Multi-objective Bayesian optimization has been widely adopted in scientific experiment design, including drug discovery and hyperparameter optimization. In practice, regulatory or safety concerns often impose additional thresholds on certain attributes of the experimental outcomes. Previous work has primarily focused on constrained single-objective optimization tasks or active search under constraints. The existing constrained multi-objective algorithms address the issue with heuristics and approximations, posing challenges to the analysis of the sample efficiency. We propose a novel constrained multi-objective Bayesian optimization algorithm \textbf{COMBOO} that balances active learning of the level-set defined on multiple unknowns with multi-objective optimization within the feasible region. We provide both theoretical analysis and empirical evidence, demonstrating the efficacy of our approach on various synthetic benchmarks and real-world applications.

Multi-fidelity Bayesian optimization (MFBO) is a powerful approach that utilizes low-fidelity, cost-effective sources to expedite the exploration and exploitation of a high-fidelity objective function. Existing MFBO methods with theoretical foundations either lack justification for performance improvements over single-fidelity optimization or rely on strong assumptions about the relationships between fidelity sources to construct surrogate models and direct queries to low-fidelity sources. To mitigate the dependency on cross-fidelity assumptions while maintaining the advantages of low-fidelity queries, we introduce a random sampling and partition-based MFBO framework with deep kernel learning. This framework is robust to cross-fidelity model misspecification and explicitly illustrates the benefits of low-fidelity queries. Our results demonstrate that the proposed algorithm effectively manages complex cross-fidelity relationships and efficiently optimizes the target fidelity function.

Optimizing objectives under constraints, where both the objectives and constraints are black box functions, is a common challenge in real-world applications such as medical therapy design, industrial process optimization, and hyperparameter optimization. Bayesian Optimization (BO) is a popular approach for tackling these complex scenarios. However, constrained Bayesian Optimization (CBO) often relies on heuristics, approximations, or relaxation of objectives, which can lead to weaker theoretical guarantees compared to canonical BO. In this paper, we address this gap by focusing on identifying the interior optimum of the constrained objective, deliberately excluding boundary candidates susceptible to noise perturbations. Our approach leverages the insight that jointly optimizing the objective and learning the constraints can help pinpoint high-confidence **regions of interest** (ROI) likely to contain the interior optimum. We introduce an efficient CBO framework, which intersects these ROIs within a discretized search space to determine a general ROI. Within this ROI, we optimize the acquisition functions, balancing constraints learning and objective optimization. We showcase the efficiency and robustness of our proposed framework by deriving high-probability regret bounds and validating its performance through extensive empirical evaluations.

This paper investigates the challenge of learning in black-box games, where the underlying utility function is unknown to any of the agents. While there is an extensive body of literature on the theoretical analysis of algorithms for computing the Nash equilibrium with *complete information* about the game, studies on Nash equilibrium in *black-box* games are less common. In this paper, we focus on learning the Nash equilibrium when the only available information about an agent’s payoff comes in the form of empirical queries. We provide a no-regret learning algorithm that utilizes Gaussian processes to identify equilibria in such games. Our approach not only ensures a theoretical convergence rate but also demonstrates effectiveness across a variety collection of games through experimental validation.

We study Bayesian optimization (BO) in high-dimensional and non-stationary scenarios. Existing algorithms for such scenarios typically require extensive hyperparameter tuning, which limits their practical effectiveness. We propose a framework, called BALLET, which adaptively filters for a high-confidence region of interest (ROI) as a superlevel-set of a nonparametric probabilistic model such as a Gaussian process (GP). Our approach is easy to tune, and is able to focus on local region of the optimization space that can be tackled by existing BO methods. The key idea is to use two probabilistic models: a coarse GP to identify the ROI, and a localized GP for optimization within the ROI. We show theoretically that BALLET can efficiently shrink the search space, and can exhibit a tighter regret bound than standard BO without ROI filtering. We demonstrate empirically the effectiveness of BALLET on both synthetic and real-world optimization tasks.