We consider Bayesian optimization using Gaussian Process models, also referred to as kernel-based bandit optimization. We study the methodology of exploring the domain using random samples drawn from a distribution. We show that this random exploration approach achieves the optimal error rates. Our analysis is based on novel concentration bounds in an infinite dimensional Hilbert space established in this work, which may be of independent interest. We further develop an algorithm based on random exploration with domain shrinking and establish its order-optimal regret guarantees under both noise-free and noisy settings. In the noise-free setting, our analysis closes the existing gap in regret performance and thereby resolves a COLT open problem. The proposed algorithm also enjoys a computational advantage over prevailing methods due to the random exploration that obviates the expensive optimization of a non-convex acquisition function for choosing the query points at each iteration.
The power system of the future will be governed by complex interactions and non-linear phenomena at small time-scales, that should be studied more and more through computationally expensive software simulations. To solve the abovementioned problems, power system engineers face problems with following characteristics: (i) a computationally expensive simulator, (ii) non-linear functions to optimize and (iii) lack of abundance of data. Existing optimization settings involving EMT-type simulations have been developed, but mainly use a deterministic model and optimizer, which may be computationally inefficient and do not guarantee finding a global optimum. Furthermore, the main focus has been on optimization routines, and less attention has been paid to other tasks such as classification. In this paper, an automation framework based on Bayesian Optimization is introduced, and applied to two case studies involving optimization and classification. It is found that the framework has the potential to reduce computational effort, outperform deterministic optimizers and is applicable to a multitude of problems. Nevertheless, it was found that the output of the Bayesian Optimization depends on the number of samples used for initialization, and in addition, careful selection of surrogate models, which should be subject to future investigation.
The increasing global demand for clean and environmentally friendly energy resources has caused increased interest in harnessing solar power through photovoltaic (PV) systems for smart grids and homes. However, the inherent unpredictability of PV generation poses problems associated with smart grid planning and management, energy trading and market participation, demand response, reliability, etc. Therefore, solar irradiance forecasting is essential for optimizing PV system utilization. This study proposes the next-generation machine learning algorithms such as random forests, Extreme Gradient Boosting (XGBoost), Light Gradient Boosted Machine (lightGBM) ensemble, CatBoost, and Multilayer Perceptron Artificial Neural Networks (MLP-ANNs) to forecast solar irradiance. Besides, Bayesian optimization is applied to hyperparameter tuning. Unlike tree-based ensemble algorithms that select the features intrinsically, MLP-ANN needs feature selection as a separate step. The simulation results indicate that the performance of the MLP-ANNs improves when feature selection is applied. Besides, the random forest outperforms the other learning algorithms.
In a previous work we introduced, in the context of gravitational wave science, an initial study on an automated domain-decomposition approach for reduced basis through hp-greedy refinement. The approach constructs local reduced bases of lower dimensionality than global ones, with the same or higher accuracy. These ``light'' local bases should imply both faster evaluations when predicting new waveforms and faster data analysis, in particular faster statistical inference (the forward and inverse problems, respectively). In this approach, however, we have previously found important dependence on several hyperparameters, which do not appear in global reduced basis. This naturally leads to the problem of hyperparameter optimization (HPO), which is the subject of this paper. We tackle the problem through a Bayesian optimization, and show its superiority when compared to grid or random searches. We find that for gravitational waves from the collision of two spinning but non-precessing black holes, for the same accuracy, local hp-greedy reduced bases with HPO have a lower dimensionality of up to $4 \times$ for the cases here studied, depending on the desired accuracy. This factor should directly translate in a parameter estimation speedup, for instance. Such acceleration might help in the near real-time requirements for electromagnetic counterparts of gravitational waves from compact binary coalescences. In addition, we find that the Bayesian approach used in this paper for HPO is two orders of magnitude faster than, for example, a grid search, with about a $100 \times$ acceleration. The code developed for this project is available as open source from public repositories.
Dynamic environments pose great challenges for expensive optimization problems, as the objective functions of these problems change over time and thus require remarkable computational resources to track the optimal solutions. Although data-driven evolutionary optimization and Bayesian optimization (BO) approaches have shown promise in solving expensive optimization problems in static environments, the attempts to develop such approaches in dynamic environments remain rarely unexplored. In this paper, we propose a simple yet effective meta-learning-based optimization framework for solving expensive dynamic optimization problems. This framework is flexible, allowing any off-the-shelf continuously differentiable surrogate model to be used in a plug-in manner, either in data-driven evolutionary optimization or BO approaches. In particular, the framework consists of two unique components: 1) the meta-learning component, in which a gradient-based meta-learning approach is adopted to learn experience (effective model parameters) across different dynamics along the optimization process. 2) the adaptation component, where the learned experience (model parameters) is used as the initial parameters for fast adaptation in the dynamic environment based on few shot samples. By doing so, the optimization process is able to quickly initiate the search in a new environment within a strictly restricted computational budget. Experiments demonstrate the effectiveness of the proposed algorithm framework compared to several state-of-the-art algorithms on common benchmark test problems under different dynamic characteristics.
Bayesian optimization (BO) is widely used for black-box optimization problems, and have been shown to perform well in various real-world tasks. However, most of the existing BO methods aim to learn the optimal solution, which may become infeasible when the parameter space is extremely large or the problem is time-sensitive. In these contexts, switching to a satisficing solution that requires less information can result in better performance. In this work, we focus on time-sensitive black-box optimization problems and propose satisficing Thompson sampling-based parallel Bayesian optimization (STS-PBO) approaches, including synchronous and asynchronous versions. We shift the target from an optimal solution to a satisficing solution that is easier to learn. The rate-distortion theory is introduced to construct a loss function that balances the amount of information that needs to be learned with sub-optimality, and the Blahut-Arimoto algorithm is adopted to compute the target solution that reaches the minimum information rate under the distortion limit at each step. Both discounted and undiscounted Bayesian cumulative regret bounds are theoretically derived for the proposed STS-PBO approaches. The effectiveness of the proposed methods is demonstrated on a fast-charging design problem of Lithium-ion batteries. The results are accordant with theoretical analyses, and show that our STS-PBO methods outperform both sequential counterparts and parallel BO with traditional Thompson sampling in both synchronous and asynchronous settings.
Robots need to estimate the material and dynamic properties of objects from observations in order to simulate them accurately. We present a Bayesian optimization approach to identifying the material property parameters of objects based on a set of observations. Our focus is on estimating these properties based on observations of scenes with different sets of interacting objects. We propose an approach that exploits the structure of the reward function by modeling the reward for each observation separately and using only the parameters of the objects in that scene as inputs. The resulting lower-dimensional models generalize better over the parameter space, which in turn results in a faster optimization. To speed up the optimization process further, and reduce the number of simulation runs needed to find good parameter values, we also propose partial evaluations of the reward function, wherein the selected parameters are only evaluated on a subset of real world evaluations. The approach was successfully evaluated on a set of scenes with a wide range of object interactions, and we showed that our method can effectively perform incremental learning without resetting the rewards of the gathered observations.
Bayesian Optimization is a popular approach for optimizing expensive black-box functions. Its key idea is to use a surrogate model to approximate the objective and, importantly, quantify the associated uncertainty that allows a sequential search of query points that balance exploitation-exploration. Gaussian process (GP) has been a primary candidate for the surrogate model, thanks to its Bayesian-principled uncertainty quantification power and modeling flexibility. However, its challenges have also spurred an array of alternatives whose convergence properties could be more opaque. Motivated by these, we study in this paper an axiomatic framework that elicits the minimal requirements to guarantee black-box optimization convergence that could apply beyond GP-based methods. Moreover, we leverage the design freedom in our framework, which we call Pseudo-Bayesian Optimization, to construct empirically superior algorithms. In particular, we show how using simple local regression, and a suitable "randomized prior" construction to quantify uncertainty, not only guarantees convergence but also consistently outperforms state-of-the-art benchmarks in examples ranging from high-dimensional synthetic experiments to realistic hyperparameter tuning and robotic applications.
The discovery of therapeutic molecules is fundamentally a multi-objective optimization problem. One formulation of the problem is to identify molecules that simultaneously exhibit strong binding affinity for a target protein, minimal off-target interactions, and suitable pharmacokinetic properties. Inspired by prior work that uses active learning to accelerate the identification of strong binders, we implement multi-objective Bayesian optimization to reduce the computational cost of multi-property virtual screening and apply it to the identification of ligands predicted to be selective based on docking scores to on- and off-targets. We demonstrate the superiority of Pareto optimization over scalarization across three case studies. Further, we use the developed optimization tool to search a virtual library of over 4M molecules for those predicted to be selective dual inhibitors of EGFR and IGF1R, acquiring 100% of the molecules that form the library's Pareto front after exploring only 8% of the library. This workflow and associated open source software can reduce the screening burden of molecular design projects and is complementary to research aiming to improve the accuracy of binding predictions and other molecular properties.
Optimizing objectives under constraints, where both the objectives and constraints are black box functions, is a common scenario in real-world applications such as scientific experimental design, design of medical therapies, and industrial process optimization. One popular approach to handling these complex scenarios is Bayesian Optimization (BO). In terms of theoretical behavior, BO is relatively well understood in the unconstrained setting, where its principles have been well explored and validated. However, when it comes to constrained Bayesian optimization (CBO), the existing framework often relies on heuristics or approximations without the same level of theoretical guarantees. In this paper, we delve into the theoretical and practical aspects of constrained Bayesian optimization, where the objective and constraints can be independently evaluated and are subject to noise. By recognizing that both the objective and constraints can help identify high-confidence regions of interest (ROI), we propose an efficient CBO framework that intersects the ROIs identified from each aspect to determine the general ROI. The ROI, coupled with a novel acquisition function that adaptively balances the optimization of the objective and the identification of feasible regions, enables us to derive rigorous theoretical justifications for its performance. We showcase the efficiency and robustness of our proposed CBO framework through empirical evidence and discuss the fundamental challenge of deriving practical regret bounds for CBO algorithms.
Ultrasound is a commonly used medical imaging modality that requires expert sonographers to manually maneuver the ultrasound probe based on the acquired image. Autonomous Robotic Ultrasound (A-RUS) is an appealing alternative to this manual procedure in order to reduce sonographers' workload. The key challenge to A-RUS is optimizing the ultrasound image quality for the region of interest across different patients. This requires knowledge of anatomy, recognition of error sources and precise probe position, orientation and pressure. Sample efficiency is important while optimizing these parameters associated with the robotized probe controller. Bayesian Optimization (BO), a sample-efficient optimization framework, has recently been applied to optimize the 2D motion of the probe. Nevertheless, further improvements are needed to improve the sample efficiency for high-dimensional control of the probe. We aim to overcome this problem by using a neural network to learn a low-dimensional kernel in BO, termed as Deep Kernel (DK). The neural network of DK is trained using probe and image data acquired during the procedure. The two image quality estimators are proposed that use a deep convolution neural network and provide real-time feedback to the BO. We validated our framework using these two feedback functions on three urinary bladder phantoms. We obtained over 50% increase in sample efficiency for 6D control of the robotized probe. Furthermore, our results indicate that this performance enhancement in BO is independent of the specific training dataset, demonstrating inter-patient adaptability.
Complex system design problems, such as those involved in aerospace engineering, require the use of numerically costly simulation codes in order to predict the performance of the system to be designed. In this context, these codes are often embedded into an optimization process to provide the best design while satisfying the design constraints. Recently, new approaches, called Quality-Diversity, have been proposed in order to enhance the exploration of the design space and to provide a set of optimal diversified solutions with respect to some feature functions. These functions are interesting to assess trade-offs. Furthermore, complex design problems often involve mixed continuous, discrete, and categorical design variables allowing to take into account technological choices in the optimization problem. Existing Bayesian Quality-Diversity approaches suited for intensive high-fidelity simulations are not adapted to mixed variables constrained optimization problems. In order to overcome these limitations, a new Quality-Diversity methodology based on mixed variables Bayesian optimization strategy is proposed in the context of limited simulation budget. Using adapted covariance models and dedicated enrichment strategy for the Gaussian processes in Bayesian optimization, this approach allows to reduce the computational cost up to two orders of magnitude, with respect to classical Quality-Diversity approaches while dealing with discrete choices and the presence of constraints. The performance of the proposed method is assessed on a benchmark of analytical problems as well as on two aerospace system design problems highlighting its efficiency in terms of speed of convergence. The proposed approach provides valuable trade-offs for decision-markers for complex system design.
Sequential maximization of expected improvement (EI) is one of the most widely used policies in Bayesian optimization because of its simplicity and ability to handle noisy observations. In particular, the improvement function often uses the best posterior mean as the best incumbent in noisy settings. However, the uncertainty associated with the incumbent solution is often neglected in many analytic EI-type methods: a closed-form acquisition function is derived in the noise-free setting, but then applied to the setting with noisy observations. To address this limitation, we propose a modification of EI that corrects its closed-form expression by incorporating the covariance information provided by the Gaussian Process (GP) model. This acquisition function specializes to the classical noise-free result, and we argue should replace that formula in Bayesian optimization software packages, tutorials, and textbooks. This enhanced acquisition provides good generality for noisy and noiseless settings. We show that our method achieves a sublinear convergence rate on the cumulative regret bound under heteroscedastic observation noise. Our empirical results demonstrate that our proposed acquisition function can outperform EI in the presence of noisy observations on benchmark functions for black-box optimization, as well as on parameter search for neural network model compression.
In the field of global optimization, many existing algorithms face challenges posed by non-convex target functions and high computational complexity or unavailability of gradient information. These limitations, exacerbated by sensitivity to initial conditions, often lead to suboptimal solutions or failed convergence. This is true even for Metaheuristic algorithms designed to amalgamate different optimization techniques to improve their efficiency and robustness. To address these challenges, we develop a sequence of multidimensional integration-based methods that we show to converge to the global optima under some mild regularity conditions. Our probabilistic approach does not require the use of gradients and is underpinned by a mathematically rigorous convergence framework anchored in the nuanced properties of nascent optima distribution. In order to alleviate the problem of multidimensional integration, we develop a latent slice sampler that enjoys a geometric rate of convergence in generating samples from the nascent optima distribution, which is used to approximate the global optima. The proposed Probabilistic Global Optimizer (ProGO) provides a scalable unified framework to approximate the global optima of any continuous function defined on a domain of arbitrary dimension. Empirical illustrations of ProGO across a variety of popular non-convex test functions (having finite global optima) reveal that the proposed algorithm outperforms, by order of magnitude, many existing state-of-the-art methods, including gradient-based, zeroth-order gradient-free, and some Bayesian Optimization methods, in term regret value and speed of convergence. It is, however, to be noted that our approach may not be suitable for functions that are expensive to compute.
Bayesian Optimization (BO), guided by Gaussian process (GP) surrogates, has proven to be an invaluable technique for efficient, high-dimensional, black-box optimization, a critical problem inherent to many applications such as industrial design and scientific computing. Recent contributions have introduced reinforcement learning (RL) to improve the optimization performance on both single function optimization and \textit{few-shot} multi-objective optimization. However, even few-shot techniques fail to exploit similarities shared between closely related objectives. In this paper, we combine recent developments in Deep Kernel Learning (DKL) and attention-based Transformer models to improve the modeling powers of GP surrogates with meta-learning. We propose a novel method for improving meta-learning BO surrogates by incorporating attention mechanisms into DKL, empowering the surrogates to adapt to contextual information gathered during the BO process. We combine this Transformer Deep Kernel with a learned acquisition function trained with continuous Soft Actor-Critic Reinforcement Learning to aid in exploration. This Reinforced Transformer Deep Kernel (RTDK-BO) approach yields state-of-the-art results in continuous high-dimensional optimization problems.
Climate-controlled cabins have for decades been standard in vehicles. Model Predictive Controllers (MPCs) have shown promising results in achieving temperature tracking in vehicle cabins and may improve upon model-free control performance. However, for the multi-zone climate control case, proper controller tuning is challenging, as externally, e.g., passenger-triggered changes in compressor setting and thus mass flow lead to degraded control performance. This paper presents a tuning method to automatically determine robust MPC parameters, as a function of the blower mass flow. Constrained contextual Bayesian optimization (BO) is used to derive policies minimizing a high-level cost function subject to constraints in a defined scenario. The proposed method leverages random disturbances and model-plant mismatch within the training episodes to generate controller parameters achieving robust disturbance rejection. The method contains a postprocessing step to achieve smooth policies that can be utilized in real-world applications. First, simulation results show that the mass flow-dependent policy outperforms a constant parametrization, while achieving the desired closed-loop behavior. Second, the robust tuning method greatly reduces worst-case overshoot and produces consistent closed-loop behavior under varying operating conditions.
Bayesian optimization is an effective method for optimizing expensive-to-evaluate black-box functions. High-dimensional problems are particularly challenging as the surrogate model of the objective suffers from the curse of dimensionality, which makes accurate modeling difficult. We propose a group testing approach to identify active variables to facilitate efficient optimization in these domains. The proposed algorithm, Group Testing Bayesian Optimization (GTBO), first runs a testing phase where groups of variables are systematically selected and tested on whether they influence the objective. To that end, we extend the well-established theory of group testing to functions of continuous ranges. In the second phase, GTBO guides optimization by placing more importance on the active dimensions. By exploiting the axis-aligned subspace assumption, GTBO is competitive against state-of-the-art methods on several synthetic and real-world high-dimensional optimization tasks. Furthermore, GTBO aids in the discovery of active parameters in applications, thereby enhancing practitioners' understanding of the problem at hand.
The one of the significant challenges faced by autonomous robotic ultrasound systems is acquiring high-quality images across different patients. The proper orientation of the robotized probe plays a crucial role in governing the quality of ultrasound images. To address this challenge, we propose a sample-efficient method to automatically adjust the orientation of the ultrasound probe normal to the point of contact on the scanning surface, thereby improving the acoustic coupling of the probe and resulting image quality. Our method utilizes Bayesian Optimization (BO) based search on the scanning surface to efficiently search for the normalized probe orientation. We formulate a novel objective function for BO that leverages the contact force measurements and underlying mechanics to identify the normal. We further incorporate a regularization scheme in BO to handle the noisy objective function. The performance of the proposed strategy has been assessed through experiments on urinary bladder phantoms. These phantoms included planar, tilted, and rough surfaces, and were examined using both linear and convex probes with varying search space limits. Further, simulation-based studies have been carried out using 3D human mesh models. The results demonstrate that the mean ($\pm$SD) absolute angular error averaged over all phantoms and 3D models is $\boldsymbol{2.4\pm0.7^\circ}$ and $\boldsymbol{2.1\pm1.3^\circ}$, respectively.
Multi-fidelity (MF) methods are gaining popularity for enhancing surrogate modeling and design optimization by incorporating data from various low-fidelity (LF) models. While most existing MF methods assume a fixed dataset, adaptive sampling methods that dynamically allocate resources among fidelity models can achieve higher efficiency in the exploring and exploiting the design space. However, most existing MF methods rely on the hierarchical assumption of fidelity levels or fail to capture the intercorrelation between multiple fidelity levels and utilize it to quantify the value of the future samples and navigate the adaptive sampling. To address this hurdle, we propose a framework hinged on a latent embedding for different fidelity models and the associated pre-posterior analysis to explicitly utilize their correlation for adaptive sampling. In this framework, each infill sampling iteration includes two steps: We first identify the location of interest with the greatest potential improvement using the high-fidelity (HF) model, then we search for the next sample across all fidelity levels that maximize the improvement per unit cost at the location identified in the first step. This is made possible by a single Latent Variable Gaussian Process (LVGP) model that maps different fidelity models into an interpretable latent space to capture their correlations without assuming hierarchical fidelity levels. The LVGP enables us to assess how LF sampling candidates will affect HF response with pre-posterior analysis and determine the next sample with the best benefit-to-cost ratio. Through test cases, we demonstrate that the proposed method outperforms the benchmark methods in both MF global fitting (GF) and Bayesian Optimization (BO) problems in convergence rate and robustness. Moreover, the method offers the flexibility to switch between GF and BO by simply changing the acquisition function.
Large language models (LLMs) have shown remarkable instruction-following capabilities and achieved impressive performances in various applications. However, the performances of LLMs depend heavily on the instructions given to them, which are typically manually tuned with substantial human efforts. Recent work has used the query-efficient Bayesian optimization (BO) algorithm to automatically optimize the instructions given to black-box LLMs. However, BO usually falls short when optimizing highly sophisticated (e.g., high-dimensional) objective functions, such as the functions mapping an instruction to the performance of an LLM. This is mainly due to the limited expressive power of the Gaussian process (GP) which is used by BO as a surrogate to model the objective function. Meanwhile, it has been repeatedly shown that neural networks (NNs), especially pre-trained transformers, possess strong expressive power and can model highly complex functions. So, we adopt a neural bandit algorithm which replaces the GP in BO by an NN surrogate to optimize instructions for black-box LLMs. More importantly, the neural bandit algorithm allows us to naturally couple the NN surrogate with the hidden representation learned by a pre-trained transformer (i.e., an open-source LLM), which significantly boosts its performance. These motivate us to propose our INSTruction optimization usIng Neural bandits Coupled with Transformers (INSTINCT) algorithm. We perform instruction optimization for ChatGPT and use extensive experiments to show that INSTINCT consistently outperforms baselines in different tasks, e.g., various instruction induction tasks and the task of improving zero-shot chain-of-thought instructions. Our code is available at https://github.com/xqlin98/INSTINCT.