Antifragile and Robust Heteroscedastic Bayesian Optimisation

1 INTRODUCTION . Bayesian Optimisation is already being utilised to make decisions in high-stakes applications such as drug discovery [ 1 , 2 , 3 , 4 , 5 ] , materials discovery [ 6 , 7 , 8 ] , robotics [ 9 ] , sensor placement [ 10 ] and tissue engineering [ 11 ] . In these problems heteroscedastic or input-dependent noise is rarely accounted for and the assumption of homoscedastic noise is often inappropriate . As a case study , heteroscedastic noise is the rule rather than the exception in the majority of scientific datasets . This is the case , not only in experimental datasets , but also in datasets where properties are predicted computationally . We illustrate this for molecular hydration free energies in Figure 1 using the dataset of [ 12 ] where there is a distribution of noise values and in general the noise function might be expected to grow in proportion to chemical complexity [ 13 ] . The consequences of neglecting heteroscedastic noise are illustrated using a second example in Figure 2 . The homoscedastic model will underestimate noise in certain regions which could induce a Bayesian Optimisation scheme to suggest values possessing large aleatoric noise . In an application such as high-throughput virtual screening the cost of misrepresenting the noise during the screening process could amount to a year wasted in the physical synthesis of a drug [ 14 ] . In materials discovery , we may derive benefit from or be antifragile [ 15 ] towards high aleatoric uncertainty . In an application such as the search for high-performing perovskite solar cells , we are faced with an extremely large compositional space , with millions of potential candidates possessing high aleatoric noise for identical reproductions [ 16 ] . In this instance we may want to guide search towards a candidate possessing a high photoluminescence quantum efficiency with high aleatoric noise . If the cost of repeating material syntheses is small relative to the cost of the search , the large aleatoric noise will present opportunities to synthesise materials possessing efficiencies far above their mean values . In this paper we present a heteroscedastic Bayesian Optimisation scheme capable of both representing and optimising aleatoric noise in the suggestions . Our contributions are : 1 . The introduction of a novel combination of surrogate model and acquisition function designed to optimise heteroscedastic aleatoric uncertainty to be used in situations where one wants to minimise ( be robust to ) aleatoric uncertainty as well as situations where one wants to maximise ( be antifragile towards ) aleatoric uncertainty . 2 . A demonstration of our scheme ’ s ability to outperform naive schemes based on homoscedastic Bayesian Optimisation on toy problems in addition to a real-world optimisation problem . 3 . The provision of an open-source implementation of the model . 2 RELATED WORK . The most similar work to our own is that of [ 17 ] where experiments are reported on a heteroscedastic Branin-Hoo function using the variational heteroscedastic GP approach of [ 18 ] although to the best of our knowledge this work does not consider sequential evaluations . A modification to EI , expected risk improvement is introduced in [ 19 ] and is applied to problems in robotics where robustness to aleatoric noise is desirable . [ 20 , 21 ] implement heteroscedastic Bayesian Optimisation but don ’ t introduce an acquisition function that penalises aleatoric uncertainty . [ 22 , 23 ] consider the related problem of safe Bayesian Optimisation through implementing constraints in parameter space . In this instance , the goal of the algorithm is to enforce a performance threshold for each evaluation of the black-box function and so is unrelated to our problem definition . In terms of acquisition functions , [ 24 , 25 ] propose principled approaches to handling aleatoric noise in the homoscedastic setting that could be extended to the heteroscedastic setting . Our primary focus in this work however , is to highlight that heteroscedasticity in the surrogate model is beneficial and so an examination of a subset of acquisition functions is sufficient for this purpose . 3 BACKGROUND . Bayesian Optimisation features a surrogate model for the black-box objective . The surrogate model should maintain calibrated uncertainty estimates in order to guide the acquisition of new data points . GPs are often a popular choice of surrogate model in BO applications because of their ability to represent uncertainty . 3.1 GAUSSIAN PROCESSES . In the terminology of stochastic processes we may formally define a GP as follows : Definition 1 . A Gaussian Process is a collection of random variables , any finite number of which have a joint Gaussian distribution . The random variables consist of function values f ( x ) at different locations x within the design space . The GP is characterised by a mean function m ( x ) = E [ f ( x ) ] ( 1 ) and a covariance function k ( x , x′ ) = E [ ( f ( x−m ( x ) ) ( f ( x′ ) −m ( x′ ) ) ] . ( 2 ) The process is written as follows f ( x ) ∼ GP ( m ( x ) , k ( x , x′ ) ) . ( 3 ) In our experiments , the prior mean function will be set to the data mean after normalisation of the inputs and standardisation of the outputs . As such , m ( x ) = 0 will be assumed henceforth . The covariance function computes the pairwise covariance between two random variables ( function values ) . The covariance between a pair of output values f ( x ) and f ( x′ ) is a function of an input pair x and x′ . As such , the kernel encodes smoothness assumptions about the latent function being modelled ; similarity in input space yields outputs that are close [ 26 ] . The most widely-utilised kernel is the squared exponential ( SE ) kernel kSQE ( x , x ′ ) = σ2f · exp ( −‖x− x′‖2 2 ` 2 ) ( 4 ) where σ2f is the signal amplitude hyperparameter ( vertical lengthscale ) and ` is the ( horizontal ) lengthscale hyperparameter . We use the squared exponential kernel in all experiments . For further information on Gaussian Processes the reader is referred to [ 27 ] . 3.2 BAYESIAN OPTIMISATION . Problem Statement The global optimisation problem is defined as x∗ = arg min x∈X f ( x ) ( 5 ) where x∗ is the global optimiser of a black-box function f : X → Y . X is the design space and is typically a compact subset of Rd . What makes this optimisation problem practically relevant in applications are the following properties : 1 . Black-box Objective : We do not have the analytic form of f . We can however evaluate f pointwise anywhere in the design space X . 2 . Expensive Evaluations : Choosing an input location x and evaluating f ( x ) takes a very long time . 3 . Noise : The evaluation of a given x is a noisy process . In addition , this noise may vary across X , making the underlying process heteroscedastic . We have a dataset of past observations D = { ( xi , ti ) } ni=1 consisting of observations of the black-box function f and fit a surrogate model to these datapoints . We then leverage the predictive mean as well as the uncertainty estimates of the surrogate model to guide the acquisition of the next data point xn+1 according to a heuristic known as an acquisition function . 4 HETEROSCEDASTIC BAYESIAN OPTIMISATION . We wish to perform Bayesian Optimisation whilst optimising input-dependent aleatoric noise . In order to represent input-dependent aleatoric noise , a heteroscedastic surrogate model is required . We take the most likely heteroscedastic GP approach of [ 28 ] , adopting the notation presented there for consistency . We have a dataset D = { ( xi , ti ) } ni=1 in which the target values ti have been generated according to ti = f ( xi ) + i . We assume independent Gaussian noise terms i ∼ N ( 0 , σi ) with variances given by σi = r ( xi ) . In the heteroscedastic setting r is typically a non-constant function over the input domain x . In order to perform Bayesian Optimisation , we wish to model the predictive distribution P ( t∗ | x∗1 , . . . , x∗q ) at the query points x∗1 , . . . , x∗q . Placing a GP prior on f and taking r ( x ) as the assumed noise rate function , the predictive distribution is multivariate GaussianN ( µ∗ , Σ∗ ) with mean µ∗ = E [ t∗ ] = K∗ ( K +R ) −1t ( 6 ) and covariance matrix Σ∗ = var [ t∗ ] = K∗∗ +R∗ −K∗ ( K +R ) −1K∗T , ( 7 ) where K ∈ Rn×n , Kij = k ( xi , xj ) , K∗ ∈ Rq×n , K∗ij = k ( x∗i , xj ) , K∗∗ ∈ Rq×q , K∗∗ij = k ( x ∗ i , x ∗ j ) , t = ( t1 , t2 , . . . , tn ) T , R = diag ( r ) with r = ( r ( x1 ) , r ( x2 ) , . . . , r ( xn ) ) T , and R∗ = diag ( r∗ ) with r∗ = ( r ( x∗1 ) , r ( x ∗ 2 ) , . . . , r ( x ∗ q ) ) T . The most likely heteroscedastic GP algorithm [ 28 ] executes the following steps : 1 . Estimate a homoscedastic GP , G1 on the dataset D = { ( xi , ti ) } ni=1 2 . Given G1 , we estimate the empirical noise levels for the training data using var [ ti , G1 ( xi , D ) ] = 0.5 ( ti − E [ x ] ) 2 forming a new dataset D′ = { ( xi , zi ) } ni=1 . A note on the form of this variance estimator is give in Appendix B . 3 . Estimate a second GP , G2 on D′ . 4 . Estimate a combined GP , G3 on D using G2 to predict the logarithmic noise levels ri . 5 . If not converged , set G3 to G1 and repeat . The Bayesian Optimisation problem may be framed as x∗ = arg min x∈χ f ( x ) , ( 8 ) where the black-box objective f , to be minimised has the form f ( x ) = g ( x ) + s ( x ) . ( 9 ) s ( x ) is , in this instance , the true noise rate function . We investigate extensions of the expected improvement [ 29 ] acquisition criterion , the principal form of which may be written in terms of the targets t and the incumbent best objective function value , η , found so far as EI ( x ) = E [ ( η − t ) + ] = ∫ ∞ −∞ ( η − t ) + p ( t |x ) dt ( 10 ) where p ( t |x ) is the posterior predictive marginal density of the objective function evaluated at x . ( η − t ) + ≡ max ( 0 , η − t ) is the improvement over the incumbent best objective function value η. Evaluations of the objective are noisy in all of the problems we consider and so we use expected improvement with plug-in [ 30 ] , the plug-in value being the GP predictive mean [ 31 ] . We propose two extensions to the expected improvement criterion . The first is an extension of the augmented expected improvement criterion AEI ( x ) = E [ ( η − t ) + ] ( 1− σn√ var [ t∗ ] + σ2n ) , ( 11 ) of [ 32 ] where σn is the fixed aleatoric noise level . We extend AEI to the heteroscedastic setting by exchanging the fixed aleaotric noise level with the input-dependent one : het-AEI ( x ) = E [ ( η − t ) + ] ( 1− √ r ( x ) √ var [ t∗ ] + r ( x ) ) . ( 12 ) where r ( x ) is the predicted aleatoric uncertainty at input x under the most likely heteroscedastic GP and var [ t∗ ] is the predictive variance of the heteroscedastic GP incorporating both aleatoric and epistemic components of the uncertainty . We also propose a simple modification to the expected improvement acquisition function that explicitly penalises regions of the input space with large aleatoric uncertainty . We call this acquisition function aleatoric noise-penalised expected improvement ( ANPEI ) and denote it ANPEI = αEI ( x ) − ( 1− α ) √ r ( x ) ( 13 ) where α is a scalarisation constant which we set to 0.5 for the experiments in this paper . In the multiobjective optimisation setting a particular value of α will correspond to a point on the Pareto frontier . We use both the modification to AEI ( het-AEI ) and ANPEI acquisition function in conjunction with the most likely heteroscedastic GP surrogate model in the experiments that follow . For the experiments on antifragility to aleatoric uncertainty we propose the antifragile augmented expected improvement ( AAEI ) acquisition function AAEI ( x ) = E [ ( η − t ) + ] ( 1− √ var [ t∗ ] + r ( x ) √ r ( x ) ) . ( 14 ) which has the effect of promoting solutions with high aleatoric noise .

Summary: This paper discusses an interesting problem of BO in the cases of robustness and antifragility to aleatoric noise/uncertainty. To tackle this problem, the authors replace the conventional homoscedastic GP model with a heteroscedastic GP model. In the case of robustness to aleatoric noise/uncertainty, the authors have modified EI by simply either scaling down [32] or subtracting from its value more when the aleatoric noise increases. In the case of antifragility to aleatoric noise/uncertainty, they do the opposite.