Finding Decision Tree Splits in Streaming and Massively Parallel Models

In this work, we provide data stream algorithms that compute optimal splits in decision tree learning. In particular, given a data stream of observations $x_i$ and their labels $y_i$, the goal is to find the optimal split point $j$ that divides the data into two sets such that the mean squared error (for regression) or misclassification rate (for classification) is minimized. We provide various fast streaming algorithms that use sublinear space and a small number of passes for these problems. These algorithms can also be extended to the massively parallel computation model. Our work, while not directly comparable, complements the seminal work of Domingos and Hulten (KDD 2000).

Optimizing Polynomial Graph Filters: A Novel Adaptive Krylov Subspace Approach

Graph Neural Networks (GNNs), known as spectral graph filters, find a wide range of applications in web networks. To bypass eigendecomposition, polynomial graph filters are proposed to approximate graph filters by leveraging various polynomial bases for filter training. However, no existing studies have explored the diverse polynomial graph filters from a unified perspective for optimization. In this paper, we first unify polynomial graph filters, as well as the optimal filters of identical degrees into the Krylov subspace of the same order, thus providing equivalent expressive power theoretically. Next, we investigate the asymptotic convergence property of polynomials from the unified Krylov subspace perspective, revealing their limited adaptability in graphs with varying heterophily degrees. Inspired by those facts, we design a novel adaptive Krylov subspace approach to optimize polynomial bases with provable controllability over the graph spectrum so as to adapt various heterophily graphs. Subsequently, we propose AdaptKry, an optimized polynomial graph filter utilizing bases from the adaptive Krylov subspaces. Meanwhile, in light of the diverse spectral properties of complex graphs, we extend AdaptKry by leveraging multiple adaptive Krylov bases without incurring extra training costs. As a consequence, extended AdaptKry is able to capture the intricate characteristics of graphs and provide insights into their inherent complexity. We conduct extensive experiments across a series of real-world datasets. The experimental results demonstrate the superior filtering capability of AdaptKry, as well as the optimized efficacy of the adaptive Krylov basis.