Robert M. Lee

Alexander Thebelt, Calvin Tsay, Robert M. Lee et al.

Tree ensembles can be well-suited for black-box optimization tasks such as algorithm tuning and neural architecture search, as they achieve good predictive performance with little or no manual tuning, naturally handle discrete feature spaces, and are relatively insensitive to outliers in the training data. Two well-known challenges in using tree ensembles for black-box optimization are (i) effectively quantifying model uncertainty for exploration and (ii) optimizing over the piece-wise constant acquisition function. To address both points simultaneously, we propose using the kernel interpretation of tree ensembles as a Gaussian Process prior to obtain model variance estimates, and we develop a compatible optimization formulation for the acquisition function. The latter further allows us to seamlessly integrate known constraints to improve sampling efficiency by considering domain-knowledge in engineering settings and modeling search space symmetries, e.g., hierarchical relationships in neural architecture search. Our framework performs as well as state-of-the-art methods for unconstrained black-box optimization over continuous/discrete features and outperforms competing methods for problems combining mixed-variable feature spaces and known input constraints.

Johannes P. Dürholt, Thomas S. Asche, Johanna Kleinekorte et al.

Our open-source Python package BoFire combines Bayesian Optimization (BO) with other design of experiments (DoE) strategies focusing on developing and optimizing new chemistry. Previous BO implementations, for example as they exist in the literature or software, require substantial adaptation for effective real-world deployment in chemical industry. BoFire provides a rich feature-set with extensive configurability and realizes our vision of fast-tracking research contributions into industrial use via maintainable open-source software. Owing to quality-of-life features like JSON-serializability of problem formulations, BoFire enables seamless integration of BO into RESTful APIs, a common architecture component for both self-driving laboratories and human-in-the-loop setups. This paper discusses the differences between BoFire and other BO implementations and outlines ways that BO research needs to be adapted for real-world use in a chemistry setting.

Becky Langdon, Gabriel D. Patrón, Chrysoula D. Kappatou et al.

The optimisation of fed-batch (bio)chemical process recipes is subject to inherent, underlying, and unmeasurable fluctuations across batches, whose trajectories are difficult to model and costly to measure. Bayesian Optimisation (BayesOpt) is a powerful tool for sampling and optimisation of expensive-to-measure functions. Gaussian Processes (GPs), the surrogate models used in BayesOpt, are static, forecast poorly, and lack generalisation across experiments, limiting their applicability to time-varying batch processes with stochastic parameters, i.e., process fluctuations. This work investigates System-Aware Neural ODE Processes (SANODEP) as a meta-learning model to overcome the limitations of GPs and increase few-shot optimisation performance in BayesOpt. Using a penicillin batch production case study, we find that SANODEP outperforms GP-based BayesOpt in the low-data regime, resulting in improved objectives when few experimental runs are performed. These improvements are observed in both on- and off-distribution batches, highlighting the generalisation capabilities of SANODEP. Using this approach, batch process operators can accelerate the initial optimisation steps in BayesOpt by deploying meta-learning or optimise the process with fewer experiments when the experimental cost is high.

Ben Tu, Nikolas Kantas, Robert M. Lee et al.

The goal of multi-objective optimisation is to identify the Pareto front surface which is the set obtained by connecting the best trade-off points. Typically this surface is computed by evaluating the objectives at different points and then interpolating between the subset of the best evaluated trade-off points. In this work, we propose to parameterise the Pareto front surface using polar coordinates. More precisely, we show that any Pareto front surface can be equivalently represented using a scalar-valued length function which returns the projected length along any positive radial direction. We then use this representation in order to rigorously develop the theory and applications of stochastic Pareto front surfaces. In particular, we derive many Pareto front surface statistics of interest such as the expectation, covariance and quantiles. We then discuss how these can be used in practice within a design of experiments setting, where the goal is to both infer and use the Pareto front surface distribution in order to make effective decisions. Our framework allows for clear uncertainty quantification and we also develop advanced visualisation techniques for this purpose. Finally we discuss the applicability of our ideas within multivariate extreme value theory and illustrate our methodology in a variety of numerical examples, including a case study with a real-world air pollution data set.

Ben Tu, Nikolas Kantas, Robert M. Lee et al.

Robust optimisation is a well-established framework for optimising functions in the presence of uncertainty. The inherent goal of this problem is to identify a collection of inputs whose outputs are both desirable for the decision maker, whilst also being robust to the underlying uncertainties in the problem. In this work, we study the multi-objective case of this problem. We identify that the majority of all robust multi-objective algorithms rely on two key operations: robustification and scalarisation. Robustification refers to the strategy that is used to account for the uncertainty in the problem. Scalarisation refers to the procedure that is used to encode the relative importance of each objective to a scalar-valued reward. As these operations are not necessarily commutative, the order that they are performed in has an impact on the resulting solutions that are identified and the final decisions that are made. The purpose of this work is to give a thorough exposition on the effects of these different orderings and in particular highlight when one should opt for one ordering over the other. As part of our analysis, we showcase how many existing risk concepts can be integrated into the specification and solution of a robust multi-objective optimisation problem. Besides this, we also demonstrate how one can principally define the notion of a robust Pareto front and a robust performance metric based on our ``robustify and scalarise'' methodology. To illustrate the efficacy of these new ideas, we present two insightful case studies which are based on real-world data sets.

Ben Tu, Nikolas Kantas, Robert M. Lee et al.

The goal of multi-objective optimisation is to identify a collection of points which describe the best possible trade-offs between the multiple objectives. In order to solve this vector-valued optimisation problem, practitioners often appeal to the use of scalarisation functions in order to transform the multi-objective problem into a collection of single-objective problems. This set of scalarised problems can then be solved using traditional single-objective optimisation techniques. In this work, we formalise this convention into a general mathematical framework. We show how this strategy effectively recasts the original multi-objective optimisation problem into a single-objective optimisation problem defined over sets. An appropriate class of objective functions for this new problem are the R2 utilities, which are utility functions that are defined as a weighted integral over the scalarised optimisation problems. As part of our work, we show that these utilities are monotone and submodular set functions which can be optimised effectively using greedy optimisation algorithms. We then analyse the performance of these greedy algorithms both theoretically and empirically. Our analysis largely focusses on Bayesian optimisation, which is a popular probabilistic framework for black-box optimisation.

Alexander Thebelt, Calvin Tsay, Robert M. Lee et al.

Energy systems optimization problems are complex due to strongly non-linear system behavior and multiple competing objectives, e.g. economic gain vs. environmental impact. Moreover, a large number of input variables and different variable types, e.g. continuous and categorical, are challenges commonly present in real-world applications. In some cases, proposed optimal solutions need to obey explicit input constraints related to physical properties or safety-critical operating conditions. This paper proposes a novel data-driven strategy using tree ensembles for constrained multi-objective optimization of black-box problems with heterogeneous variable spaces for which underlying system dynamics are either too complex to model or unknown. In an extensive case study comprised of synthetic benchmarks and relevant energy applications we demonstrate the competitive performance and sampling efficiency of the proposed algorithm compared to other state-of-the-art tools, making it a useful all-in-one solution for real-world applications with limited evaluation budgets.

Alexander Thebelt, Jan Kronqvist, Miten Mistry et al.

Gradient boosted trees and other regression tree models perform well in a wide range of real-world, industrial applications. These tree models (i) offer insight into important prediction features, (ii) effectively manage sparse data, and (iii) have excellent prediction capabilities. Despite their advantages, they are generally unpopular for decision-making tasks and black-box optimization, which is due to their difficult-to optimize structure and the lack of a reliable uncertainty measure. ENTMOOT is our new framework for integrating (already trained) tree models into larger optimization problems. The contributions of ENTMOOT include: (i) explicitly introducing a reliable uncertainty measure that is compatible with tree models, (ii) solving the larger optimization problems that incorporate these uncertainty aware tree models, (iii) proving that the solutions are globally optimal, i.e. no better solution exists. In particular, we show how the ENTMOOT approach allows a simple integration of tree models into decision-making and black-box optimization, where it proves as a strong competitor to commonly-used frameworks.

Miten Mistry, Dimitrios Letsios, Gerhard Krennrich et al.

Decision trees usefully represent sparse, high dimensional and noisy data. Having learned a function from this data, we may want to thereafter integrate the function into a larger decision-making problem, e.g., for picking the best chemical process catalyst. We study a large-scale, industrially-relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pre-trained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models, or they may wish to optimize a discrete model that particularly well-represents a data set. We develop several heuristic methods to find feasible solutions, and an exact, branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on concrete mixture design instance and a chemical catalysis industrial instance.