Bayesian Optimization with Gradients
Bayesian optimization has been successful at global optimization of expensive-to-evaluate multimodal objective functions. However, unlike most optimization methods, Bayesian optimization typically does not use derivative information. In this paper we show how Bayesian optimization can exploit derivative information to find good solutions with fewer objective function evaluations. In particular, we develop a novel Bayesian optimization algorithm, the derivative-enabled knowledge-gradient (d-KG), which is one-step Bayes-optimal, asymptotically consistent, and provides greater one-step value of information than in the derivative-free setting. d-KG accommodates noisy and incomplete derivative information, comes in both sequential and batch forms, and can optionally reduce the computational cost of inference through automatically selected retention of a single directional derivative. We also compute the d-KG acquisition function and its gradient using a novel fast discretization-free technique. We show d-KG provides state-of-the-art performance compared to a wide range of optimization procedures with and without gradients, on benchmarks including logistic regression, deep learning, kernel learning, and k-nearest neighbors.
Bayesian Optimization with Gradients
Jian Wu Matthias Poloczek Andrew Gordon Wilson Peter I. Frazier Cornell University, University of Arizona
noticebox[b]31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\end@float
Bayesian optimization (Brochu et al., 2010; Jones et al., 1998) is able to find global optima with a remarkably small number of potentially noisy objective function evaluations. Bayesian optimization has thus been particularly successful for automatic hyperparameter tuning of machine learning algorithms (Snoek et al., 2012; Swersky et al., 2013; Gelbart et al., 2014; Gardner et al., 2014), where objectives can be extremely expensive to evaluate, noisy, and multimodal.
Bayesian optimization supposes that the objective function (e.g., the predictive performance with respect to some hyperparameters) is drawn from a prior distribution over functions, typically a Gaussian process (GP), maintaining a posterior as we observe the objective in new places. Acquisition functions, such as expected improvement (Jones et al., 1998; Huang et al., 2006; Picheny et al., 2013), upper confidence bound (Srinivas et al., 2010), predictive entropy search (Hernández-Lobato et al., 2014) or the knowledge gradient (Scott et al., 2011), determine a balance between exploration and exploitation, to decide where to query the objective next. By choosing points with the largest acquisition function values, one seeks to identify a global optimum using as few objective function evaluations as possible.
Bayesian optimization procedures do not generally leverage derivative information, beyond a few exceptions described in Sect. 2. By contrast, other types of continuous optimization methods (Snyman, 2005) use gradient information extensively. The broader use of gradients for optimization suggests that gradients should also be quite useful in Bayesian optimization: (1) Gradients inform us about the objective’s relative value as a function of location, which is well-aligned with optimization. (2) In -dimensional problems, gradients provide distinct pieces of information about the objective’s relative value in each direction, constituting values per query together with the objective value itself. This advantage is particularly significant for high-dimensional problems. (3) Derivative information is available in many applications at little additional cost. Recent work (e.g., Maclaurin et al., 2015) makes gradient information available for hyperparameter tuning. Moreover, in the optimization of engineering systems modeled by partial differential equations, which pre-dates most hyperparameter tuning applications (Forrester et al., 2008), adjoint methods provide gradients cheaply (Plessix, 2006; Jameson, 1999). And even when derivative information is not readily available, we can compute approximative derivatives in parallel through finite differences.
In this paper, we explore the “what, when, and why” of Bayesian optimization with derivative information. We also develop a Bayesian optimization algorithm that effectively leverages gradients in hyperparameter tuning to outperform the state of the art. This algorithm accommodates incomplete and noisy gradient observations, can be used in both the sequential and batch settings, and can optionally reduce the computational overhead of inference by selecting the single most valuable directional derivatives to retain. For this purpose, we develop a new acquisition function, called the derivative-enabled knowledge-gradient (d-KG). d-KG generalizes the previously proposed batch knowledge gradient method of Wu and Frazier (2016) to the derivative setting, and replaces its approximate discretization-based method for calculating the knowledge-gradient acquisition function by a novel faster exact discretization-free method. We note that this discretization-free method is also of interest beyond the derivative setting, as it can be used to improve knowledge-gradient methods for other problem settings. We also provide a theoretical analysis of the d-KG algorithm, showing (1) it is one-step Bayes-optimal by construction when derivatives are available; (2) that it provides one-step value greater than in the derivative-free setting, under mild condition; and (3) that its estimator of the global optimum is asymptotically consistent.
In numerical experiments we compare with state-of-the-art batch Bayesian optimization algorithms with and without derivative information, and the gradient-based optimizer BFGS with full gradients.
We assume familiarity with GPs and Bayesian optimization, for which we recommend Rasmussen and Williams (2006) and Shahriari et al. (2016) as a review. In Sect. 2 we begin by describing related work. In Sect. 3 we describe our Bayesian optimization algorithm exploiting derivative information. In Sect. 4 we compare the performance of our algorithm with several competing methods on a collection of synthetic and real problems.
The code in this paper is available at https://github.com/wujian16/Cornell-MOE.
2 Related Work
Osborne et al. (2009) proposes fully Bayesian optimization procedures that use derivative observations to improve the conditioning of the GP covariance matrix. Samples taken near previously observed points use only the derivative information to update the covariance matrix. Unlike our current work, derivative information is not fully utilized for optimization in this previous work in the sense that derivation information does not affect the acquisition function. We directly compare with Osborne et al. (2009) within the KNN benchmark in Sect. 4.2.
Lizotte (2008, Sect. 4.2.1 and Sect. 5.2.4) incorporates derivatives into Bayesian optimization, modeling the derivatives of a GP as in Rasmussen and Williams (2006, Sect. 9.4). Lizotte (2008) shows that Bayesian optimization with the expected improvement (EI) acquisition function and complete gradient information at each sample can outperform BFGS. Our approach has six key differences: (i) we allow for noisy and incomplete derivative information; (ii) we develop a novel acquisition function that outperforms EI with derivatives; (iii) we enable batch evaluations; (iv) we implement and compare batch Bayesian optimization with derivatives across several acquisition functions, on benchmarks and new applications such as kernel learning, logistic regression, deep learning and k-nearest neighbors, further revealing empirically where gradient information will be most valuable; (v) we provide a theoretical analysis of Bayesian optimization with derivatives; (vi) we develop a scalable implementation.
Very recently, Koistinen et al. (2016) uses GPs with derivative observations for minimum energy path calculations of atomic rearrangements and Ahmed et al. (2016) studies expected improvement with gradient observations. In Ahmed et al. (2016), a randomly selected directional derivative is retained in each iteration for computational reasons, which is similar to our approach of retaining a single directional derivative, though differs in its random selection in contrast with our value-of-information-based selection. Our approach is complementary to these works.
For batch Bayesian optimization, several recent algorithms have been proposed that choose a set of points to evaluate in each iteration (Snoek et al., 2012; Wang et al., 2016; Marmin et al., 2016; Contal et al., 2013; Desautels et al., 2014; Kathuria et al., 2016; Shah and Ghahramani, 2015; Gonzalez et al., 2016). Within this area, our approach to handling batch observations is most closely related to the batch knowledge gradient (KG) of Wu and Frazier (2016). We generalize this approach to the derivative setting, and provide a novel exact method for computing the knowledge-gradient acquisition function that avoids the discretization used in Wu and Frazier (2016). This generalization improves speed and accuracy, and is also applicable to other knowledge gradient methods in continuous search spaces.
Recent advances improving both access to derivatives and computational tractability of GPs make Bayesian optimization with gradients increasingly practical and timely for discussion.
3 Knowledge Gradient with Derivatives
Sect. 3.1 reviews a general approach to incorporating derivative information into GPs for Bayesian optimization. Sect. 3.2 introduces a novel acquisition function d-KG, based on the knowledge gradient approach, which utilizes derivative information. Sect. 3.3 computes this acquisition function and its gradient efficiently using a novel fast discretization-free approach. Sect. 3.4 shows that this algorithm provides greater value of information than in the derivative-free setting, is one-step Bayes-optimal, and is asymptotically consistent when used over a discretized feasible space.
3.1 Derivative Information
Given an expensive-to-evaluate function , we wish to find , where is the domain of optimization. We place a GP prior over , which is specified by its mean function and kernel function . We first suppose that for each sample of we observe the function value and all partial derivatives, possibly with independent normally distributed noise, and then later discuss relaxation to observing only a single directional derivative.
Since the gradient is a linear operator, the gradient of a GP is also a GP (see also Sect. 9.4 in Rasmussen and Williams (2006)), and the function and its gradient follows a multi-output GP with mean function and kernel function defined below:
where and is the Hessian of .
When evaluating at a point , we observe the noise-obscured function value and gradient . Jointly, these observations form a -dimensional vector with conditional distribution
where gives the variance of the observational noise. If is not known, we may estimate it from data. The posterior distribution is again a GP. We refer to the mean function of this posterior GP after samples as and its kernel function as . Suppose that we have sampled at points and observed , where each observation consists of the function value and the gradient at , then and are given by
The rows and columns in , , and of Eq. (3.3) corresponding to partial derivatives (or function values) that were not observed are to be omitted. Furthermore, if we can only observe some directional derivatives, then each observation in , and of Eq. (3.3) only consists of the function value and the directional derivatives at , which can be obtained by linearly transforming the original terms. We can then perform inference similarly to the procedure described above for full gradient observations.
3.2 The d-KG Acquisition Function
We propose a novel Bayesian optimization algorithm to exploit available derivative information, based on the knowledge gradient approach (Frazier et al., 2009). We call this algorithm the derivative-enabled knowledge gradient (d-KG).
The algorithm proceeds iteratively, selecting in each iteration a batch of points in that has a maximum value of information (VOI). Suppose we have observed points, and recall from Sect. 3.1 that is the -dimensional vector giving the posterior mean for and its partial derivatives at . Sect. 3.1 discusses how to remove the assumption that all values are provided.
The expected value of under the posterior distribution is . If after samples we were to make an irrevocable (risk-neutral) decision now about the solution to our overarching optimization problem and receive a loss equal to the value of at the chosen point, we would choose and suffer conditional expected loss . Similarly, if we made this decision after samples our conditional expected loss would be . Therefore, we define the d-KG factor for a given set of candidate points as
where is the expectation taken with respect to the posterior distribution after evaluations, and the distribution of under this posterior marginalizes over the observations upon which it depends. We subsequently refer to Eq. (LABEL:eqn:multiKG) as the inner optimization problem.
The d-KG algorithm then seeks to evaluate the batch of points next that maximizes the d-KG factor,
The d-KG acquisition function differs from the batch knowledge gradient acquisition function in Wu and Frazier (2016) because here the posterior mean at time depends on . This in turn requires calculating the distribution of these gradient observations under the time- posterior and marginalizing over them. Thus, the d-KG algorithm differs from KG not just in that gradient observations change the posterior, but also in that the prospect of future gradient observations changes the acquisition function. An additional major distinction from Wu and Frazier (2016) is that d-KG employs a novel discretization-free method for computing the acquisition function (see Sect. 3.3).
Fig. 1 illustrates the behavior of d-KG and d-EI on a -d example. d-EI generalizes expected improvement (EI) to batch-acquisition with derivative information Lizotte (2008). d-KG clearly chooses a better point to evaluate than d-EI.
Including all partial derivatives can be computationally prohibitive since GP inference scales as . To overcome this challenge while retaining the value of derivative observations, we can include only one directional derivative from each iteration in our inference. d-KG can naturally decide which derivative to include, and can adjust our choice of where to best sample given that we observe more limited information. We define the d-KG acquisition function for observing only the function value and the derivative with direction at as
where conditioning on is here understood to mean that is the conditional mean of given and . The full algorithm is as follows.
3.3 Efficient Exact Computation of d-KG
Calculating and maximizing d-KG is difficult when is continuous because the term in Eq. (3.6) requires optimizing over a continuous domain, and then we must integrate this optimal value through its dependence on and . Previous work on the knowledge gradient in continuous domains (Scott et al., 2011; Wu and Frazier, 2016; Poloczek et al., 2017) approach this computation by taking minima within expectations not over the full domain but over a discretized finite approximation. This approach supports analytic integration in Scott et al. (2011) and Poloczek et al. (2017), and a sampling-based scheme in Wu and Frazier (2016). However, the discretization in this approach introduces an error and scales poorly with the dimension of .
Here we propose a novel method for calculating an unbiased estimator of the gradient of d-KG which we then use within stochastic gradient ascent to maximize d-KG. This method avoids discretization, and thus is exact. It also improves speed significantly over a discretization-based scheme.
In Sect. A of the supplement we show that the d-KG factor can be expressed as
where is the mean function of after evaluations, is a dimensional standard normal random column vector and is the first row of a dimensional matrix, which is related to the kernel function of after evaluations with an exact form specified in (A.2) of the supplement.
Under sufficient regularity conditions L’Ecuyer (1995), one can interchange the gradient and expectation operators,
If is continuously differentiable and is compact, the envelope theorem (Milgrom and Segal, 2002) implies
where . To find , one can utilize a multi-start gradient descent method since the gradient is analytically available for the objective . Practically, we find that the learning rate of is robust for finding .
This expression implies that is an unbiased estimator of . We can use this unbiased gradient estimator within stochastic gradient ascent (Harold et al., 2003), optionally with multiple starts, to solve the outer optimization problem (3.5). For the outer optimization problem, we find that the learning rate of performs well over all the benchmarks we tested.
Bayesian Treatment of Hyperparameters.
We adopt a fully Bayesian treatment of hyperparameters similar to Snoek et al. (2012). We draw samples of hyperparameters for via the emcee package (Foreman-Mackey et al., 2013) and average our acquisition function across them to obtain
where the additional argument in d-KG indicates that the computation is performed conditioning on hyperparameters . In our experiments, we found this method to be computationally efficient and robust, although a more principled treatment of unknown hyperparameters within the knowledge gradient framework would instead marginalize over them when computing and .
3.4 Theoretical Analysis
Here we present three theoretical results giving insight into the properties of d-KG, with proofs in the supplementary material. For the sake of simplicity, we suppose all partial derivatives are provided to d-KG. Note that similar results hold for d-KG with relevant directional derivative detection. We begin by stating that the value of information (VOI) obtained by d-KG exceeds the VOI that can be achieved in the derivative-free setting.
Next, we show that d-KG is one-step Bayes-optimal by construction.
If only one iteration is left and we can observe both function values and its partial derivatives, then d-KG is Bayes-optimal among all feasible policies.
As a complement to the one-step optimality, we show that d-KG is asymptotically consistent if the feasible set is finite. Asymptotic consistency means that d-KG will choose the correct solution when the number of samples goes to infinity.
If the function is sampled from a GP with known hyperparameters, the d-KG algorithm is asymptotically consistent, i.e.
almost surely, where is the point recommended by d-KG after iterations.
We evaluate the performance of the proposed algorithm d-KG with relevant directional derivative detection (Algorithm 1) on six standard synthetic benchmarks (see Fig. 2). Moreover, we examine its ability to tune the hyperparameters for the weighted k-nearest neighbor metric, logistic regression, deep learning, and for a spectral mixture kernel (see Fig. 3).
We provide an easy-to-use Python package with the core written in C++. The code in this paper is available at https://github.com/wujian16/Cornell-MOE.
We compare d-KG to several state-of-the-art methods: (1) The batch expected improvement method (EI) of Wang et al. (2016) that does not utilize derivative information and an extension of EI that incorporates derivative information denoted by d-EI. Note that d-EI is similar to Lizotte (2008) but handles incomplete gradients and supports batches. (2) The batch GP-UCB-PE method of Contal et al. (2013) that does not utilize derivative information, and an extension of this method that does. (3) The batch knowledge gradient algorithm without derivative information (KG) of Wu and Frazier (2016). Moreover, we generalize the method of Osborne et al. (2009) to batches and evaluate it on the KNN benchmark. We stress that all of the above algorithms can be run even if not all partial derivatives are given. In benchmarks that provide the full gradient, we additionally compare to the gradient-based method L-BFGS-B provided in scipy. We suppose that the objective function is drawn from a Gaussian process , where is a constant mean function and is the squared exponential kernel. We sample sets of hyperparameters by the emcee package (Foreman-Mackey et al., 2013).
Recall that the immediate regret is defined as the loss with respect to a global optimum. The plots for synthetic benchmark functions, shown in Fig. 2, report the log10 scale of immediate regret of the solution that each algorithm would pick as a function of the number of function evaluations. For the other experiments the plots depict the objective value of the solution instead of the immediate regret. The error bars give the mean value plus and minus one standard deviation. The number of replications is stated in each benchmark’s description.
4.1 Synthetic Test Functions
We evaluate all methods on six test functions chosen from Bingham (2015). In order to demonstrate the ability to benefit from noisy derivative information, we sample additive normally distributed noise with zero mean and standard deviation for both the objective function and its partial derivatives. Note that is not known to the algorithms but has to be estimated from observations. Moreover, we investigate how the performance of the algorithms is affected if partial derivatives are not given for all parameters. We also experiment with two different batch sizes: we use a batch size for the Branin, Rosenbrock, and Ackley functions; otherwise, we use a batch size . The experimental results are summarized in Fig. 2.
Functions with Full Gradient Information.
For 2d Branin on domain , 5d Ackley on , and 6d Hartmann function on , we assume that the full gradient is available.
Looking at the results for the Branin function in Fig. 2, d-KG outperforms its competitors after 40 function evaluations and obtains the best solution overall (within the limit of function evaluations). BFGS makes faster progress than the Bayesian optimization methods during the first 20 evaluations, but subsequently stalls and fails to obtain a competitive solution. On the Ackley function d-EI makes fast progress during the first evaluations but also fails to make any subsequent progress. Conversely, d-KG requires about evaluations to improve on the performance of d-EI, after which d-KG achieves the best overall performance again. For the Hartmann function d-KG clearly dominates its competitors over all function evaluations.
Functions with Incomplete Derivative Information.
For the 3d Rosenbrock function on we only provide a noisy observation of the third partial derivative. Both EI and d-EI get stuck early. d-KG on the other hand finds a near optimal solution after about function evaluations; KG, without derivatives, catches up after about evaluations and has a comparable performance afterwards. The 4d Levy benchmark on , where the fourth partial derivative is observable with noise, shows a different ordering of the algorithms: here EI has the best performance, beating even its formulation that utilizes derivative information. A possible explanation could be that the smoothness and regularized shape of the function surface benefits this acquisition criterion. For the 8d Cosine mixture function on we provide two noisy partial derivatives. d-KG and UCB with derivatives perform better than EI-type criterion, and achieve the best performances, with d-KG beating UCB with derivatives slightly.
In general, we see that d-KG successfully exploits noisy derivative information and has the best overall performance.
4.2 Real-World Test Functions
Weighted k-Nearest Neighbor.
Suppose a cab company wishes to predict the duration of trips. Clearly, the duration not only depends on the endpoints of the trip, but also on the day and time. In this benchmark we tune a weighted k-nearest neighbor (KNN) metric to optimize predictions of these durations, based on historical data. A trip is described by the pick-up time , the pick-up location , and the drop-off point . Then the estimate of the duration is obtained as a weighted average over all trips in our database that happened in the time interval minutes, where is a tunable hyperparameter: The weight of trip in this prediction is given by ,
where are the respective parameter values for trip , and are tunable hyperparameters. Thus, we have hyperparameters to tune: . We choose in , in , and each in .
We use the yellow cab NYC public data set from June 2016, sampling records from June 1 – 25 as training data and trip records from June – as validation data. Our test criterion is the root mean squared error (RMSE), for which we compute the partial derivatives on the validation dataset with respect to the hyperparameters , while the hyperparameter is not differentiable. In Fig. 3 we see that d-KG overtakes the alternatives, and that UCB and KG acquisition functions also benefit from exploiting derivative information.
Spectral mixture kernels (Wilson and Adams, 2013) can be used for flexible kernel learning to enable long-range extrapolation. These kernels are obtained by modeling a spectral density by a mixture of Gaussians. While any stationary kernel can be described by a spectral mixture kernel with a particular setting of its hyperparameters, initializing and learning these parameters can be difficult. Although we have access to an analytic closed form of the (marginal likelihood) objective, this function is (i) expensive to evaluate and (ii) highly multimodal. Moreover, (iii) derivative information is available. Thus, learning flexible kernel functions is a perfect candidate for our approach.
The task is to train a -component spectral mixture kernel on an airline data set Wilson and Adams (2013). We have to determine the mixture weights, means, and variances, for each of the two Gaussians. Fig. 3 summarizes the experimental performances for batch size . BFGS turns out to be sensitive to its initialization and human intervention and is often trapped in local optima. d-KG, on other hand, more consistently finds a good solution, and obtains the best solution of all algorithms (within the step limit). Overall, we observe that gradient information is highly valuable in performing this kernel learning task.
Logistic Regression and Deep Learning.
We tune logistic regression and a feedforward neural network with hidden layers on the MNIST dataset (LeCun et al., 1998), a standard classification task for handwritten digits. The training set contains images, the test set . We tune hyperparameters for logistic regression: the regularization parameter from to , learning rate from to , mini batch size from to and training epochs from to . The first derivatives of the first two parameters can be obtained via the technique of Maclaurin et al. (2015). For the neural network, we additionally tune the number of hidden units in .
Fig. 3 reports the mean and standard deviation of the mean cross-entropy loss (or its log scale) on the test set for replications. d-KG outperforms the other approaches, which suggests that derivative information is helpful. Our algorithm proves its value in tuning a deep neural network, which harmonizes with research computing the gradient of hyperparameters (Maclaurin et al., 2015; Pedregosa, 2016).
Bayesian optimization is successfully applied to low dimensional problems where we wish to find a good solution with a very small number of objective function evaluations. We considered several such benchmarks, as well as logistic regression, deep learning, kernel learning, and k-nearest neighbor applications. We have shown that in this context derivative information can be extremely useful: we can greatly decrease the number of objective function evaluations, especially when building upon the knowledge gradient acquisition function, even when derivative information is noisy and only available for some variables.
Bayesian optimization is increasingly being used to automate parameter tuning in machine learning, where objective functions can be extremely expensive to evaluate. For example, the parameters to learn through Bayesian optimization could even be the hyperparameters of a deep neural network. We expect derivative information with Bayesian optimization to help enable such promising applications, moving us towards fully automatic and principled approaches to statistical machine learning.
In the future, one could combine derivative information with flexible deep projections (Wilson et al., 2016), and recent advances in scalable Gaussian processes for training and test time predictions (Wilson and Nickisch, 2015; Wilson et al., 2015). These steps would help make Bayesian optimization applicable to a much wider range of problems, wherever standard gradient based optimizers are used – even when we have analytic objective functions that are not expensive to evaluate – while retaining faster convergence and robustness to multimodality.
Wilson was partially supported by NSF IIS-1563887. Frazier, Poloczek, and Wu were partially supported by NSF CAREER CMMI-1254298, NSF CMMI-1536895, NSF IIS-1247696, AFOSR FA9550-12-1-0200, AFOSR FA9550-15-1-0038, and AFOSR FA9550-16-1-0046.
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Appendix A The Computation of d-KG and its Gradient: Additional Details
In this section, we show additional details in Sect. 3.3 of the main document: how to provide unbiased estimators of and its gradient. It is well-known that if and are the mean and the kernel function respectively of the posterior of after evaluating points, then follows a bivariate Gaussian process with the mean function and the kernel function as follows
Analogously, the is also subject to noise,
where . Following Wu and Frazier (2016), we express as
Conditioning on and the knowledge after evaluations, we have is normally distributed with mean and covariance matrix where the function maps the sample to its function and the directional derivative observation at direction . Thus, we can rewrite as
where is a -dimensional standard normal vector and
Here is the Cholesky factor of the covariance matrix . Now we can follow Sect. 3.3 of the main document to estimate and its gradient.
Proof of Proposition 1.
Recall that we start with the same posterior , then
where recall that is the observed function value at , and are the derivative observations at accordingly. The inequality above holds due to Jensen’s inequality.
Now we will show that the inequality is strict when is an inconstant function of where for a given . The equality holds only if there exist sets and such that: (1) and ; (2) for any given , is a linear function of on .
By (3.3), we can express as
Then condition (2) holds only if is constant for all , which holds only if are the same for all . It is usually impossible in practice that remains unchanged for all , so the inequality is strict in settings where affects . ∎
Next we analyze the Bayesian optimization problem under the dynamic programming (DP) framework and show that d-KG is one-step Bayes-optimal.
Proof of Proposition 2.
Suppose that we are given a budget of samples, i.e. we may run the algorithm for iterations. Our goal is to choose sampling decisions and the implementation decision that minimizes . We assume that is drawn from the prior , then follows the posterior process after iterations, so we have . Thus, letting be the set of feasible policies , we can formulate our problem as follows
We analyze this problem under the DP framework. We define our state space as after iteration as it completely characterizes our belief on . Under the DP framework, we will define the value function as follows
for every . The bellman equation tells us that the value function can be written recursively as
At the same time, we also know that any policy whose decision satisfy
is optimal. If we were to stop at iteration , then and (B.3) reduces to
which is exactly the d-KG algorithm. This proves that d-KG is one-step Bayes-optimal. ∎
Appendix C Proof of Theorem 1
Recall that we define the value function in Eq. (B.2). Similarly, we can define the value function for a specific policy as
Since we are varying the number of iterations , we define as the optimal value function when the number of iteration budgets is . Additionally, we define . Similarly, we define and for a specific policy .
Next we will state two lemmas concerning the benefits of additional samples, which will be useful in the latter proofs. First we have the following result for any stationary policy . A policy is called stationary if the decision of the policy only depends on the current state (not related to which iteration it is after, i.e. ). d-KG is stationary.
For any stationary policy and state , .
This lemma states that for any stationary policy, one additional iteration helps on average.
Proof of Lemma 1.
We prove by induction on . When , by Jensen’s inequality,
Then by the induction hypothesis,
We concludes the proof. ∎
The following lemma is related to the optimal policy. It says that if allowed an extra fixed batch of samples, the optimal policy performs better on average than if no extra samples allowed.
For any state and , .
As a direct corollary, we have for any state .
Proof of Lemma 2.
The lemma below shows that is well defined and bounded below.
For any state , exists and