To mitigate this issue, various surrogate models have been proposed, including Random Forest (RF) [17], Gaussian process (GP) [18], Bayesian Neural Networks (BNNs) [19,20], Tree-structured Parzen Estimator (TPE) [8], and Mondrian trees [21]. These surrogate models focus on constructing useful prior distributions for objective functions and computing posterior beliefs based on observations. Soon afterward, the surrogate model is combined with an acquisition function to guide the selection of the next query for the unknown objective function. In general, the one-step lookahead approach is used in acquisition functions, where the direct effect of the evaluation on an immediate measure of solution quality is considered. For example, Probability of Improvement (PI) [22], EI [23], Knowledge Gradient (KG) [24], UCB [25], Entropy Search (ES) [26], Predictive Entropy Search (PES) [27], and Maxvalue Entropy Search (MES) [28] are used to myopically maximize the immediate improvement in solution quality. In addition, Wu et al. [29] proposed the non-myopic method to improve sampling accuracy. However, the above-mentioned methods in BO suffer from low evaluation efficiency due to limited data in the initial stage.

To address this question, we propose PMCBO, a prior-informed multi-task collaborative Bayesian optimization method. In the design of PMCBO, the following key challenges need to be addressed: (1) how to mitigate the issue of expensive function evaluations in the distributed Bayesian framework; (2) how to facilitate collaborative optimization among clients while maximizing the protection of private information for each client; and (3) how to better integrate expert prior knowledge. To this end, we utilize an expert prior that is represented as a probability distribution on the location of the optimum over the objective function to mitigate the issue of expensive evaluation costs. Moreover, we introduce the consensus mechanism to information exchange among clients to facilitate collaborative optimization. In addition, we consider the prior decay technique to avoid erroneous acquisition due to incorrect prior and the temporally dynamic information-sharing approach to gain consolidated insights within the DBO framework without divulging client-specific information due to privacy or security apprehensions. Therefore, PMCBO can cooperatively construct a meaningful model over the objective function separately. Concretely, the comparisons between existing methods and PMCBO are summarized in Table 1. Our main contributions are threefold: