We study the node classification problem on feature-decorated graphs in the sparse setting, i.e., when the expected degree of a node is $O(1)$ in the number of nodes, in the fixed-dimensional asymptotic regime, i.e., the dimension of the feature data is fixed while the number of nodes is large. Such graphs are typically known to be locally tree-like. We introduce a notion of Bayes optimality for node classification tasks, called asymptotic local Bayes optimality, and compute the optimal classifier according to this criterion for a fairly general statistical data model with arbitrary distributions of the node features and edge connectivity. The optimal classifier is implementable using a message-passing graph neural network architecture. We then compute the generalization error of this classifier and compare its performance against existing learning methods theoretically on a well-studied statistical model with naturally identifiable signal-to-noise ratios (SNRs) in the data. We find that the optimal message-passing architecture interpolates between a standard MLP in the regime of low graph signal and a typical convolution in the regime of high graph signal. Furthermore, we prove a corresponding non-asymptotic result.

We study the node classification problem on feature-decorated graphs in the sparse setting, i.e., when the expected degree of a node is O(1) in the number of nodes, in the fixed-dimensional asymptotic regime, i.e., the dimension of the feature data is fixed while the number of nodes is large. Such graphs are typically known to be locally tree-like. We introduce a notion of Bayes optimality for node classification tasks, called asymptotic local Bayes optimality, and compute the optimal classifier according to this criterion for a fairly general statistical data model with arbitrary distributions of the node features and edge connectivity. The optimal classifier is implementable using a message-passing graph neural network architecture. We then compute the generalization error of this classifier and compare its performance against existing learning methods theoretically on a well-studied statistical model with naturally identifiable signal-to-noise ratios (SNRs) in the data.

Distributed optimization has experienced a significant surge in interest due to its wide-ranging applications in distributed learning and adaptation. While various scenarios, such as shared-memory, local-memory, and consensus-based approaches, have been extensively studied in isolation, there remains a need for further exploration of their interconnections. This paper specifically concentrates on a scenario where agents collaborate toward a unified mission while potentially having distinct tasks. Each agent's actions can potentially impact other agents through interactions. Within this context, the objective for the agents is to optimize their local parameters based on the aggregate of local reward functions, where only local zeroth-order oracles are available. Notably, the learning process is asynchronous, meaning that agents update and query their zeroth-order oracles asynchronously while communicating with other agents subject to bounded but possibly random communication delays. This paper presents theoretical convergence analyses and establishes a convergence rate for the proposed approach. Furthermore, it addresses the relevant issue of deep learning-based resource allocation in communication networks and conducts numerical experiments in which agents, acting as transmitters, collaboratively train their individual (possibly unique) policies to maximize a common performance metric.

We use neural graph networks with a message-passing architecture and an attention mechanism to enhance the branching heuristic in two SATsolving algorithms. We report improvements of learned neural heuristics compared with two standard human-designed heuristics. We compare the performance in terms of number of branching decisions and show the possibility of enhancing the performance of SAT solvers with the help of learned heuristics. A similar graph representation, but more general in order to accommodate for higher-order logic is used in FormulaNet presented in [WTWD17]. To the best of our knowledge the FormulaNet architecture was never used for neural guidance.