Bayesian Optimization of the PC Algorithm for Learning Gaussian Bayesian Networks

Córdoba, Irene; Garrido-Merchán, Eduardo C.; Hernández-Lobato, Daniel; Bielza, Concha; Larrañaga, Pedro

doi:10.1007/978-3-030-00374-6_5

Cited by 10 publications

(7 citation statements)

References 12 publications

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“…3 Black-Box Bayesian Optimization BO has been used in a plethora of scenarios: From Hyperparameter Tuning of ML Algorithms [Snoek et al, 2012] [Córdoba et al, 2018] to renewable energies [Cornejo-Bueno et al, 2018] and a wide variety of applications [Shahriari et al, 2016]. BO process follows an iterative scheme where it uses a probabilistic model as a surrogate model, typically a Gaussian Process (GP) which is a prior over functions.…”

Section: Related Workmentioning

confidence: 99%

Suggesting Cooking Recipes Through Simulation and Bayesian Optimization

Garrido-Merchán

Albarca-Molina

2018

Intelligent Data Engineering and Automated Learning – IDEAL 2018

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Cooking typically involves a plethora of decisions about ingredients and tools that need to be chosen in order to write a good cooking recipe. Cooking can be modelled in an optimization framework, as it involves a search space of ingredients, kitchen tools, cooking times or temperatures. If we model as an objective function the quality of the recipe, several problems arise. No analytical expression can model all the recipes, so no gradients are available. The objective function is subjective, in other words, it contains noise. Moreover, evaluations are expensive both in time and human resources. Bayesian Optimization (BO) emerges as an ideal methodology to tackle problems with these characteristics. In this paper, we propose a methodology to suggest recipe recommendations based on a Machine Learning (ML) model that fits real and simulated data and BO. We provide empirical evidence with two experiments that support the adequacy of the methodology.

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Section: Related Workmentioning

confidence: 99%

Suggesting Cooking Recipes Through Simulation and Bayesian Optimization

Garrido-Merchán

Albarca-Molina

2018

Intelligent Data Engineering and Automated Learning – IDEAL 2018

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“…For example, the estimation of the generalization error of ML algorithms is considered to be a black-box function. We find other applications in structure learning of probabilistic graphical models [7], astrophysics [3] or even subjective tasks as suggesting better recipes [11].…”

Section: Introductionmentioning

confidence: 99%

Many Objective Bayesian Optimization

Martín¹,

Garrido-Merchán²

2021

Preprint

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Some real problems require the evaluation of expensive and noisy objective functions. Moreover, the analytical expression of these objective functions, and hence its gradients, may be unknown. These functions that are known as black-boxes, for example, estimating the generalization error of a machine learning algorithm and computing its prediction time in terms of its hyper-parameters. Multi-objective Bayesian optimization (MOBO) is a set of methods that has been successfully applied for the simultaneous optimization of black-boxes. Concretely, BO methods rely on a probabilistic model of the objective functions, typically a Gaussian process (GP). This model generates a predictive distribution of the objectives, capturing the uncertainty about their potential values. However, MOBO methods have problems when the number of objectives in a multi-objective optimization problem are 3 or more, which is the many objective setting. In particular, the BO process is more costly as more objectives are considered, computing the quality of the solution via the hyper-volume is also more costly and, most importantly, we have to evaluate every objective function, wasting expensive computational, economic or other resources. However, as more objectives are involved in the optimization problem, it is highly probable that some of them are redundant and not add information about the problem solution. A measure that represents how similar are GP predictive distributions is proposed. We also propose a many objective Bayesian optimization algorithm that uses this metric to determine whether two objectives are redundant. The algorithm stops evaluating one of them if the similarity is found, saving resources and not hurting the performance of the multi-objective BO algorithm. We show empirical evidence in a set of toy, synthetic, benchmark and real experiments that GPs predictive distributions of the effectiveness of the metric and the algorithm.

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“…The estimation of the generalization error of ML algorithms is considered to be a black-box function. We find other applications in structure learning of probabilistic graphical models [2] or even subjective tasks as suggesting better recipes [4]. When not only one but several black-boxes are optimized, we deal with the Multi-objective BO scenario [9].…”

Section: Introductionmentioning

confidence: 99%

A Similarity Measure of Gaussian Process Predictive Distributions

Asencio-Martín¹,

Garrido-Merchán²

2021

Preprint

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Some scenarios require the computation of a predictive distribution of a new value evaluated on an objective function conditioned on previous observations. We are interested on using a model that makes valid assumptions on the objective function whose values we are trying to predict. Some of these assumptions may be smoothness or stationarity. Gaussian process (GPs) are probabilistic models that can be interpreted as flexible distributions over functions. They encode the assumptions through covariance functions, making hypotheses about new data through a predictive distribution by being fitted to old observations. We can face the case where several GPs are used to model different objective functions. GPs are non-parametric models whose complexity is cubic on the number of observations. A measure that represents how similar is one GP predictive distribution with respect to another would be useful to stop using one GP when they are modelling functions of the same input space. We are really inferring that two objective functions are correlated, so one GP is enough to model both of them by performing a transformation of the prediction of the other function in case of inverse correlation. We show empirical evidence in a set of synthetic and benchmark experiments that GPs predictive distributions can be compared and that one of them is enough to predict two correlated functions in the same input space. This similarity metric could be extremely useful used to discard objectives in Bayesian many-objective optimization.

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Bayesian Optimization of the PC Algorithm for Learning Gaussian Bayesian Networks

Cited by 10 publications

References 12 publications

Suggesting Cooking Recipes Through Simulation and Bayesian Optimization

Suggesting Cooking Recipes Through Simulation and Bayesian Optimization

Many Objective Bayesian Optimization

A Similarity Measure of Gaussian Process Predictive Distributions

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