text: Bayesian optimization (BO) is a powerful paradigm for derivative-free global optimization of a black-box objective function (BOF) that is expensive to evaluate. However, the overhead of BO can still be prohibitive if the maximum number of allowed function evaluations is less than required. In this paper, we investigate how to reduce the required number of function evaluations for BO without compromise in solution quality. We explore the idea of posterior regularization for harnessing low fidelity (LF) data within the Gaussian process upper confidence bound (GP-UCB) framework. The LF data are assumed to arise from previous evaluations of an LF approximation of the BOF. An extra GP expert called LF-GP is trained to fit the LF data. We develop a dynamic weighted product of experts (DW-POE) fusion operator. The regularization is induced from this operator on the posterior of the BOF. The impact of the LF-GP expert on the resulting regularized posterior is adaptively adjusted via Bayesian formalism. Extensive experimental results on benchmark BOF optimization tasks demonstrate the superior performance of the proposed algorithm over state-of-the-art.
  @cite_14: Design decisions for complex systems often can be made or informed by a variety of information sources. When optimizing such a system, the evaluation of a quantity of interest is typically required at many different input configurations. For systems with expensive to evaluate available information sources, the optimization task can potentially be computationally prohibitive using traditional techniques. This paper presents an information-economic approach to the constrained optimization of a system with multiple available information sources. The approach rigorously quantifies the correlation between the discrepancies of different information sources, which enables the overcoming of information source bias. All information is exploited efficiently by fusing newly acquired information with that previously evaluated. Independent decision-makings are achieved by developing a two-step look-ahead utility policy and an information gain policy for objective function and constraints respectively. The approach is demonstrated on a one-dimensional example test problem and an aerodynamic design problem, where it is shown to perform well in comparison to traditional multi-information source techniques.
  @cite_22: Standard practice when analyzing data from different types of experiments is to treat data from each type separately. By borrowing strength across multiple sources, an integrated analysis can produce better results. Careful adjustments must be made to incorporate the systematic differences among various experiments. Toward this end, some Bayesian hierarchical Gaussian process models are proposed. The heterogeneity among different sources is accounted for by performing flexible location and scale adjustments. The approach tends to produce prediction closer to that from the high-accuracy experiment. The Bayesian computations are aided by the use of Markov chain Monte Carlo and sample average approximation algorithms. The proposed method is illustrated with two examples, one with detailed and approximate finite elements simulations for mechanical material design and the other with physical and computer experiments for modeling a food processor.
  @cite_8: 
  @cite_9: In many situations across computational science and engineering, multiple computational models are available that describe a system of interest. These different models have varying evaluation costs...
  @cite_1: Recently there has been an increasing interest in regression methods that deal with multiple outputs. This has been motivated partly by frameworks like multitask learning, multisensor networks or structured output data. From a Gaussian processes perspective, the problem reduces to specifying an appropriate covariance function that, whilst being positive semi-definite, captures the dependencies between all the data points and across all the outputs. One approach to account for non-trivial correlations between outputs employs convolution processes. Under a latent function interpretation of the convolution transform we establish dependencies between output variables. The main drawbacks of this approach are the associated computational and storage demands. In this paper we address these issues. We present different efficient approximations for dependent output Gaussian processes constructed through the convolution formalism. We exploit the conditional independencies present naturally in the model. This leads to a form of the covariance similar in spirit to the so called PITC and FITC approximations for a single output. We show experimental results with synthetic and real data, in particular, we show results in school exams score prediction, pollution prediction and gene expression data.
  @cite_6: 
  @cite_0: 
  @cite_24: 
  @cite_19: Model selection and hyperparameter optimization is crucial in applying machine learning to a novel dataset. Recently, a sub-community of machine learning has focused on solving this problem with Sequential Model-based Bayesian Optimization (SMBO), demonstrating substantial successes in many applications. However, for expensive algorithms the computational overhead of hyperparameter optimization can still be prohibitive. In this paper we explore the possibility of speeding up SMBO by transferring knowledge from previous optimization runs on similar datasets; specifically, we propose to initialize SMBO with a small number of configurations suggested by a met alearning procedure. The resulting simple MI-SMBO technique can be trivially applied to any SMBO method, allowing us to perform experiments on two quite different SMBO methods with complementary strengths applied to optimize two machine learning frameworks on 57 classification datasets. We find that our initialization procedure mildly improves the state of the art in low-dimensional hyperparameter optimization and substantially improves the state of the art in the more complex problem of combined model selection and hyperparameter optimization.
  @cite_2: We present a computational framework for model inversion based on multi-fidelity information fusion and Bayesian optimization. The proposed methodology targets the accurate construction of response surfaces in parameter space, and the efficient pursuit to identify global optima while keeping the number of expensive function evaluations at a minimum. We train families of correlated surrogates on available data using Gaussian processes and auto-regressive stochastic schemes, and exploit the resulting predictive posterior distributions within a Bayesian optimization setting. This enables a smart adaptive sampling procedure that uses the predictive posterior variance to balance the exploration versus exploitation trade-off, and is a key enabler for practical computations under limited budgets. The effectiveness of the proposed framework is tested on three parameter estimation problems. The first two involve the calibration of outflow boundary conditions of blood flow simulations in arterial bifurcations using multi-fidelity realizations of one- and three-dimensional models, whereas the last one aims to identify the forcing term that generated a particular solution to an elliptic partial differential equation.
  @cite_10: 
  @cite_12: We consider Bayesian methods for multi-information source optimization (MISO), in which we seek to optimize an expensive-to-evaluate black-box objective function while also accessing cheaper but biased and noisy approximations ("information sources"). We present a novel algorithm that outperforms the state of the art for this problem by using a Gaussian process covariance kernel better suited to MISO than those used by previous approaches, and an acquisition function based on a one-step optimality analysis supported by efficient parallelization. We also provide a novel technique to guarantee the asymptotic quality of the solution provided by this algorithm. Experimental evaluations demonstrate that this algorithm consistently finds designs of higher value at less cost than previous approaches.
summary: Multi-fidelity optimization has recently attracted considerable research interests. Techniques such as hierarchical partitioning @cite_6 , hierarchical modeling @cite_22 and ensemble methods @cite_9 , are used to incorporate multiple fidelities cheap approximations of the BOF. Most relevant to this paper is the line of work on Bayesian optimization with multi-fidelity data such as the MF-GP-UCB method in @cite_0 and the multi-fidelity BO (MFBO) algorithm in @cite_2 . Research topics that are close to MFBO in concept include multi-information source optimization @cite_14 @cite_12 , multi-task BO @cite_8 , multi-output GP @cite_1 @cite_24 , meta-learning based BO @cite_10 @cite_19 .