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Recently, subgraph GNNs have emerged as an important direction for developing expressive graph neural networks (GNNs). While numerous architectures have been proposed, so far there is still a limited understanding of how various design paradigms differ in terms of expressive power, nor is it clear what design principle achieves maximal expressiveness with minimal architectural complexity. To address these fundamental questions, this paper conducts a systematic study of general node-based subgraph GNNs through the lens of Subgraph Weisfeiler-Lehman Tests (SWL). Our central result is to build a complete hierarchy of SWL with strictly growing expressivity. Concretely, we prove that any node-based subgraph GNN falls into one of the six SWL equivalence classes, among which $\mathsf{SSWL}$ achieves the maximal expressive power. We also study how these equivalence classes differ in terms of their practical expressiveness such as encoding graph distance and biconnectivity. Furthermore, we give a tight expressivity upper bound of all SWL algorithms by establishing a close relation with localized versions of WL and Folklore WL (FWL) tests. Our results provide insights into the power of existing subgraph GNNs, guide the design of new architectures, and point out their limitations by revealing an inherent gap with the 2-FWL test. Finally, experiments demonstrate that $\mathsf{SSWL}$-inspired subgraph GNNs can significantly outperform prior architectures on multiple benchmarks despite great simplicity.

A distributed system is permissionless when participants can join and leave the network without permission from a central authority. Many modern distributed systems are naturally permissionless, in the sense that a central permissioning authority would defeat their design purpose: this includes blockchains, filesharing protocols, some voting systems, and more. By their permissionless nature, such systems are heterogeneous: participants may only have a partial view of the system, and they may also have different goals and beliefs. Thus, the traditional notion of consensus -- i.e. system-wide agreement -- may not be adequate, and we may need to generalise it. This is a challenge: how should we understand what heterogeneous consensus is; what mathematical framework might this require; and how can we use this to build understanding and mathematical models of robust, effective, and secure permissionless systems in practice? We analyse heterogeneous consensus using semitopology as a framework. This is like topology, but without the restriction that intersections of opens be open. Semitopologies have a rich theory which is related to topology, but with its own distinct character and mathematics. We introduce novel well-behavedness conditions, including an anti-Hausdorff property and a new notion of `topen set', and we show how these structures relate to consensus. We give a restriction of semitopologies to witness semitopologies, which are an algorithmically tractable subclass corresponding to Horn clause theories, having particularly good mathematical properties. We introduce and study several other basic notions that are specific and novel to semitopologies, and study how known quantities in topology, such as dense subsets and closures, display interesting and useful new behaviour in this new semitopological context.

A one-dimensional sequence $u_0, u_1, u_2, \ldots \in [0, 1)$ is said to be completely uniformly distributed (CUD) if overlapping $s$-blocks $(u_i, u_{i+1}, \ldots , u_{i+s-1})$, $i = 0, 1, 2, \ldots$, are uniformly distributed for every dimension $s \geq 1$. This concept naturally arises in Markov chain quasi-Monte Carlo (QMC). However, the definition of CUD sequences is not constructive, and thus there remains the problem of how to implement the Markov chain QMC algorithm in practice. Harase (2021) focused on the $t$-value, which is a measure of uniformity widely used in the study of QMC, and implemented short-period Tausworthe generators (i.e., linear feedback shift register generators) over the two-element field $\mathbb{F}_2$ that approximate CUD sequences by running for the entire period. In this paper, we generalize a search algorithm over $\mathbb{F}_2$ to that over arbitrary finite fields $\mathbb{F}_b$ with $b$ elements and conduct a search for Tausworthe generators over $\mathbb{F}_b$ with $t$-values zero (i.e., optimal) for dimension $s = 3$ and small for $s \geq 4$, especially in the case where $b = 3, 4$, and $5$. We provide a parameter table of Tausworthe generators over $\mathbb{F}_4$, and report a comparison between our new generators over $\mathbb{F}_4$ and existing generators over $\mathbb{F}_2$ in numerical examples using Markov chain QMC.

We consider the problem of identification of linear dynamical systems from a single trajectory. Recent results have predominantly focused on the setup where no structural assumption is made on the system matrix $A^* \in \mathbb{R}^{n \times n}$, and have consequently analyzed the ordinary least squares (OLS) estimator in detail. We assume prior structural information on $A^*$ is available, which can be captured in the form of a convex set $\mathcal{K}$ containing $A^*$. For the solution of the ensuing constrained least squares estimator, we derive non-asymptotic error bounds in the Frobenius norm which depend on the local size of the tangent cone of $\mathcal{K}$ at $A^*$. To illustrate the usefulness of this result, we instantiate it for the settings where, (i) $\mathcal{K}$ is a $d$ dimensional subspace of $\mathbb{R}^{n \times n}$, or (ii) $A^*$ is $k$-sparse and $\mathcal{K}$ is a suitably scaled $\ell_1$ ball. In the regimes where $d, k \ll n^2$, our bounds improve upon those obtained from the OLS estimator.

Due to mutual interference between users, power allocation problems in wireless networks are often non-convex and computationally challenging. Graph neural networks (GNNs) have recently emerged as a promising approach to tackling these problems and an approach that exploits the underlying topology of wireless networks. In this paper, we propose a novel graph representation method for wireless networks that include full-duplex (FD) nodes. We then design a corresponding FD Graph Neural Network (F-GNN) with the aim of allocating transmit powers to maximise the network throughput. Our results show that our F-GNN achieves state-of-art performance with significantly less computation time. Besides, F-GNN offers an excellent trade-off between performance and complexity compared to classical approaches. We further refine this trade-off by introducing a distance-based threshold for inclusion or exclusion of edges in the network. We show that an appropriately chosen threshold reduces required training time by roughly 20% with a relatively minor loss in performance.

With more and more data being collected, data-driven modeling methods have been gaining in popularity in recent years. While physically sound, classical gray-box models are often cumbersome to identify and scale, and their accuracy might be hindered by their limited expressiveness. On the other hand, classical black-box methods, typically relying on Neural Networks (NNs) nowadays, often achieve impressive performance, even at scale, by deriving statistical patterns from data. However, they remain completely oblivious to the underlying physical laws, which may lead to potentially catastrophic failures if decisions for real-world physical systems are based on them. Physically Consistent Neural Networks (PCNNs) were recently developed to address these aforementioned issues, ensuring physical consistency while still leveraging NNs to attain state-of-the-art accuracy. In this work, we scale PCNNs to model building temperature dynamics and propose a thorough comparison with classical gray-box and black-box methods. More precisely, we design three distinct PCNN extensions, thereby exemplifying the modularity and flexibility of the architecture, and formally prove their physical consistency. In the presented case study, PCNNs are shown to achieve state-of-the-art accuracy, even outperforming classical NN-based models despite their constrained structure. Our investigations furthermore provide a clear illustration of NNs achieving seemingly good performance while remaining completely physics-agnostic, which can be misleading in practice. While this performance comes at the cost of computational complexity, PCNNs on the other hand show accuracy improvements of 17-35% compared to all other physically consistent methods, paving the way for scalable physically consistent models with state-of-the-art performance.

Existing analysis of AdaGrad and other adaptive methods for smooth convex optimization is typically for functions with bounded domain diameter. In unconstrained problems, previous works guarantee an asymptotic convergence rate without an explicit constant factor that holds true for the entire function class. Furthermore, in the stochastic setting, only a modified version of AdaGrad, different from the one commonly used in practice, in which the latest gradient is not used to update the stepsize, has been analyzed. Our paper aims at bridging these gaps and developing a deeper understanding of AdaGrad and its variants in the standard setting of smooth convex functions as well as the more general setting of quasar convex functions. First, we demonstrate new techniques to explicitly bound the convergence rate of the vanilla AdaGrad for unconstrained problems in both deterministic and stochastic settings. Second, we propose a variant of AdaGrad for which we can show the convergence of the last iterate, instead of the average iterate. Finally, we give new accelerated adaptive algorithms and their convergence guarantee in the deterministic setting with explicit dependency on the problem parameters, improving upon the asymptotic rate shown in previous works.

In this paper we study the finite sample and asymptotic properties of various weighting estimators of the local average treatment effect (LATE), several of which are based on Abadie's (2003) kappa theorem. Our framework presumes a binary treatment and a binary instrument, which may only be valid after conditioning on additional covariates. We argue that one of the Abadie estimators, which is weight normalized, is preferable in many contexts. Several other estimators, which are unnormalized, do not generally satisfy the properties of scale invariance with respect to the natural logarithm and translation invariance, thereby exhibiting sensitivity to the units of measurement when estimating the LATE in logs and the centering of the outcome variable more generally. On the other hand, when noncompliance is one-sided, certain unnormalized estimators have the advantage of being based on a denominator that is bounded away from zero. To reconcile these findings, we demonstrate that when the instrument propensity score is estimated using an appropriate covariate balancing approach, the resulting normalized estimator also shares this advantage. We use a simulation study and three empirical applications to illustrate our findings. In two cases, the unnormalized estimates are clearly unreasonable, with "incorrect" signs, magnitudes, or both.

Relational verification encompasses information flow security, regression verification, translation validation for compilers, and more. Effective alignment of the programs and computations to be related facilitates use of simpler relational invariants and relational procedure specs, which in turn enables automation and modular reasoning. Alignment has been explored in terms of trace pairs, deductive rules of relational Hoare logics (RHL), and several forms of product automata. This article shows how a simple extension of Kleene Algebra with Tests (KAT), called BiKAT, subsumes prior formulations, including alignment witnesses for forall-exists properties, which brings to light new RHL-style rules for such properties. Alignments can be discovered algorithmically or devised manually but, in either case, their adequacy with respect to the original programs must be proved; an explicit algebra enables constructive proof by equational reasoning. Furthermore our approach inherits algorithmic benefits from existing KAT-based techniques and tools, which are applicable to a range of semantic models.

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.