Abstract:We establish nearly optimal upper and lower bounds for approximating decision tree splits in data streams. For regression with labels in the range $\{0,1,\ldots,M\}$, we give a one-pass algorithm using $\tilde{O}(M^2/\epsilon)$ space that outputs a split within additive $\epsilon$ error of the optimal split, improving upon the two-pass algorithm of Pham et al. (ISIT 2025). Furthermore, we provide a matching one-pass lower bound showing that $\Omega(M^2/\epsilon)$ space is indeed necessary.
For classification, we also obtain a one-pass algorithm using $\tilde{O}(1/\epsilon)$ space for approximating the optimal Gini split, improving upon the previous $\tilde{O}(1/\epsilon^2)$-space algorithm. We complement these results with matching space lower bounds: $\Omega(1/\epsilon)$ for Gini impurity and $\Omega(1/\epsilon)$ for misclassification (which matches the upper bound obtained by sampling).
Our algorithms exploit the Lipschitz property of the loss functions and use reservoir sampling along with Count--Min sketches with range queries. Our lower bounds follow from careful reductions from the INDEX problem.
From: Hoa Vu [view email]
[v1]
Wed, 22 Apr 2026 09:56:59 UTC (229 KB)
[v2]
Tue, 7 Jul 2026 01:08:27 UTC (232 KB)
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