Lower Bounds for Frank-Wolfe on Strongly Convex Sets

Bartolomeo Stellato; Christophe Roux; Daniel Deza; Jannis Halbey; Max Zimmer; Sebastian Pokutta

arxiv: 2602.04378 · v2 · submitted 2026-02-04 · 🧮 math.OC

Lower Bounds for Frank-Wolfe on Strongly Convex Sets

Jannis Halbey , Daniel Deza , Max Zimmer , Christophe Roux , Bartolomeo Stellato , Sebastian Pokutta This is my paper

Pith reviewed 2026-05-16 07:30 UTC · model grok-4.3

classification 🧮 math.OC

keywords Frank-Wolfelower boundsstrongly convex setsconvergence ratessmooth convex optimizationtrajectory construction

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The pith

Frank-Wolfe requires Ω(1/√ε) iterations even when both the objective and the constraint set are smooth and strongly convex.

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

The paper shows that the Frank-Wolfe algorithm cannot improve beyond the known O(1/√ε) rate when both the objective function and the feasible set are smooth and strongly convex. This is established by constructing explicit worst-case instances where the optimizer lies on the boundary of the Euclidean unit ball and proving that convergence to accuracy ε still demands Ω(1/√ε) steps. The result matters because it rules out the possibility that strong convexity of the set alone would yield uniformly faster rates independent of where the solution sits. The authors develop both computational constructions of hard trajectories and an analytical proof to confirm the bound is tight.

Core claim

For the problem of minimizing a strongly convex quadratic over the Euclidean unit ball with the optimizer on the boundary, Frank-Wolfe requires Ω(1/√ε) iterations to reach ε-accuracy, matching the existing uniform upper bound and showing that strong convexity of the set does not improve the worst-case rate without further assumptions on the optimizer location.

What carries the argument

Minimizing a strongly convex quadratic over the Euclidean unit ball (with optimizer on the boundary) together with a computational method for constructing worst-case Frank-Wolfe trajectories.

Load-bearing premise

The quadratic-over-unit-ball problem with boundary optimum is representative of the worst case for general smooth strongly convex sets.

What would settle it

An explicit smooth strongly convex set and objective where Frank-Wolfe reaches ε-accuracy in o(1/√ε) iterations without any extra assumption on the optimizer position.

read the original abstract

We present a constructive lower bound of $\Omega(1/\sqrt{\varepsilon})$ for Frank-Wolfe (FW) when both the objective and the constraint set are smooth and strongly convex, showing that the known uniform $\mathcal{O}(1/\sqrt{\varepsilon})$ guarantees in this regime are tight. It is known that under additional assumptions on the position of the optimizer, FW can converge linearly. However, it remained unclear whether strong convexity of the set can yield rates uniformly faster than $\mathcal{O}(1/\sqrt{\varepsilon})$, i.e., irrespective of the position of the optimizer. To investigate this question, we focus on a simple yet representative problem class: minimizing a strongly convex quadratic over the Euclidean unit ball, with the optimizer on the boundary. We analyze the dynamics of FW for this problem in detail and develop a novel computational approach to construct worst-case FW trajectories, which is of independent interest. Guided by these constructions, we develop an analytical proof establishing the lower bound.

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 shows Frank-Wolfe cannot improve on Ω(1/√ε) uniformly even when both objective and set are strongly convex and smooth, via an explicit construction on the unit ball.

read the letter

The main takeaway is that Frank-Wolfe requires Ω(1/√ε) iterations in the worst case even when both the objective and the constraint set are smooth and strongly convex. This makes the known uniform O(1/√ε) upper bounds tight and shows that strong convexity of the set by itself does not deliver faster rates independent of optimizer location. They focus on minimizing a strongly convex quadratic over the Euclidean unit ball with the optimum on the boundary, analyze the algorithm dynamics there, and build an analytical proof from it. The new piece is their computational method for constructing worst-case trajectories, which they use to guide the proof and which could be useful for other analyses. This combination of numerical exploration and formal argument is a clear strength. The main limitation is the choice of this specific quadratic-over-ball instance. It sits inside the target class, so it suffices to rule out any better uniform rate, but it is not obvious how far the behavior generalizes to arbitrary smooth strongly convex sets. The paper treats the instance as representative for the purpose of tightness, which holds up for the claim they make. No circularity appears in the approach, and the argument rests on explicit trajectory construction rather than fitted parameters. This paper is for readers who work on convergence rates for projection-free methods. Someone studying Frank-Wolfe or related algorithms would find the lower-bound construction and the computational technique worth seeing. It deserves peer review because it directly addresses whether strong convexity of the set can improve uniform rates and supplies both a concrete example and a proof.

Referee Report

0 major / 2 minor

Summary. The paper establishes a constructive lower bound of Ω(1/√ε) on the number of iterations needed for the Frank-Wolfe algorithm to reach ε-accuracy when both the objective and the constraint set are smooth and strongly convex. The authors focus on the specific instance of minimizing a strongly convex quadratic over the Euclidean unit ball (with the optimizer on the boundary), analyze the algorithm dynamics in detail, introduce a computational method to construct worst-case trajectories, and then provide an analytical proof that this rate is tight, showing that known O(1/√ε) upper bounds cannot be improved uniformly without further assumptions on the optimizer location.

Significance. If the central construction and proof hold, the result is significant: it resolves whether strong convexity of the set alone can yield uniformly faster rates than O(1/√ε) irrespective of optimizer position, and demonstrates tightness via an explicit worst-case instance. The novel computational approach for generating worst-case FW trajectories is a strength of independent interest that could apply to other first-order methods. The work ships an explicit construction rather than a non-constructive argument, which strengthens the contribution.

minor comments (2)

[Computational construction section] In the description of the computational construction (around the trajectory analysis), include a brief pseudocode or high-level algorithm box to make the method reproducible without reading the full implementation details.
[Preliminaries] The notation for the strong-convexity parameters (μ for the function and the set) should be introduced with a single consolidated definition early in the preliminaries to avoid later cross-references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. We are pleased that the significance of the constructive lower bound for Frank-Wolfe under strong convexity of both objective and set, as well as the independent interest of the computational method for generating worst-case trajectories, was recognized. The recommendation to accept is appreciated.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit construction

full rationale

The paper establishes the Ω(1/√ε) lower bound by direct construction and analysis of worst-case Frank-Wolfe trajectories on the specific instance of minimizing a strongly convex quadratic over the Euclidean unit ball (optimizer on boundary). This instance is chosen to exhibit tightness within the broader class of smooth strongly convex problems, but the bound itself follows from explicit dynamical analysis and a novel computational trajectory-construction method rather than from any fitted parameter, self-referential definition, or load-bearing self-citation. No equation reduces to its own input by construction, and the central claim remains independent of the result it proves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the representativeness of the quadratic-over-unit-ball instance and on standard properties of strongly convex smooth functions; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard properties of smooth strongly convex functions and sets hold for the chosen quadratic and unit ball
Invoked when restricting to this representative problem class

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