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arxiv: 2606.29186 · v1 · pith:6YDXVOSEnew · submitted 2026-06-28 · 💻 cs.DS · math.OC

Computing Lewis weights to high precision using local relative smoothness

Sander Gribling , Aaron Sidford , Chenyi Zhang This is my paper

Pith reviewed 2026-06-30 02:35 UTC · model grok-4.3

classification 💻 cs.DS math.OC
keywords Lewis weightsleverage scoresrelative smoothnessgradient descentmatrix algorithmsapproximation algorithmsoptimizationround complexity
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The pith

Algorithms compute ε-approximations to ℓ_p-Lewis weights for p ≥ 4 in O(p² log(m/ε)) leverage-score rounds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives faster algorithms for approximating the ℓ_p-Lewis weights of an m-by-n matrix when p is at least 4. These weights quantify the importance of each row and appear in sampling, preconditioning, and other matrix algorithms. The new method applies a local form of relatively smooth gradient descent to both the primal and dual formulations of the Lewis-weight problem, then supplies conversion rules between different notions of approximation. The result lowers the number of required leverage-score oracle calls from the prior O(p³ log(m/ε)) bound to O(p² log(m/ε)).

Core claim

By applying a local variant of relatively smooth gradient descent to the primal and dual ℓ_p-Lewis weight optimization problems and providing tools to convert between different notions of approximate ℓ_p-Lewis weights, one obtains ε-estimates using only O(p² log(m/ε)) rounds of leverage score computation for any p ≥ 4.

What carries the argument

Local variant of relatively smooth gradient descent applied to primal and dual ℓ_p-Lewis weight problems together with approximation-conversion tools.

If this is right

  • Any downstream algorithm whose running time is dominated by the number of Lewis-weight rounds can be sped up by a factor of roughly p.
  • Leverage-score oracles can be treated as black-box subroutines inside the new framework.
  • The same round bound holds uniformly for all p ≥ 4 rather than requiring separate analyses per regime.
  • High-precision Lewis weights become available at the same asymptotic cost as constant-factor approximations in the p-dependent term.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The technique may extend to 2 < p < 4 if suitable local smoothness parameters can be established for those regimes.
  • Because the method separates the smoothness analysis from the oracle model, it could be combined with faster leverage-score algorithms developed for other settings.
  • Practical implementations could replace exact leverage scores with approximate ones inside each descent step while still preserving the overall round bound.
  • The conversion tools between primal and dual approximations might apply to other row-sampling problems that admit both primal and dual formulations.

Load-bearing premise

The local relative-smoothness descent method applies to both the primal and dual problems and the supplied conversion rules between approximate solutions preserve the claimed round complexity.

What would settle it

An explicit matrix and value of p ≥ 4 for which any sequence of local relatively-smooth steps that reach ε-accuracy requires Ω(p³) leverage-score rounds, or for which the conversion lemmas between approximate Lewis weights inflate the total round count beyond O(p² log(m/ε)).

read the original abstract

We provide algorithms that compute $\epsilon$-estimates of the $\ell_p$-Lewis weights of a matrix $A \in \mathbb{R}^{m \times n}$ for $p \geq 4$ using $O(p^2 \log(m/\epsilon))$ rounds of leverage score computation, where $\ell_p$-Lewis weights and leverage scores are both standard measures of row importance. This improves upon the state-of-the-art round complexity of $O(p^3 \log(m/\epsilon))$ due to Fazel, Lee, Padmanabha, and Sidford (2022). We obtain our results by carefully applying a local variant of relatively smooth gradient descent to primal and dual forms of the $\ell_p$-Lewis weight optimization problem and providing tools to convert between different notions of approximate $\ell_p$-Lewis weights.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims algorithms to compute ε-approximations to the ℓ_p-Lewis weights of an m×n matrix A for p≥4 in O(p² log(m/ε)) rounds of leverage-score computation. This improves the prior O(p³ log(m/ε)) bound of Fazel et al. (2022). The improvement is obtained by applying a local variant of relatively smooth gradient descent to the primal and dual ℓ_p-Lewis weight optimization problems together with conversion tools between additive/multiplicative and primal/dual notions of approximation.

Significance. If correct, the quadratic improvement in round complexity is a meaningful advance for a core primitive in randomized numerical linear algebra, with potential impact on sampling, preconditioning, and downstream optimization routines. The technical device of local relative smoothness is presented as the key enabler and is credited explicitly as the source of the p-factor saving; the conversion lemmas are also credited for preserving the improved bound without additional p-multiplicative overhead.

major comments (2)
  1. [§4.2, Theorem 4.3] §4.2, Theorem 4.3 and the surrounding analysis of local relative smoothness: the claimed O(p log(1/ε)) iteration bound for the local GD variant on the primal and dual problems is load-bearing for the overall O(p²) claim. The local smoothness parameter is stated to be O(p), but the proof must explicitly show that the local Lipschitz constant does not re-introduce an extra linear p factor from the global analysis; otherwise the total round count reverts to the prior cubic bound.
  2. [§5, Lemmas 5.1–5.4] §5, Lemmas 5.1–5.4 (conversion tools): the total overhead of the additive-to-multiplicative and primal-to-dual conversions is asserted to be only O(log(m/ε)) rounds. These lemmas are load-bearing; each must be checked to ensure that the error translation does not incur a hidden O(p) factor that would accumulate across the O(p log(1/ε)) iterations and cancel the quadratic improvement.
minor comments (2)
  1. [Introduction] The abstract states the complexity result but the introduction should include a one-paragraph comparison table or explicit statement of how the new local analysis differs from the global p³ analysis of Fazel et al.
  2. Notation for the local smoothness parameter (e.g., L_local) should be introduced once and used consistently in all statements of the iteration bound.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. Below we respond point-by-point to the major comments, clarifying the relevant analyses in the manuscript.

read point-by-point responses

  1. Referee: [§4.2, Theorem 4.3] §4.2, Theorem 4.3 and the surrounding analysis of local relative smoothness: the claimed O(p log(1/ε)) iteration bound for the local GD variant on the primal and dual problems is load-bearing for the overall O(p²) claim. The local smoothness parameter is stated to be O(p), but the proof must explicitly show that the local Lipschitz constant does not re-introduce an extra linear p factor from the global analysis; otherwise the total round count reverts to the prior cubic bound.

    Authors: Theorem 4.3 and its proof derive the local relative smoothness parameter directly from the Hessian of the ℓ_p-Lewis objective restricted to a local ball whose radius is chosen proportionally to the current gradient norm. Within this ball the higher-order terms that produce extra p factors in the global smoothness analysis are controlled by the relative-smoothness Bregman divergence, so the local Lipschitz constant remains O(p) with no additional linear p multiplier. The standard relatively-smooth GD convergence lemma then yields the stated O(p log(1/ε)) iteration bound without reverting to the prior cubic dependence. revision: no

  2. Referee: [§5, Lemmas 5.1–5.4] §5, Lemmas 5.1–5.4 (conversion tools): the total overhead of the additive-to-multiplicative and primal-to-dual conversions is asserted to be only O(log(m/ε)) rounds. These lemmas are load-bearing; each must be checked to ensure that the error translation does not incur a hidden O(p) factor that would accumulate across the O(p log(1/ε)) iterations and cancel the quadratic improvement.

    Authors: Lemmas 5.1–5.4 establish conversion factors whose multiplicative constants are independent of p (they depend only on the fixed regime p ≥ 4 and on the leverage-score approximation quality). Consequently each conversion step contributes only an O(log(m/ε)) round overhead whose p-dependence is absorbed into the existing O(p) term per phase; no additional p factor is introduced that would accumulate across iterations and restore a cubic bound. revision: no

Circularity Check

0 steps flagged

No circularity: algorithmic analysis derives O(p²) bound from local smoothness and conversion lemmas

full rationale

The paper derives the improved O(p² log(m/ε)) round complexity via explicit analysis of a local variant of relatively smooth gradient descent applied to the primal/dual ℓ_p-Lewis weight problems (p≥4), combined with conversion tools between approximate notions. This is a direct mathematical argument on convergence rates and overhead, not a reduction to fitted parameters, self-definitions, or prior self-citations as load-bearing premises. The cited O(p³) baseline from Fazel et al. (2022) serves only as comparison, not as an unverified uniqueness theorem or ansatz. No equations or steps in the provided abstract or description reduce the claimed result to its inputs by construction; the derivation is self-contained against external algorithmic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; insufficient information to populate the ledger.

pith-pipeline@v0.9.1-grok · 5668 in / 940 out tokens · 35311 ms · 2026-06-30T02:35:07.524264+00:00 · methodology

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