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arxiv: 2510.16468 · v3 · submitted 2025-10-18 · 🧮 math.OC

Frank-Wolfe Algorithms for (L0, L1)-smooth functions

A.A. Vyguzov , F.S. Stonyakin This is my paper

Pith reviewed 2026-05-18 06:15 UTC · model grok-4.3

classification 🧮 math.OC
keywords Frank-Wolfe algorithm(L0, L1)-smoothnessconvergence ratesadaptive algorithmconvex optimizationnumerical experimentsoptimization methods
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The pith

A new Frank-Wolfe algorithm for (L0, L1)-smooth objectives achieves superior convergence rates.

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

The paper presents the (L0, L1)-Frank-Wolfe algorithm designed specifically for optimization problems with (L0, L1)-smooth objective functions. This approach is shown to have better theoretical convergence properties than the traditional Frank-Wolfe method. Additionally, an adaptive variant is introduced that automatically tunes the smoothness parameters during optimization. Experiments demonstrate that these methods perform better in practice as well.

Core claim

The (L0, L1)-Frank-Wolfe algorithm exploits the (L0, L1)-smoothness condition to attain improved convergence rates compared to the classical Frank-Wolfe method on the same class of problems. The adaptive version dynamically adjusts parameters to enhance stability and efficiency.

What carries the argument

The (L0, L1)-Frank-Wolfe algorithm that incorporates the two-parameter smoothness condition into the step selection and linear subproblem solving process.

Load-bearing premise

The objective function must satisfy the (L0, L1)-smoothness condition that the algorithm is designed to use.

What would settle it

Finding an (L0, L1)-smooth function where the new algorithm converges no faster than the classical Frank-Wolfe method would disprove the superior rate claim.

Figures

Figures reproduced from arXiv: 2510.16468 by A.A. Vyguzov, F.S. Stonyakin.

Figure 1
Figure 1. Figure 1: ℓ2-ball set: increasing number of data points (log scale on the Y-axis). Acknowledgments ARC 09-2025 001 This work was supported by the Ministry of Science and Higher Education of the Russian Federation within state assignment no. 075-03-2024- [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: ℓ2-ball set: increasing dimension (log scale on the Y-axis). 074 under the project “Study of Asymptotic Characteristics of Fluctuations of Differential Equations and Systems, and Optimization Methods.” [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: 1-simplex set: increasing dimension (log scale on the Y-axis). References 1. Aivazian, G., Stonyakin, F.S., Pasechnyk, D., Alkousa, M.S., Raigorodsky, A., Baran, I.: Adaptive variant of the frank–wolfe algorithm for convex optimization [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: ℓ∞-ball set: increasing dimension (log scale on the Y-axis). problems. Programming and Computer Software 49(6), 493–504 (2023) 2. Bomze, I.M., Rinaldi, F., Zeffiro, D.: Frank–wolfe and friends: a journey into projection-free first-order optimization methods. 4OR 19, 313–345 (2021) [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
read the original abstract

We propose a new version of the Frank-Wolfe method, called the (L0, L1)-Frank-Wolfe algorithm, developed for optimization problems with (L0, L1)-smooth objectives. We establish that this algorithm achieves superior theoretical convergence rates compared to the classical Frank-Wolfe method. In addition, we introduce a novel adaptive procedure, termed the Adaptive (L0, L1)-Frank-Wolfe algorithm, which dynamically adjusts the smoothness parameters to further improve performance and stability. Comprehensive numerical experiments confirm the theoretical results and demonstrate the clear practical advantages of both proposed algorithms over existing Frank-Wolfe variants.

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

0 major / 3 minor

Summary. The manuscript proposes the (L0, L1)-Frank-Wolfe algorithm for constrained optimization of objectives satisfying a generalized (L0, L1)-smoothness condition. It derives a modified curvature constant and step-size rule that yields a strictly improved dependence on the iteration counter relative to the classical Frank-Wolfe O(1/k) bound. An adaptive variant is introduced that estimates L0 and L1 on the fly while recovering the same rate, and the claims are supported by numerical experiments.

Significance. If the analysis holds, the result meaningfully extends Frank-Wolfe theory to a broader function class without requiring stronger boundedness assumptions. The explicit construction of a better rate via the new smoothness parameters, together with the adaptive procedure that avoids manual tuning, represents a concrete advance for first-order methods on non-standard smooth problems.

minor comments (3)
  1. [§2.2] §2.2: The definition of the (L0, L1)-smoothness condition is clear, but the subsequent derivation of the modified curvature constant C_{L0,L1} would benefit from an explicit side-by-side comparison with the classical curvature constant to quantify the improvement in the rate.
  2. [§4] §4, Algorithm 2: The adaptive procedure for estimating L0 and L1 is described, yet the analysis of its convergence does not explicitly bound the number of iterations needed for the estimates to stabilize; adding this would strengthen the practical claim.
  3. [Figure 3] Figure 3: The plots compare the proposed methods but omit error bars or multiple random seeds; including these would make the numerical advantage over classical Frank-Wolfe more convincing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the (L0, L1)-Frank-Wolfe algorithm and its adaptive variant, as well as for recommending minor revision. We appreciate the recognition that the results extend Frank-Wolfe theory to a broader function class with improved rates and practical advantages.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines a new (L0, L1)-smoothness condition and derives modified curvature constants plus step-size rules for a variant of the Frank-Wolfe algorithm. The resulting convergence bound improves on the classical O(1/k) rate under this assumption, with the adaptive version estimating the parameters without presupposing the target rate. No equation reduces to its own input by construction, no load-bearing self-citation chain appears, and the analysis remains independent of the claimed improvement. The derivation is therefore self-contained against the stated smoothness hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; no specific free parameters, axioms, or invented entities can be identified from the provided information.

pith-pipeline@v0.9.0 · 5635 in / 1098 out tokens · 37649 ms · 2026-05-18T06:15:35.242813+00:00 · methodology

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