Researchers and practitioners often wish to measure treatment effects in settings where units interact via markets and recommendation systems. In these settings, units are affected by certain shared states, like prices, algorithmic recommendations or social signals. We formalize this structure, calling it shared-state interference, and argue that our formulation captures many relevant applied settings. Our key modeling assumption is that individuals' potential outcomes are independent conditional on the shared state. We then prove an extension of a double machine learning (DML) theorem providing conditions for achieving efficient inference under shared-state interference. We also instantiate our general theorem in several models of interest where it is possible to efficiently estimate the average direct effect (ADE) or global average treatment effect (GATE).

Negative distance kernels $K(x,y) := - \|x-y\|$ were used in the definition of maximum mean discrepancies (MMDs) in statistics and lead to favorable numerical results in various applications. In particular, so-called slicing techniques for handling high-dimensional kernel summations profit from the simple parameter-free structure of the distance kernel. However, due to its non-smoothness in $x=y$, most of the classical theoretical results, e.g. on Wasserstein gradient flows of the corresponding MMD functional do not longer hold true. In this paper, we propose a new kernel which keeps the favorable properties of the negative distance kernel as being conditionally positive definite of order one with a nearly linear increase towards infinity and a simple slicing structure, but is Lipschitz differentiable now. Our construction is based on a simple 1D smoothing procedure of the absolute value function followed by a Riemann-Liouville fractional integral transform. Numerical results demonstrate that the new kernel performs similarly well as the negative distance kernel in gradient descent methods, but now with theoretical guarantees.

Learning node representations is a fundamental problem in graph machine learning. While existing embedding methods effectively preserve local similarity measures, they often fail to capture global functions like graph distances. Inspired by Bourgain's seminal work on Hilbert space embeddings of metric spaces (1985), we study the performance of local distance-preserving node embeddings. Known as landmark-based algorithms, these embeddings approximate pairwise distances by computing shortest paths from a small subset of reference nodes (i.e., landmarks). Our main theoretical contribution shows that random graphs, such as Erd\H{o}s-R\'enyi random graphs, require lower dimensions in landmark-based embeddings compared to worst-case graphs. Empirically, we demonstrate that the GNN-based approximations for the distances to landmarks generalize well to larger networks, offering a scalable alternative for graph representation learning.

Using both observational and experimental data, a causal discovery process can identify the causal relationships between variables. A unique adaptive intervention design paradigm is presented in this work, where causal directed acyclic graphs (DAGs) are for effectively recovered with practical budgetary considerations. In order to choose treatments that optimize information gain under these considerations, an iterative integer programming (IP) approach is proposed, which drastically reduces the number of experiments required. Simulations over a broad range of graph sizes and edge densities are used to assess the effectiveness of the suggested approach. Results show that the proposed adaptive IP approach achieves full causal graph recovery with fewer intervention iterations and variable manipulations than random intervention baselines, and it is also flexible enough to accommodate a variety of practical constraints.

In this paper, we introduce a non-crossing quantile (NQ) network for conditional distribution learning. By leveraging non-negative activation functions, the NQ network ensures that the learned distributions remain monotonic, effectively addressing the issue of quantile crossing. Furthermore, the NQ network-based deep distributional learning framework is highly adaptable, applicable to a wide range of applications, from classical non-parametric quantile regression to more advanced tasks such as causal effect estimation and distributional reinforcement learning (RL). We also develop a comprehensive theoretical foundation for the deep NQ estimator and its application to distributional RL, providing an in-depth analysis that demonstrates its effectiveness across these domains. Our experimental results further highlight the robustness and versatility of the NQ network.

Bayesian optimization (BO) is an effective technique for black-box optimization. However, its applicability is typically limited to moderate-budget problems due to the cubic complexity in computing the Gaussian process (GP) surrogate model. In large-budget scenarios, directly employing the standard GP model faces significant challenges in computational time and resource requirements. In this paper, we propose a novel approach, gradient-based sample selection Bayesian Optimization (GSSBO), to enhance the computational efficiency of BO. The GP model is constructed on a selected set of samples instead of the whole dataset. These samples are selected by leveraging gradient information to maintain diversity and representation. We provide a theoretical analysis of the gradient-based sample selection strategy and obtain explicit sublinear regret bounds for our proposed framework. Extensive experiments on synthetic and real-world tasks demonstrate that our approach significantly reduces the computational cost of GP fitting in BO while maintaining optimization performance comparable to baseline methods.

Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these models have been derived using principles of variational inference, denoising, statistical signal processing, and stochastic differential equations. In contrast to the conventional presentation, in this study we derive diffusion models using ideas from linear partial differential equations and demonstrate that this approach has several benefits that include a constructive derivation of the forward and reverse processes, a unified derivation of multiple formulations and sampling strategies, and the discovery of a new class of models. We also apply the conditional version of these models to solving canonical conditional density estimation problems and challenging inverse problems. These problems help establish benchmarks for systematically quantifying the performance of different formulations and sampling strategies in this study, and for future studies. Finally, we identify and implement a mechanism through which a single diffusion model can be applied to measurements obtained from multiple measurement operators. Taken together, the contents of this manuscript provide a new understanding and several new directions in the application of diffusion models to solving physics-based inverse problems.

This paper addresses the task of learning convex regularizers to guide the reconstruction of images from limited data. By imposing that the reconstruction be amplitude-equivariant, we narrow down the class of admissible functionals to those that can be expressed as a power of a seminorm. We then show that such functionals can be approximated to arbitrary precision with the help of polyhedral norms. In particular, we identify two dual parameterizations of such systems: (i) a synthesis form with an $\ell_1$-penalty that involves some learnable dictionary; and (ii) an analysis form with an $\ell_\infty$-penalty that involves a trainable regularization operator. After having provided geometric insights and proved that the two forms are universal, we propose an implementation that relies on a specific architecture (tight frame with a weighted $\ell_1$ penalty) that is easy to train. We illustrate its use for denoising and the reconstruction of biomedical images. We find that the proposed framework outperforms the sparsity-based methods of compressed sensing, while it offers essentially the same convergence and robustness guarantees.

Power grid operation is becoming increasingly complex due to the rising integration of renewable energy sources and the need for more adaptive control strategies. Reinforcement Learning (RL) has emerged as a promising approach to power network control (PNC), offering the potential to enhance decision-making in dynamic and uncertain environments. The Learning To Run a Power Network (L2RPN) competitions have played a key role in accelerating research by providing standardized benchmarks and problem formulations, leading to rapid advancements in RL-based methods. This survey provides a comprehensive and structured overview of RL applications for power grid topology optimization, categorizing existing techniques, highlighting key design choices, and identifying gaps in current research. Additionally, we present a comparative numerical study evaluating the impact of commonly applied RL-based methods, offering insights into their practical effectiveness. By consolidating existing research and outlining open challenges, this survey aims to provide a foundation for future advancements in RL-driven power grid optimization.

We are interested in the estimation of a rank-one tensor signal when only a portion $\varepsilon$ of its noisy observation is available. We show that the study of this problem can be reduced to that of a random matrix model whose spectral analysis gives access to the reconstruction performance. These results shed light on and specify the loss of performance induced by an artificial reduction of the memory cost of a tensor via the deletion of a random part of its entries.