Operator learning is a recently developed generalization of regression to mappings between functions. It promises to drastically reduce expensive numerical integration of PDEs to fast evaluations of mappings between functional states of a system, i.e., surrogate and reduced-order modeling. Operator learning has already found applications in several areas such as modeling sea ice, combustion, and atmospheric physics. Recent approaches towards integrating uncertainty quantification into the operator models have relied on likelihood based methods to infer parameter distributions from noisy data. However, stochastic operators may yield actions from which a likelihood is difficult or impossible to construct. In this paper, we introduce, GenUQ, a measure-theoretic approach to UQ that avoids constructing a likelihood by introducing a generative hyper-network model that produces parameter distributions consistent with observed data. We demonstrate that GenUQ outperforms other UQ methods in three example problems, recovering a manufactured operator, learning the solution operator to a stochastic elliptic PDE, and modeling the failure location of porous steel under tension.

Sparse Bayesian learning (SBL) associates to each weight in the underlying linear model a hyperparameter by assuming that each weight is Gaussian distributed with zero mean and precision (inverse variance) equal to its associated hyperparameter. The method estimates the hyperparameters by marginalizing out the weights and performing (marginalized) maximum likelihood (ML) estimation. SBL returns many hyperparameter estimates to diverge to infinity, effectively setting the estimates of the corresponding weights to zero (i.e., pruning the corresponding weights from the model) and thereby yielding a sparse estimate of the weight vector. In this letter, we analyze the marginal likelihood as function of a single hyperparameter while keeping the others fixed, when the Gaussian assumptions on the noise samples and the weight distribution that underlies the derivation of SBL are weakened. We derive sufficient conditions that lead, on the one hand, to finite hyperparameter estimates and, on the other, to infinite ones. Finally, we show that in the Gaussian case, the two conditions are complementary and coincide with the pruning condition of fast SBL (F-SBL), thereby providing additional insights into this algorithm.

Learning, whether natural or artificial, is a process of selection. It starts with a set of candidate options and selects the more successful ones. In the case of machine learning the selection is done based on empirical estimates of prediction accuracy of candidate prediction rules on some data. Due to randomness of data sampling the empirical estimates are inherently noisy, leading to selection under uncertainty. The book provides statistical tools to obtain theoretical guarantees on the outcome of selection under uncertainty. We start with concentration of measure inequalities, which are the main statistical instrument for controlling how much an empirical estimate of expectation of a function deviates from the true expectation. The book covers a broad range of inequalities, including Markov's, Chebyshev's, Hoeffding's, Bernstein's, Empirical Bernstein's, Unexpected Bernstein's, kl, and split-kl. We then study the classical (offline) supervised learning and provide a range of tools for deriving generalization bounds, including Occam's razor, Vapnik-Chervonenkis analysis, and PAC-Bayesian analysis. The latter is further applied to derive generalization guarantees for weighted majority votes. After covering the offline setting, we turn our attention to online learning. We present the space of online learning problems characterized by environmental feedback, environmental resistance, and structural complexity. A common performance measure in online learning is regret, which compares performance of an algorithm to performance of the best prediction rule in hindsight, out of a restricted set of prediction rules. We present tools for deriving regret bounds in stochastic and adversarial environments, and under full information and bandit feedback.

We study the problem of causal structure learning from a combination of observational and interventional data generated by a linear non-Gaussian structural equation model that might contain cycles. Recent results show that using mere observational data identifies the causal graph only up to a permutation-equivalence class. We obtain a combinatorial characterization of this class by showing that each graph in an equivalence class corresponds to a perfect matching in a bipartite graph. This bipartite representation allows us to analyze how interventions modify or constrain the matchings. Specifically, we show that each atomic intervention reveals one edge of the true matching and eliminates all incompatible causal graphs. Consequently, we formalize the optimal experiment design task as an adaptive stochastic optimization problem over the set of equivalence classes with a natural reward function that quantifies how many graphs are eliminated from the equivalence class by an intervention. We show that this reward function is adaptive submodular and provide a greedy policy with a provable near-optimal performance guarantee. A key technical challenge is to efficiently estimate the reward function without having to explicitly enumerate all the graphs in the equivalence class. We propose a sampling-based estimator using random matchings and analyze its bias and concentration behavior. Our simulation results show that performing a small number of interventions guided by our stochastic optimization framework recovers the true underlying causal structure.

In regression and causal inference, controlled subgroup selection aims to identify, with inferential guarantees, a subgroup (defined as a subset of the covariate space) on which the average response or treatment effect is above a given threshold. E.g., in a clinical trial, it may be of interest to find a subgroup with a positive average treatment effect. However, existing methods either lack inferential guarantees, heavily restrict the search for the subgroup, or sacrifice efficiency by naive data splitting. We propose a novel framework called chiseling that allows the analyst to interactively refine and test a candidate subgroup by iteratively shrinking it. The sole restriction is that the shrinkage direction only depends on the points outside the current subgroup, but otherwise the analyst may leverage any prior information or machine learning algorithm. Despite this flexibility, chiseling controls the probability that the discovered subgroup is null (e.g., has a non-positive average treatment effect) under minimal assumptions: for example, in randomized experiments, this inferential validity guarantee holds under only bounded moment conditions. When applied to a variety of simulated datasets and a real survey experiment, chiseling identifies substantially better subgroups than existing methods with inferential guarantees.

This paper proposes a novel method for analyzing food retail processes with a focus on reducing food waste. The approach integrates object-centric process mining (OCPM) with stochastic process discovery and analysis. First, a stochastic process in the form of a continuous-time Markov chain is discovered from grocery store sales data. This model is then extended with supply activities. Finally, a what-if analysis is conducted to evaluate how the quantity of products in the store evolves over time. This enables the identification of an optimal balance between customer purchasing behavior and supply strategies, helping to prevent both food waste due to oversupply and product shortages.

In recent years, diversity has emerged as a useful mechanism to enhance the efficiency of multi-agent reinforcement learning (MARL). However, existing methods predominantly focus on designing policies based on individual agent characteristics, often neglecting the interplay and mutual influence among agents during policy formation. To address this gap, we propose Competitive Diversity through Constructive Conflict (CoDiCon), a novel approach that incorporates competitive incentives into cooperative scenarios to encourage policy exchange and foster strategic diversity among agents. Drawing inspiration from sociological research, which highlights the benefits of moderate competition and constructive conflict in group decision-making, we design an intrinsic reward mechanism using ranking features to introduce competitive motivations. A centralized intrinsic reward module generates and distributes varying reward values to agents, ensuring an effective balance between competition and cooperation. By optimizing the pa-rameterized centralized reward module to maximize environmental rewards, we reformulate the constrained bilevel optimization problem to align with the original task objectives. We evaluate our algorithm against state-of-the-art methods in the SMAC and GRF environments. Experimental results demonstrate that CoDiCon achieves superior performance, with competitive intrinsic rewards effectively promoting diverse and adaptive strategies among cooperative agents.

Model-based reinforcement learning (MBRL) has shown significant potential in robotics due to its high sample efficiency and planning capability. However, extending MBRL to multi-robot cooperation remains challenging due to the complexity of joint dynamics and the reliance on synchronous communication. SeqWM employs independent, autoregressive agent-wise world models to represent joint dynamics, where each agent generates its future trajectory and plans its actions based on the predictions of its predecessors. This design lowers modeling complexity, alleviates the reliance on communication synchronization, and enables the emergence of advanced cooperative behaviors through explicit intention sharing. Experiments in challenging simulated environments (Bi-DexHands and Multi-Quad) demonstrate that SeqWM outperforms existing state-of-the-art model-based and model-free baselines in both overall performance and sample efficiency, while exhibiting advanced cooperative behaviors such as predictive adaptation, temporal alignment, and role division. Furthermore, SeqWM has been success fully deployed on physical quadruped robots, demonstrating its effectiveness in real-world multi-robot systems. Demos and code are available at: https://sites.google.com/view/seqwm-marl

Multimodal electronic health record (EHR) data is useful for disease risk prediction based on medical domain knowledge. However, general medical knowledge must be adapted to specific healthcare settings and patient populations to achieve practical clinical use. Additionally, risk prediction systems must handle uncertainty from incomplete data and non-deterministic health outcomes while remaining explainable. These challenges can be alleviated by the integration of knowledge graphs (KGs) and Bayesian networks (BNs). We present a novel approach for constructing BNs from ontology-based KGs and multimodal EHR data for explainable disease risk prediction. Through an application use case of atrial fibrillation and real-world EHR data, we demonstrate that the approach balances generalised medical knowledge with patient-specific context, effectively handles uncertainty, is highly explainable, and achieves good predictive performance.

This work presents a forward-only diffusion (FoD) approach for generative modelling. In contrast to traditional diffusion models that rely on a coupled forward-backward diffusion scheme, FoD directly learns data generation through a single forward diffusion process, yielding a simple yet efficient generative framework. The core of FoD is a state-dependent stochastic differential equation that involves a mean-reverting term in both the drift and diffusion functions. This mean-reversion property guarantees the convergence to clean data, naturally simulating a stochastic interpolation between source and target distributions. More importantly, FoD is analytically tractable and is trained using a simple stochastic flow matching objective, enabling a few-step non-Markov chain sampling during inference. The proposed FoD model, despite its simplicity, achieves state-of-the-art performance on various image restoration tasks. Its general applicability on image-conditioned generation is also demonstrated via qualitative results on image-to-image translation. Our code is available at https://github.com/Algolzw/FoD.