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Optimizing objectives under constraints, where both the objectives and constraints are black box functions, is a common scenario in real-world applications such as the design of medical therapies, industrial process optimization, and hyperparameter optimization. One popular approach to handle these complex scenarios is Bayesian Optimization (BO). However, when it comes to the theoretical understanding of constrained Bayesian optimization (CBO), the existing framework often relies on heuristics, approximations, or relaxation of objectives and, therefore, lacks the same level of theoretical guarantees as in canonical BO. In this paper, we exclude the boundary candidates that could be compromised by noise perturbation and aim to find the interior optimum of the black-box-constrained objective. We rely on the insight that optimizing the objective and learning the constraints can both help identify the high-confidence regions of interest (ROI) that potentially contain the interior optimum. We propose an efficient CBO framework that intersects the ROIs identified from each aspect on a discretized search space to determine the general ROI. Then, on the ROI, we optimize the acquisition functions, balancing the learning of the constraints and the optimization of the objective. We showcase the efficiency and robustness of our proposed CBO framework through the high probability regret bounds for the algorithm and extensive empirical evidence. 

Bayesian optimization (BO) is a powerful paradigm for optimizing expensive black-box functions. Traditional BO methods typically rely on separate hand-crafted acquisition functions and surrogate models for the underlying function, and often operate in a myopic manner. In this paper, we propose a novel direct regret optimization approach that jointly learns the optimal model and non-myopic acquisition by distilling from a set of candidate models and acquisitions, and explicitly targets minimizing the multi-step regret. Our framework leverages an ensemble of Gaussian Processes (GPs) with varying hyperparameters to generate simulated BO trajectories, each guided by an acquisition function chosen from a pool of conventional choices, until a Bayesian early stop criterion is met. These simulated trajectories, capturing multi-step exploration strategies, are used to train an end-to-end decision transformer that directly learns to select next query points aimed at improving the ultimate objective. We further adopt a dense training–sparse learning paradigm: The decision transformer is trained offline with abundant simulated data sampled from ensemble GPs and acquisitions, while a limited number of real evaluations refine the GPs online. Experimental results on synthetic and real-world benchmarks suggest that our method consistently outperforms BO baselines, achieving lower simple regret and demonstrating more robust exploration in high-dimensional or noisy settings. 

As prompts become central to Large Language Models (LLMs), optimizing them is vital. Textual Stochastic Gradient Descent (TSGD) offers a data-driven approach by iteratively refining prompts using LLM-suggested updates over minibatches. We empirically show that increasing training data initially improves but can later degrade TSGD's performance across NLP tasks, while also raising computational costs. To address this, we propose Textual Stochastic Gradient Descent with Momentum (TSGD-M)—a scalable method that reweights prompt sampling based on past batches. Evaluated on 9 NLP tasks across three domains, TSGD-M outperforms TSGD baselines for most tasks and reduces performance variance. 

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. 

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 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. 

Active Learning (AL) has gained prominence in integrating data-intensive machine learning (ML) models into domains with limited labeled data. However, its effectiveness diminishes significantly when the labeling budget is low. In this paper, we first empirically observe the performance degradation of existing AL algorithms in the low-budget settings, and then introduce Direct Acquisition Optimization (DAO), a novel AL algorithm that optimizes sample selections based on expected true loss reduction. Specifically, DAO utilizes influence functions to update model parameters and incorporates an additional acquisition strategy to mitigate bias in loss estimation. This approach facilitates a more accurate estimation of the overall error reduction, without extensive computations or reliance on labeled data. Experiments demonstrate DAO's effectiveness in low budget settings, outperforming state-of-the-arts approaches across seven benchmarks. 

Active learning is a promising paradigm to reduce the labeling cost by strategically requesting labels to improve model performance. However, existing active learning methods often rely on expensive acquisition function to compute, extensive modeling retraining and multiple rounds of interaction with annotators. To address these limitations, we propose a novel approach for active learning, which aims to select batches of unlabeled instances through a learned surrogate model for data acquisition. A key challenge in this approach is developing an acquisition function that generalizes well, as the history of data, which forms part of the utility function’s input, grows over time. Our novel algorithmic contribution is a multi-task bilevel optimization framework that predicts the relative utility, measured by the validation accuracy, of different training sets, and ensures the learned acquisition function generalizes effectively. For cases where validation accuracy is expensive to evaluate, we introduce efficient interpolation-based surrogate models to estimate the utility function, reducing the evaluation cost. We demonstrate the performance of our approach through extensive experiments on standard active classification benchmarks.