Distributional Regression for Data Analysis

Flexible modeling of how an entire distribution changes with covariates is an important yet challenging generalization of mean-based regression that has seen growing interest over the past decades in both the statistics and machine learning literature. This review outlines selected state-of-the-art statistical approaches to distributional regression, complemented with alternatives from machine learning. Topics covered include the similarities and differences between these approaches, extensions, properties and limitations, estimation procedures, and the availability of software. In view of the increasing complexity and availability of large-scale data, this review also discusses the scalability of traditional estimation methods, current trends, and open challenges. Illustrations are provided using data on childhood malnutrition in Nigeria and Australian electricity prices.Comment: Accepted for publication in Annual Review of Statistics and its Applicatio

Cover Tree Bayesian Reinforcement Learning

This paper proposes an online tree-based Bayesian approach for reinforcement learning. For inference, we employ a generalised context tree model. This defines a distribution on multivariate Gaussian piecewise-linear models, which can be updated in closed form. The tree structure itself is constructed using the cover tree method, which remains efficient in high dimensional spaces. We combine the model with Thompson sampling and approximate dynamic programming to obtain effective exploration policies in unknown environments. The flexibility and computational simplicity of the model render it suitable for many reinforcement learning problems in continuous state spaces. We demonstrate this in an experimental comparison with least squares policy iteration

Hybrid Models for Mixed Variables in Bayesian Optimization

This paper presents a new type of hybrid models for Bayesian optimization (BO) adept at managing mixed variables, encompassing both quantitative (continuous and integer) and qualitative (categorical) types. Our proposed new hybrid models merge Monte Carlo Tree Search structure (MCTS) for categorical variables with Gaussian Processes (GP) for continuous ones. Addressing efficiency in searching phase, we juxtapose the original (frequentist) upper confidence bound tree search (UCTS) and the Bayesian Dirichlet search strategies, showcasing the tree architecture's integration into Bayesian optimization. Central to our innovation in surrogate modeling phase is online kernel selection for mixed-variable BO. Our innovations, including dynamic kernel selection, unique UCTS (hybridM) and Bayesian update strategies (hybridD), position our hybrid models as an advancement in mixed-variable surrogate models. Numerical experiments underscore the hybrid models' superiority, highlighting their potential in Bayesian optimization.Comment: 32 pages, 8 Figure

Are Random Decompositions all we need in High Dimensional Bayesian Optimisation?

Learning decompositions of expensive-to-evaluate black-box functions promises to scale Bayesian optimisation (BO) to high-dimensional problems. However, the success of these techniques depends on finding proper decompositions that accurately represent the black-box. While previous works learn those decompositions based on data, we investigate data-independent decomposition sampling rules in this paper. We find that data-driven learners of decompositions can be easily misled towards local decompositions that do not hold globally across the search space. Then, we formally show that a random tree-based decomposition sampler exhibits favourable theoretical guarantees that effectively trade off maximal information gain and functional mismatch between the actual black-box and its surrogate as provided by the decomposition. Those results motivate the development of the random decomposition upper-confidence bound algorithm (RDUCB) that is straightforward to implement - (almost) plug-and-play - and, surprisingly, yields significant empirical gains compared to the previous state-of-the-art on a comprehensive set of benchmarks. We also confirm the plug-and-play nature of our modelling component by integrating our method with HEBO, showing improved practical gains in the highest dimensional tasks from Bayesmark

High-dimensional Bayesian optimization with intrinsically low-dimensional response surfaces

Bayesian optimization is a powerful technique for the optimization of expensive black-box functions. It is used in a wide range of applications such as in drug and material design and training of machine learning models, e.g. large deep networks. We propose to extend this approach to high-dimensional settings, that is where the number of parameters to be optimized exceeds 10--20. In this thesis, we scale Bayesian optimization by exploiting different types of projections and the intrinsic low-dimensionality assumption of the objective function. We reformulate the problem in a low-dimensional subspace and learn a response surface and maximize an acquisition function in this low-dimensional projection. Contributions include i) a probabilistic model for axis-aligned projections, such as the quantile-Gaussian process and ii) a probabilistic model for learning a feature space by means of manifold Gaussian processes. In the latter contribution, we propose to learn a low-dimensional feature space jointly with (a) the response surface and (b) a reconstruction mapping. Finally, we present empirical results against well-known baselines in high-dimensional Bayesian optimization and provide possible directions for future research in this field.Open Acces