Physics-inspired graph neural networks (PI-GNNs) have been utilized as an efficient unsupervised framework for relaxing combinatorial optimization problems encoded through a specific graph structure and loss, reflecting dependencies between the problem's variables. While the framework has yielded promising results in various combinatorial problems, we show that the performance of PI-GNNs systematically plummets with an increasing density of the combinatorial problem graphs. Our analysis reveals an interesting phase transition in the PI-GNNs' training dynamics, associated with degenerate solutions for the denser problems, highlighting a discrepancy between the relaxed, real-valued model outputs and the binary-valued problem solutions. To address the discrepancy, we propose principled alternatives to the naive strategy used in PI-GNNs by building on insights from fuzzy logic and binarized neural networks. Our experiments demonstrate that the portfolio of proposed methods significantly improves the performance of PI-GNNs in increasingly dense settings.

Dynamic treatment regimes are sequential decision rules that adapt treatment according to individual time-varying characteristics and outcomes to achieve optimal effects, with applications in precision medicine, personalized recommendations, and dynamic marketing. Estimating optimal dynamic treatment regimes via sequential randomized trials might face costly and ethical hurdles, often necessitating the use of historical observational data. In this work, we utilize proximal causal inference framework for learning optimal dynamic treatment regimes when the unconfoundedness assumption fails. Our contributions are four-fold: (i) we propose three nonparametric identification methods for optimal dynamic treatment regimes; (ii) we establish the semiparametric efficiency bound for the value function of a given regime; (iii) we propose a (K+1)-robust method for learning optimal dynamic treatment regimes, where K is the number of stages; (iv) as a by-product for marginal structural models, we establish identification and estimation of counterfactual means under a static regime. Numerical experiments validate the efficiency and multiple robustness of our proposed methods.

Instrumental variable methods are fundamental to causal inference when treatment assignment is confounded by unobserved variables. In this article, we develop a general nonparametric framework for identification and learning with multi-categorical or continuous instrumental variables. Specifically, we propose an additive instrumental variable framework to identify mean potential outcomes and the average treatment effect with a weighting function. Leveraging semiparametric theory, we derive efficient influence functions and construct consistent, asymptotically normal estimators via debiased machine learning. Extensions to longitudinal data, dynamic treatment regimes, and multiplicative instrumental variables are further developed. We demonstrate the proposed method by employing simulation studies and analyzing real data from the Job Training Partnership Act program.

Expectile regression neural networks (ERNNs) are powerful tools for capturing heterogeneity and complex nonlinear structures in data. However, most existing research has primarily focused on fully observed data, with limited attention paid to scenarios involving censored observations. In this paper, we propose a data augmentation-based ERNNs algorithm, termed DAERNN, for modeling heterogeneous censored data. The proposed DAERNN is fully data-driven, requires minimal assumptions, and offers substantial flexibility. Simulation studies and real-data applications demonstrate that DAERNN outperforms existing censored ERNNs methods and achieves predictive performance comparable to models trained on fully observed data. Moreover, the algorithm provides a unified framework for handling various censoring mechanisms without requiring explicit parametric model specification, thereby enhancing its applicability to practical censored data analysis. Introduction Expectile regression (ER), first proposed by Newey and Powell (1987), has been extensively studied as a flexible alternative to quantile regression (QR) for modeling heterogeneous distributions.

We propose an approach for dynamic efficiency evaluation across multiple organizational dimensions using data envelopment analysis (DEA). The method generates both dimension-specific and aggregate efficiency scores, incorporates desirable and undesirable outputs, and is suitable for large-scale problem settings. Two regularized DEA models are introduced: a slack-based measure (SBM) and a linearized version of a nonlinear goal programming model (GP-SBM). While SBM estimates an aggregate efficiency score and then distributes it across dimensions, GP-SBM first estimates dimension-level efficiencies and then derives an aggregate score. Both models utilize a regularization parameter to enhance discriminatory power while also directly integrating both desirable and undesirable outputs. We demonstrate the computational efficiency and validity of our approach on multiple datasets and apply it to a case study of twelve hospitals in Ontario, Canada, evaluating three theoretically grounded dimensions of organizational effectiveness over a 24-month period from January 2018 to December 2019: technical efficiency, clinical efficiency, and patient experience. Our numerical results show that SBM and GP-SBM better capture correlations among input/output variables and outperform conventional benchmarking methods that separately evaluate dimensions before aggregation.

In networked environments, users frequently share recommendations about content, products, services, and courses of action with others. The extent to which such recommendations are successful and adopted is highly contextual, dependent on the characteristics of the sender, recipient, their relationship, the recommended item, and the medium, which makes peer influence probabilities highly heterogeneous. Accurate estimation of these probabilities is key to understanding information diffusion processes and to improving the effectiveness of viral marketing strategies. However, learning these probabilities from data is challenging; static data may capture correlations between peer recommendations and peer actions but fails to reveal influence relationships. Online learning algorithms can learn these probabilities from interventions but either waste resources by learning from random exploration or optimize for rewards, thus favoring exploration of the space with higher influence probabilities. In this work, we study learning peer influence probabilities under a contextual linear bandit framework. We show that a fundamental trade-off can arise between regret minimization and estimation error, characterize all achievable rate pairs, and propose an uncertainty-guided exploration algorithm that, by tuning a parameter, attains any pair within this trade-off. Our experiments on semi-synthetic network datasets show the advantages of our method over static methods and contextual bandits that ignore this trade-off.

In this work, we establish non-asymptotic Berry-Esseen bounds for linear functionals of online least-squares SGD, thereby providing a Gaussian Central Limit Theorem (CL T) in a growing-dimensional regime. To render the theory practically applicable, we further develop an online variance estimator for the asymptotic variance appearing in the CL T and establish high-probability deviation bounds for this estimator. Stochastic gradient descent [56] is a popular optimization algorithm widely used in data science. It is a stochastic iterative method for minimizing the expected loss function by updating model parameters based on the (stochastic) gradient of the loss with respect to the parameters obtained from a random sample. SGD is widely used for training linear and logistic regression models, support vector machines, deep neural networks, and other such machine learning models on large-scale datasets. Because of its simplicity and effectiveness, SGD has become a staple of modern data science and machine learning, and has been continuously improved and extended to handle more complex scenarios. Despite its wide-spread applicability for prediction and point estimation, quantifying the uncertainty associated with SGD is not well-understood. Indeed, uncertainty quantification is a key component of decision making systems, ensuring the credibility and validity of data-driven findings; see, for e.g., [17], for a concrete medical application where it is not enough to just optimize SGD to obtain prediction performance but is more important to quantify the associated uncertainty.

Understanding the dynamics of feature learning in neural networks (NNs) remains a significant challenge. The work of (Mousavi-Hosseini et al., 2023) analyzes a multiple index teacher-student setting and shows that a two-layer student attains a low-rank structure in its first-layer weights when trained with stochastic gradient descent (SGD) and a strong regularizer. This structural property is known to reduce sample complexity of generalization. Indeed, in a second step, the same authors establish algorithm-specific learning guarantees under additional assumptions. In this paper, we focus exclusively on the structure discovery aspect and study it under weaker assumptions, more specifically: we allow (a) NNs of arbitrary size and depth, (b) with all parameters trainable, (c) under any smooth loss function, (d) tiny regularization, and (e) trained by any method that attains a second-order stationary point (SOSP), e.g.\ perturbed gradient descent (PGD). At the core of our approach is a key $\textit{derandomization}$ lemma, which states that optimizing the function $\mathbb{E}_{\mathbf{x}} \left[g_θ(\mathbf{W}\mathbf{x} + \mathbf{b})\right]$ converges to a point where $\mathbf{W} = \mathbf{0}$, under mild conditions. The fundamental nature of this lemma directly explains structure discovery and has immediate applications in other domains including an end-to-end approximation for MAXCUT, and computing Johnson-Lindenstrauss embeddings.

Directed graphs naturally model systems with asymmetric, ordered relationships, essential to applications in biology, transportation, social networks, and visual understanding. Generating such graphs enables tasks such as simulation, data augmentation and novel instance discovery; however, directed graph generation remains underexplored. We identify two key factors limiting progress in this direction: first, modeling edge directionality introduces a substantially larger dependency space, making the underlying distribution harder to learn; second, the absence of standardized benchmarks hinders rigorous evaluation. Addressing the former requires more expressive models that are sensitive to directional topologies. We propose Directo, the first generative model for directed graphs built upon the discrete flow matching framework. Our approach combines: (i) principled positional encodings tailored to asymmetric pairwise relations, (ii) a dual-attention mechanism capturing both incoming and outgoing dependencies, and (iii) a robust, discrete generative framework. To support evaluation, we introduce a benchmark suite covering synthetic and real-world datasets. It shows that our method performs strongly across diverse settings and even competes with specialized models for particular classes, such as directed acyclic graphs. Our results highlight the effectiveness and generality of our approach, establishing a solid foundation for future research in directed graph generation.

This paper establishes a continuous time approximation, a piece-wise continuous differential equation, for the discrete Heavy-Ball (HB) momentum method with explicit discretization error. Investigating continuous differential equations has been a promising approach for studying the discrete optimization methods. Despite the crucial role of momentum in gradient-based optimization methods, the gap between the original discrete dynamics and the continuous time approximations due to the discretization error has not been comprehensively bridged yet. In this work, we study the HB momentum method in continuous time while putting more focus on the discretization error to provide additional theoretical tools to this area. In particular, we design a first-order piece-wise continuous differential equation, where we add a number of counter terms to account for the discretization error explicitly. As a result, we provide a continuous time model for the HB momentum method that allows the control of discretization error to arbitrary order of the step size. As an application, we leverage it to find a new implicit regularization of the directional smoothness and investigate the implicit bias of HB for diagonal linear networks, indicating how our results can be used in deep learning. Our theoretical findings are further supported by numerical experiments.