Bayesian optimization (BO) is increasingly employed in critical applications to find the optimal design with minimal cost. While BO is known for its sample efficiency, relying solely on costly high-fidelity data can still result in high costs. This is especially the case in constrained search spaces where BO must not only optimize but also ensure feasibility. A related issue in the BO literature is the lack of a systematic stopping criterion. To solve these challenges, we develop a constrained cost-aware multi-fidelity BO (CMFBO) framework whose goal is to minimize overall sampling costs by utilizing inexpensive low-fidelity sources while ensuring feasibility. In our case, the constraints can change across the data sources and may be even black-box functions. We also introduce a systematic stopping criterion that addresses the long-lasting issue associated with BO's convergence assessment. Our framework is publicly available on GitHub through the GP+ Python package and herein we validate it's efficacy on multiple benchmark problems.

Variational inference (VI) is a popular method to estimate statistical and econometric models. The key to VI is the selection of a tractable density to approximate the Bayesian posterior. For large and complex models a common choice is to assume independence between multivariate blocks in a partition of the parameter space. While this simplifies the problem it can reduce accuracy. This paper proposes using vector copulas to capture dependence between the blocks parsimoniously. Tailored multivariate marginals are constructed using learnable cyclically monotone transformations. We call the resulting joint distribution a ``dependent block posterior'' approximation. Vector copula models are suggested that make tractable and flexible variational approximations. They allow for differing marginals, numbers of blocks, block sizes and forms of between block dependence. They also allow for solution of the variational optimization using fast and efficient stochastic gradient methods. The efficacy and versatility of the approach is demonstrated using four different statistical models and 16 datasets which have posteriors that are challenging to approximate. In all cases, our method produces more accurate posterior approximations than benchmark VI methods that either assume block independence or factor-based dependence, at limited additional computational cost.

We introduce a novel kernel-based framework for learning differential equations and their solution maps that is efficient in data requirements, in terms of solution examples and amount of measurements from each example, and computational cost, in terms of training procedures. Our approach is mathematically interpretable and backed by rigorous theoretical guarantees in the form of quantitative worst-case error bounds for the learned equation. Numerical benchmarks demonstrate significant improvements in computational complexity and robustness while achieving one to two orders of magnitude improvements in terms of accuracy compared to state-of-the-art algorithms. Significance statement We present a novel algorithm inspired by kernel methods and Gaussian processes for learning differential equations and their solution operators in scarce data regimes. Our approach: (a) is significantly more efficient than state-of-the-art methods, including neural networks, in terms of required data and computational time. In fact, we obtain one to two orders of magnitude improvement in accuracy on a number of benchmarks; (b) is supported by rigorous theory featuring the first quantitative worst-case error bounds for equation learning; and (c) can solve previously intractable scientific computing problems such as one-shot operator learning and learning of variable-coefficient PDEs in extremely scarce data regimes.

We provide a general framework for learning diffusion bridges that transport prior to target distributions. It includes existing diffusion models for generative modeling, but also underdamped versions with degenerate diffusion matrices, where the noise only acts in certain dimensions. Extending previous findings, our framework allows to rigorously show that score matching in the underdamped case is indeed equivalent to maximizing a lower bound on the likelihood. Motivated by superior convergence properties and compatibility with sophisticated numerical integration schemes of underdamped stochastic processes, we propose \emph{underdamped diffusion bridges}, where a general density evolution is learned rather than prescribed by a fixed noising process. We apply our method to the challenging task of sampling from unnormalized densities without access to samples from the target distribution. Across a diverse range of sampling problems, our approach demonstrates state-of-the-art performance, notably outperforming alternative methods, while requiring significantly fewer discretization steps and no hyperparameter tuning.

Preferential Bayesian Optimization (PBO) is a sample-efficient method to learn latent user utilities from preferential feedback over a pair of designs. It relies on a statistical surrogate model for the latent function, usually a Gaussian process, and an acquisition strategy to select the next candidate pair to get user feedback on. Due to the non-conjugacy of the associated likelihood, every PBO step requires a significant amount of computations with various approximate inference techniques. This computational overhead is incompatible with the way humans interact with computers, hindering the use of PBO in real-world cases. Building on the recent advances of amortized BO, we propose to circumvent this issue by fully amortizing PBO, meta-learning both the surrogate and the acquisition function. Our method comprises a novel transformer neural process architecture, trained using reinforcement learning and tailored auxiliary losses. On a benchmark composed of synthetic and real-world datasets, our method is several orders of magnitude faster than the usual Gaussian process-based strategies and often outperforms them in accuracy.

Existing contextual multi-armed bandit (MAB) algorithms fail to effectively capture both long-term trends and local patterns across all arms, leading to suboptimal performance in environments with rapidly changing reward structures. They also rely on static exploration rates, which do not dynamically adjust to changing conditions. To overcome these limitations, we propose LNUCB-TA, a hybrid bandit model integrating a novel nonlinear component (adaptive k-Nearest Neighbors (k-NN)) for reducing time complexity, alongside a global-and-local attention-based exploration mechanism. Our approach uniquely combines linear and nonlinear estimation techniques, with the nonlinear module dynamically adjusting k based on reward variance to enhance spatiotemporal pattern recognition. This reduces the likelihood of selecting suboptimal arms while improving reward estimation accuracy and computational efficiency. The attention-based mechanism ranks arms by past performance and selection frequency, dynamically adjusting exploration and exploitation in real time without requiring manual tuning of exploration rates. By integrating global attention (assessing all arms collectively) and local attention (focusing on individual arms), LNUCB-TA efficiently adapts to temporal and spatial complexities. Empirical results show LNUCB-TA significantly outperforms state-of-the-art linear, nonlinear, and hybrid bandits in cumulative and mean reward, convergence, and robustness across different exploration rates. Theoretical analysis further confirms its reliability with a sub-linear regret bound.

Diffusion models optimized via variational inference (VI) have emerged as a promising tool for generating samples from unnormalized target densities. These models create samples by simulating a stochastic differential equation, starting from a simple, tractable prior, typically a Gaussian distribution. However, when the support of this prior differs greatly from that of the target distribution, diffusion models often struggle to explore effectively or suffer from large discretization errors. Moreover, learning the prior distribution can lead to mode-collapse, exacerbated by the mode-seeking nature of reverse Kullback-Leibler divergence commonly used in VI. To address these challenges, we propose end-to-end learnable Gaussian mixture priors (GMPs). GMPs offer improved control over exploration, adaptability to target support, and increased expressiveness to counteract mode collapse. We further leverage the structure of mixture models by proposing a strategy to iteratively refine the model by adding mixture components during training. Our experimental results demonstrate significant performance improvements across a diverse range of real-world and synthetic benchmark problems when using GMPs without requiring additional target evaluations.

We study the stochastic linear bandits with heavy-tailed noise. Two principled strategies for handling heavy-tailed noise, truncation and median-of-means, have been introduced to heavy-tailed bandits. Nonetheless, these methods rely on specific noise assumptions or bandit structures, limiting their applicability to general settings. The recent work [Huang et al.2024] develops a soft truncation method via the adaptive Huber regression to address these limitations. However, their method suffers undesired computational cost: it requires storing all historical data and performing a full pass over these data at each round. In this paper, we propose a \emph{one-pass} algorithm based on the online mirror descent framework. Our method updates using only current data at each round, reducing the per-round computational cost from $\widetilde{\mathcal{O}}(t \log T)$ to $\widetilde{\mathcal{O}}(1)$ with respect to current round $t$ and the time horizon $T$, and achieves a near-optimal and variance-aware regret of order $\widetilde{\mathcal{O}}\big(d T^{\frac{1-\epsilon}{2(1+\epsilon)}} \sqrt{\sum_{t=1}^T \nu_t^2} + d T^{\frac{1-\epsilon}{2(1+\epsilon)}}\big)$ where $d$ is the dimension and $\nu_t^{1+\epsilon}$ is the $(1+\epsilon)$-th central moment of reward at round $t$.

Heatwaves, intensified by climate change and rapid urbanisation, pose significant threats to urban systems, particularly in the Global South, where adaptive capacity is constrained. This study investigates the relationship between heatwaves and nighttime light (NTL) radiance, a proxy of nighttime economic activity, in four hyperdense cities: Delhi, Guangzhou, Cairo, and Sao Paulo. We hypothesised that heatwaves increase nighttime activity. Using a double machine learning (DML) framework, we analysed data from 2013 to 2019 to quantify the impact of heatwaves on NTL while controlling for local climatic confounders. Results revealed a statistically significant increase in NTL intensity during heatwaves, with Cairo, Delhi, and Guangzhou showing elevated NTL on the third day, while S\~ao Paulo exhibits a delayed response on the fourth day. Sensitivity analyses confirmed the robustness of these findings, indicating that prolonged heat stress prompts urban populations to shift activities to night. Heterogeneous responses across cities highlight the possible influence of urban morphology and adaptive capacity to heatwave impacts. Our findings provide a foundation for policymakers to develop data-driven heat adaptation strategies, ensuring that cities remain liveable and economically resilient in an increasingly warming world.

The analysis of sales information, is a vital step in designing an effective marketing strategy. This work proposes a novel approach to analyse the shopping behaviour of customers to identify their purchase patterns. An extended version of the Multi-Cluster Overlapping k-Means Extension (MCOKE) algorithm with weighted k-Means algorithm is utilized to map customers to the garments of interest. The age & gender traits of the customer; the time spent and the expressions exhibited while selecting garments for purchase, are utilized to associate a customer or a group of customers to a garments they are interested in. Such study on the customer base of a retail business, may help in inferring the products of interest of their consumers, and enable them in developing effective business strategies, thus ensuring customer satisfaction, loyalty, increased sales and profits.