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

A.A. Vyguzov; F.S. Stonyakin

arxiv: 2510.16468 · v4 · pith:ZKSWBLSLnew · 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 2026-05-21 20:15 UTC · model grok-4.3

classification 🧮 math.OC

keywords Frank-Wolfe algorithm(L0, L1)-smoothnessconvergence ratesadaptive optimizationconvex optimizationprojection-free methodsnumerical experiments

0 comments

The pith

A (L0, L1)-Frank-Wolfe algorithm achieves faster convergence than the classical version on objectives meeting a combined smoothness condition.

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

The paper introduces the (L0, L1)-Frank-Wolfe algorithm for optimization problems whose objectives obey an (L0, L1)-smoothness property. This property supports refined step-size choices that yield stronger theoretical convergence guarantees than those of the standard Frank-Wolfe method. The authors further present an adaptive variant that updates the smoothness parameters during iterations to improve stability. Readers would care because many practical objectives in machine learning and engineering satisfy this mixed smoothness condition, so faster convergence directly reduces the number of expensive function evaluations needed to reach a solution.

Core claim

The (L0, L1)-Frank-Wolfe algorithm is developed for optimization problems with (L0, L1)-smooth objectives and establishes superior theoretical convergence rates compared to the classical Frank-Wolfe method. In addition, the Adaptive (L0, L1)-Frank-Wolfe algorithm 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.

What carries the argument

The (L0, L1)-smoothness property of the objective, which combines two smoothness parameters to derive improved step-size rules and convergence bounds within the Frank-Wolfe iteration.

If this is right

The (L0, L1)-Frank-Wolfe algorithm attains strictly better convergence rates than classical Frank-Wolfe when the objective meets the combined smoothness condition.
The adaptive version automatically tunes the two smoothness parameters to maintain performance across varying problem instances.
Numerical tests show both variants require fewer iterations and exhibit greater stability than prior Frank-Wolfe implementations.
The approach remains projection-free and therefore applicable to the same large-scale constrained problems as the classical method.

Where Pith is reading between the lines

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

The same smoothness condition could be applied to accelerate other projection-free first-order methods beyond Frank-Wolfe.
In high-dimensional settings where standard L-smoothness is too strong an assumption, the (L0, L1) relaxation may enable practical speed-ups without sacrificing theoretical guarantees.
Parameter-free or line-search variants of the adaptive algorithm could further reduce the need for manual tuning in deployed optimization pipelines.

Load-bearing premise

The objective function satisfies the (L0, L1)-smoothness property that is used to obtain the improved step sizes and bounds.

What would settle it

An explicit (L0, L1)-smooth function on which the new algorithm requires at least as many iterations as classical Frank-Wolfe to reach a given accuracy would disprove the claimed rate improvement.

Figures

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

**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: 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: 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: 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.

Desk Editor's Note private letter to a colleague

The paper gives a targeted Frank-Wolfe variant for (L0, L1)-smooth objectives that improves the convergence rate over the classical L-smooth case, with proofs that check out and supporting experiments.

read the letter

The main point is that this work defines a (L0, L1)-Frank-Wolfe algorithm and an adaptive version that exploit a two-parameter smoothness condition to obtain a strictly better iteration dependence than standard Frank-Wolfe. The proofs telescope the usual gap inequality under the new smoothness bound and recover the improved rate without extra assumptions like bounded diameter. The adaptive procedure updates the parameters on the fly and appears to preserve the guarantees. Numerical tests show practical speed-ups on the chosen problems compared with earlier FW variants. That is the concrete advance: a direct, assumption-driven refinement rather than a wholesale new framework. The math in sections 3 and 4 holds together cleanly once the (L0, L1) inequality is substituted, and the experiments are run at a scale that lets the reader see the difference. The soft spot is that the gain only materializes when the objective really satisfies the (L0, L1) property with finite constants; for many standard FW applications this may not be the case or may require extra verification. The test problems are reasonable but could have included a wider range of dimensions or conditioning to strengthen the practical claim. Overall the paper is aimed at researchers who already work with conditional gradient methods and want tighter rates under structured smoothness. A reader looking for incremental but verifiable progress in convex optimization will get value from it. It is coherent on its own terms and deserves a serious referee.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes the (L0, L1)-Frank-Wolfe algorithm for optimization problems with (L0, L1)-smooth objectives. It establishes superior theoretical convergence rates compared to the classical Frank-Wolfe method via a tailored step-size rule and curvature bound derived from the new smoothness assumption. An adaptive variant is introduced that dynamically adjusts the parameters L0 and L1, supported by numerical experiments confirming the theory and demonstrating practical advantages.

Significance. If the results hold, the work is significant because it relaxes global L-smoothness to the weaker (L0, L1)-smoothness condition while recovering strictly better dependence on iteration count. Credit is due to the proofs in Sections 3 and 4, which correctly telescope the standard FW gap inequality under this assumption and obtain the improved rate when both L0 and L1 are finite. The adaptive procedure adds practical stability without introducing hidden circularity or unstated assumptions.

minor comments (2)

[Abstract] The abstract states 'superior theoretical convergence rates' but does not briefly indicate the improved dependence on iteration count (e.g., the exponent achieved).
[Numerical Experiments] In the numerical experiments, the test problems could more explicitly verify or report the values of L0 and L1 used to confirm the (L0, L1)-smoothness assumption.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. We appreciate the recognition that the (L0, L1)-smoothness assumption relaxes global L-smoothness while yielding strictly improved convergence rates, as well as the credit given to the proofs in Sections 3 and 4 and the practical value of the adaptive variant.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central claim derives improved convergence rates for the (L0, L1)-Frank-Wolfe algorithm directly from the assumed (L0, L1)-smoothness inequality applied to the standard Frank-Wolfe gap bound. Sections 3 and 4 telescope this inequality with a tailored step-size rule to obtain a strictly better iteration dependence than the classical L-smooth case. The (L0, L1) parameters enter as an explicit assumption rather than being fitted or defined in terms of the output rates. No self-citation is load-bearing for the main theorem, no ansatz is smuggled, and no prediction reduces to its input by construction. The adaptive variant similarly updates parameters from observed quantities without circular redefinition. This is the normal case of an independent extension of a known framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the work appears to rest on standard convex optimization assumptions such as a compact convex domain and the (L0, L1)-smoothness definition itself.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel / Jcost uniqueness unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1. A function f is said to be (L0,L1)-smooth if … ||∇²f(x)|| ≤ L0 + L1||∇f(x)|| … αk := min{1, −∇f(xk)⊤dk / ((L0 + L1||∇f(xk)||) ||dk||² e)}
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 1 … f(y) ≤ f(x) + ⟨∇f(x),y−x⟩ + (L0 + L1||∇f(x)||)/2 * exp(L1||x−y||) ||x−y||²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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work page 2023
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4OR19, 313–345 (2021) Frank-Wolfe Algorithms for(L0, L1)-smooth functions 21

Bomze, I.M., Rinaldi, F., Zeffiro, D.: Frank–wolfe and friends: a journey into projection-free first-order optimization methods. 4OR19, 313–345 (2021) Frank-Wolfe Algorithms for(L0, L1)-smooth functions 21

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Cambridge University Press (2004)

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Braun, G., Carderera, A., Combettes, C.W., Hassani, H., Karbasi, A., Mokhtari, A., Pokutta, S.: Conditional gradient methods. arXiv preprint arXiv:2211.14103 (2022)

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Vyguzov et al

Aivazian, G., Stonyakin, F.S., Pasechnyk, D., Alkousa, M.S., Raigorodsky, A., Baran, I.: Adaptive variant of the frank–wolfe algorithm for convex optimization 20 A. Vyguzov et al. Fig. 4.ℓ ∞-ball set: increasing dimension (log scale on the Y-axis). problems. Programming and Computer Software49(6), 493–504 (2023)

work page 2023

[2] [2]

4OR19, 313–345 (2021) Frank-Wolfe Algorithms for(L0, L1)-smooth functions 21

Bomze, I.M., Rinaldi, F., Zeffiro, D.: Frank–wolfe and friends: a journey into projection-free first-order optimization methods. 4OR19, 313–345 (2021) Frank-Wolfe Algorithms for(L0, L1)-smooth functions 21

work page 2021

[3] [3]

Cambridge University Press (2004)

Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004)

work page 2004

[4] [4]

arXiv preprint arXiv:2211.14103 (2022)

Braun, G., Carderera, A., Combettes, C.W., Hassani, H., Karbasi, A., Mokhtari, A., Pokutta, S.: Conditional gradient methods. arXiv preprint arXiv:2211.14103 (2022)

work page arXiv 2022

[5] [5]

In: International Conference on Machine Learning

Chen, Z., Zhou, Y., Liang, Y., Lu, Z.: Generalized-smooth nonconvex optimization is as efficient as smooth nonconvex optimization. In: International Conference on Machine Learning. pp. 5396–5427. PMLR (2023)

work page 2023

[6] [6]

Operations Research Letters49(4), 565–571 (2021)

Combettes, C.W., Pokutta, S.: Complexity of linear minimization and projection on some sets. Operations Research Letters49(4), 565–571 (2021)

work page 2021

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Naval re- search logistics quarterly3(1-2), 95–110 (1956)

Frank, M., Wolfe, P., et al.: An algorithm for quadratic programming. Naval re- search logistics quarterly3(1-2), 95–110 (1956)

work page 1956

[8] [8]

arXiv preprint arXiv:2409.14989 (2024)

Gorbunov, E., Tupitsa, N., Choudhury, S., Aliev, A., Richtárik, P., Horváth, S., Takáč, M.: Methods for convex(l_0, l_1)-smooth optimization: Clipping, acceler- ation, and adaptivity. arXiv preprint arXiv:2409.14989 (2024)

work page arXiv 2024

[9] [9]

In: Joint European con- ference on machine learning and knowledge discovery in databases

Karimi, H., Nutini, J., Schmidt, M.: Linear convergence of gradient and proximal- gradient methods under the polyak-łojasiewicz condition. In: Joint European con- ference on machine learning and knowledge discovery in databases. pp. 795–811. Springer (2016)

work page 2016

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arXiv preprint arXiv:2510.11440 (2025)

Khademi, A., Silveti-Falls, A.: Adaptive conditional gradient descent. arXiv preprint arXiv:2510.11440 (2025)

work page arXiv 2025

[11] [11]

USSR Computa- tional mathematics and mathematical physics6(5), 1–50 (1966)

Levitin, E.S., Polyak, B.T.: Constrained minimization methods. USSR Computa- tional mathematics and mathematical physics6(5), 1–50 (1966)

work page 1966

[12] [12]

Mathematical programming140(1), 125–161 (2013)

Nesterov, Y.: Gradient methods for minimizing composite functions. Mathematical programming140(1), 125–161 (2013)

work page 2013

[13] [13]

Nesterov, Y., et al.: Lectures on convex optimization, vol. 137. Springer (2018)

work page 2018

[14] [14]

In: International conference on artificial intelligence and statistics

Pedregosa, F., Negiar, G., Askari, A., Jaggi, M.: Linearly convergent frank-wolfe with backtracking line-search. In: International conference on artificial intelligence and statistics. pp. 1–10. PMLR (2020)

work page 2020

[15] [15]

arXiv preprint arXiv:2506.01913 (2025)

Pethick, T., Xie, W., Erdogan, M., Antonakopoulos, K., Silveti-Falls, T., Cevher, V.: Generalized gradient norm clipping & non-euclidean(l_0, l_1)-smoothness. arXiv preprint arXiv:2506.01913 (2025)

work page arXiv 2025

[16] [16]

Nauka, Moscow (1983)

Polyak, B.T.: Vvedenie v optimizatsiyu [Introduction to Optimization]. Nauka, Moscow (1983)

work page 1983

[17] [17]

Zhurnal vychislitel’noi matematiki i matematicheskoi fiziki3(4), 643–653 (1963)

Polyak, B.T.: Gradient methods for minimizing functionals. Zhurnal vychislitel’noi matematiki i matematicheskoi fiziki3(4), 643–653 (1963)

work page 1963

[18] [18]

arXiv preprint arXiv:2504.04330 (2025)

Takahashi, S., Pokutta, S., Takeda, A.: Fast frank–wolfe algorithms with adaptive bregman step-size for weakly convex functions. arXiv preprint arXiv:2504.04330 (2025)

work page arXiv 2025

[19] [19]

Computational Mathematics and Mathematical Physics65(3), 591–602 (2025)

Vyguzov, A.A., Stonyakin, F.S.: An adaptive variant of the frank–wolfe method for relative smooth convex optimization problems. Computational Mathematics and Mathematical Physics65(3), 591–602 (2025)

work page 2025

[20] [20]

Advances in Neural Information Processing Systems 33, 15511–15521 (2020)

Zhang, B., Jin, J., Fang, C., Wang, L.: Improved analysis of clipping algorithms for non-convex optimization. Advances in Neural Information Processing Systems 33, 15511–15521 (2020)

work page 2020

[21] [21]

arXiv preprint arXiv:1905.11881 (2019)

Zhang, J., He, T., Sra, S., Jadbabaie, A.: Why gradient clipping accelerates train- ing: A theoretical justification for adaptivity. arXiv preprint arXiv:1905.11881 (2019)

work page arXiv 1905