Delayed Assignments in Online Non-Centroid Clustering with Stochastic Arrivals

Clustering is a fundamental problem, aiming to partition a set of elements, like agents or data points, into clusters such that elements in the same cluster are closer to each other than to those in other clusters. In this paper, we present a new framework for studying online non-centroid clustering with delays, where elements, that arrive one at a time as points in a finite metric space, should be assigned to clusters, but assignments need not be immediate. Specifically, upon arrival, each point's location is revealed, and an online algorithm has to irrevocably assign it to an existing cluster or create a new one containing, at this moment, only this point. However, we allow decisions to be postponed at a delay cost, instead of following the more common assumption of immediate decisions upon arrival. This poses a critical challenge: the goal is to minimize both the total distance costs between points in each cluster and the overall delay costs incurred by postponing assignments. In the classic worst-case arrival model, where points arrive in an arbitrary order, no algorithm has a competitive ratio better than sublogarithmic in the number of points. To overcome this strong impossibility, we focus on a stochastic arrival model, where points' locations are drawn independently across time from an unknown and fixed probability distribution over the finite metric space. We offer hope for beyond worst-case adversaries: we devise an algorithm that is constant competitive in the sense that, as the number of points grows, the ratio between the expected overall costs of the output clustering and an optimal offline clustering is bounded by a constant.

Convexified Message-Passing Graph Neural Networks

Graph Neural Networks (GNNs) have become prominent methods for graph representation learning, demonstrating strong empirical results on diverse graph prediction tasks. In this paper, we introduce Convexified Message Passing Graph Neural Networks (CGNNs), a novel and general framework that combines the power of message-passing GNNs with the tractability of convex optimization. By mapping their nonlinear filters into a reproducing kernel Hilbert space, CGNNs transform training into a convex optimization problem, which can be solved efficiently and optimally by projected gradient methods. This convexity further allows the statistical properties of CGNNs to be analyzed accurately and rigorously. For two-layer CGNNs, we establish rigorous generalization guarantees, showing convergence to the performance of the optimal GNN. To scale to deeper architectures, we adopt a principled layer-wise training strategy. Experiments on benchmark datasets show that CGNNs significantly exceed the performance of leading GNN models, achieving 10 to 40 percent higher accuracy in most cases, underscoring their promise as a powerful and principled method with strong theoretical foundations. In rare cases where improvements are not quantitatively substantial, the convex models either slightly exceed or match the baselines, stressing their robustness and wide applicability. Though over-parameterization is often employed to enhance performance in nonconvex models, we show that our CGNNs framework yields shallow convex models that can surpass these models in both accuracy and resource efficiency.