To this end, we first establish a new class of GNNs that can solve a strictly wider variety of problems than existing GNNs. Then, we bridge the gap between GNN theory and the theory of distributed local algorithms.

Descriptive set theorists study the niche mathematics of infinity. Now, they've shown that their problems can be rewritten in the concrete language of algorithms. All of modern mathematics is built on the foundation of set theory, the study of how to organize abstract collections of objects. But in general, research mathematicians don't need to think about it when they're solving their problems. They can take it for granted that sets behave the way they'd expect, and carry on with their work. Descriptive set theorists are an exception. This small community of mathematicians never stopped studying the fundamental nature of sets--particularly the strange infinite ones that other mathematicians ignore. Their field just got a lot less lonely. In 2023, a mathematician named Anton Bernshteyn published a deep and surprising connection between the remote mathematical frontier of descriptive set theory and modern computer science.

In this paper, from a theoretical perspective, we study how powerful graph neural networks (GNNs) can be for learning approximation algorithms for combinatorial problems. To this end, we first establish a new class of GNNs that can solve a strictly wider variety of problems than existing GNNs. Then, we bridge the gap between GNN theory and the theory of distributed local algorithms. We theoretically demonstrate that the most powerful GNN can learn approximation algorithms for the minimum dominating set problem and the minimum vertex cover problem with some approximation ratios with the aid of the theory of distributed local algorithms. We also show that most of the existing GNNs such as GIN, GAT, GCN, and GraphSAGE cannot perform better than with these ratios. This paper is the first to elucidate approximation ratios of GNNs for combinatorial problems. Furthermore, we prove that adding coloring or weak-coloring to each node feature improves these approximation ratios. This indicates that preprocessing and feature engineering theoretically strengthen model capabilities.

To this end, we first establish a new class of GNNs that can solve a strictly wider variety of problems than existing GNNs. Then, we bridge the gap between GNN theory and the theory of distributed local algorithms.

However, in real-world applications, the network structure is typically dynamic, with nodes that join over time. In this setting, we would like a detection algorithm to perform only a limited number of updates at each node arrival. While standard voting approaches satisfy this constraint, it is unclear whether they exploit the network information optimally. We introduce a simple model for networks growing over time which we refer to as streaming stochastic block model (StSBM). Within this model, we prove that voting algorithms have fundamental limitations.

Graph clustering is an important algorithmic technique for analysing massive graphs, and has been widely applied in many research fields of data science. While the objective of most graph clustering algorithms is to find a vertex set of low conductance, a sequence of recent studies highlights the importance of the inter-connection between vertex sets when analysing real-world datasets. Following this line of research, in this work we study bipartite-like clusters and present efficient and online sparsification algorithms that find such clusters in both undirected graphs and directed ones. We conduct experimental studies on both synthetic and real-world datasets, and show that our algorithms significantly speedup the running time of existing clustering algorithms while preserving their effectiveness.

Weaknesses: Main criticism: 1) The paper claims two main contributions, one of which is "The first contribution of this paper is to analyze some past procedures, and show that even in the favorable setting of deterministic updates (i.e., no stochastic approximation used), these methods typically fail to preserve solutions of the original optimization problem as fixed points " I believe the text above is misleading. In fact, it was already well known for the "past procedures" to not have the correct fixed points; one alternative approach to deal with such an issue was to incorporate the "drift"; see https://arxiv.org/abs/1910.06378 for example. Therefore, believe it would be more appropriate to not claim the contribution for showing the wrong fixed point of the local algorithms. Specifically, in strongly convex case (same arguments apply for weakly convex), the communication complexity of FedSplit is O(sqrt(kappa)log(1/epsilon)), which is identical to the communication complexity of AGD. In fact, AGD is favorable (in terms of the rate) as is requires a single gradient evaluation instead of evaluating the prox with high enough precision so that the inexactness does not drive the rate.

In this paper, from a theoretical perspective, we study how powerful graph neural networks (GNNs) can be for learning approximation algorithms for combinatorial problems. To this end, we first establish a new class of GNNs that can solve a strictly wider variety of problems than existing GNNs. Then, we bridge the gap between GNN theory and the theory of distributed local algorithms. We theoretically demonstrate that the most powerful GNN can learn approximation algorithms for the minimum dominating set problem and the minimum vertex cover problem with some approximation ratios with the aid of the theory of distributed local algorithms. We also show that most of the existing GNNs such as GIN, GAT, GCN, and GraphSAGE cannot perform better than with these ratios.

We study the oracle complexity of nonsmooth nonconvex optimization, with the algorithm assumed to have access only to local function information. It has been shown by Davis, Drusvyatskiy, and Jiang (2023) that for nonsmooth Lipschitz functions satisfying certain regularity and strictness conditions, perturbed gradient descent converges to local minimizers asymptotically. Motivated by this result and by other recent algorithmic advances in nonconvex nonsmooth optimization concerning Goldstein stationarity, we consider the question of obtaining a non-asymptotic rate of convergence to local minima for this problem class. We provide the following negative answer to this question: Local algorithms acting on regular Lipschitz functions cannot, in the worst case, provide meaningful local guarantees in terms of function value in sub-exponential time, even when all near-stationary points are global minima. This sharply contrasts with the smooth setting, for which it is well-known that standard gradient methods can do so in a dimension-independent rate. Our result complements the rich body of work in the theoretical computer science literature that provide hardness results conditional on conjectures such as $\mathsf{P}\neq\mathsf{NP}$ or cryptographic assumptions, in that ours holds unconditional of any such assumptions.