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1. Bayesian optimization with adaptive surrogate models for automated experimental design

   https://doi.org/10.1038/s41524-021-00662-x  

   Lei, Bowen; Kirk, Tanner Quinn; Bhattacharya, Anirban; Pati, Debdeep; Qian, Xiaoning; Arroyave, Raymundo; Mallick, Bani K. (December 2021, npj Computational Materials)

   Abstract Bayesian optimization (BO) is an indispensable tool to optimize objective functions that either do not have known functional forms or are expensive to evaluate. Currently, optimal experimental design is always conducted within the workflow of BO leading to more efficient exploration of the design space compared to traditional strategies. This can have a significant impact on modern scientific discovery, in particular autonomous materials discovery, which can be viewed as an optimization problem aimed at looking for the maximum (or minimum) point for the desired materials properties. The performance of BO-based experimental design depends not only on the adopted acquisition function but also on the surrogate models that help to approximate underlying objective functions. In this paper, we propose a fully autonomous experimental design framework that uses more adaptive and flexible Bayesian surrogate models in a BO procedure, namely Bayesian multivariate adaptive regression splines and Bayesian additive regression trees. They can overcome the weaknesses of widely used Gaussian process-based methods when faced with relatively high-dimensional design space or non-smooth patterns of objective functions. Both simulation studies and real-world materials science case studies demonstrate their enhanced search efficiency and robustness. 

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2. Uncertainty Quantification for Bayesian Optimization

   Tuo, R.; Wang, W. (January 2022, Proceedings of Machine Learning Research)

   Bayesian optimization is a class of global optimization techniques. In Bayesian optimization, the underlying objective function is modeled as a realization of a Gaussian process. Although the Gaussian process assumption implies a random distribution of the Bayesian optimization outputs, quantification of this uncertainty is rarely studied in the literature. In this work, we propose a novel approach to assess the output uncertainty of Bayesian optimization algorithms, which proceeds by constructing confidence regions of the maximum point (or value) of the objective function. These regions can be computed efficiently, and their confidence levels are guaranteed by the uniform error bounds for sequential Gaussian process regression newly developed in the present work. Our theory provides a unified uncertainty quantification framework for all existing sequential sampling policies and stopping criteria. 

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3. Noise-Adaptive Confidence Sets for Linear Bandits and Application to Bayesian Optimization

   Jun, Kwang-Sung; Kim, Jungtaek (July 2024, Proceedings of Machine Learning Research)

   Adapting to a priori unknown noise level is a very important but challenging problem in sequential decision-making as efficient exploration typically requires knowledge of the noise level, which is often loosely specified. We report significant progress in addressing this issue in linear bandits in two respects. First, we propose a novel confidence set that is ’semi-adaptive’ to the unknown sub-Gaussian parameter $$\sigma_*^2$$ in the sense that the (normalized) confidence width scales with $$\sqrt{d\sigma_*^2 + \sigma_0^2}$$ where $$d$$ is the dimension and $$\sigma_0^2$$ is the specified sub-Gaussian parameter (known) that can be much larger than $$\sigma_*^2$$. This is a significant improvement over $$\sqrt{d\sigma_0^2}$$ of the standard confidence set of Abbasi-Yadkori et al. (2011), especially when $$d$$ is large. We show that this leads to an improved regret bound in linear bandits. Second, for bounded rewards, we propose a novel variance-adaptive confidence set that has a much improved numerical performance upon prior art. We then apply this confidence set to develop, as we claim, the first practical variance-adaptive linear bandit algorithm via an optimistic approach, which is enabled by our novel regret analysis technique. Both of our confidence sets rely critically on ‘regret equality’ from online learning. Our empirical evaluation in Bayesian optimization tasks shows that our algorithms demonstrate better or comparable performance compared to existing methods. 

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4. Bayesian Optimization with High-Dimensional Outputs

   https://doi.org/10.48550/arXiv.2106.12997  

   Maddox, W. J.; Balandat, M.; Wilson, A. G.; Bakshy, E (January 2021, Advances in neural information processing systems)

   Bayesian optimization is a sample-efficient black-box optimization procedure that is typically applied to a small number of independent objectives. However, in practice we often wish to optimize objectives defined over many correlated outcomes (or “tasks”). For example, scientists may want to optimize the coverage of a cell tower network across a dense grid of locations. Similarly, engineers may seek to balance the performance of a robot across dozens of different environments via constrained or robust optimization. However, the Gaussian Process (GP) models typically used as probabilistic surrogates for multi-task Bayesian optimization scale poorly with the number of outcomes, greatly limiting applicability. We devise an efficient technique for exact multi-task GP sampling that combines exploiting Kronecker structure in the covariance matrices with Matheron’s identity, allowing us to perform Bayesian optimization using exact multi-task GP models with tens of thousands of correlated outputs. In doing so, we achieve substantial improvements in sample efficiency compared to existing approaches that model solely the outcome metrics. We demonstrate how this unlocks a new class of applications for Bayesian optimization across a range of tasks in science and engineering, including optimizing interference patterns of an optical interferometer with 65,000 outputs. 

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5. Robust and Tuning-Free Sparse Linear Regression via Square-Root Slope

   https://doi.org/10.1137/23M1608690  

   Minsker, Stanislav; Ndaoud, Mohamed; Wang, Lang (June 2024, SIAM Journal on Mathematics of Data Science)

   We consider the high-dimensional linear regression model and assume that a fraction of the measurements are altered by an adversary with complete knowledge of the data and the underlying distribution. We are interested in a scenario where dense additive noise is heavy-tailed while the measurement vectors follow a sub-Gaussian distribution. Within this framework, we establish minimax lower bounds for the performance of an arbitrary estimator that depend on the the fraction of corrupted observations as well as the tail behavior of the additive noise. Moreover, we design a modification of the so-called Square-Root Slope estimator with several desirable features: (a) it is provably robust to adversarial contamination, and satisfies performance guarantees in the form of sub-Gaussian deviation inequalities that match the lower error bounds, up to logarithmic factors; (b) it is fully adaptive with respect to the unknown sparsity level and the variance of the additive noise, and (c) it is computationally tractable as a solution of a convex optimization problem. To analyze performance of the proposed estimator, we prove several properties of matrices with sub-Gaussian rows that may be of independent interest. 

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