InthecaseofSEkernel, the regularity assumption implies the existence ofall weak derivativesoff. The third equation follows from the definition of Zn(x). The first inequality holds by Assumption 2. We utilize Proposition 1 to conclude thatkZn(x)k2 σ Thesecond inequality holds bydefinition oflight-tailed distributions. Notice thatthecareful choice ofτ andθ ensures θζi(x) h0, which will be validated next. The seventh line is obtained by replacing the valueof θ.

This paper studies optimization on networks modeled as metric graphs. Motivated by applications where the objective function is expensive to evaluate or only available as a black box, we develop Bayesian optimization algorithms that sequentially update a Gaussian process surrogate model of the objective to guide the acquisition of query points. To ensure that the surrogates are tailored to the network's geometry, we adopt Whittle-Matérn Gaussian process prior models defined via stochastic partial differential equations on metric graphs. In addition to establishing regret bounds for optimizing sufficiently smooth objective functions, we analyze the practical case in which the smoothness of the objective is unknown and the Whittle-Matérn prior is represented using finite elements. Numerical results demonstrate the effectiveness of our algorithms for optimizing benchmark objective functions on a synthetic metric graph and for Bayesian inversion via maximum a posteriori estimation on a telecommunication network.

The issue of failed weaning is a critical concern in the intensive care unit (ICU) setting. This scenario occurs when a patient experiences difficulty maintaining spontaneous breathing and ensuring a patent airway within the first 48 hours after the withdrawal of mechanical ventilation. Approximately 20 of ICU patients experience this phenomenon, which has severe repercussions on their health. It also has a substantial impact on clinical evolution and mortality, which can increase by 25 to 50. To address this issue, we propose a medical support system that uses a convolutional neural network (CNN) to assess a patients suitability for disconnection from a mechanical ventilator after a spontaneous breathing test (SBT). During SBT, respiratory flow and electrocardiographic activity were recorded and after processed using time-frequency analysis (TFA) techniques. Two CNN architectures were evaluated in this study: one based on ResNet50, with parameters tuned using a Bayesian optimization algorithm, and another CNN designed from scratch, with its structure also adapted using a Bayesian optimization algorithm. The WEANDB database was used to train and evaluate both models. The results showed remarkable performance, with an average accuracy 98 when using CNN from scratch. This model has significant implications for the ICU because it provides a reliable tool to enhance patient care by assisting clinicians in making timely and accurate decisions regarding weaning. This can potentially reduce the adverse outcomes associated with failed weaning events.

We consider sequential optimization of an unknown function in a reproducing kernel Hilbert space. We propose a Gaussian process-based algorithm and establish its order-optimal regret performance (up to a poly-logarithmic factor). This is the first GP-based algorithm with an order-optimal regret guarantee. The proposed algorithm is rooted in the methodology of domain shrinking realized through a sequence of tree-based region pruning and refining to concentrate queries in increasingly smaller high-performing regions of the function domain. The search for high-performing regions is localized and guided by an iterative estimation of the optimal function value to ensure both learning efficiency and computational efficiency.

Extending Bayesian optimization to batch evaluation can enable the designer to make the most use of parallel computing technology. Most of current batch approaches use artificial functions to simulate the sequential Bayesian optimization algorithm's behavior to select a batch of points for parallel evaluation. However, as the batch size grows, the accumulated error introduced by these artificial functions increases rapidly, which dramatically decreases the optimization efficiency of the algorithm. In this work, we propose a simple and efficient approach to extend Bayesian optimization to batch evaluation. Different from existing batch approaches, the idea of the new approach is to draw a batch of subspaces of the original problem and select one acquisition point from each subspace. To achieve this, we propose the expected subspace improvement criterion to measure the amount of the improvement that a candidate point can achieve within a certain subspace. By optimizing these expected subspace improvement functions simultaneously, we can get a batch of query points for expensive evaluation. Numerical experiments show that our proposed approach can achieve near-linear speedup when compared with the sequential Bayesian optimization algorithm, and performs very competitively when compared with eight state-of-the-art batch algorithms. This work provides a simple yet efficient approach for batch Bayesian optimization. A Matlab implementation of our approach is available at https://github.com/zhandawei/Expected_Subspace_Improvement_Batch_Bayesian_Optimization

We introduce new algorithms rooted in scattered data approximation that rely on a random exploration step to ensure that the fill-distance of query points decays at a near-optimal rate. Our algorithms retain the ease of implementation of the classical GP-UCB algorithm and satisfy cumulative regret bounds that nearly match those conjectured in [Vak22], hence solving a COLT open problem. Furthermore, the new algorithms outperform GP-UCB and other popular Bayesian optimization strategies in several examples.

Bayesian optimization has become a powerful tool for safe online optimization of systems, due to its high sample efficiency and noise robustness. For further speed-up reduced physical models of the system can be incorporated into the optimization to accelerate the process, since the models are able to offer an approximation of the actual system, and sampling from them is significantly cheaper. The similarity between model and reality is represented by additional hyperparameters and learned within the optimization process. Safety is an important criteria for online optimization methods like Bayesian optimization, which has been addressed by recent literature, which provide safety guarantees under the assumption of known hyperparameters. However, in practice this is not applicable. Therefore, we extend the robust Gaussian process uniform error bounds to meet the multi-task setting, which involves the calculation of a confidence region from the hyperparameter posterior distribution utilizing Markov chain Monte Carlo methods. Then, using the robust safety bounds, Bayesian optimization is applied to safely optimize the system while incorporating measurements of the models. Simulations show that the optimization can be significantly accelerated compared to other state-of-the-art safe Bayesian optimization methods depending on the fidelity of the models.

Bayesian optimization provides an effective method to optimize expensive-to-evaluate black box functions. It has been widely applied to problems in many fields, including notably in computer science, e.g. in machine learning to optimize hyperparameters of neural networks, and in engineering, e.g. in fluid dynamics to optimize control strategies that maximize drag reduction. This paper empirically studies and compares the performance and the robustness of common Bayesian optimization algorithms on a range of synthetic test functions to provide general guidance on the design of Bayesian optimization algorithms for specific problems. It investigates the choice of acquisition function, the effect of different numbers of training samples, the exact and Monte Carlo based calculation of acquisition functions, and both single-point and multi-point optimization. The test functions considered cover a wide selection of challenges and therefore serve as an ideal test bed to understand the performance of Bayesian optimization to specific challenges, and in general. To illustrate how these findings can be used to inform a Bayesian optimization setup tailored to a specific problem, two simulations in the area of computational fluid dynamics are optimized, giving evidence that suitable solutions can be found in a small number of evaluations of the objective function for complex, real problems. The results of our investigation can similarly be applied to other areas, such as machine learning and physical experiments, where objective functions are expensive to evaluate and their mathematical expressions are unknown.

Consider the sequential optimization of a continuous, possibly non-convex, and expensive to evaluate objective function $f$. The problem can be cast as a Gaussian Process (GP) bandit where $f$ lives in a reproducing kernel Hilbert space (RKHS). The state of the art analysis of several learning algorithms shows a significant gap between the lower and upper bounds on the simple regret performance. When $N$ is the number of exploration trials and $\gamma_N$ is the maximal information gain, we prove an $\tilde{\mathcal{O}}(\sqrt{\gamma_N/N})$ bound on the simple regret performance of a pure exploration algorithm that is significantly tighter than the existing bounds. We show that this bound is order optimal up to logarithmic factors for the cases where a lower bound on regret is known. To establish these results, we prove novel and sharp confidence intervals for GP models applicable to RKHS elements which may be of broader interest.