Alma Rahat

Alma Rahat, Tinkle Chugh, Jonathan Fieldsend et al.

Many methods for performing multi-objective optimisation of computationally expensive problems have been proposed recently. Typically, a probabilistic surrogate for each objective is constructed from an initial dataset. The surrogates can then be used to produce predictive densities in the objective space for any solution. Using the predictive densities, we can compute the expected hypervolume improvement (EHVI) due to a solution. Maximising the EHVI, we can locate the most promising solution that may be expensively evaluated next. There are closed-form expressions for computing the EHVI, integrating over the multivariate predictive densities. However, they require partitioning the objective space, which can be prohibitively expensive for more than three objectives. Furthermore, there are no closed-form expressions for a problem where the predictive densities are dependent, capturing the correlations between objectives. Monte Carlo approximation is used instead in such cases, which is not cheap. Hence, the need to develop new accurate but cheaper approximation methods remains. Here we investigate an alternative approach toward approximating the EHVI using Gauss-Hermite quadrature. We show that it can be an accurate alternative to Monte Carlo for both independent and correlated predictive densities with statistically significant rank correlations for a range of popular test problems.

Andy Gray, Alma Rahat, Tom Crick et al.

Assessment is a crucial part of education. Traditional marking is a source of inconsistencies and unconscious bias, placing a high cognitive load on the assessors. An approach to address these issues is comparative judgement (CJ). In CJ, the assessor is presented with a pair of items and is asked to select the better one. Following a series of comparisons, a rank is derived using a ranking model, for example, the BTM, based on the results. While CJ is considered a reliable method for marking, there are concerns around transparency, and the ideal number of pairwise comparisons to generate a reliable estimation of the rank order is not known. Additionally, there have been attempts to generate a method of selecting pairs that should be compared next in an informative manner, but some existing methods are known to have created their own bias within results inflating the reliability metric used. As a result, a random selection approach is usually deployed. We propose a novel Bayesian approach to CJ (BCJ) for determining the ranks of compared items alongside a new way to select the pairs to present to the marker(s) using active learning (AL), addressing the key shortcomings of traditional CJ. Furthermore, we demonstrate how the entire approach may provide transparency by providing the user insights into how it is making its decisions and, at the same time, being more efficient. Results from our experiments confirm that the proposed BCJ combined with entropy-driven AL pair-selection method is superior to other alternatives. We also find that the more comparisons done, the more accurate BCJ becomes, which solves the issue the current method has of the model deteriorating if too many comparisons are performed. As our approach can generate the complete predicted rank distribution for an item, we also show how this can be utilised in devising a predicted grade, guided by the assessor.

Xietao Wang Lin, Juan Ungredda, Max Butler et al.

Bayesian optimisation has proven to be a powerful tool for expensive global black-box optimisation problems. In this paper, we propose new Bayesian optimisation variants of the popular Knowledge Gradient acquisition functions for problems with \emph{decoupled} black-box constraints, in which subsets of the objective and constraint functions may be evaluated independently. In particular, our methods aim to take into account that often only a handful of the constraints may be binding at the optimum, and hence we should evaluate only relevant constraints when trying to optimise a function. We empirically benchmark these methods against existing methods and demonstrate their superiority over the state-of-the-art.

Sean P. Walton, Ben J. Evans, Alma A. M. Rahat et al.

As AI systems increasingly shape decision making in creative design contexts, understanding how humans engage with these tools has become a critical challenge for interactive intelligent systems research. This paper contributes a challenge to rethink how to evaluate human--AI collaborative systems, advocating for a more nuanced and multidimensional approach. Findings from one of the largest field studies to date (n = 808) of a human--AI co-creative system, The Genetic Car Designer, complemented by a controlled lab study (n = 12) are presented. The system is based on an interactive evolutionary algorithm where participants were tasked with designing a simple two dimensional representation of a car. Participants were exposed to galleries of design suggestions generated by an intelligent system, MAP--Elites, and a random control. Results indicate that exposure to galleries generated by MAP--Elites significantly enhanced both cognitive and behavioural engagement, leading to higher-quality design outcomes. Crucially for the wider community, the analysis reveals that conventional evaluation methods, which often focus on solely behavioural and design quality metrics, fail to capture the full spectrum of user engagement. By considering the human--AI design process as a changing emotional, behavioural and cognitive state of the designer, we propose evaluating human--AI systems holistically and considering intelligent systems as a core part of the user experience -- not simply a back end tool.

Andy Gray, Alma Rahat, Stephen Lindsay et al.

Ensuring transparency in educational assessment is increasingly critical, particularly post-pandemic, as demand grows for fairer and more reliable evaluation methods. Comparative Judgement (CJ) offers a promising alternative to traditional assessments, yet concerns remain about its perceived opacity. This paper examines how Bayesian Comparative Judgement (BCJ) enhances transparency by integrating prior information into the judgement process, providing a structured, data-driven approach that improves interpretability and accountability. BCJ assigns probabilities to judgement outcomes, offering quantifiable measures of uncertainty and deeper insights into decision confidence. By systematically tracking how prior data and successive judgements inform final rankings, BCJ clarifies the assessment process and helps identify assessor disagreements. Multi-criteria BCJ extends this by evaluating multiple learning outcomes (LOs) independently, preserving the richness of CJ while producing transparent, granular rankings aligned with specific assessment goals. It also enables a holistic ranking derived from individual LOs, ensuring comprehensive evaluations without compromising detailed feedback. Using a real higher education dataset with professional markers in the UK, we demonstrate BCJ's quantitative rigour and ability to clarify ranking rationales. Through qualitative analysis and discussions with experienced CJ practitioners, we explore its effectiveness in contexts where transparency is crucial, such as high-stakes national assessments. We highlight the benefits and limitations of BCJ, offering insights into its real-world application across various educational settings.

George De Ath, Tinkle Chugh, Alma A. M. Rahat

Optimisation problems often have multiple conflicting objectives that can be computationally and/or financially expensive. Mono-surrogate Bayesian optimisation (BO) is a popular model-based approach for optimising such black-box functions. It combines objective values via scalarisation and builds a Gaussian process (GP) surrogate of the scalarised values. The location which maximises a cheap-to-query acquisition function is chosen as the next location to expensively evaluate. While BO is an effective strategy, the use of GPs is limiting. Their performance decreases as the problem input dimensionality increases, and their computational complexity scales cubically with the amount of data. To address these limitations, we extend previous work on BO by density-ratio estimation (BORE) to the multi-objective setting. BORE links the computation of the probability of improvement acquisition function to that of probabilistic classification. This enables the use of state-of-the-art classifiers in a BO-like framework. In this work we present MBORE: multi-objective Bayesian optimisation by density-ratio estimation, and compare it to BO across a range of synthetic and real-world benchmarks. We find that MBORE performs as well as or better than BO on a wide variety of problems, and that it outperforms BO on high-dimensional and real-world problems.

Clym Stock-Williams, Tinkle Chugh, Alma Rahat et al.

Performing multi-objective Bayesian optimisation by scalarising the objectives avoids the computation of expensive multi-dimensional integral-based acquisition functions, instead of allowing one-dimensional standard acquisition functions\textemdash such as Expected Improvement\textemdash to be applied. Here, two infill criteria based on hypervolume improvement\textemdash one recently introduced and one novel\textemdash are compared with the multi-surrogate Expected Hypervolume Improvement. The reasons for the disparities in these methods' effectiveness in maximising the hypervolume of the acquired Pareto Front are investigated. In addition, the effect of the surrogate model mean function on exploration and exploitation is examined: careful choice of data normalisation is shown to be preferable to the exploration parameter commonly used with the Expected Improvement acquisition function. Finally, the effectiveness of all the methodological improvements defined here is demonstrated on a real-world problem: the optimisation of a wind turbine blade aerofoil for both aerodynamic performance and structural stiffness. With effective scalarisation, Bayesian optimisation finds a large number of new aerofoil shapes that strongly dominate standard designs.

Alma Rahat, Michael Wood

We are often interested in identifying the feasible subset of a decision space under multiple constraints to permit effective design exploration. If determining feasibility required computationally expensive simulations, the cost of exploration would be prohibitive. Bayesian search is data-efficient for such problems: starting from a small dataset, the central concept is to use Bayesian models of constraints with an acquisition function to locate promising solutions that may improve predictions of feasibility when the dataset is augmented. At the end of this sequential active learning approach with a limited number of expensive evaluations, the models can accurately predict the feasibility of any solution obviating the need for full simulations. In this paper, we propose a novel acquisition function that combines the probability that a solution lies at the boundary between feasible and infeasible spaces (representing exploitation) and the entropy in predictions (representing exploration). Experiments confirmed the efficacy of the proposed function.

George De Ath, Richard M. Everson, Alma A. M. Rahat et al.

The performance of acquisition functions for Bayesian optimisation to locate the global optimum of continuous functions is investigated in terms of the Pareto front between exploration and exploitation. We show that Expected Improvement (EI) and the Upper Confidence Bound (UCB) always select solutions to be expensively evaluated on the Pareto front, but Probability of Improvement is not guaranteed to do so and Weighted Expected Improvement does so only for a restricted range of weights. We introduce two novel $ε$-greedy acquisition functions. Extensive empirical evaluation of these together with random search, purely exploratory, and purely exploitative search on 10 benchmark problems in 1 to 10 dimensions shows that $ε$-greedy algorithms are generally at least as effective as conventional acquisition functions (e.g., EI and UCB), particularly with a limited budget. In higher dimensions $ε$-greedy approaches are shown to have improved performance over conventional approaches. These results are borne out on a real world computational fluid dynamics optimisation problem and a robotics active learning problem. Our analysis and experiments suggest that the most effective strategy, particularly in higher dimensions, is to be mostly greedy, occasionally selecting a random exploratory solution.

Nicholas D. Sanders, Richard M. Everson, Jonathan E. Fieldsend et al.

Many expensive black-box optimisation problems are sensitive to their inputs. In these problems it makes more sense to locate a region of good designs, than a single-possibly fragile-optimal design. Expensive black-box functions can be optimised effectively with Bayesian optimisation, where a Gaussian process is a popular choice as a prior over the expensive function. We propose a method for robust optimisation using Bayesian optimisation to find a region of design space in which the expensive function's performance is relatively insensitive to the inputs whilst retaining a good quality. This is achieved by sampling realisations from a Gaussian process that is modelling the expensive function, and evaluating the improvement for each realisation. The expectation of these improvements can be optimised cheaply with an evolutionary algorithm to determine the next location at which to evaluate the expensive function. We describe an efficient process to locate the optimum expected improvement. We show empirically that evaluating the expensive function at the location in the candidate uncertainty region about which the model is most uncertain, or at random, yield the best convergence in contrast to exploitative schemes. We illustrate our method on six test functions in two, five, and ten dimensions, and demonstrate that it is able to outperform two state-of-the-art approaches from the literature. We also demonstrate our method one two real-world problems in 4 and 8 dimensions, which involve training robot arms to push objects onto targets.