Optimization problems are ubiquitous in various fields, ranging from computer science and engineering to finance and healthcare. Whether the focus is on minimizing costs or improving efficiency, these challenges frequently involve finding the best outcome from a large pool of feasible solutions within defined constraints. Thanks to its ability to efficiently navigate this search space, Bayesian Optimization (BO) has emerged as a go-to solution to tackle these problems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], especially in cases where running the objective function is expensive or time-consuming. In the typical BO setup [11], a surrogate model - often a Gaussian Process (GP) regression - is leveraged to estimate the target response function from given input data. An acquisition function is then used to suggest strategic new test points based on the uncertainty level of this model. If selected carefully, this sampling strategy allows BO to explore the parameter space to uncover promising regions with high uncertainty or exploit known favorable regions to refine the search toward the global optimum.

Thompson sampling (TS) serves as a solution for addressing the exploitation-exploration dilemma in Bayesian optimization (BO). While it prioritizes exploration by randomly generating and maximizing sample paths of Gaussian process (GP) posteriors, TS weakly manages its exploitation by gathering information about the true objective function after each exploration is performed. In this study, we incorporate the epsilon-greedy ($\varepsilon$-greedy) policy, a well-established selection strategy in reinforcement learning, into TS to improve its exploitation. We first delineate two extremes of TS applied for BO, namely the generic TS and a sample-average TS. The former and latter promote exploration and exploitation, respectively. We then use $\varepsilon$-greedy policy to randomly switch between the two extremes. A small value of $\varepsilon \in (0,1)$ prioritizes exploitation, and vice versa. We empirically show that $\varepsilon$-greedy TS with an appropriate $\varepsilon$ is better than one of its two extremes and competes with the other.

In the field of global optimization, many existing algorithms face challenges posed by non-convex target functions and high computational complexity or unavailability of gradient information. These limitations, exacerbated by sensitivity to initial conditions, often lead to suboptimal solutions or failed convergence. This is true even for Metaheuristic algorithms designed to amalgamate different optimization techniques to improve their efficiency and robustness. To address these challenges, we develop a sequence of multidimensional integration-based methods that we show to converge to the global optima under some mild regularity conditions. Our probabilistic approach does not require the use of gradients and is underpinned by a mathematically rigorous convergence framework anchored in the nuanced properties of nascent optima distribution. In order to alleviate the problem of multidimensional integration, we develop a latent slice sampler that enjoys a geometric rate of convergence in generating samples from the nascent optima distribution, which is used to approximate the global optima. The proposed Probabilistic Global Optimizer (ProGO) provides a scalable unified framework to approximate the global optima of any continuous function defined on a domain of arbitrary dimension. Empirical illustrations of ProGO across a variety of popular non-convex test functions (having finite global optima) reveal that the proposed algorithm outperforms, by order of magnitude, many existing state-of-the-art methods, including gradient-based, zeroth-order gradient-free, and some Bayesian Optimization methods, in term regret value and speed of convergence. It is, however, to be noted that our approach may not be suitable for functions that are expensive to compute.

Bayesian optimization (BO) is widely adopted in black-box optimization problems and it relies on a surrogate model to approximate the black-box response function. With the increasing number of black-box optimization tasks solved and even more to solve, the ability to learn from multiple prior tasks to jointly pre-train a surrogate model is long-awaited to further boost optimization efficiency. In this paper, we propose a simple approach to pre-train a surrogate, which is a Gaussian process (GP) with a kernel defined on deep features learned from a Transformerbased encoder, using datasets from prior tasks with possibly heterogeneous input spaces. In addition, we provide a simple yet effective mix-up initialization strategy for input tokens corresponding to unseen input variables and therefore accelerate new tasks' convergence. Experiments on both synthetic and real benchmark problems demonstrate the effectiveness of our proposed pre-training and transfer BO strategy over existing methods. In black-box optimization problems, one could only observe outputs of the function being optimized based on some given inputs, and can hardly access the explicit form of the function. These kinds of optimization problems are ubiquitous in practice (e.g., (Mahapatra et al., 2015; Korovina et al., 2020; Griffiths & Lobato, 2020)). Among black-box optimization problems, some are particularly challenging since their function evaluations are expensive, in the sense that the evaluation either takes a substantial amount of time or requires a considerable monetary cost.

Bayesian optimization (BO) is widely used to optimize black-box functions. It works by first building a surrogate for the objective and quantifying the uncertainty in that surrogate. It then decides where to sample by maximizing an acquisition function defined by the surrogate model. Prior approaches typically use randomly generated raw samples to initialize the acquisition function maximizer. However, this strategy is ill-suited for high-dimensional BO. Given the large regions of high posterior uncertainty in high dimensions, a randomly initialized acquisition function maximizer is likely to focus on areas with high posterior uncertainty, leading to overly exploring areas that offer little gain. This paper provides the first comprehensive empirical study to reveal the importance of the initialization phase of acquisition function maximization. It proposes a better initialization approach by employing multiple heuristic optimizers to leverage the knowledge of already evaluated samples to generate initial points to be explored by an acquisition function maximizer. We evaluate our approach on widely used synthetic test functions and real-world applications. Experimental results show that our techniques, while simple, can significantly enhance the standard BO and outperforms state-of-the-art high-dimensional BO techniques by a large margin in most test cases.

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.

In this paper we consider a continuous description based on stochastic differential equations of the popular particle swarm optimization (PSO) process for solving global optimization problems and derive in the large particle limit the corresponding mean-field approximation based on Vlasov-Fokker-Planck-type equations. The disadvantage of memory effects induced by the need to store the local best position is overcome by the introduction of an additional differential equation describing the evolution of the local best. A regularization process for the global best permits to formally derive the respective mean-field description. Subsequently, in the small inertia limit, we compute the related macroscopic hydrodynamic equations that clarify the link with the recently introduced consensus based optimization (CBO) methods. Several numerical examples illustrate the mean field process, the small inertia limit and the potential of this general class of global optimization methods.

In Bayesian optimization, accounting for the importance of the output relative to the input is a crucial yet challenging exercise, as it can considerably improve the final result but often involves inaccurate and cumbersome entropy estimations. We approach the problem from the perspective of importance-sampling theory, and advocate the use of the likelihood ratio to guide the search algorithm towards regions of the input space where the objective function to be minimized assumes abnormally small values. The likelihood ratio acts as a sampling weight and can be computed at each iteration without severely deteriorating the overall efficiency of the algorithm. In particular, it can be approximated in a way that makes the approach tractable in high dimensions. The "likelihood-weighted" acquisition functions introduced in this work are found to outperform their unweighted counterparts in a number of applications.

We present the implementation of a new stochastic Kuramoto-Vicsek-type model for global optimization of nonconvex functions on the sphere. This model belongs to the class of Consensus-Based Optimization. In fact, particles move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of particle locations, weighted by the cost function according to Laplace's principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached the stochastic component vanishes. The main results of this paper are about the proof of convergence of the numerical scheme to global minimizers provided conditions of well-preparation of the initial datum. The proof combines previous results of mean-field limit with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. We present several numerical experiments, which show that the algorithm proposed in the present paper scales well with the dimension and is extremely versatile. To quantify the performances of the new approach, we show that the algorithm is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning, namely the phase retrieval problem and the robust subspace detection.

We propose a novel sparse spectrum approximation of Gaussian process (GP) tailored for Bayesian optimisation. Whilst the current sparse spectrum methods provide good approximations for regression problems, it is observed that this particular form of sparse approximations generates an overconfident GP, i.e. it predicts less variance than the original GP. Since the balance between predictive mean and the predictive variance is a key determinant in the success of Bayesian optimisation, the current sparse spectrum methods are less suitable. We derive a regularised marginal likelihood for finding the optimal frequencies in optimisation problems. The regulariser trades the accuracy in the model fitting with the targeted increase in the variance of the resultant GP. We first consider the entropy of the distribution over the maxima as the regulariser that needs to be maximised. Later we show that the Expected Improvement acquisition function can also be used as a proxy for that, thus making the optimisation less computationally expensive. Experiments show an increase in the Bayesian optimisation convergence rate over the vanilla sparse spectrum method.