Multi-Objective Bayesian Global Optimization using expected hypervolume improvement gradient

The Expected Hypervolume Improvement (EHVI) is a frequently used infill criterion in Multi-Objective Bayesian Global Optimization (MOBGO), due to its good ability to lead the exploration. Recently, the computational complexity of EHVI calculation is reduced to O(n log n) for both 2-D and 3-D cases. However, the optimizer in MOBGO still requires a significant amount of time, because the calculation of EHVI is carried out in each iteration and usually tens of thousands of the EHVI calculations are required. This paper derives a formula for the Expected Hypervolume Improvement Gradient (EHVIG) and proposes an efficient algorithm to calculate EHVIG. The new criterion (EHVIG) is utilized by two different strategies to improve the efficiency of the optimizer discussed in this paper. Firstly, it enables gradient ascent methods to be used in MOBGO. Moreover, since the EHVIG of an optimal solution should be a zero vector, it can be regarded as a stopping criterion in global optimization, e.g., in Evolution Strategies. Empirical experiments are performed on seven benchmark problems. The experimental results show that the second proposed strategy, using EHVIG as a stopping criterion for local search, can outperform the normal MOBGO on problems where the optimal solutions are located in the interior of the search space. For the ZDT series test problems, EHVIG still can perform better when gradient projection is applied.Algorithms and the Foundations of Software technolog

Multi-objective Bayesian global optimization for continuous problems and applications

A common method to solve expensive function evaluation problem is using Bayesian Global Optimization, instead of Evolutionary Algorithms. However, the execution time of multi-objective Bayesian Global Optimization (MOBGO) itself is still too long, even though it only requires a few function evaluations. The reason for the high cost of MOBGO is two-fold: on the one hand, MOBGO requires an infill criterion to be calculated many times, but the computational complexity of an infill criterion has so far been very high. Another reason is that the optimizer, which aims at searching for an optimal solution according to the surrogate models, is not sufficiently efficient. The main contributions of this thesis consist of 1. Decreased the computational complexity of a well-known infill criteria, Expected Hypervolume Improvement, into $n log (n)$ both in 2-D and 3-D cases; 2. Proposed a new criterion, Truncated Expected Hypervolume Improvement, to make full use of a-priori knowledge of objective functions, whenever it is available; 3. Proposed another infill criterion, Expected Hypervolume Improvement Gradient, to improve the convergence of the optimizer in MOBGO. China Scholarship CouncilAlgorithms and the Foundations of Software technolog

Hypervolume-based Multi-objective Bayesian Optimization with Student-t Processes

Student-$t$ processes have recently been proposed as an appealing alternative non-parameteric function prior. They feature enhanced flexibility and predictive variance. In this work the use of Student-$t$ processes are explored for multi-objective Bayesian optimization. In particular, an analytical expression for the hypervolume-based probability of improvement is developed for independent Student-$t$ process priors of the objectives. Its effectiveness is shown on a multi-objective optimization problem which is known to be difficult with traditional Gaussian processes.Comment: 5 pages, 3 figure

Bayesian optimization in adverse scenarios

Optimization problems with expensive-to-evaluate objective functions are ubiquitous in scientific and industrial settings. Bayesian optimization has gained widespread acclaim for optimizing expensive (and often black box) functions due to its theoretical performance guarantees and empirical sample efficiency in a variety of settings. Nevertheless, many practical scenarios remain where prevailing Bayesian optimization techniques fall short. We consider four such scenarios. First, we formalize the optimization problem where the goal is to identify robust designs with respect to multiple objective functions that are subject to input noise. Such robust design problems frequently arise, for example, in manufacturing settings where fabrication can only be performed with limited precision. We propose a method that identifies a set of optimal robust designs, where each design provides probabilistic guarantees jointly on multiple objectives. Second, we consider sample-efficient high-dimensional multi-objective optimization. This line of research is motivated by the challenging task of designing optical displays for augmented reality to optimize visual quality and efficiency, where the designs are specified by high-dimensional parameterizations governing complex geometries. Our proposed trust-region based algorithm yields order-of-magnitude improvements in sample complexity on this problem. Third, we consider multi-objective optimization of expensive functions with variable-cost, decoupled, and/or multi-fidelity evaluations and propose a Bayes-optimal, non-myopic acquisition function, which significantly improves sample efficiency in scenarios with incomplete information. We apply this to hardware-aware neural architecture search where the objective, on-device latency and model accuracy, can often be evaluated independently. Fourth, we consider the setting where the search space consists of discrete (and potentially continuous) parameters. We propose a theoretically grounded technique that uses a probabilistic reparameterization to transform the discrete or mixed inner optimization problem into a continuous one leading to more effective Bayesian optimization policies. Together, this thesis provides a playbook for Bayesian optimization in several practical adverse scenarios

Closed-loop automatic gradient design for liquid chromatography using Bayesian optimization

Contemporary complex samples require sophisticated methods for full analysis. This work describes the development of a Bayesian optimization algorithm for automated and unsupervised development of gradient programs. The algorithm was tailored to LC using a Gaussian process model with a novel covariance kernel. To facilitate unsupervised learning, the algorithm was designed to interface directly with the chromatographic system. Single-objective and multi-objective Bayesian optimization strategies were investigated for the separation of two complex (n>18, and n>80) dye mixtures. Both approaches found satisfactory optima in under 35 measurements. The multi-objective strategy was found to be powerful and flexible in terms of exploring the Pareto front. The performance difference between the single-objective and multi-objective strategy was further investigated using a retention modeling example. One additional advantage of the multi-objective approach was that it allows for a trade-off to be made between multiple objectives without prior knowledge. In general, the Bayesian optimization strategy was found to be particularly suitable, but not limited to, cases where retention modelling is not possible, although its scalability might be limited in terms of the number of parameters that can be simultaneously optimized