Bayesian Optimal Experimental Design (BOED) provides a rigorous framework for decision-making tasks in which data acquisition is often the critical bottleneck, especially in resource-constrained settings. Traditionally, BOED typically selects designs by maximizing expected information gain (EIG), commonly defined through the Kullback-Leibler (KL) divergence. However, classical evaluation of EIG often involves challenging nested expectations, and even advanced variational methods leave the underlying log-density-ratio objective unchanged. As a result, support mismatch, tail underestimation, and rare-event sensitivity remain intrinsic concerns for KL-based BOED. To address these fundamental bottlenecks, we introduce an IPM-based BOED framework that replaces density-based divergences with integral probability metrics (IPMs), including the Wasserstein distance, Maximum Mean Discrepancy, and Energy Distance, resulting in a highly flexible plug-and-play BOED framework. We establish theoretical guarantees showing that IPM-based utilities provide stronger geometry-aware stability under surrogate-model error and prior misspecification than classical EIG-based utilities. We also validate the proposed framework empirically, demonstrating that IPM-based designs yield highly concentrated credible sets. Furthermore, by extending the same sample-based BOED template in a plug-and-play manner to geometry-aware discrepancies beyond the IPM class, illustrated by a neural optimal transport estimator, we achieve accurate optimal designs in high-dimensional settings where conventional nested Monte Carlo estimators and advanced variational methods fail.

The proposed algorithm seeks to provide a novel data-driven framework for the discovery of stochastic differential equations (SDEs) by application of the Weak-formulation to stochastic SINDy. This Weak formulation of the algorithm provides a noise-robust methodology that avoids traditional noisy derivative computation using finite differences. An additional novelty is the adoption of spatial Gaussian test functions in place of temporal test functions, wherein the use of the kernel weight $K_j(X_{t_n})$ guarantees unbiasedness in expectation and prevents the structural regression bias that is otherwise pertinent with temporal test functions. The proposed framework converts the SDE identification problem into two SINDy based linear sparse identification problems. We validate the algorithm on three SDEs, for which we recover all active non-linear terms with coefficient errors below 4%, stationary-density total-variation distances below 0.01, and autocorrelation functions that reproduce true relaxation timescales across all three benchmarks faithfully.

In this work, we present a new deterministic partition-based global optimization algorithm, HALO (Hybrid Adaptive Lipschitzian Optimization), which uses estimates of the local Lipschitz constants associated with different sub-regions of the objective function's domain to compute lower bounds and guide the search toward global minimizers. These estimates are obtained by adaptively balancing the global and local information collected from the algorithm, based on absolute slopes. HALO is hyperparameter-free, eliminating the need for manual tuning, and it highlights the most important variables to help interpret the optimization problem. We also introduce a coupling strategy with local optimization algorithms, both gradient-based and derivative-free, to accelerate convergence. We compare HALO with popular global optimization algorithms on hundreds of test functions. The numerical results are very promising and demonstrate that HALO can expand our arsenal of efficient procedures of efficient procedures for challenging real-world black-box optimization problems. The Python code of HALO is publicly available on GitHub. https://github.com/dannyzx/HALO

Global optimization of expensive, potentially gradient-free functions has long been a critical component of many complex problems in science and engineering.

Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.

We provide the definitions of important terms used throughout the paper. Assumption 2.3 when the demand distribution is exponential. Note that Lemma B.1 implies that In the following result, we show that there exist appropriate constants such that prior distribution satisfies Assumption 2.3 when the demand distribution is a multivariate Gaussian with unknown The proof is a direct consequence of Theorem 3.2, Lemmas B.6, B.7, B.8, B.9, and Proposition 3.2. Theorem 6.19] the prior induced by Assumption 2.2 is a direct consequence of Assumption 2.4 and 2.5 are straightforward to satisfy since the model risk function Lemma B.13. F or a given Using the result above together with Proposition 3.2 implies that the RSVB posterior converges at C.1 Alternative derivation of LCVB We present the alternative derivation of LCVB. We prove our main result after a series of important lemmas.

We compute the convergence rates of the RSVB approximate posterior and the corresponding optimal value.

The results of our main method in Section 4.1 is reported in column Main. Testdenotes the variant of using the estimated reward function as the test function when trainingtheMIWω. Thismayberelatedtotheunstable estimation ofKL-dual discussed in Section3.2. Removing rollout data in the policy learning generally leads to worse performance and larger standard deviations. From Eq. (22), the MIWω can be optimized via two alternativeapproaches.(1)Wecan