Near-Optimal Algorithm for Non-Stationary Kernelized Bandits

This paper studies a non-stationary kernelized bandit (KB) problem, also called time-varying Bayesian optimization, where one seeks to minimize the regret under an unknown reward function that varies over time. In particular, we focus on a near-optimal algorithm whose regret upper bound matches the regret lower bound. For this goal, we show the first algorithm-independent regret lower bound for non-stationary KB with squared exponential and Mat\'ern kernels, which reveals that an existing optimization-based KB algorithm with slight modification is near-optimal. However, this existing algorithm suffers from feasibility issues due to its huge computational cost. Therefore, we propose a novel near-optimal algorithm called restarting phased elimination with random permutation (R-PERP), which bypasses the huge computational cost. A technical key point is the simple permutation procedures of query candidates, which enable us to derive a novel tighter confidence bound tailored to the non-stationary problems.

Regret Analysis for Randomized Gaussian Process Upper Confidence Bound

Gaussian process upper confidence bound (GP-UCB) is a theoretically established algorithm for Bayesian optimization (BO), where we assume the objective function $f$ follows GP. One notable drawback of GP-UCB is that the theoretical confidence parameter $\beta$ increased along with the iterations is too large. To alleviate this drawback, this paper analyzes the randomized variant of GP-UCB called improved randomized GP-UCB (IRGP-UCB), which uses the confidence parameter generated from the shifted exponential distribution. We analyze the expected regret and conditional expected regret, where the expectation and the probability are taken respectively with $f$ and noises and with the randomness of the BO algorithm. In both regret analyses, IRGP-UCB achieves a sub-linear regret upper bound without increasing the confidence parameter if the input domain is finite. Finally, we show numerical experiments using synthetic and benchmark functions and real-world emulators.

Distributionally Robust Safe Sample Screening

In this study, we propose a machine learning method called Distributionally Robust Safe Sample Screening (DRSSS). DRSSS aims to identify unnecessary training samples, even when the distribution of the training samples changes in the future. To achieve this, we effectively combine the distributionally robust (DR) paradigm, which aims to enhance model robustness against variations in data distribution, with the safe sample screening (SSS), which identifies unnecessary training samples prior to model training. Since we need to consider an infinite number of scenarios regarding changes in the distribution, we applied SSS because it does not require model training after the change of the distribution. In this paper, we employed the covariate shift framework to represent the distribution of training samples and reformulated the DR covariate-shift problem as a weighted empirical risk minimization problem, where the weights are subject to uncertainty within a predetermined range. By extending the existing SSS technique to accommodate this weight uncertainty, the DRSSS method is capable of reliably identifying unnecessary samples under any future distribution within a specified range. We provide a theoretical guarantee for the DRSSS method and validate its performance through numerical experiments on both synthetic and real-world datasets.

Multi-Objective Bayesian Optimization with Active Preference Learning

There are a lot of real-world black-box optimization problems that need to optimize multiple criteria simultaneously. However, in a multi-objective optimization (MOO) problem, identifying the whole Pareto front requires the prohibitive search cost, while in many practical scenarios, the decision maker (DM) only needs a specific solution among the set of the Pareto optimal solutions. We propose a Bayesian optimization (BO) approach to identifying the most preferred solution in the MOO with expensive objective functions, in which a Bayesian preference model of the DM is adaptively estimated by an interactive manner based on the two types of supervisions called the pairwise preference and improvement request. To explore the most preferred solution, we define an acquisition function in which the uncertainty both in the objective functions and the DM preference is incorporated. Further, to minimize the interaction cost with the DM, we also propose an active learning strategy for the preference estimation. We empirically demonstrate the effectiveness of our proposed method through the benchmark function optimization and the hyper-parameter optimization problems for machine learning models.

Posterior Sampling-Based Bayesian Optimization with Tighter Bayesian Regret Bounds

Among various acquisition functions (AFs) in Bayesian optimization (BO), Gaussian process upper confidence bound (GP-UCB) and Thompson sampling (TS) are well-known options with established theoretical properties regarding Bayesian cumulative regret (BCR). Recently, it has been shown that a randomized variant of GP-UCB achieves a tighter BCR bound compared with GP-UCB, which we call the tighter BCR bound for brevity. Inspired by this study, this paper first shows that TS achieves the tighter BCR bound. On the other hand, GP-UCB and TS often practically suffer from manual hyperparameter tuning and over-exploration issues, respectively. To overcome these difficulties, we propose yet another AF called a probability of improvement from the maximum of a sample path (PIMS). We show that PIMS achieves the tighter BCR bound and avoids the hyperparameter tuning, unlike GP-UCB. Furthermore, we demonstrate a wide range of experiments, focusing on the effectiveness of PIMS that mitigates the practical issues of GP-UCB and TS.

Randomized Gaussian Process Upper Confidence Bound with Tight Bayesian Regret Bounds

Gaussian process upper confidence bound (GP-UCB) is a theoretically promising approach for black-box optimization; however, the confidence parameter $\beta$ is considerably large in the theorem and chosen heuristically in practice. Then, randomized GP-UCB (RGP-UCB) uses a randomized confidence parameter, which follows the Gamma distribution, to mitigate the impact of manually specifying $\beta$. This study first generalizes the regret analysis of RGP-UCB to a wider class of distributions, including the Gamma distribution. Furthermore, we propose improved RGP-UCB (IRGP-UCB) based on a two-parameter exponential distribution, which achieves tight Bayesian regret bounds. IRGP-UCB does not require an increase in the confidence parameter in terms of the number of iterations, which avoids over-exploration in the later iterations. Finally, we demonstrate the effectiveness of IRGP-UCB through extensive experiments.

Towards Practical Preferential Bayesian Optimization with Skew Gaussian Processes

We study preferential Bayesian optimization (BO) where reliable feedback is limited to pairwise comparison called duels. An important challenge in preferential BO, which uses the preferential Gaussian process (GP) model to represent flexible preference structure, is that the posterior distribution is a computationally intractable skew GP. The most widely used approach for preferential BO is Gaussian approximation, which ignores the skewness of the true posterior. Alternatively, Markov chain Monte Carlo (MCMC) based preferential BO is also proposed. In this work, we first verify the accuracy of Gaussian approximation, from which we reveal the critical problem that the predictive probability of duels can be inaccurate. This observation motivates us to improve the MCMC-based estimation for skew GP, for which we show the practical efficiency of Gibbs sampling and derive the low variance MC estimator. However, the computational time of MCMC can still be a bottleneck in practice. Towards building a more practical preferential BO, we develop a new method that achieves both high computational efficiency and low sample complexity, and then demonstrate its effectiveness through extensive numerical experiments.

Bayesian Optimization for Distributionally Robust Chance-constrained Problem

In black-box function optimization, we need to consider not only controllable design variables but also uncontrollable stochastic environment variables. In such cases, it is necessary to solve the optimization problem by taking into account the uncertainty of the environmental variables. Chance-constrained (CC) problem, the problem of maximizing the expected value under a certain level of constraint satisfaction probability, is one of the practically important problems in the presence of environmental variables. In this study, we consider distributionally robust CC (DRCC) problem and propose a novel DRCC Bayesian optimization method for the case where the distribution of the environmental variables cannot be precisely specified. We show that the proposed method can find an arbitrary accurate solution with high probability in a finite number of trials, and confirm the usefulness of the proposed method through numerical experiments.

Bayesian Optimization for Cascade-type Multi-stage Processes

Complex processes in science and engineering are often formulated as multi-stage decision-making problems. In this paper, we consider a type of multi-stage decision-making process called a cascade process. A cascade process is a multi-stage process in which the output of one stage is used as an input for the next stage. When the cost of each stage is expensive, it is difficult to search for the optimal controllable parameters for each stage exhaustively. To address this problem, we formulate the optimization of the cascade process as an extension of Bayesian optimization framework and propose two types of acquisition functions (AFs) based on credible intervals and expected improvement. We investigate the theoretical properties of the proposed AFs and demonstrate their effectiveness through numerical experiments. In addition, we consider an extension called suspension setting in which we are allowed to suspend the cascade process at the middle of the multi-stage decision-making process that often arises in practical problems. We apply the proposed method in the optimization problem of the solar cell simulator, which was the motivation for this study.

Sequential- and Parallel- Constrained Max-value Entropy Search via Information Lower Bound

Recently, several Bayesian optimization (BO) methods have been extended to the expensive black-box optimization problem with unknown constraints, which is an important problem that appears frequently in practice. We focus on an information-theoretic approach called Max-value Entropy Search (MES) whose superior performance has been repeatedly shown in BO literature. Since existing MES-based constrained BO is restricted to only one constraint, we first extend it to multiple constraints, but we found that this approach can cause negative approximate values for the mutual information, which can result in unreasonable decisions. In this paper, we employ a different approximation strategy that is based on a lower bound of the mutual information, and propose a novel constrained BO method called Constrained Max-value Entropy Search via Information lower BOund (CMES-IBO). Our approximate mutual information derived from the lower bound has a simple closed-form that is guaranteed to be nonnegative, and we show that irrational behavior caused by the negative value can be avoided. Furthermore, by using conditional mutual information, we extend our methods to the parallel setting in which multiple queries can be issued simultaneously. Finally, we demonstrate the effectiveness of our proposed methods by benchmark functions and real-world applications to materials science.