Abstracts Engineering

Active Learning for Solving Expensive Optimization Problems

by Siwei Fu

Many practical engineering and computer science problems can be translated into a global otimization problem where the task is to find the best (optimal) vector 𝒙𝒙∗ that gives the smallest possible function value 𝑓𝑓 = 𝑓𝑓(𝒙𝒙∗) for some function 𝑓𝑓() of interest. Often the function 𝑓𝑓() is expensive in the sense that it cannot be easily evaluated computationally or experimentally in terms of time and/or resources required. Moreover, often the function values are not directly observed but embedded in additive experimental/numerical noise. For such problems, it is important that the required step-wise search towards the global optimum 𝒙𝒙∗ is made in such a way that the number of function evaluations is minimized. In order to systematically evaluate different optimization methods for such expensive optimization problems, one needs a flexible and developer-friendly modular toolbox written for example in Python that enables comparison of the state-of-the-art approach, known as Bayesian Optimization, with other alternatives. The primary goal of this thesis project was therefore to build a prototype of a such a developer-friendly and well-documented modular toolbox, and validate its usefulness based on standard test examples using objective functions of 1-3 variables observed in additive noise, followed by more realistic industrial problems with objective functions of 12-24 variables. The toolbox developed is centered on comparing the so called query-by-committee (QBC) methods with one of the state-of-the-art methods often referred to as Gaussian Process based Bayesian Optimization. While this state-of-the-art method uses the current training dataset 𝐷𝐷 to derive a posterior density distribution 𝑝𝑝(𝑦𝑦|𝒙𝒙, 𝐷𝐷) that reflects the current uncertainty about the unknown value 𝑓𝑓(𝒙𝒙) of the objective function, in the QBC approach instead a panel (committee) of prediction models is created and the uncertainty about 𝑓𝑓(𝒙𝒙) is quantified based on the diversity in the predictions these models produce for the input 𝒙𝒙 of interest. The toolbox developed makes it possible to compare this state-of-the-art method with QBCs that are based on standard Feedforward Neural Networks, conventional Regression Trees, or Gaussian Process regression models. The toolbox also includes a random baseline method where each new point (experiment) 𝒙𝒙 is selected randomly from the relevant search space. The toolbox also offers several additional features, for example: (i) Estimating uncertainty using a kernel density estimator as an alternative to assuming a conventional normal distribution. (ii) Particle Swarm Optimization as a tool to search for global minima of the surrogate models of the objective function created during the optimization process. (iii) Several alternative acquisition functions used to estimate the expected improvement provided by different candidate points. (iv) Ensuring extra exploration of the search space by means of an "ε-mechanism" that with probability ε makes a random but partly biased selection of a new candidate point at…