Beyond Averaging in John Ellipsoid Approximation: High-Accuracy Algorithms in the Leverage-Score Model

2026-06-18

AI Summary

This work addresses the high computational cost of traditional leverage-score-based algorithms for approximating the John ellipsoid to within a $(1+\varepsilon)$ factor, whose iteration complexity scales with $\varepsilon^{-1}$ and thus hinders high-precision efficiency. By reframing the problem as D-optimal design, the paper reveals that the $\varepsilon^{-1}$ dependence stems solely from conventional certification mechanisms. It proposes a novel strategy that focuses exclusively on the final iteration and avoids solution averaging. Leveraging leverage scores as a first-order gradient oracle, the method combines warm-start acceleration with a damped Newton algorithm exploiting the self-concordance of the barrier function to accurately recover the Hessian. This approach achieves a $(1+\varepsilon)$-approximation using only $O(d^2 \log\log(1/\varepsilon))$ oracle queries and incurs a preprocessing cost $C(A)$ independent of $\varepsilon$, substantially reducing the overhead for high-accuracy computation.

Abstract

The John ellipsoid of a symmetric polytope $P=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{A}\mathbf{x}\|_\infty\le1\}$, $\mathbf{A}\in\mathbb{R}^{n\times d}$, is computed by a long line of leverage-score algorithms, from Cohen, Cousins, Lee and Yang (COLT 2019) to its successors [WY24, CLS+25], all reaching a $(1+\varepsilon)$-approximation in $Θ(\varepsilon^{-1}\log(n/d))$ iterations. We separate this complexity into three costs the modern line conflates (certification, identification, and accuracy) and locate the historical $\varepsilon^{-1}$ in the first alone. In the equivalent D-optimal-design form $\min_{\mathbf{p}\inΔ_n}-\log\det(\sum_i p_i\mathbf{a}_i\mathbf{a}_i^\top)$, the leverage-score oracle is exactly the first-order oracle and the $(1+\varepsilon)$-John guarantee the Frank-Wolfe gap $g(\mathbf{p})\le\varepsilon d$; through this dictionary the costs come apart. The $\varepsilon^{-1}$ is a certification artifact: the uniform average of the iterates, the certificate used throughout the line, has gap exactly $Θ(1/T)$, however cheap each iteration is made. Pointed instead at the last iterate the same oracle is fast: a warm-started accelerated method reaches the guarantee in $C(\mathbf{A})+O(\sqrtκ\log(1/\varepsilon))$ queries after an $\varepsilon$-independent setup $C(\mathbf{A})$, and once the optimal face is identified the facial problem is an unconstrained self-concordant minimization whose Hessian the oracle recovers exactly, so damped Newton needs only $O(\log\log(1/\varepsilon))$ steps, for a total of $C(\mathbf{A})+O(d^2\log\log(1/\varepsilon))$ queries. The accuracy dependence is thus doubly logarithmic after an $\varepsilon$-independent, condition-dependent setup; the open problem is the remaining identification cost (a condition-free bound on reaching the optimal face) and lower bounds. Accuracy is not the obstruction.

Problem

• John ellipsoid
• leverage-score
• D-optimal design
• Frank-Wolfe gap
• high-accuracy approximation

Innovation

• John ellipsoid
• leverage-score oracle
• D-optimal design
• accelerated optimization
• self-concordant minimization