Linesearch-free adaptive Bregman proximal gradient for convex minimization under local relative smoothness

Andreas Themelis; Hongjia Ou; Puya Latafat

REVIEW 3 major objections 4 minor 2 cited by

Linesearch-free adaptive Bregman proximal gradient for convex minimization under local relative smoothness

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proposes two linesearch-free adaptive stepsize rules for the Bregman proximal gradient method and proves that, under only local relative smoothness, the objective values converge to the infimum and, under mild extra…

desk verdict A genuinely new adaptive Bregman proximal gradient method with a mostly sound analysis, one fixable gap in Lemma 4.2, and honest numerics. read the letter →

arxiv 2508.01353 v2 pith:NIVTDPRY submitted 2025-08-02 math.OC

Hongjia Ou , Puya Latafat , Andreas Themelis This is my paper

classification math.OC MSC 65K0590C0690C2590C3049M29

keywords convexcompositeoptimizationBregmanproximalgradientrelativesmoothnessadaptivestepsizeslinesearch-freedistanceBregman-YounginequalityLegendrefunctions

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

This paper presents two adaptive, linesearch-free variants of the Bregman proximal gradient method for convex composite minimization. The central claim is that the iterates reach the optimal value in the limit, inf_k φ(x_k) = inf_C φ, assuming only that the smooth term is locally relatively smooth with respect to a Legendre kernel, with no global relative smoothness modulus, no global strong convexity of the kernel, and no backtracking linesearch. If a minimizer lies inside the kernel's domain, the whole sequence converges to a solution, and under a mild zone condition the sequence is bounded with a unique optimal limit point. The proof rests on a Bregman generalization of Young's inequality that controls the inner-product term which had blocked such extensions. This matters because the usual global assumptions are hard to verify and force conservative stepsizes, while the new rules generate stepsizes from local data only.

What carries the argument

The central object is the Bregman-Young inequality of Lemma 3.2, ⟨x−y, v⟩ ≤ (1/δ)D_φ(x,y) + (1/δ)D_{φ*}(∇φ(y)+δv, ∇φ(y)), derived from the three-point identity and Legendre duality; it replaces Euclidean Young and Cauchy-Schwarz bounds that are unavailable in Bregman geometry. Around it the paper builds three data-driven estimates: the local relative smoothness ℓ_k between consecutive iterates, the curvature Λ_{k,δ} of the forward operator H_k = ∇φ − γ_k∇f measured in conjugate Bregman distance, and the local symmetry ratio α_k. These feed multiplicative stepsize updates ρ_{k+1} = γ_{k+1}/γ_k, given in (2.6) and (2.7), which keep a merit function U_k decreasing. The descent of this merit function, combined with Lemma 4.1 and the no-vanishing-stepsizes argument of Claim 2, yields the convergence statements.

What would settle it

Run B-adaPG on the polynomial-Hessian problem (5.1) with the quartic kernel (5.2) and on the KL nonnegative regression problem (5.3), recording the stepsizes γ_k and iterates x_k. If any run shows γ_k → 0 while (x_k) stays inside a compact subset of int dom φ yet fails to converge, for example oscillating between two cluster points, then Lemma 4.2's assertion is false and the proof of Claim 2 would need replacement.

Watch

Extended reading notes

Core claim

In the paper's own formulation, the discovery is Theorem 2.5: for B-adaPG, and for B-adaPGα when the kernel has a positive symmetry coefficient, under local relative smoothness one always has inf_k φ(x_k) = inf_C φ. If the solution set meets the interior of the kernel's domain, the sequence converges to a minimizer; under Assumption 2.3 the iterates are bounded and admit exactly one optimal limit point. The stepsizes come from three local quantities—the relative smoothness estimate ℓ_k, the forward-operator curvature estimate Λ_{k,δ}, and the local symmetry ratio α_k—so no trial-and-error inner loop is needed. The engine is a Bregman version of Young's inequality, which bounds the troublesome inner product B_{k+1} directly in Bregman distances and lets the analysis proceed with only local moduli.

Load-bearing premise

The proof rests on the unproved assertion in Lemma 4.2 that if stepsizes tend to zero while iterates remain in a compact region inside the kernel's domain, the iterates converge; if that assertion fails, the argument that stepsizes stay bounded away from zero collapses.

Editorial extensions

If this is right

Bregman proximal gradient can be run with no backtracking linesearch and no knowledge of any global smoothness constant, as long as the smooth term is locally relatively smooth.
Stepsizes are computed from the current and previous iterates, so the per-iteration cost stays comparable to plain BPG while stepsizes can grow orders of magnitude beyond the conservative 1/L constant.
Even when all minimizers lie on the boundary of the kernel's domain, such as with relative-entropy barriers, the objective values still satisfy inf_k φ(x_k) = inf_C φ; under Assumption 2.3 the sequence has a unique optimal limit point.
When the kernel is the Euclidean quadratic, the rules reduce to slightly conservative versions of the existing Euclidean adaptive proximal gradient updates, recovering that theory as a special case.
The Bregman-Young inequality is stated as a standalone tool and can be reused wherever inner products need to be controlled by Bregman distances.

Reading between the lines

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

If the unproved convergence assertion in Lemma 4.2 turns out to require extra conditions, the proof that stepsizes stay bounded away from zero would need a different argument; Theorem 2.5 might still be true, just not proven by the current route.
The Bregman-Young inequality likely applies beyond proximal gradient, for example to Bregman proximal point, Douglas-Rachford, or accelerated Bregman variants where inner products appear; testing it there is a natural next step.
The numerical observation that stepsizes stay bounded away from zero even near boundary points suggests a stronger theorem under Assumption 2.3 might hold, a question the paper explicitly leaves open.
One direct test would be to run B-adaPG on a problem with no global relative smoothness modulus at all, such as f(x) = 1/x relative to the Boltzmann-Shannon entropy, and check that the convergence behavior and stepsize trajectories match the theory.

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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

The genuinely new piece is the Bregman-Young inequality (Lemma 3.2) and the two adaptive stepsize rules built on it. They yield the first Bregman proximal gradient method that provably attains the infimum under local relative smoothness, without global strong convexity of the kernel and without backtracking linesearch. That is a real extension of the Euclidean adaPG line, and the conservatism relative to [12, 11] is openly acknowledged and shown to be minor in practice.

The analysis is mostly rigorous: Lemma 3.1 checks out, Theorem 2.5 follows from explicit assumptions, and the numerical experiments cover a sensible range of problems with an honest comparison against linesearch and BaGRAAL. The authors also mark their own open questions (e.g., stepsizes staying bounded away from zero on the boundary, Section 5.4) instead of overclaiming.

The soft spots are fixable rather than structural. Lemma 4.2 invokes 'standard results ensure that (x_k) converges' to get Lambda_{k,delta}->1 from gamma_k->0. That assertion is not standard and is false in general. Since Claim 2 uses it to rule out gamma_k->0, the proof of Theorem 4.4 needs the repair: show consecutive differences vanish directly from gamma_k->0 via the limit prox_{gamma g}(y)->y, which the authors did not include. I expect that argument to work, so I would call this a gap, not a collapse. Minor issues: Remark 2.6(i) states an O(1/(K+1)) rate without proof (either prove it or delete it), there is a small indexing typo in Section 4.1 (v_k should involve H_k, not H_{k-1}), and the figures are single trajectories without error bars or code release. None of these change my recommendation.

This paper deserves a serious referee. I would send it out, with an explicit request to fix Lemma 4.2 and either justify or cut the rate claim in Remark 2.6. It will be of value to researchers working on adaptive Bregman methods and relative-smooth optimization.

Referee Report

3 major / 4 minor

Summary. The paper studies the composite convex minimization problem (P) in which the differentiable term f is only locally smooth relative to a Legendre kernel phi (Assumption 2.1). It proposes two adaptive Bregman proximal gradient algorithms, B-adaPG and B-adaPG-alpha, whose stepsizes are updated from local quantities l_k, Lambda_{k,delta}, and the local symmetry coefficient alpha_k, with no backtracking linesearch. The main result, Theorem 2.5, states that inf_k phi(x_k) = inf_C phi; under an interior-minimizer condition the iterates converge to a solution, and under Assumption 2.3 they are bounded with a unique optimal limit point. The analysis rests on a Bregman generalization of Young's inequality (Lemma 3.2) and a merit-function identity (Lemma 3.1). Numerical experiments cover polynomial-growth Hessians, KL regression, simplex barrier problems, and Euclidean problems, with comparisons to linesearch methods, BaGRAAL, ABPG-g, and Euclidean adaPG variants.

Significance. If the main results hold, this is a substantial advance in adaptive Bregman methods: it removes global relative smoothness, global strong convexity of the kernel, and backtracking linesearch while keeping a self-contained convergence proof from explicit assumptions. The Bregman-Young inequality is a clean and potentially reusable tool, and the local moduli are defined from iterates rather than fitted to the conclusion, so the analysis does not appear circular. The numerical study is broad and the paper is honest about its remaining open questions. However, the contribution is conditional: one load-bearing lemma is proved by an unjustified convergence assertion, and the displayed stepsize updates appear to have a missing reciprocal. These issues must be resolved before the central claims are fully supported.

major comments (3)

[Lemma 4.2 (Section 4.2)] The proof asserts that if gamma_k -> 0, then 'standard results ensure that (x_k) converges to some point x in K'. This is not a standard consequence of the assumptions: gamma_k -> 0 does not by itself imply ||x_k - x_{k-1}|| -> 0, and even that weaker condition would not force convergence of a sequence in a compact set. The assertion is load-bearing, because Claim 2 of Theorem 4.4 uses Lemma 4.2 to conclude that the second term in (2.6)/(2.7) eventually forces gamma_{k+1} = rho-hat_{k+1} gamma_k, contradicting gamma_k -> 0. Without a valid proof of Lambda_{k,delta} -> 1, the argument for Theorem 4.4 and hence Eq. (2.8) collapses. The proof should be repaired by a direct argument that does not require convergence of (x_k), for example by showing that the Bregman proximal step with vanishing stepsize satisfies Delta_phi(x_k, x_{k-1}) -> 0 and then estimating the ratio defining Lambda_{k,delta} via the mean-value theorem and continuity of nabla^2 phi* on the compact set nabla phi(K).
[Section 2.2, Eqs. (2.6)-(2.7)] As displayed, the second argument of the minimum is proportional to [Lambda_{k,delta} - (1 - gamma_k l_k)]_+ and not to its reciprocal. Consequently, whenever Lambda_{k,delta} <= 1 - gamma_k l_k, the update would give rho_{k+1} = 0, contradicting Remark 2.4 (which states that in this case all updates reduce to rho_{k+1} = rho-hat_{k+1}), contradicting Claim 2 in Section 4.1, and contradicting the requirement that all stepsizes remain positive. The derivation in Section 3.3 leading to (3.16), together with the logic of Claim 2, indicates that the bracket should appear in the denominator, i.e. rho_{k+1} = min(rho-hat_{k+1}, alpha_k/(1+alpha_k) * 1/(2 rho-hat_{k+1} [Lambda_{k,delta} - (1 - gamma_k l_k)]_+)), with the analogous correction in (2.7). The displayed formulas must be corrected and all subsequent uses checked against the corrected update.
[Section 4.1, paragraph following Eq. (4.8)] The definition v_k := H_{k-1}(x_{k-1}) - H_{k-1}(x_k) is not an element of gamma_k partial phi(x_k). From (3.4b) the correct relation is gamma_k tilde{nabla}phi(x_k) = H_k(x_{k-1}) - H_k(x_k). With the printed definition, the inequality gamma_k P_{k-1}(x_star) <= rho_k <x_star - x_{k-1}, -v_{k-1}> and the subsequent Bregman-Young bound do not follow. This indexing error is local and fixable by setting v_k := H_k(x_{k-1}) - H_k(x_k) (equivalently, v_{k-1} = H_{k-1}(x_{k-2}) - H_{k-1}(x_{k-1})), after which the displayed chain of inequalities becomes consistent with the definition of Lambda_{k-1,2 rho-hat_k}.

minor comments (4)

[Remark 2.6(i)] The claimed O(1/(K+1)) rate is asserted without proof. Lemma 4.1(ii) alone yields P_min^K(x) <= U_0(x) / sum_{k=1}^{K+1} gamma_k, and a rate requires a lower bound gamma_k >= gamma_min > 0; the remark says such a bound can be derived but provides no details. Either supply the argument or explicitly label the rate as a conjecture/open problem.
[Lemma 3.5] The displayed equality B_{k+1} = Delta_phi(x_{k+1}, x_k) + gamma_{k+1} tilde D_g(x_{k+1}, x_k) appears to omit a second nonnegative term. The correct identity is Delta_phi(x_{k+1}, x_k) + gamma_{k+1}(tilde D_g(x_{k+1}, x_k) + tilde D_g(x_k, x_{k+1})), or alternatively one can invoke monotonicity of partial g directly. The inequality conclusion remains valid, so this is a typographical issue rather than a mathematical one.
[Section 5] The numerical comparisons are reported as single trajectories, without multiple runs or error bars. Given the claim of robustness across problem classes, reporting averaged behavior over several seeds, or at least a small number of repeated trials, would substantially strengthen the empirical conclusions.
[Section 5.1] There are several minor typos, including 'B-adaPGand is compared' and 'the linear variant'. The paper would also benefit from a sentence noting explicitly that B-adaPG-alpha is tested only in settings where alpha(phi) > 0, consistent with Table 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Bregman-Young inequality and merit-function descent are proved in-paper; step-size rules derive from local iterate data and are not fitted to the convergence conclusion.

full rationale

The convergence analysis is self-contained and does not fit any constant to the target conclusions. The local quantities ℓ_k, Λ_{k,δ}, and α_k in (2.3) are exact algebraic ratios of iterate data; they enter the proof through the identity of Lemma 3.1 (proved in Appendix A), the Bregman-Young inequality of Lemma 3.2 (proved from the external three-point identity, Fact 1.1 [8]), and the identity (2.4). The step-size rules (2.6)–(2.7) are derived, not assumed, as conditions making the coefficients in (3.14) nonpositive, and Lemma 3.6 verifies the rules satisfy them; the authors explicitly address in Remark 2.7 the potential circular dependency between γ_{k+1} and Λ_{k,δ} in the Bregman setting and resolve it with the overestimator ρ̂_{k+1} rather than exploiting it. Results cited from prior work are either external or re-proved: Lemma 3.1 parallels [12, Lem. 2.2] but its proof is reproduced in the appendix; Legendre-function properties come from [4,8]; the ∇ϕ-firm nonexpansiveness of the Bregman proximal operator used in Lemma 4.3 is a standard property with independent support [26,27]. Self-citations to [12,11] provide the Euclidean blueprint, comparative context, and an initialization heuristic, but do not carry the proof of Theorem 2.5. The flagged assertion in Lemma 4.2 — that 'standard results ensure that (x_k) converges' when γ_k → 0 — is a genuine correctness gap (a sequence in a compact set with consecutive differences tending to zero need not converge) and, together with the open problem stated in Remark 2.6(ii), it weakens the rigor of the step-size nonvanishing argument in Claim 2; however, it is a missing argument, not a reduction of a conclusion to its inputs, so it falls under correctness risk rather than circularity. No fitted parameter is renamed as a prediction, no load-bearing conclusion is imported from the authors' own uniqueness results, and no known result is repackaged under new coordinates.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numbers are fitted to data; the algorithms depend only on local estimates computed from iterates. The central proofs rest on standard Legendre-Bregman facts and on the problem assumptions listed, and no invented entities are introduced.

assumptions (7)

domain assumption Assumption 2.1(i): phi is proper, convex, 1-coercive, Legendre, twice differentiable with nabla^2 phi > 0 on int dom phi.
Defines the Bregman geometry and supplies the three-point identity, coercivity, and conjugacy facts used in Lemmas 3.1 to 3.3 and Lemma 4.2.
domain assumption Assumption 2.1(ii): f is proper, convex, lsc, and locally smooth relative to phi on every compact convex K subset of int dom phi.
This is the core smoothness assumption replacing global relative smoothness; it is used to bound ell_k and Lambda_{k,delta} in the convergence proof.
domain assumption Assumption 2.1(iii): g is proper, lsc, and convex with dom g intersecting int dom phi.
Ensures the Bregman proximal subproblem has solutions and the optimality conditions in (3.4) hold.
domain assumption Assumption 2.1(iv): arg min_C phi is nonempty.
Guarantees the problem has a solution, as required for all convergence statements.
domain assumption Assumption 2.3: the Bregman distance has zone C, meaning D_phi(x, x_k) tends to 0 when x_k tends to x and D_phi(x, .) is level bounded for x in C minus C.
Used in Theorem 2.5(ii) to handle minimizers on the boundary of the domain and to prove boundedness and a unique optimal limit point.
standard math Standard Legendre-Bregman facts: three-point identity, nabla phi* = (nabla phi)^{-1}, D_phi(x, .) is 1-coercive, and D_phi(x, y) = D_phi*(nabla phi(y), nabla phi(x)).
These cited facts from Fact 1.1 and Fact 1.2 are used throughout the derivations; the paper does not prove them.
domain assumption Local relative smoothness implies differentiability of f on int dom phi and the bound D_f(x, y) <= L_K D_phi(x, y) on compact K.
Used in Lemma 4.2 to control the term v_k = nabla f(x_k) - nabla f(x_{k-1}) via the local relative smoothness modulus L_{f,K}.

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Pith. "Pith review of Linesearch-free adaptive Bregman proximal gradient for convex minimization under local relative smoothness." pith.science (2026). https://pith.science/paper/NIVTDPRY

@misc{pith2026250801353,
  author       = {Pith},
  title        = {Pith review of: Linesearch-free adaptive Bregman proximal gradient for convex minimization under local relative smoothness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NIVTDPRY}},
  note         = {Machine review of arXiv:2508.01353}
}

read the original abstract

This paper introduces adaptive Bregman proximal gradient algorithms for solving convex composite minimization problems without relying on global relative smoothness or strong convexity assumptions. Building upon recent advances in adaptive stepsize selections, the proposed methods generate stepsizes based on local curvature estimates, entirely eliminating the need for backtracking linesearch. A key tool in our analysis is a Bregman generalization of Young's inequality, which allows the control of a critical inner product in terms of the same Bregman distances used in the updates. Our theory applies to problems where the differentiable term is merely \emph{locally} smooth relative to a distance-generating function, without requiring the existence of global moduli or symmetry coefficients. Numerical experiments demonstrate their competitive performance compared to existing approaches across various problem classes.

Figures

Figures reproduced from arXiv: 2508.01353 by the authors.

**Figure 5.2.** 1 [PITH_FULL_IMAGE:figures/full_fig_p026_5_2.png] view at source ↗

**Figure 5.3.** 1: KL-divergence nonnegative regression. Top: convergence in terms [PITH_FULL_IMAGE:figures/full_fig_p028_5_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 5.4.** Figure 5.4: 1: Relative entropy barrier minimization on the simplex using LIB [PITH_FULL_IMAGE:figures/full_fig_p029_5_4.png]

**Figure 5.5.** Figure 5.5: 1 [PITH_FULL_IMAGE:figures/full_fig_p030_5_5.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

A Breg- man forward-backward linesearch algorithm for nonconvex composite op- timization: Superlinear convergence to nonisolated local minima

Masoud Ahookhosh, Andreas Themelis, and Panagiotis Patrinos. A Breg- man forward-backward linesearch algorithm for nonconvex composite op- timization: Superlinear convergence to nonisolated local minima. SIAM Journal on Optimization, 31(1):653–685, 2021

2021

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work page 1988

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