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On The Linear Convergence of Bregman Proximal Gradient Methods with Applications to Kullback--Leibler regression

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Pith's one-line read A restricted relative strong-convexity condition yields linear rates for Bregman proximal gradient methods, and a smoothed Burg entropy makes that condition hold for Kullback–Leibler regression even when solutions lie on the boundary of the

desk verdict Clean RRSC theory for BPGM plus a practical mirror map that actually restores linear rates for KL/Poisson problems, including boundary solutions. read the letter →

arxiv 2607.05539 v1 pith:QIFULESC submitted 2026-07-06 math.OC cs.NAmath.NA

Jonathan Chirinos-Rodríguez , Christian Daniele , Cédric Févotte , Emmanuel Soubies This is my paper · ORCID
classification math.OCcs.NAmath.NA MSC 90C2565K0549M37
keywords BregmanproximalgradientrestrictedrelativestrongconvexityKullback–LeiblerregressionlinearconvergencesmoothedBurgentropysmoothnessPoissoninverseproblems
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

Standard linear-rate proofs for first-order methods demand strong convexity everywhere. Many practical problems, including Kullback–Leibler (KL) regression, only look strongly convex near their solution set. The paper defines Restricted Relative Strong Convexity (RRSC): the smooth term needs only relative strong convexity between a point and its Euclidean projection onto the solution set. Together with a positive restricted symmetry coefficient of the Bregman divergence and relative smoothness, RRSC implies linear decay of both the Bregman distance to the solution set and the objective gap for Bregman proximal gradient methods, with constant or back-tracking steps. For KL regression the classical Burg entropy fails to guarantee RRSC once a solution touches the coordinate hyperplanes, while a simple smoothed Burg entropy restores a uniform RRSC constant on the whole nonnegative orthant except a thin conical neighbourhood of the kernel of the measurement matrix. Numerical experiments on synthetic Poisson inverse problems confirm that the smoothed geometry produces linear rates precisely where the theory predicts them, often outperforming both Burg entropy and the classical Richardson–Lucy iteration.

What carries the argument

Restricted Relative Strong Convexity (RRSC): the inner-product inequality ⟨∇F(x)-∇F(x̄_ℓ_{2}),x-x̄_ℓ_{2}⟩≥μ⟨∇φ(x)-∇φ(x̄_ℓ_{2}),x-x̄_ℓ_{2}⟩ is required only between an arbitrary feasible point and its Euclidean projection onto the solution set; this single inequality, combined with the restricted symmetry coefficient α_S(φ)>0, produces the linear-rate lemmas for BPGM.

What would settle it

Run BPGM with the smoothed Burg entropy on a rank-deficient ℓ_{1}-regularised KL problem whose unique solution lies on the boundary; if the objective gap decays only sublinearly for a large fraction of random starts, the claimed guarantee fails.

Watch

Extended reading notes

Core claim

Under Restricted Relative Strong Convexity, a positive restricted symmetry coefficient of the Bregman divergence, and relative L-smoothness, Bregman proximal gradient iteration (constant or back-tracking step-size) converges linearly: the Bregman distance of the iterates to the solution set contracts by the factor 1/(1+α_S τ μ) at every step, and the objective gap obeys the same contraction scaled by 1/τ. For KL regression the smoothed Burg entropy makes RRSC hold outside a thin conical set around ker(A), guaranteeing the linear rates even when solutions lie on the boundary of the nonnegative orthant.

Load-bearing premise

The iterates must stay inside a region where RRSC holds with a uniform positive constant; if they remain forever on a fibre of the measurement matrix the linear-rate proof does not apply.

Editorial extensions

If this is right

  • Any Bregman proximal gradient scheme whose smooth term satisfies RRSC and whose mirror map has positive restricted symmetry inherits an explicit linear rate, constant or back-tracking.
  • For KL regression the smoothed Burg entropy guarantees linear convergence on the whole nonnegative orthant except a thin conical set around ker(A), covering both unique and non-unique, regularised and unregularised cases.
  • Classical Burg entropy yields linear rates only when at least one solution lies in the strict interior; otherwise the theory predicts—and experiments confirm—sublinear behaviour.
  • The same RRSC framework immediately specialises to ordinary proximal gradient (φ = ½‖·‖_{2}^{2}) and recovers known restricted-strong-convexity rates as a special case.

Reading between the lines

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

  • The same RRSC argument should apply verbatim to other β-divergences once a suitable smoothed mirror map is identified, extending the linear-rate guarantee beyond Poisson models.
  • Because the linear rate depends only on the trajectory remaining outside a measure-zero cone, a mild random perturbation of the step-size or of the initial point is likely sufficient to restore the rate in the edge cases the paper leaves open.
  • The explicit dependence of the RRSC constant on the smoothing parameter ξ suggests an automatic line-search or continuation scheme that balances μ and L without manual tuning.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper introduces Restricted Relative Strong Convexity (RRSC), a Bregman analogue of restricted strong convexity that only requires relative strong convexity between a point and its Euclidean projection onto the solution set S. Under RRSC, a positive restricted symmetry coefficient α_S(φ), and relative L-smoothness, Theorem 2.4 and Corollary 2.6 establish linear rates for BPGM (constant step-size and backtracking) on both the Bregman distance to S and the objective gap. The framework is applied to (regularized) Kullback–Leibler regression with three generating functions: Burg entropy, a smoothed Burg entropy φ_sB, and the squared Euclidean norm. Propositions 3.4–3.6 characterize the regions where RRSC, α_S>0 and relative smoothness hold; the analysis shows that φ_sB guarantees the linear rates even when solutions lie on the boundary of the nonnegative orthant, whereas plain Burg entropy does not. Synthetic Poisson experiments corroborate every regime summarized in Table 1.

Significance. The work supplies a clean, self-contained extension of restricted strong convexity to the Bregman setting and gives the first systematic linear-rate theory for BPGM on KL regression that covers both unique and non-unique solutions, regularized and unregularized formulations, and boundary solutions. The introduction of the smoothed Burg entropy as a distance-generating function is practically useful: it restores linear convergence in regimes where the classical Burg entropy and Richardson–Lucy degrade to sublinear rates, while remaining closed-form. The proofs rely only on standard convex analysis, the constants are derived analytically (not fitted), and the numerical study is reproducible and matches the theory across all predicted cases. These contributions are of clear interest to the first-order optimization and Poisson imaging communities.

minor comments (5)
  1. In the statement of Lemma 2.3 the integral representation of J(x)−J(x̄_ℓ₂) is written with an arbitrary subgradient s_J(γ(t)); a short remark that the subsequent monotonicity argument is independent of the choice would improve readability.
  2. Figure 1 caption: the green/red shading is clear, but adding a one-line legend that “green = RRSC holds for some heta,ε>0” would help readers who only glance at the figure.
  3. Proposition 3.6 (relative smoothness for φ_sB): the constant L≥∑_m ω_m^{2} y_m with ω_m=max{1,[A1]_m ξ/b_m} is correct, yet a brief comparison with the classical Burg constant ∑ y_m would make the price of smoothing more transparent.
  4. Section 3.2.1: the empirical trade-off for ξ is well illustrated, but a short sentence recalling the analytic lower bound μ≥ min{c1 ξ^{2},c2 ξ/(c3+ξ)} already derived in the proof of Proposition 3.4 would link the plot more tightly to the theory.
  5. A few typographical slips remain (e.g., “hierarchichal” → hierarchical, “Lojasiewicz” consistently with diacritics, missing space before some citations). A final proof-reading pass would remove them.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity; main rates under RRSC are self-contained first-order analysis. One minor self-citation completes part of the RRSC verification for smoothed Burg entropy.

  1. self citation load bearing [Appendix A.1.1 (Proof for φ_sB of Proposition 3.4)]
    "The rest of the proof follows exactly the same as the one of [20, Theorem 2.7]. For this reason, we omit it and only recall here the behaviour of µ as a function of ξ: we have that µ≥min{c1ξ2,c2ξ/(c3+ξ)}, for some c1,c2,c3>0."

    Existence of a uniform µ>0 for the smoothed-Burg geometry is finished by deferring the remaining estimates to the authors’ own prior preprint [20] rather than reproducing them. While the key new lower bound (33) is proved in the present paper, the final positivity claim for µ partially rests on that overlapping-author citation; the step is minor because the general RRSC theory and the qualitative superiority claim for φ_sB do not depend on the omitted algebra.

full rationale

The core derivation (Definition 2.1 of RRSC, Lemma 2.3 growth, Theorem 2.4 / Corollary 2.6 linear rates for BPGM under relative L-smoothness + RRSC + α_S(φ)>0) is standard convex analysis using three-point identities, monotonicity of subdifferentials, and the fundamental theorem of calculus along the Euclidean line segment; it does not reduce to its own inputs by construction, nor does it rely on fitted parameters or uniqueness theorems imported from the authors. The KL application (Propositions 3.4–3.6) analytically lower-bounds the relative strong-convexity and symmetry quantities via singular-value estimates, compactness of S, and elementary bounds on the Hessian of the smoothed KL; the resulting μ(ξ) lower bound is derived, not fitted, and remains positive for any fixed ξ>0. The single self-citation appears only in the appendix completion of the φ_sB case of Proposition 3.4 and is not load-bearing for the general rates or for the qualitative claim that φ_sB places iterates in a region where RRSC holds. Empirical tuning of ξ affects observed speed but is never used to ‘prove’ existence of μ. No fitted-input-as-prediction, self-definitional, or ansatz-smuggling patterns are present. The paper is therefore essentially free of circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 2 invented entities

The paper rests on standard convex-analysis axioms (Legendre functions, Bregman three-point identity, monotonicity of subdifferentials) plus the new RRSC and restricted-symmetry definitions. The only free parameter that affects practical performance is the smoothing length xi, which is chosen by grid search; all other constants (L, mu, alpha) are derived, not fitted.

free parameters (2)
  • smoothing parameter xi of phi_sB
    Chosen by evaluating 30 candidate values and retaining the one that yields the fastest observed linear rate; the theoretical lower bound on mu depends on xi but remains positive for any fixed xi>0.
  • back-tracking factor beta
    Arbitrary in (0,1); only affects the concrete constant in the rate, not the existence of linear convergence.
assumptions (5)
  • standard math phi is a Legendre function (essentially smooth and strictly convex on int(dom phi))
    Required for uniqueness of Bregman projections and well-posedness of BPGM (Assumption 2.1).
  • domain assumption F is L-Lipschitz relative to phi (LC condition)
    Standard relative-smoothness hypothesis that replaces classical Lipschitz gradient; verified for KL with each of the three phi's.
  • ad hoc to paper F satisfies Restricted Relative Strong Convexity with constant mu>0
    The novel sufficient condition introduced in Definition 2.1; its validity region is characterized for KL in Proposition 3.4.
  • ad hoc to paper restricted symmetry coefficient alpha_S(phi)>0
    Definition 2.2; proved positive for the three generating functions under the stated restrictions (Proposition 3.5).
  • domain assumption solution set S is nonempty and compact
    Proved for KL + bounded-below regularizer by coercivity (Lemma 3.2).
invented entities (2)
  • Restricted Relative Strong Convexity (RRSC)
    purpose: Weaker substitute for global relative strong convexity that still yields linear BPGM rates
    Definition 2.1; the central new analytic tool of the paper.
  • restricted symmetry coefficient alpha_D(phi)
    purpose: Quantifies local symmetry of the Bregman divergence needed to convert Euclidean growth into Bregman growth
    Definition 2.2; used in the key growth inequality of Lemma 2.3.

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Cite this review

Pith. "Pith review of On The Linear Convergence of Bregman Proximal Gradient Methods with Applications to Kullback--Leibler regression." pith.science (2026). https://pith.science/paper/QIFULESC

@misc{pith2026260705539,
  author       = {Pith},
  title        = {Pith review of: On The Linear Convergence of Bregman Proximal Gradient Methods with Applications to Kullback--Leibler regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIFULESC}},
  note         = {Machine review of arXiv:2607.05539}
}
read the original abstract

Bregman Proximal Gradient methods (BPGM) exploit the underlying geometry of the objective function through a carefully chosen mirror map. In this work, we introduce a novel notion of strong convexity, termed Restricted Relative Strong Convexity, and establish linear convergence rates for BPGM under this condition. We then exploit the proposed theoretical framework to provide an in-depth analysis of the convergence of BPGM for (regularized) Kullback--Leibler regression problems, covering scenarios with both unique and non-unique minimizers, as well as regularized and unregularized formulations. Specifically, we demonstrate that using the popular Burg's entropy as a distance-generating function may only yield linear convergence for certain KL regression problems. In contrast, we show that employing a smoothed version of the Burg's entropy induces the suitable geometry required to guarantee linear convergence. We conclude with numerical experiments that nicely align with our theoretical findings.

Figures

Figures reproduced from arXiv: 2607.05539 by the authors.

Figure 1
Figure 1. Validity regions in which (RRSC) holds depending on 1) the rank of the matrix A, and 2) the choice of φ. In each plot, the green shaded areas represent regions in which (RRSC) holds (for some fixed ϵ > 0 and θ ∈ (0, π/2)), and the red lines in the third row indicate that (RRSC) is not satisfied therein (cf. Lemma 3.3). Note that ε and θ can be chosen arbitrarily close to 0 at the cost of worse constants µ and αS. Fr… view at source ↗
Figure 2
Figure 2. Convergence plots of (BPGM) with φsB for different values of ξ. Observe that the linear convergence gets faster when increasing ξ until the optimal value ξ ∼ 0.6 in this case. After, the convergence of (BPGM) degrades when continuing to increase ξ. the final iteration. Code details: All simulations were implemented in Python and run on a MacBook Pro equipped with an Apple M4 Pro chip and 24 GB of RAM The code is pub… view at source ↗
Figure 3
Figure 3. On the left, we plot the convergence of (BPGM) with backtrack￾ing and the three choices of φ, as well as the RL algorithm, for unregularized KL in the case N = 500, M = 400. On the right, we focus on φB, where we separate the cases (among the 30 initial points considered to make the left graph) in which x∞ ∈ ∂C or x∞ ∈ int(C): while only sublinear con￾vergence is observed when x∞ ∈ ∂C, we always observe linear conve… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Convergence plots of (BPGM) with backtracking and the three choices of φ, as well as the RL algorithm, for unregularized KL in the case N = M = 500. On the left (x¯ ∈ int(C)), we observe linear convergence both for φB and φsB (and RL). On the right (x¯ ∈ ∂C), linear co…
Figure 5
Figure 5. Figure 5: Convergence of (BPGM) with backtracking and the three choices of φ, as well as the RL algorithm, for ℓ1-regularized KL in the case N = 500, M = 400 and x¯ ∈ ∂C. We only observe linear converge for φsB. 0.0 0.5 1.0 1.5 2.0 2.5 Iterations ×10 5 10 14 10 11 10 8 10 5 10 2…
Figure 6
Figure 6. Figure 6: Convergence of (BPGM) with backtracking and the three choices of φ, as well as the RL algorithm, for ℓ1-regularized KL in the case N = M = 500. On the left (x¯ ∈ int(C)), we observe linear convergence both for φB and φsB (and RL). On the right (x¯ ∈ ∂C), linear converg…

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    math.OC 2026-08 conditional novelty 8.0 of 10

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