One popular choice is to estimate the prior parameters using empirical Bayes with, e.g., the maximum likelihood estimator [44] .
Despite the vast literature that shows many empirical Bayes approaches have wellfounded theoretical guarantees such as consistency [40] and admissibility [26], it is difficult to analyze a version of BO that uses empirical Bayes because of the circular dependencies between the estimated parameters and the data acquisition strategies. The requirement to select the prior model and estimate its parameters leads to a BO version of the chickenandegg dilemma: the prior model selection depends on the data collected and the data collection strategy depends on having a “correct” prior. Theoretically, there is little evidence that BO with unknown parameters in the prior can work well. Empirically, there is evidence showing it works well in some situations, but not others [33, 23], which is not surprising in light of no free lunch results [56, 22].
In this paper, we propose a simple yet effective strategy for learning a prior in a metalearning setting where training data on functions from the same Gaussian process prior are available. We use a variant of empirical Bayes that gives unbiased estimates for both the parameters in the prior and the posterior given observations of the function we wish to optimize. We analyze the regret bounds in two settings: (1) finite input space, and (2) compact input space in . We clarify additional assumptions on the training data and form of Gaussian processes of both settings in Sec. 4.1 and Sec. 4.2. We prove theorems that show a nearzero regret bound for variants of GPUCB [2, 51] and probability of improvement (PI) [29, 53]. The regret bound decreases to a constant proportional to the observational noise as online evaluations and offline data size increase.
From a more pragmatic perspective on Bayesian optimization for important areas such as robotics, we further explore how our approach works for problems in task and motion planning domains [27], and we explain why the assumptions in our theorems make sense for these problems in Sec. 5. Indeed, assuming a common kernel, such as squared exponential or Matérn, is very limiting for robotic problems that involve discontinuity and nonstationarity. However, with our approach of setting the prior and posterior parameters, BO outperforms all other methods in the task and motion planning benchmark problems.
The contributions of this paper are (1) a standalone BO module that takes in only a multitask training data set as input and then actively selects inputs to efficiently optimize a new function and (2) analysis of the regret of this module. The analysis is constructive, and determines appropriate hyperparameter settings for the GPUCB acquisition function. Thus, we make a step forward to resolving the problem that, despite being used for hyperparameter tuning, BO algorithms themselves have hyperparameters.
2 Background and related work
BO optimizes a blackbox objective function through sequential queries. We usually assume knowledge of a Gaussian process [44]
prior on the function, though other priors such as Bayesian neural networks and their variants
[17, 30] are applicable too. Then, given possibly noisy observations and the prior distribution, we can do Bayesian posterior inference and construct acquisition functions [29, 38, 2] to search for the function optimizer.However, in practice, we do not know the prior and it must be estimated. One of the most popular methods of prior estimation in BO is to optimize mean/kernel hyperparameters by maximizing datalikelihood of the current observations [44, 19]. Another popular approach is to put a prior on the mean/kernel hyperparameters and obtain a distribution of such hyperparameters to adapt the model given observations [20, 50]. These methods require a predetermined form of the mean function and the kernel function. In the existing literature, mean functions are usually set to be 0 or linear and the popular kernel functions include Matérn kernels, Gaussian kernels, linear kernels [44] or additive/product combinations of the above [11, 24].
Meta BO
aims to improve the optimization of a given objective function by learning from past experiences with other similar functions. Meta BO can be viewed as a special case of transfer learning or multitask learning. One wellstudied instance of meta BO is the machine learning (ML) hyperparameter tuning problem on a dataset, where, typically, the validation errors are the functions to optimize
[14]. The key question is how to transfer the knowledge from previous experiments on other datasets to the selection of ML hyperparameters for the current dataset.To determine the similarity between validation error functions on different datasets, metafeatures of datasets are often used [6]. With those metafeatures of datasets, one can use contextual Bayesian optimization approaches [28] that operate with a probabilistic functional model on both the dataset metafeatures and ML hyperparameters [3]. Feurer et al. [16]
, on the other hand, used metafeatures of datasets to construct a distance metric, and to sort hyperparameters that are known to work for similar datasets according to their distances to the current dataset. The best k hyperparameters are then used to initialize a vanilla BO algorithm. If the function metafeatures are not given, one can estimate the metafeatures, such as the mean and variance of all observations, using Monte Carlo methods
[52], maximum likelihood estimates [57] or maximum a posteriori estimates [43, 42].As an alternative to using metafeatures of functions, one can construct a kernel between functions. For functions that are represented by GPs, Malkomes et al. [36] studied a “kernel kernel”, a kernel for kernels, such that one can use BO with a “kernel kernel” to select which kernel to use to model or optimize an objective function [35] in a Bayesian way. However, [36] requires an initial set of kernels to select from. Instead, Golovin et al. [18] introduced a setting where the functions come in sequence and the posterior of the former function becomes the prior of the current function. Removing the assumption that functions come sequentially, Feurer et al. [15] proposed a method to learn an additive ensemble of GPs that are known to fit all of those past “training functions”.
Theoretically, it has been shown that meta BO methods that use information from similar functions may result in an improvement for the cumulative regret bound [28, 47] or the simple regret bound [42] with the assumptions that the GP priors are given. If the form of the GP kernel is given and the prior mean function is 0 but the kernel hyperparameters are unknown, it is possible to obtain a regret bound given a range of these hyperparameters [54]. In this paper, we prove a regret bound for meta BO where the GP prior is unknown; this means, neither the range of GP hyperparameters nor the form of the kernel or mean function is given.
A more ambitious approach to solving meta BO is to train an endtoend system, such as a recurrent neural network
[21], that takes the history of observations as an input and outputs the next point to evaluate [8]. Though it has been demonstrated that the method in [8] can learn to tradeoff exploration and exploitation for a short horizon, it is unclear how many “training instances”, in the form of observations of BO performed on similar functions, are necessary to learn the optimization strategies for any given horizon of optimization. In this paper, we show both theoretically and empirically how the number of “training instances” in our method affects the performance of BO.Our methods are most similar to the BOX algorithm [27], which uses evaluations of previous functions to make point estimates of a mean and covariance matrix on the values over a discrete domain. Our methods for the discrete setting (described in Sec. 4.1) directly improve on BOX by choosing the exploration parameters in GPUCB more effectively. This general strategy is extended to the continuousdomain setting in Sec. 4.2, in which we extend a method for learning the GP prior [41] and the use the learned prior in GPUCB and PI.
Learning how to learn, or “meta learning”, has a long history in machine learning [46]. It was argued that learning how to learn is “learning the prior” [4] with “point sets” [37], a set of iid sets of potentially noniid points. We follow this simple intuition and present a meta BO approach that learns its GP prior from the data collected on functions that are assumed to have been drawn from the same prior distribution.
is a standard methodology for estimating unknown parameters of a Bayesian model. Our approach is a variant of empirical Bayes. We can view our computations as the construction of a sequence of estimators for a Bayesian model. The key difference from traditional empirical Bayes methods is that we are able to prove a regret bound for a BO method that uses estimated parameters to construct priors and posteriors. In particular, we use frequentist concentration bounds to analyze Bayesian procedures, which is one way to certify empirical Bayes in statistics
[49, 13].3 Problem formulation and notations
Unlike the standard BO setting, we do not assume knowledge of the mean or covariance in the GP prior, but we do assume the availability of a dataset of iid sets of potentially noniid observations on functions sampled from the same GP prior. Then, given a new, unknown function sampled from that same distribution, we would like to find its maximizer.
More formally, we assume there exists a distribution , and both the mean and the kernel are unknown. Nevertheless, we are given a dataset , where is drawn independently from and is drawn independently from . The noise level is unknown as well. We will specify inputs in Sec. 4.1 and Sec. 4.2.
Given a new function sampled from , our goal is to maximize it by sequentially querying the function and constructing , . We study two evaluation criteria: (1) the bestsample simple regret which indicates the value of the best query in hindsight, and (2) the simple regret, which measures how good the inferred maximizer is.
Notation
We use
to denote a multivariate Gaussian distribution with mean
and variance and use to denote a Wishart distribution with degrees of freedom and scale matrix . We also use to denote. We overload function notation for evaluations on vectors
by denoting the output column vector as , and the output matrix as , and we overload the kernel function .4 Meta BO and its theoretical guarantees
Instead of handcrafting the mean and kernel , we estimate them using the training dataset . Our approach is fairly straightforward: in the offline phase, the training dataset is collected and we obtain estimates of the mean function and kernel ; in the online phase, we treat as the Bayesian “prior” to do Bayesian optimization. We illustrate the two phases in Fig. 1. In Alg. 1, we depict our algorithm, assuming the dataset has been collected. We use Estimate() to denote the “prior” estimation and Infer the “posterior” inference, both of which we will introduce in Sec. 4.1 and Sec. 4.2. For acquisition functions, we consider special cases of probability of improvement (PI) [53, 29] and upper confidence bound (GPUCB) [51, 2]:
Here, PI assumes additional information^{1}^{1}1Alternatively, an upper bound can be estimated adaptively [53]. Note that here we are maximizing the PI acquisition function and hence is a negative version of what was defined in [53]. in the form of the upper bound on function value . For GPUCB, we set its hyperparameter to be
where is the size of the dataset and . With probability , the regret bound in Thm. 2 or Thm. 4 holds with these special cases of GPUCB and PI. Under two different settings of the search space , finite and compact , we show how our algorithm works in detail and why it works via regret analyses on the bestsample simple regret. Finally in Sec. 4.3 we show how the simple regret can be bounded. The proofs of the analyses can be found in the appendix.
4.1 is a finite set
We first study the simplest case, where the function domain is a finite set with cardinality . For convenience, we treat this set as an ordered vector of items indexed by . We collect the training dataset , where are independently drawn from , are drawn independently from and . Because the training data can be collected offline by querying the functions in parallel, it is not unreasonable to assume that such a dataset is available. If it means the th entry of the dataset is missing, perhaps as a result of a failed experiment.
Estimating GP parameters
If , we have missing entries in the observation matrix . Under additional assumptions specified in [7], including that and the total number of valid observations , we can use matrix completion [7] to fully recover the matrix with high probability. In the following, we proceed by considering completed observations only.
Let the completed observation matrix be . We use an unbiased sample mean and covariance estimator for and ; that is, and , where is an by 1 vector of ones. It is well known that and are independent and [1].
Constructing estimators of the posterior
Given noisy observations , we can do Bayesian posterior inference to obtain . By the GP assumption, we get
(1)  
(2) 
where , [44]. The problem is that neither the posterior mean nor the covariance are computable because the Bayesian prior mean , the kernel and the noise parameter are all unknown. How to estimate and without knowing those prior parameters?
We introduce the following unbiased estimators for the posterior mean and covariance,
(3)  
(4) 
Notice that unlike Eq. (1) and Eq. (2), our estimators and do not depend on any unknown values or an additional estimate of the noise parameter . In Lemma 1, we show that our estimators are indeed unbiased and we derive their concentration bounds.
Regret bounds
We show a nearzero upper bound on the bestsample simple regret of meta BO with GPUCB and PI that uses specific parameter settings in Thm. 2. In particular, for both GPUCB and PI, the regret bound converges to a residual whose scale depends on the noise level in the observations.
Theorem 2.
Assume there exists constant and a training dataset is available whose size is . Then, with probability at least , the bestsample simple regret in iterations of meta BO with special cases of either GPUCB or PI satisfies
where , , , are constants, and .
This bound reflects how training instances and BO iterations affect the bestsample simple regret. The coefficients and both converge to constants (more details in the appendix), with components converging at rate . The convergence of the shared term depends on , the maximum information gain between function and up to observations . If, for example, each input has dimension and , then [51], in which case converges to the observational noise level at rate . Together, the bounds indicate that the bestsample simple regret of both our settings of GPUCB and PI decreases to a constant proportional to noise level .
4.2 is compact
For compact , we consider the primal form of GPs. We further assume that there exist basis functions , mean parameter and covariance parameter such that and . Notice that is a column vector and for any . This means, for any input , the observation satisfies , where and the linear operator [39]. In the following analyses, we assume the basis functions are given.
We assume that a training dataset is given, where , are independently drawn from , are drawn independently from and .
Estimating GP parameters
Because the basis functions are given, learning the mean function and the kernel in the GP is equivalent to learning the mean parameter and the covariance parameter that parameterize distribution of the linear operator . Notice that ,
where , and . If the matrix has linearly independent rows, one unbiased estimator of is
Let . We use the estimator and to the estimate GP parameters. Again, and are independent and
[1].
Constructing estimators of the posterior
We assume the total number of evaluations . Given noisy observations , we have and , where the posterior of satisfies
(5)  
(6) 
Similar to the strategy used in Sec. 4.1, we construct an estimator for the posterior of to be
(7)  
(8) 
We can compute the conditional mean and variance of the observation on to be and . For convenience of notation, we define .
Lemma 3.
Pick probability . Assume has full row rank. For any nonnegative integer , , conditioned on the observations , . Moreover, if the size of the training dataset satisfies , then for any input , with probability at least , both
hold, where and .
Regret bounds
Similar to the finite case, we can also show a nearzero regret bound for compact . The following theorem clarifies our results. The convergence rates are the same as Thm. 2. Note that converges to instead of in Thm. 2 and is proportional to .
Theorem 4.
Assume all the assumptions in Thm. 2 and that has full row rank. With probability at least , the bestsample simple regret in iterations of meta BO with either GPUCB or PI satisfies
where , , , are constants, and .
4.3 Bounding the simple regret by the bestsample simple regret
Once we have the observations , we can infer where the of the function is. For all the cases in which is discrete or compact and the acquisition function is GPUCB or PI, we choose the inferred to be where . We show in Lemma 5 that with high probability, the difference between the simple regret and the bestsample simple regret is proportional to the observation noise .
Lemma 5.
With probability at least , .
5 Experiments
We evaluate our algorithm in four different blackbox function optimization problems, involving discrete or continuous function domains. One problem is optimizing a synthetic function in , and the rest are optimizing decision variables in robotic task and motion planning problems that were used in [27]^{2}^{2}2 Our code is available at https://github.com/beomjoonkim/MetaLearnBO..
At a high level, our task and motion planning benchmarks involve computing kinematically feasible collisionfree motions for picking and placing objects in a scene cluttered with obstacles. This problem has a similar setup to experimental design: the robot can “experiment” by assigning values to decision variables including grasps, base poses, and object placements until it finds a feasible plan. Given the assigned values for these variables, the robot program makes a call to a planner^{3}^{3}3We use Rapidlyexploring random tree (RRT) [32] with predefined random seed, but other choices are possible. which then attempts to find a sequence of motions that achieve these grasps and placements. We score the variable assignment based on the results of planning, assigning a very low score if the problem was infeasible and otherwise scoring based on plan length or obstacle clearance. An example problem is given in Figure 2.
Planning problem instances are characterized by arrangements of obstacles in the scene and the shape of the target object to be manipulated, and each problem instance defines a different score function. Our objective is to optimize the score function for a new problem instance, given sets of decisionvariable and score pairs from a set of previous planning problem instances as training data.
In two robotics domains, we discretize the original function domain using samples from the past planning experience, by extracting the values of the decision variables and their scores from successful plans. This is inspired by the previous successful use of BO in a discretized domain [9] to efficiently solve an adaptive locomotion problem.
We compare our approach, called point estimate meta Bayesian optimization (PEMBO), to three baseline methods. The first is a plain Bayesian optimization method that uses a kernel function to represent the covariance matrix, which we call Plain. Plain optimizes its GP hyperparameters by maximizing the data likelihood. The second is a transfer learning sequential modelbased optimization [57] method, that, like PEMBO, uses past function evaluations, but assumes that functions sampled from the same GP have similar response surface values. We call this method TLSMBO. The third is random selection, which we call Random. We present the results on the UCB acquisition function in the paper and results on the PI acquisition function are available in the appendix.
In all domains, we use the value as specified in Sec. 4. For continuous domains, we use as our basis functions. In order to train the weights and , we represent the function
with a 1hiddenlayer neural network with cosine activation function and a linear output layer with functionspecific weights
. We then train this network on the entire dataset . Then, fixing , for each set of pairs, we analytically solve the linear regression problem
as described in Sec. 4.2.Optimizing a continuous synthetic function
In this problem, the objective is to optimize a blackbox function sampled from a GP, whose domain is , given a set of evaluations of different functions from the same GP. Specifically, we consider a GP with a squared exponential kernel function. The purpose of this problem is to show that PEMBO, which estimates mean and covariance matrix based on , would perform similarly to BO methods that start with an appropriate prior. We have training data from functions with sample points each.
Figure 3(a) shows the learning curve, when we have different portions of data. The xaxis represents the percentage of the dataset used to train the basis functions, , and from the training dataset, and the yaxis represents the best function value found after 10 evaluations on a new function. We can see that even with just ten percent of the training data points, PEMBO performs just as well as Plain, which uses the appropriate kernel for this particular problem. Compared to PEMBO, which can efficiently use all of the dataset, we had to limit the number of training data points for TLSMBO to 1000, because even performing inference requires time. This leads to its noticeably worse performance than Plain and PEMBO.
Figure 3(d) shows the how evolves, where . As we can see, PEMBO using the UCB acquisition function performs similarly to Plain with the same acquisition function. TLSMBO again suffers because we had to limit the number of training data points.
Optimizing a grasp
In the robotplanning problem shown in Figure 2, the robot has to choose a grasp for picking the target object in a cluttered scene. A planning problem instance is defined by the poses of obstacles and the target objects, which changes the feasibility of a grasp across different instances.
The reward function is the negative of the length of the picking motion if the motion is feasible, and otherwise, where is a suitably lower number than the lengths of possible trajectories. We construct the discrete set of grasps by using grasps that worked in the past planning problem instances. The original space of grasps is , which describes position, direction, roll, and depth of a robot gripper with respect to the object, as used in [10]. For both Plain and TLSMBO, we use squared exponential kernel function on this original grasp space to represent the covariance matrix. We note that this is a poor choice of kernel, because the grasp space includes angles, making it a nonvector space. These methods also choose a grasp from the discrete set. We train on dataset with previous problems, and let .
Figure 3(b) shows the learning curve with . The xaxis is the percentage of the dataset used for training, ranging from one percent to ten percent. Initially, when we just use one percent of the training data points, PEMBO performs as poorly as TLSMBO, which again, had only 1000 training data points. However, PEMBO outperforms both TLSMBO and Plain after that. The main reason that PEMBO outperforms these approaches is because their prior, which is defined by the squared exponential kernel, is not suitable for this problem. PEMBO, on the other hand, was able to avoid this problem by estimating a distribution over values at the discrete sample points that commits only to their joint normality, but not to any metric on the underlying space. These trends are also shown in Figure 3(e), where we plot for . PEMBO outperforms the baselines significantly.
Optimizing a grasp, base pose, and placement
We now consider a more difficult task that involves both picking and placing objects in a cluttered scene. A planning problem instance is defined by the poses of obstacles and the poses and shapes of the target object to be pick and placed. The reward function is again the negative of the length of the picking motion if the motion is feasible, and otherwise. For both Plain and TLSMBO, we use three different squared exponential kernels on the original spaces of grasp, base pose, and object placement pose respectively and then add them together to define the kernel for the whole set. For this domain, , and .
Figure 3(c) shows the learning curve, when . The xaxis is the percentage of the dataset used for training, ranging from one percent to ten percent. Initially, when we just use one percent of the training data points, PEMBO does not perform well. Similar to the previous domain, it then significantly outperforms both TLSMBO and Plain after increasing the training data. This is also reflected in Figure 3(f), where we plot for . PEMBO outperforms baselines. Notice that Plain and TLSMBO perform worse than Random, as a result of making inappropriate assumptions on the form of the kernel.
6 Conclusion
We proposed a new framework for meta BO that estimates its Gaussian process prior based on past experience with functions sampled from the same prior. We established regret bounds for our approach without the reliance on a known prior and showed its good performance on task and motion planning benchmark problems.
Acknowledgments
We would like to thank Stefanie Jegelka, Tamara Broderick, Trevor Campbell, Tomás LozanoPérez for discussions and comments. We would like to thank Sungkyu Jung and Brian Axelrod for discussions on Wishart distributions. We gratefully acknowledge support from NSF grants 1420316, 1523767 and 1723381, from AFOSR grant FA95501710165, from Honda Research and Draper Laboratory. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of our sponsors.
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Appendix A Discussions and conclusions
In this section, we discuss related topics to our approach. Both theoreticians and practitioners may find this section useful in terms of clarifying theoretical insights and precautions.
a.1 Connections and differences to empirical Bayes
In classic empirical Bayes [45, 26], we estimate the unknown parameters of the Bayesian model and usually use a point estimate to proceed any Bayesian computations. One very popular approach to estimate those unknown parameters is by maximizing the data likelihood. There also exit other variants of empirical Bayes; for example, oracle Bayes, which “shows empirical Bayes in its most frequentist mode” [13].
In this paper, we use a variant of empirical Bayes that constructs estimators for both the prior distribution and the posterior distribution. For the estimators of the posterior, we do not use a plugin estimate like classic empirical Bayes but we construct them through Lemma. 11, which establishes the unbiasedness and concentration bounds for those estimates.
a.2 Connections and differences to hierarchical Bayes
Hierarchical Bayes is a Bayesian hierarchical model that places priors on priors. For both of our finite case and continuous and compact case, we can write down a hierarchical Bayes model that puts a normal inverse Wishart prior on or .
Our approach can be viewed as a special case of the hierarchical Bayes model using point estimates to approximate the posterior. Neither our estimators nor our regret analyses depend on the prior parameters of those hierarchical Bayes models. But one may analyze the regret of BO with a better approximation from a full Bayesian perspective using hierarchical Bayes.
a.3 Future directions
Due to the limited space, we only give the formulation of meta BO in its simple and basic settings. Our setting restricts the evaluated inputs in the training data to follow certain norms, such as where they are and how many they are, but one may certainly extend our analyses to less restrictive scenarios.
Missing entries
We did not consider any bounds in matrix completion [7] in our regret analyses, and proceeded with the assumption that there is no missing entry in the training data. But if missing data is a concern, one should definitely consider adapting bounds from [7] or use better estimators [34] that take into account missing entries when bounding the estimates.
a.4 Broader impact
We developed a statistically sound approach for meta BO with an unknown Gaussian process prior. We verified our approach on simulated task and motion planning problems. We showed that our approach is able to guide task and motion planning with good action recommendations, such that the resulting plans are better and faster to compute. We believe the theoretical guarantees may support better explanations for more practical BO approaches. In particular, our method can serve as a building block of artificial intelligence systems, and our analyses can be combined with the theoretical guarantees of other parts of the system to analyze an integrated system.
a.5 Caveats
We did not expand the experiment sections to include applications other than task and motion planning in simulation. But there are many more scenarios that this meta BO approach will be useful. For example, our finite formulation can be used to adaptively recommend advertisements, movies or songs to Internet users, by learning a mean and kernel for those discrete items.
Optimization objectives
Like other bandit algorithms, our approach only treats objective functions or any metrics to be optimized as given. Practitioners need to be very careful about what exactly they are optimizing with our approach or other optimization algorithms. For example, maximizing number of advertisement clicks or corporation profits may not be a good metric in recommendation systems; maximizing a poorly designed reward function for robotic systems may result in unexpected hazards.
Guarantees with assumptions
In realworld applications, practitioners need to be extra cautious with our algorithm. We provided detailed assumptions and analyses, that are only based those assumptions, in Section 3 and Section 4. Outside those assumptions, we do not claim that our analyses will hold in any way. For example, in robotics applications, it may not be true that the underlying reward/cost functions are actually sampled from a GP, in which case using our method may harm the physical robot; even if those objective functions are in fact from a GP, because our regret bounds only hold with high probability, meta BO may still give dangerous actions with certain probabilities (as in frequency).
In addition, please notice that we did not provide any theoretical guarantees for using basis functions trained with neural networks. We assume those basis functions are given, which is usually not the case in practice. To the best of our knowledge, proving bounds for neural networks is very hard [25].
Appendix B Proofs for Section 4.1
Recall that we assume is a finite set. The posterior given observations is where
We use the following estimators to approximate :
(9)  
(10) 
We will prove a bound on the bestsample simple regret . The evaluated inputs are selected either by a special case of GPUCB using the acquisition function
(11)  
(12) 
or by a special case of PI using the acquisition function
This special case of PI assumes additional information of the upper bound on function value .
Corollary 6 ([51]).
Let . For any Gaussian variable
where .
Proof.
Let . We have
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