Fast, Parallel, Query-Efficient Binary Classification

We study the fundamental classification problem of computing a separating hyperplane for a binary-labeled dataset of size $n$ with normalized $d$-dimensional features. Letting $\Phi \in \mathbb{R}^{n \times d}$ denote the feature matrix and $\gamma$ the margin of the maximum-margin separating hyperplane, we present a randomized algorithm that solves this problem in $\tilde{O}(\gamma^{-2/3}\, \operatorname{nnz}(\Phi) + \gamma^{-2(\omega+1)/3})$-sequential running time (work), $\tilde{O}(\gamma^{-2/3})$-parallel (computational) depth, and accesses $\Phi$ only through $\tilde{O}(\gamma^{-2/3})$-matrix-vector queries (matvecs). We also present a second, faster randomized algorithm with a $\tilde{O}(\gamma^{-2/3}\, \operatorname{nnz}(\Phi) + \gamma^{-2})$-sequential running time that uses $\tilde{O}(\gamma^{-2/3})$-matvecs to $\Phi$, but achieves only $\tilde{O}(\gamma^{-4/3})$-parallel depth. Both algorithms match the near-optimal deterministic matvec complexity recently established by Kornowski and Shamir [2025], Karmarkar et al. [2026] and achieve improved sequential runtime and parallel depth, albeit at the expense of using randomness.