Convergence of linesearch-based generalized conditional gradient methods without smoothness assumptions

Shotaro Yagishita

REVIEW 2 major objections 4 minor 30 references

Convergence of linesearch-based generalized conditional gradient methods without smoothness assumptions

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Two linesearch-based generalized conditional gradient methods converge to stationary points of a composite objective even when the gradient of the smooth part is merely continuous, with no Lipschitz or Hölder condition assumed.

desk verdict Useful convergence proofs for generalized conditional gradient methods without gradient smoothness; the main theorems hold, but Lemma 3.2's termination claim needs a small patch at stationary points. read the letter →

arxiv 2505.01092 v1 pith:ATFHVJ5K submitted 2025-05-02 math.OC

Shotaro Yagishita This is my paper · ORCID

classification math.OC MSC 90C3090C2565K05

keywords generalizedconditionalgradientmethodcompositeoptimizationnonmonotonelinesearchparameter-freealgorithmHöldercontinuityFrank-Wolfegapstationarypointconvergencewithoutsmoothness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Convergence analyses of the generalized conditional gradient method — the composite-problem solver that at each step minimizes a linearized smooth part plus an unchanged nonsmooth convex term — almost always assume the gradient of the smooth part is Lipschitz or Hölder continuous. This paper proves that two linesearch-based variants need no such assumption: with only continuous differentiability of the smooth part and convexity plus supercoercivity of the nonsmooth part, every accumulation point of the nonmonotone Armijo variant is a stationary point of the composite objective, and the parameter-free variant (Algorithm 2) converges to a stationary point under an additional bounded-level-set condition. The practical upshot is that both algorithms can be run without first certifying any Hölder exponent; the parameter-free method still yields an approximate stationary point in finitely many iterations even when no Hölder condition holds. The proof operates through the Frank-Wolfe gap, which the linesearch rejection inequality forces to zero using only continuity of the gradient.

What carries the argument

The central object is the Frank-Wolfe gap, G(x) := max_v {⟨∇f(x), x − v⟩ + g(x) − g(v)}, a nonnegative, lower-semicontinuous optimality measure that vanishes exactly at stationary points of (1). The mechanism carrying both proofs is the descent estimate of Lemma 3.1, F(x + td) ≤ F(x) − tG(x) + o(t) for t ∈ [0,1] and d a subproblem solution direction, which follows from continuous differentiability of f and convexity of g alone. Armijo-type backtracking then guarantees a function decrease proportional to the gap at every iteration — for Algorithm 1, F_{k+1} ≤ F_k − $pσβ^{{i_k}}$G(x_k), and for Algorithm 2, F(x_{k+1}) ≤ F(x_k) − (1/4)τ_k G(x_k) — so the gap is driven to zero along any subsequence where the stepsize does not degenerate. In the vanishing-stepsize case, rejection of the trial point forces inequalities (7) and (10), which bound the gap by ‖∇f(ξ_k) − ∇f(x_k)‖‖d_k‖ with ξ_k a mean-value point converging to the same limit as x_k; continuity of ∇f, rather than any quantitative modulus, sends that bound to zero, while boundedness of the directions d_k is secured by supercoercivity of g.

What would settle it

Concrete test: with g the indicator of [−1,1] and f(x) = x + x² sin(1/x²), the smooth part is differentiable everywhere with f'(0) = 1 while f' is discontinuous and unbounded at 0, so 0 is a non-stationary point. Run Algorithms 1 and 2 from several starting points and record the Frank-Wolfe gap G(x_k); iterates accumulating at 0 with a gap bounded away from zero would show where the continuous-gradient hypothesis is doing its work, while a gap that always tends to zero would suggest the theorem survives weaker assumptions. Independently, verifying inequality (7) numerically during a run on an ordinary C¹ but non-Lipschitz objective would confirm the proof's load-bearing estimate holds as claimed.

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Extended reading notes

Core claim

The paper's central claim is a pair of subsequential convergence theorems. Theorem 3.1: for the composite problem min F = f + g with g proper, closed, convex, and supercoercive and f continuously differentiable on dom g, any accumulation point of the sequence generated by the generalized conditional gradient method with average-type nonmonotone Armijo linesearch (Algorithm 1) is stationary; no Lipschitz or Hölder assumption on ∇f appears anywhere in the hypotheses. Theorem 3.2: the same conclusion holds for the parameter-free backtracking variant of [16] (Algorithm 2), giving an accumulation point that is stationary and min_{0≤l≤k} G(x_l) → 0 as k → ∞, under the sole added assumption that the lower level set {x : F(x) ≤ F(x0)} is bounded — a condition automatic when f is convex or dom g is bounded. In the delicate regime where backtracking stepsizes shrink to zero, the mean-value theorem produces a point ξ_k between x_k and the rejected trial point, and the linesearch rejection inequality yields (1−σ)G(x_k) ≤ ‖∇f(ξ_k) − ∇f(x_k)‖ ‖d_k‖; because ξ_k and x_k converge to the same limit and ∇f is continuous, the gap must vanish. The paper thus establishes convergence under a mere continuity condition on the gradient, where the prior literature required quantitative smoothness.

Load-bearing premise

The load-bearing premise is continuity of the gradient ∇f of the smooth part at the accumulation point, used to conclude that ‖∇f(ξ_k) − ∇f(x_k)‖ shrinks to zero when the trial point ξ_k and the iterate x_k approach the same limit, and, for the parameter-free variant, boundedness of the lower level set {x : F(x) ≤ F(x0)}.

Editorial extensions

If this is right

Corollary 3.1: if an accumulation point exists — e.g. when dom g is bounded — the minimum Frank-Wolfe gap over the first k iterates tends to zero, so an ε-approximate stationary point is obtained in finitely many iterations.
Theorem 3.2: the parameter-free variant converges as soon as the lower level set is bounded, a condition that holds automatically when f is convex or when dom g is bounded.
The nonmonotone Armijo result extends the smoothness-free analysis of [2] beyond compact convex constraint sets, and the monotone backtracking method of [22] (p = 1, σ ≤ 1/2) is covered as a special case.
Because the parameter-free method's Hölder-adaptive rate remains valid when ∇f is Hölder continuous, skipping a Hölder check loses the rate only in cases where no Hölder exponent actually exists.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proofs appear to consume only continuity of ∇f at the accumulation point and boundedness of the directions d_k, so the hypothesis 'continuously differentiable on dom g' could likely be weakened to differentiability with locally continuous gradient on the lower level set.
If ∇f is known only through a modulus of continuity ω(δ), the same inequalities should yield a quantitative gap decay that interpolates between the O(ε^{−1−1/ν}) rate for ω(δ) = Cδ^ν and the purely asymptotic guarantee when no modulus exists.
Because every accumulation point must be stationary, any counterexample to whole-sequence convergence under Armijo stepsizes would have to unwind along the stationary set itself, which constrains the search left open by the divergence counterexamples the paper cites.
The mean-value-plus-linesearch-rejection mechanism should transfer to linesearch-based proximal gradient methods, producing a common smoothness-free convergence theory for the two principal composite solvers.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper is a genuinely useful contribution to the convergence theory of generalized conditional gradient methods. It proves subsequential stationarity for two linesearch variants under only C^1 smoothness of the smooth part — no Lipschitz or Hölder gradient. That is a real extension of Bertsekas and of Ito et al. The o(t) descent lemma (Lemma 3.1) is the key, and it is valid under mere continuous differentiability. The proofs are complete and careful, including the vanishing-step cases and the parameter-free adaptive backtracking in Algorithm 2. The comparison with prior work is accurate, and the self-citations are not inflated.

The one soft spot is real but small. Lemma 3.2 asserts the linesearch in Algorithm 1 always terminates. The proof uses Lemma 3.1 to claim F(x_k + β^i d_k) ≤ F(x_k) − σ β^i G(x_k) for all large i. That is correct when G(x_k) > 0, because the linear term dominates the o(t). But the lemma is stated for all x_k, including stationary points where G(x_k) = 0. Then the inequality becomes F ≤ F(x_k) + o(t), and o(t) can be positive. Concretely, take f(x) = x^2, g(x) = max(0, |x|−1)^2, x_0 = 0. All assumptions hold, G(0) = 0, but the subproblem oracle may return v = 1, d = 1, and then F(β^i) > F(0) for every finite i. The loop never terminates. The well-definedness proof needs an explicit “if G(x_k) = 0, stop” in Algorithm 1, or a requirement that the oracle returns d_k = 0 when x_k is already a subproblem minimizer. This does not damage the main convergence theorems for sequences that actually run, since those would only be produced when G(x_k) > 0, but as written Lemma 3.2 is false and the pseudocode is incomplete. A referee should ask for this patch.

Also worth noting: the continuous differentiability of ∇f is load-bearing in the vanishing-step argument. That is fine; it is the intended weakening and it is stated clearly.

Bottom line: this is a competent, honest paper that deserves peer review. After the Lemma 3.2 fix it would be publishable. I would not cite it myself because it is not my subfield, but anyone working on conditional gradient methods should read it. I would send it to review, with the patch requested.

Referee Report

2 major / 4 minor

Summary. The paper analyzes two linesearch-based generalized conditional gradient methods for the composite problem min F(x)=f(x)+g(x), where f is continuously differentiable on dom g (but not assumed to have a Lipschitz or Hölder continuous gradient) and g is proper, closed, convex, and supercoercive. The main results are Theorem 3.1, asserting that every accumulation point of the iterates of a nonmonotone Armijo-type method (Algorithm 1) is stationary, and Theorem 3.2, asserting that under a bounded-level-set condition the parameter-free variant of Ito, Lu, and He (Algorithm 2) has at least one stationary accumulation point and its running minimum Frank-Wolfe gap tends to zero. The key new ingredient is Lemma 3.1, which derives the o(t) descent estimate F(x+td) <= F(x)-tG(x)+o(t) using only continuous differentiability of f and convexity of g.

Significance. If the technical gaps identified below are fixed, this is a worthwhile contribution: it removes global Lipschitz/Hölder smoothness assumptions that are standard in essentially all existing convergence analyses of generalized conditional gradient methods, and it shows that a practically attractive parameter-free backtracking variant remains convergent even when the Hölder condition is unknown or absent. The proofs are short, elementary, and mostly self-contained, and Lemma 3.1 is a clean reusable tool. The extension of Bertsekas-style subsequential convergence from constrained smooth problems to the composite, nonconvex, supercoercive setting is a genuine advance. However, the well-definedness of Algorithm 1 has a real gap at stationary points, so the paper is not yet fully sound as written.

major comments (2)

[Section 3, Lemma 3.2] Lemma 3.2 is false when G(x_k)=0 and the subproblem oracle returns a minimizer v_k different from x_k. The proof applies Lemma 3.1 and concludes F(x_{k,i}) <= F(x_k) - sigma beta^i G(x_k) for all sufficiently large i, but when G(x_k)=0 this reduces to F(x_{k,i}) <= F(x_k)+o(beta^i), which does not force the Armijo condition (4) to hold because o(beta^i) may be positive. Concretely, take E=R, f(x)=x^2, g(x)=max(0,|x|-1)^2, x_0=0; then all standing assumptions hold and x_0 is stationary with G(x_0)=0, but the subproblem minimizer v=1 gives d=1 and F(beta^i)=beta^{2i}>0=F(x_0), so (4) fails for every finite i. The algorithm therefore may never produce an infinite sequence from a stationary point. The convergence theorems would still hold for sequences that are actually generated, but the well-definedness claim in Lemma 3.2 and the pseudocode need a patch, such as an explicit stopping rule 'if G(x_k)=0, stop' or a tie-breaking rule that selects d_k=0 at stationary points.
[Section 3.1, Algorithm 2] The definition of the trial step size in Algorithm 2 is incomplete when d_k=0. The formula tau_k^(i)=min{1, G(x_k)/(2L_k^(i)||d_k||^2)} involves division by ||d_k||^2, and d_k=0 occurs exactly when v_k=x_k, i.e., when x_k is a minimizer of the linearized subproblem. Since -grad f(x_k) in partial g(x_k) is equivalent to G(x_k)=0 by Lemma 2.1, this is precisely the stationary case. The algorithm needs an explicit convention for d_k=0, e.g., tau_k^(i)=1 or termination when G(x_k)=0. Without this, the statement of Proposition 3.2 and Lemma 3.3 do not formally cover all admissible outputs of a subproblem solver.

minor comments (4)

[Abstract and Introduction] The abstract and introduction say the methods 'guarantee convergence' even without Hölder continuity, but the theorems establish subsequential convergence and convergence of the running minimum of the Frank-Wolfe gap, not convergence of the whole sequence. Please make this distinction explicit in the abstract.
[Section 3, Lemma 3.2 proof, Eq. (5)] In the displayed inequality (5), the exponent on beta appears to be i_{k-1}, but the notation is easy to misread as i_k - 1. Please add brackets or a clarifying sentence so that the index is unambiguous.
[Footnote, uniform convex structure] The footnote defining uniform convex structure states 'rho >= 21' near the end of Section 3; this is presumably rho >= 2 and should be corrected.
[References] Reference [18] contains the odd text 'Kurdyka–/suppress Lojasiewicz'; this should read 'Kurdyka–Łojasiewicz property'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence theorems are derived self-containedly from the problem definitions and the algorithm line-search rules.

full rationale

The paper's convergence proofs are direct mathematical derivations. Lemma 3.1 uses only continuous differentiability of f, convexity of g, and the definition of the Frank-Wolfe gap G(x); Lemma 2.2 is a definitional lower-semicontinuity argument. Theorems 3.1 and 3.2 then combine these lemmas with the descent/acceptance inequalities (4) and (8), monotonicity of F_k or F(x_k), boundedness of the subproblem minimizers via supercoercivity, and continuity of ∇f. No parameter is fitted to any subset of data, and no 'prediction' is derived from a quantity already used as an input; the stationarity conclusion comes from showing G(x*) = 0, which is equivalent to (2) by the cited definitional Lemma 2.1. Algorithm 2 is taken from Ito et al. [16], but Theorem 3.2 does not rely on any convergence claim from that work; it proves convergence directly under the stated assumptions and the algorithm's own backtracking definition. The only related concerns are non-circular: the title's 'without smoothness assumptions' is stronger than the actual C^1 assumption on f, and the proof of Lemma 3.2 may need a separate termination rule at stationary points where G(x_k)=0. These are correctness or assumption-strength issues, not cases where a claim reduces to its inputs by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper relies only on standard background assumptions and the previously published algorithms; it introduces no new parameters, fitted constants, or entities. The algorithmic constants (β, σ, p, L_{-1}) are not tuned to data and the results are valid for any allowed values.

assumptions (3)

domain assumption f is lower semicontinuous and continuously differentiable on dom g
Stated in Section 2, problem (1). The proofs of the vanishing-step cases in Theorems 3.1 and 3.2 use the continuity of ∇f to show that terms like ||∇f(ξ_k)-∇f(x_k)|| vanish when ξ_k and x_k converge to the same point (inequalities (7) and (10)).
domain assumption g is proper closed convex and supercoercive
Stated in Section 2. Supercoercivity is used to guarantee nonemptiness of the subproblem solution (3) and to prove boundedness of the subproblem solutions {v_k} in Propositions 3.1(iv) and 3.2(iv).
domain assumption The lower level set {x : F(x) <= F(x0)} is bounded
Added in Theorem 3.2 for Algorithm 2 to guarantee that the iterates are bounded and an accumulation point exists. The paper notes this holds when f is coercive or when g is bounded, and gives a sufficient condition f(x) >= a||x|| + b.

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Cite this review

Pith. "Pith review of Convergence of linesearch-based generalized conditional gradient methods without smoothness assumptions." pith.science (2026). https://pith.science/paper/ATFHVJ5K

@misc{pith2026250501092,
  author       = {Pith},
  title        = {Pith review of: Convergence of linesearch-based generalized conditional gradient methods without smoothness assumptions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ATFHVJ5K}},
  note         = {Machine review of arXiv:2505.01092}
}

read the original abstract

The generalized conditional gradient method is a popular algorithm for solving composite problems whose objective function is the sum of a smooth function and a nonsmooth convex function. Many convergence analyses of the algorithm rely on smoothness assumptions, such as the Lipschitz continuity of the gradient of the smooth part. This paper provides convergence results of linesearch-based generalized conditional gradient methods without smoothness assumptions. In particular, we show that a parameter-free variant, which automatically adapts to the H\"older exponent, guarantees convergence even when the gradient of the smooth part of the objective is not H\"older continuous.

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Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

First-order methods in optimization

Amir Beck. First-order methods in optimization . SIAM, 2017

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Dimitri P Bertsekas. Nonlinear Programming. Athena scientiﬁc, 2nd edition, 1999

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A H\"olderian backtracking method for min-max and min-min problems

J´ erˆ ome Bolte, Lilian Glaudin, Edouard Pauwels, and Mathieu Serrurier. Ah\” olderian backtracking method for min-max and min-min problems. arXiv preprint arXiv:2007.08810 , 2020

work page Pith review arXiv 2007

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J´ erˆ ome Bolte, Cyrille W Combettes, and Edouard Pauwels. The iterates of the frank–wolfe algorithm may not converge. Mathematics of Operations Research , 49(4):2565–2578, 2024. 9

work page 2024

[5] [5]

Conditional gradient methods

G´ abor Braun, Alejandro Carderera, Cyrille W Combettes, Hame d Hassani, Amin Karbasi, Aryan Mokhtari, and Sebastian Pokutta. Conditional gradient methods. arXiv preprint arXiv:2211.14103 , 2022

arXiv 2022

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A generalized c onditional gradient method and its connection to an iterative shrinkage method

Kristian Bredies, Dirk A Lorenz, and Peter Maass. A generalized c onditional gradient method and its connection to an iterative shrinkage method. Computational Optimization and applications , 42:173–193, 2009

work page 2009

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Coresets, sparse greedy approximation, and the frank-wolfe algorithm

Kenneth L Clarkson. Coresets, sparse greedy approximation, and the frank-wolfe algorithm. ACM Transactions on Algorithms (TALG) , 6(4):1–30, 2010

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Complexity of linear min imization and projection on some sets

Cyrille W Combettes and Sebastian Pokutta. Complexity of linear min imization and projection on some sets. Operations Research Letters, 49(4):565–571, 2021

work page 2021