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arxiv: 2605.10595 · v1 · submitted 2026-05-11 · 🧮 math.OC

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Curvature-Dependent Lower Bounds for Frank-Wolfe

Christophe Roux, Jannis Halbey, Sebastian Pokutta

Pith reviewed 2026-05-12 04:37 UTC · model grok-4.3

classification 🧮 math.OC
keywords Frank-Wolfe algorithmlower boundsp-uniformly convex setsconvergence ratesconvex optimizationHölderian error boundexact line search
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The pith

Frank-Wolfe cannot converge faster than order T to the minus p over p minus one on p-uniformly convex sets for p at least 3.

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

The paper proves that the known upper bound of order T to the minus p over p minus one for Frank-Wolfe on p-uniformly convex feasible sets is tight. It does so by constructing simple low-dimensional instances and tracking how the algorithm's linear optimization steps advance under exact line search or short steps. The lower bound of Omega of T to the minus p over p minus one holds for every p greater than or equal to 3 and extends to objectives that obey a Hölderian error bound. If this is correct, the curvature parameter p fully controls the best achievable rate for this algorithm without stronger assumptions.

Core claim

For every p at least 3 there exist smooth convex problems over p-uniformly convex domains on which the Frank-Wolfe algorithm with exact line search or short steps satisfies a lower bound of Omega of T to the minus p over p minus one. The proofs follow the evolution of iterates on low-dimensional explicit examples rather than relying on high-dimensional information-theoretic arguments, and the same lower bound applies when the objective satisfies a Hölderian error bound.

What carries the argument

Direct tracking of the progress made by successive linear optimization oracle calls on low-dimensional p-uniformly convex sets, which limits improvement per iteration according to the curvature parameter p.

If this is right

  • The previously known upper bounds of order T to the minus p over p minus one are optimal for p at least 3.
  • The lower bound applies even in low dimensions through concrete constructions.
  • The same rate lower bound holds for objectives satisfying a Hölderian error bound.
  • Further acceleration would require assumptions stronger than p-uniform convexity or changes to the algorithm steps.

Where Pith is reading between the lines

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

  • For domains whose curvature sits between strong convexity and general convexity, Frank-Wolfe may lose out to other first-order methods that exploit different geometry.
  • The low-dimensional instances could be reused to derive matching lower bounds for accelerated variants or related conditional-gradient methods.
  • If a feasible set has locally varying curvature, the global worst-case rate would be governed by the smallest local p value.

Load-bearing premise

The feasible set must be p-uniformly convex for a fixed p at least 3 and the objective must be smooth and convex, while the algorithm uses exact line search or short steps.

What would settle it

An explicit p-uniformly convex set together with a smooth convex objective on which Frank-Wolfe with exact line search or short steps reaches the optimum faster than the stated rate would falsify the lower bound.

Figures

Figures reproduced from arXiv: 2605.10595 by Christophe Roux, Jannis Halbey, Sebastian Pokutta.

Figure 1
Figure 1. Figure 1: Exact-line-search FW trajectories in the (u, w)-plane for four random initializations for the projection problem (1), i.e., minx∈Bp ∥x − e1∥ 2 with p ≥ 3. Across runs, u remains positive and controls the distance to the optimum, while w alternates sign and contracts faster. the one-step update of the scaled variable y, y ′ = Φ(u, y) def = |w ′ | (u ′) 1+α , (4) where u ′ and w ′ are the new values of u and… view at source ↗
Figure 2
Figure 2. Figure 2: Three coordinate views of FW with exact line search on Bp, p = 3, for f(x) = ∥x − e1∥ 2 2 , run from the slow-start initialization (5) with u0 = 3 4 (open circle). Left: original coordinates (x1, x2), with the optimum e1 (gold star) and ∂Bp (dashed). Middle: the recentered coordinates (u, w) = (1 − x1, x2) shift e1 to the origin and turn f into a quadratic centered at 0, but the iterates still cluster in a… view at source ↗
Figure 3
Figure 3. Figure 3: Heatmaps showing the number of FW iterations required to reach the target accuracy (10−4 ), initialized from all feasible points in the open p-unit ball in R 2 . Each panel corresponds to a different value of p. Darker colors correspond to fewer iterations. The heatmap for p = 3 shows clear strips of high iteration counts, where the widest strips contain the fixed point curve y ∗ (u). The case of p = 3.1 i… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the primal gap f(xT ) − f(x ∗ ) versus iteration T for exact-line-search FW on the minimization problem in (1) for p ∈ {3, 5}. The solid curve uses the slow initialization (5), while the faint curves are runs from generic initializations. The dashed line shows T −p/(p−1), which matches both our lower bound from Theorem 4 and the upper bound rate of Kerdreux et al. (2021a). (see Proposition 15… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the primal gap g(xT ) − g(x ∗ ) versus iteration T for exact-line-search FW for minimizing (9) on Bp for p ∈ {3, 5} and θ ∈ { 1 3 , 1 4 }. The solid curve uses the slow initialization (5), while the faint curves are runs from generic initializations. The dotted reference line shows the Ω [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
read the original abstract

The Frank-Wolfe algorithm achieves a convergence rate of $\mathcal{O}(1/T)$ for smooth convex optimization over compact convex domains, accelerating to $\mathcal{O}(1/T^2)$ when both the objective and the feasible set are strongly convex. This acceleration extends beyond strong convexity: Kerdreux et al. (2021a) proved rates of $\mathcal{O}(T^{-p/(p-1)})$ over $p$-uniformly convex feasible sets, a class that interpolates between strongly convex sets and more general curved domains such as $\ell_p$ balls. In this work, we establish a matching $\Omega(T^{-p/(p-1)})$ lower bound for every $p\ge 3$ under exact line search or short steps, and extend the lower bound to objectives satisfying a H\"olderian error bound. The proofs analyze the dynamics of Frank-Wolfe iterates on simple instances and hence are not limited to the high-dimensional setting, unlike information-theoretic lower bounds.

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 establishes matching lower bounds of Ω(T^{-p/(p-1)}) for the Frank-Wolfe algorithm on p-uniformly convex feasible sets (p ≥ 3) under exact line search or short steps. The bounds are derived via explicit analysis of the algorithm's iterate dynamics on simple low-dimensional instances with smooth convex objectives, and the result is extended to objectives satisfying a Hölderian error bound. The approach is constructive and not restricted to high-dimensional regimes.

Significance. If the central claims hold, the work is significant because it provides matching lower bounds that confirm the optimality of the upper bounds from Kerdreux et al. (2021) for curvature-dependent acceleration in Frank-Wolfe, even in low dimensions. The direct dynamic analysis on explicit instances is a clear strength, yielding constructive and falsifiable examples rather than relying on information-theoretic arguments. This completes the rate picture for Frank-Wolfe under p-uniform convexity and Hölderian error bounds.

minor comments (3)
  1. [Preliminaries] The precise definition and constants for p-uniform convexity should be recalled explicitly in the preliminaries section to ensure the simple instances are immediately verifiable against the assumption.
  2. A brief numerical example or plot illustrating the constructed lower-bound instance for p=3 would help readers confirm the dynamics without reconstructing the full proof.
  3. [Main results] The statement of the short-step rule in the main theorem should include a reference to the exact step-size formula used, to distinguish it from other variants in the literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. We appreciate the recognition that our constructive analysis on low-dimensional instances provides matching lower bounds confirming the optimality of the rates in Kerdreux et al. (2021), and that the approach extends naturally to Hölderian error bounds without relying on high-dimensional information-theoretic arguments.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes its Ω(T^{-p/(p-1)}) lower bounds via direct, explicit analysis of Frank-Wolfe iterate dynamics on low-dimensional instances that satisfy the p-uniform convexity assumption (p≥3) together with standard smoothness and convexity of the objective. This constructive approach yields the matching rate under exact line search or short steps and extends to Hölderian error bounds without reducing any quantity to a fitted parameter, a self-referential definition, or a load-bearing self-citation. The cited upper-bound result (Kerdreich et al. 2021a) is external to the lower-bound derivation and does not enter the proof equations. The argument is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard definitions of p-uniform convexity and Hölderian error bounds from convex analysis, with no free parameters, no invented entities, and no ad-hoc axioms beyond domain assumptions.

axioms (2)
  • domain assumption The feasible set is p-uniformly convex for p ≥ 3
    This curvature condition is required for the specific rate T^{-p/(p-1)} to hold.
  • domain assumption The objective function is smooth and convex
    Standard assumption for Frank-Wolfe convergence analysis.

pith-pipeline@v0.9.0 · 5471 in / 1413 out tokens · 35749 ms · 2026-05-12T04:37:33.056171+00:00 · methodology

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