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Gaussian Process based Bayesian Optimization is a well-known sample efficient sequential strategy for globally optimizing black-box, expensive, and multi-extremal functions. The role of the Gaussian Process is to provide a probabilistic approximation of the unknown function, depending on the sequentially collected observations, while an acquisition function drives the choice of the next solution to evaluate, balancing between exploration and exploitation, depending on the current Gaussian Process model. Despite the huge effort of the scientific community in defining effective exploration-exploitation mechanisms, we are still far away from the master acquisition function. This paper merges the most relevant results and insights from both algorithmic and human search strategies to propose a novel acquisition function, mastering the trade-off between explorative and exploitative choices, adaptively. We compare the proposed acquisition function on a number of test functions and against different state-of-the-art ones, which are instead based on prefixed or random scheduling between exploration and exploitation. A Pareto analysis is performed with respect to two (antagonistic) goals: convergence to the optimum and exploration capability. Results empirically prove that the proposed acquisition function is almost always Pareto optimal and also the most balanced trade-off between the two goals.

Optimal resource allocation is gaining a renewed interest due its relevance as a core problem in managing, over time, cloud and high-performance computing facilities. Semi-Bandit Feedback (SBF) is the reference method for efficiently solving this problem. In this paper we propose (i) an extension of the optimal resource allocation to a more general class of problems, specifically with resources availability changing over time, and (ii) Bayesian Optimization as a more efficient alternative to SBF. Three algorithms for Bayesian Optimization for Resource Allocation, namely BORA, are presented, working on allocation decisions represented as numerical vectors or distributions. The second option required to consider the Wasserstein distance as a more suitable metric to use into one of the BORA algorithms. Results on (i) the original SBF case study proposed in the literature, and (ii) a real-life application (i.e., the optimization of multi-channel marketing) empirically prove that BORA is a more efficient and effective learning-and-optimization framework than SBF., Comment: 31 pages, 12 figures

There is a consensus that focusing only on accuracy in searching for optimal machine learning models amplifies biases contained in the data, leading to unfair predictions and decision supports. Recently, multi-objective hyperparameter optimization has been proposed to search for machine learning models which offer equally Pareto-efficient trade-offs between accuracy and fairness. Although these approaches proved to be more versatile than fairness-aware machine learning algorithms -- which optimize accuracy constrained to some threshold on fairness -- they could drastically increase the energy consumption in the case of large datasets. In this paper we propose FanG-HPO, a Fair and Green Hyperparameter Optimization (HPO) approach based on both multi-objective and multiple information source Bayesian optimization. FanG-HPO uses subsets of the large dataset (aka information sources) to obtain cheap approximations of both accuracy and fairness, and multi-objective Bayesian Optimization to efficiently identify Pareto-efficient machine learning models. Experiments consider two benchmark (fairness) datasets and two machine learning algorithms (XGBoost and Multi-Layer Perceptron), and provide an assessment of FanG-HPO against both fairness-aware machine learning algorithms and hyperparameter optimization via a multi-objective single-source optimization algorithm in BoTorch, a state-of-the-art platform for Bayesian Optimization.

The main objective of this paper is to outline a theoretical framework to characterise humans' decision-making strategies under uncertainty, in particular active learning in a black-box optimization task and trading-off between information gathering (exploration) and reward seeking (exploitation). Humans' decisions making according to these two objectives can be modelled in terms of Pareto rationality. If a decision set contains a Pareto efficient strategy, a rational decision maker should always select the dominant strategy over its dominated alternatives. A distance from the Pareto frontier determines whether a choice is Pareto rational. To collect data about humans' strategies we have used a gaming application that shows the game field, with previous decisions and observations, as well as the score obtained. The key element in this paper is the representation of behavioural patterns of human learners as a discrete probability distribution. This maps the problem of the characterization of humans' behaviour into a space whose elements are probability distributions structured by a distance between histograms, namely the Wasserstein distance (WST). The distributional analysis gives new insights about human search strategies and their deviations from Pareto rationality. Since the uncertainty is one of the two objectives defining the Pareto frontier, the analysis has been performed for three different uncertainty quantification measures to identify which better explains the Pareto compliant behavioural patterns. Beside the analysis of individual patterns WST has also enabled a global analysis computing the barycenters and WST k-means clustering. A further analysis has been performed by a decision tree to relate non-Paretian behaviour, characterized by exasperated exploitation, to the dynamics of the evolution of the reward seeking process., Comment: 49 pages, 39 figures. arXiv admin note: substantial text overlap with arXiv:2102.07647

Automated driving systems (ADS) have undergone a significant improvement in the last years. ADS and more precisely self-driving cars technologies will change the way we perceive and know the world of transportation systems in terms of user experience, mode choices and business models. The emerging field of Deep Learning (DL) has been successfully applied for the development of innovative ADS solutions. However, the attempt to single out the best deep neural network architecture and tuning its hyperparameters are all expensive processes, both in terms of time and computational resources. In this work, Bayesian Optimization (BO) is used to optimize the hyperparameters of a Spatiotemporal-Long Short Term Memory (ST-LSTM) network with the aim to obtain an accurate model for the prediction of the steering angle in a ADS. BO was able to identify, within a limited number of trials, a model -- namely BOST-LSTM -- which resulted, on a public dataset, the most accurate when compared to classical end-to-end driving models.

Optimal sensor placement (SP) usually minimizes an impact measure, such as the amount of contaminated water or the number of inhabitants affected before detection. The common choice is to minimize the minimum detection time (MDT) averaged over a set of contamination events, with contaminant injected at a different location. Given a SP, propagation is simulated through a hydraulic software model of the network to obtain spatio-temporal concentrations and the average MDT. Searching for an optimal SP is NP-hard: even for mid-size networks, efficient search methods are required, among which evolutionary approaches are often used. A bi-objective formalization is proposed: minimizing the average MDT and its standard deviation, that is the risk to detect some contamination event too late than the average MDT. We propose a data structure (sort of spatio-temporal heatmap) collecting simulation outcomes for every SP and particularly suitable for evolutionary optimization. Indeed, the proposed data structure enabled a convergence analysis of a population-based algorithm, leading to the identification of indicators for detecting problem-specific converge issues which could be generalized to other similar problems. We used Pymoo, a recent Python framework flexible enough to incorporate our problem specific termination criterion. Results on a benchmark and a real-world network are presented., Comment: 9 pages, 15 figures

This paper addresses black-box optimization over multiple information sources whose both fidelity and query cost change over the search space, that is they are location dependent. The approach uses: (i) an Augmented Gaussian Process, recently proposed in multi-information source optimization as a single model of the objective function over search space and sources, and (ii) a Gaussian Process to model the location-dependent cost of each source. The former is used into a Confidence Bound based acquisition function to select the next source and location to query, while the latter is used to penalize the value of the acquisition depending on the expected query cost for any source-location pair. The proposed approach is evaluated on a set of Hyperparameters Optimization tasks, consisting of two Machine Learning classifiers and three datasets of different sizes.

Modelling human function learning has been the subject of in-tense research in cognitive sciences. The topic is relevant in black-box optimization where information about the objective and/or constraints is not available and must be learned through function evaluations. In this paper we focus on the relation between the behaviour of humans searching for the maximum and the probabilistic model used in Bayesian Optimization. As surrogate models of the unknown function both Gaussian Processes and Random Forest have been considered: the Bayesian learning paradigm is central in the development of active learning approaches balancing exploration/exploitation in uncertain conditions towards effective generalization in large decision spaces. In this paper we analyse experimentally how Bayesian Optimization compares to humans searching for the maximum of an unknown 2D function. A set of controlled experiments with 60 subjects, using both surrogate models, confirm that Bayesian Optimization provides a general model to represent individual patterns of active learning in humans

Sequential Model-based Bayesian Optimization has been successful-ly applied to several application domains, characterized by complex search spaces, such as Automated Machine Learning and Neural Architecture Search. This paper focuses on optimal control problems, proposing a Sequential Model-based Bayesian Optimization framework to learn optimal control strategies. A quite general formalization of the problem is provided, along with a specific instance related to optimization of pumping operations in an urban Water Distri-bution Network. Relevant results on a real-life Water Distribution Network are reported, comparing different possible choices for the proposed framework.

Bayesian Optimization has become the reference method for the global optimization of black box, expensive and possibly noisy functions. Bayesian Op-timization learns a probabilistic model about the objective function, usually a Gaussian Process, and builds, depending on its mean and variance, an acquisition function whose optimizer yields the new evaluation point, leading to update the probabilistic surrogate model. Despite its sample efficiency, Bayesian Optimiza-tion does not scale well with the dimensions of the problem. The optimization of the acquisition function has received less attention because its computational cost is usually considered negligible compared to that of the evaluation of the objec-tive function. Its efficient optimization is often inhibited, particularly in high di-mensional problems, by multiple extrema. In this paper we leverage the addition-ality of the objective function into mapping both the kernel and the acquisition function of the Bayesian Optimization in lower dimensional subspaces. This ap-proach makes more efficient the learning/updating of the probabilistic surrogate model and allows an efficient optimization of the acquisition function. Experi-mental results are presented for real-life application, that is the control of pumps in urban water distribution systems.

In this paper, the problem of safe global maximization (it should not be confused with robust optimization) of expensive noisy black-box functions satisfying the Lipschitz condition is considered. The notion "safe" means that the objective function $f(x)$ during optimization should not violate a "safety" threshold, for instance, a certain a priori given value $h$ in a maximization problem. Thus, any new function evaluation (possibly corrupted by noise) must be performed at "safe points" only, namely, at points $y$ for which it is known that the objective function $f(y) > h$. The main difficulty here consists in the fact that the used optimization algorithm should ensure that the safety constraint will be satisfied at a point $y$ before evaluation of $f(y)$ will be executed. Thus, it is required both to determine the safe region $\Omega$ within the search domain~$D$ and to find the global maximum within $\Omega$. An additional difficulty consists in the fact that these problems should be solved in the presence of the noise. This paper starts with a theoretical study of the problem and it is shown that even though the objective function $f(x)$ satisfies the Lipschitz condition, traditional Lipschitz minorants and majorants cannot be used due to the presence of the noise. Then, a $\delta$-Lipschitz framework and two algorithms using it are proposed to solve the safe global maximization problem. The first method determines the safe area within the search domain and the second one executes the global maximization over the found safe region. For both methods a number of theoretical results related to their functioning and convergence is established. Finally, numerical experiments confirming the reliability of the proposed procedures are performed., Comment: Published paper (37 pages, 44 figures, 4 tables): Yaroslav D. Sergeyev - corresponding author. Soft Computing (2020)