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Accelerated Bregman gradient methods for relatively smooth and relatively Lipschitz continuous minimization problems

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Pith's one-line read Accelerated Bregman proximal gradient methods keep their convergence rate when the oracle is inexact, up to an additive noise term, provided the Bregman divergence obeys a triangular scaling inequality with exponent between 1 and 2.

desk verdict Useful new accelerated Bregman methods with inexact oracles, but the advertised Poisson inverse problem application is not covered by the theory because its divergence has the wrong scaling exponent. read the letter →

arxiv 2411.16743 v1 pith:V3FS5UXV submitted 2024-11-23 math.OC

O. S. Savchuk , M. S. Alkousa , A. S. Shushko , A. A. Vyguzov , F. S. Stonyakin , D. A. Pasechnyuk , A. V. Gasnikov This is my paper · ORCID
classification math.OC MSC 90C2590C3065K05
keywords convexoptimizationacceleratedBregmanproximalgradientinexactoraclerelativesmoothnessrelativelyLipschitzcontinuitytriangularscalingpropertyadaptivefastmethodintermediate
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 tries to establish that accelerated Bregman proximal gradient methods still converge at their fast rates when the optimization oracle is inexact, meaning that only approximate function values and gradients are available. For a convex function that is relatively smooth with respect to a reference prox-function, it proposes an adaptive fast gradient method and two accelerated Bregman proximal gradient methods, and proves rates of the form $O(L(\gamma/(\gamma+N))^\gamma V(x_*,x_0))$ plus an additive term proportional to the noise level $\delta$. The same algorithms are shown to apply to relatively Lipschitz continuous objectives, and an adaptive intermediate method interpolates between slower, noise-tolerant and faster, noise-sensitive regimes. A sympathetic reader would care because noisy gradients are the norm in large-scale machine learning and inverse problems, and the paper's message is that acceleration need not be sacrificed in that setting.

What carries the argument

The load-bearing object is the triangular scaling property of the Bregman divergence $V(x,y)=d(x)-d(y)-\langle\nabla d(y),x-y\rangle$: the inequality $V((1-\theta)x+\theta z,(1-\theta)x+\theta \tilde z)\le \theta^\gamma V(z,\tilde z)$ for every $\theta\in[0,1]$. This inequality is what lets the proofs bound the Bregman distance between the convex combination points $y_k=(1-\theta_k)x_k+\theta_k z_k$ and $x_{k+1}=(1-\theta_k)x_k+\theta_k z_{k+1}$ by $\theta_k^\gamma V(z_{k+1},z_k)$, converting the algorithm's updates into a telescoping Lyapunov inequality. The second ingredient is the $(\delta,L)$-oracle model $0\le f(x)-\left(f_\delta(y)+\langle\nabla f_\delta(y),x-y\rangle\right)\le LV(x,y)+\delta$, and the third is relative smoothness, $f(y)\le f(x)+\langle\nabla f(x),y-x\rangle+LV(y,x)$, which replaces the usual Lipschitz-gradient upper model.

What would settle it

Evaluate the triangular scaling inequality numerically for the prox-function used in an application: draw random triples $(x,z,\tilde z)$ and compare $V((1-\theta)x+\theta z,(1-\theta)x+\theta\tilde z)$ with $\theta^\gamma V(z,\tilde z)$. If the smallest exponent that fits is $\le1$, then the proof step behind Theorem 4.5 is invalid for that problem and the accelerated guarantee cannot be certified; for the Poisson experiments with the log-barrier prox, the paper's own Example 2.9 indicates exactly this situation.

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Extended reading notes

Core claim

The central claim is that the inexact-oracle accelerated Bregman proximal gradient method, under the triangular scaling property with exponent $\gamma\in(1,2]$, satisfies $f(x_N)-f(x_*) \le 2L\left(\frac{\gamma}{\gamma+N-1}\right)^\gamma V(x_*,x_0)+(2(N-1)L+1)\delta$ after $N$ iterations; in the noiseless limit $\delta=0$ this recovers the $O(1/N^\gamma)$ rate of the exact-oracle method. The paper also proves a similar-triangles fast gradient method with rate $8LR^2/(N+1)^2 + \frac{2}{A_N}\sum_{k=0}^{N-1}A_{k+1}\delta_k$, and an adaptive intermediate Bregman method whose rate $(k+2)^{-(p-1)(\gamma-1)-1}$ interpolates between non-accelerated ($p=1$) and accelerated ($p=2$) behaviour. The universality statements extend both accelerated families to relatively Lipschitz continuous objectives, at the cost of a slightly degraded oracle in the line-search test.

Load-bearing premise

The entire argument rests on the chosen Bregman divergence satisfying the triangular scaling inequality with an exponent strictly between 1 and 2; the paper's own examples show the Kullback-Leibler divergence has exponent 1 and the Itakura-Saito divergence has exponent below 1, so the accelerated guarantees do not cover those natural measures.

Editorial extensions

If this is right

  • For any relatively smooth problem whose prox-function has a triangular scaling factor $\gamma\in(1,2]$, inexact oracles preserve the accelerated rate; noise only adds a term proportional to $\delta$.
  • The same iterations solve relatively Lipschitz continuous problems, so the methods are universal in the sense the paper states.
  • The adaptive intermediate method gives a user a dial $p\in[1,2]$ between non-accelerated and accelerated convergence, without switching algorithms.
  • In the Euclidean case $\gamma=2$, the intermediate method's rate is $(k+2)^{-p}$ plus $(k+2p)^{p-1}\delta$, making the interpolation explicit.
  • With fixed noise $\delta$, the bound in Theorem 4.5 grows linearly in $N$, so the method is best viewed as achieving accuracy of order $\delta$ rather than converging to zero.

Reading between the lines

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

  • A design lesson the paper leaves implicit is that the triangular scaling exponent should be treated as a selection criterion for the prox-function, since the Kullback-Leibler and Itakura-Saito divergences lie outside the accelerated regime.
  • One testable extension would be to estimate $\gamma$ numerically for a candidate divergence and use it as a pre-screening test before committing to an accelerated Bregman method.
  • The growth of the noise term in Theorem 4.5 suggests that when $\delta$ is fixed and not small, a user should switch to the intermediate method; this is a practical reading of the interpolation the authors construct.
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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

2 major / 6 minor

Summary. The paper proposes accelerated Bregman gradient methods with an inexact oracle for minimizing convex functions that are relatively smooth or relatively Lipschitz continuous with respect to a reference prox-function. It presents four algorithms: an adaptive fast gradient method based on similar triangles (Algorithm 1), a non-adaptive accelerated Bregman proximal gradient method (Algorithm 2), an adaptive version of it (Algorithm 3), and an adaptive intermediate Bregman method (Algorithm 4) that interpolates between accelerated and non-accelerated behavior. Convergence-rate theorems are stated for the relatively smooth case (Theorems 3.1, 4.1, 4.5, and 5.4) in the presence of an inexact oracle, and the universality of the methods for relatively Lipschitz continuous problems is argued informally. The paper concludes with numerical experiments on a Poisson inverse problem comparing the proposed algorithms with an adaptive Bregman proximal gradient method.

Significance. If the results are correct, they would constitute a useful extension of accelerated Bregman proximal methods to the inexact-oracle setting and introduce an interesting noise-robust intermediate method. The proofs are self-contained, and the non-adaptive Theorem 4.1 appears coherent. However, the actual scope is considerably narrower than advertised: the accelerated guarantees require the triangular scaling property with exponent γ strictly in (1,2], while the motivating Poisson inverse problem uses a divergence with γ<1, and numerical experiments treat γ as a freely tunable parameter. In addition, the adaptive line-search condition in Algorithm 3 is scale-inconsistent and breaks the proof of Theorem 4.5. These are load-bearing issues that prevent acceptance in the present form.

major comments (2)
  1. [§4.2, Algorithm 3, line 8] The line-search condition L_{k+1} ≥ (θ_{k+1}/θ_k)^γ in Algorithm 3 is scale-inconsistent: the right-hand side is dimensionless, while L_{k+1} scales with the relative smoothness parameter. If the true parameter L is less than 1, then for large k this condition forces L_{k+1} to be at least about 1, and the assumption L_1 < 2L need not imply L_N < 2L. The proof of Theorem 4.5 explicitly uses this condition in the chain following Eq. (47) to replace θ_k^γ L_{k+1} by θ_{k+1}^γ, so the final bound (46) is not justified. The condition should be replaced by a scale-corrected version, for example L_{k+1} ≥ (θ_{k+1}/θ_k)^γ L_k or another form that respects the scaling of L.
  2. [§6 and Examples 2.8–2.9] The accelerated convergence results in Theorems 3.1, 4.1, and 4.5 assume the triangular scaling property with γ ∈ (1,2]. The paper's own Examples 2.8 and 2.9 state that the KL divergence has γ=1 and the Itakura–Saito divergence has γ<1, while Example 2.3, which motivates the paper, uses the log-barrier prox-function d(x)=−Σ ln x_i, whose Bregman divergence is VIS. Hence γ<1 for the Poisson inverse problem, and the convergence theorems do not apply to the problem advertised in the abstract. The numerical experiments in Section 6 run the same Poisson problem with γ=2, 1.4, and 1.1, but γ is a structural property of the divergence rather than an algorithmic input; no prox-function satisfying triangular scaling with those exponent values is specified for the experiments. The claims and experiments should be restricted to divergences with γ∈(1,2], or the experimental setup must be justified by explicit prox-functions for each γ.
minor comments (6)
  1. [Example 2.9] The name 'Itakuro–Saito divergence' is a typo; it should be 'Itakura–Saito divergence'.
  2. [Algorithm 1, line 4] The update L_{k+1}=2^{i_k−1}L_k gives L_{k+1}=L_k/2 when i_k=0; clarify whether the exponent is intended to be i_k or explain why halving is allowed, and state that the line search terminates in finite time.
  3. [Algorithm 4, lines 3–5] The while condition at line 3 refers to y0 before y0 is computed in line 5; the pseudocode should be reordered so that y0 is initialized before the condition is evaluated, or a repeat-until form should be used.
  4. [Theorem 5.4 and Corollary 5.3] The statement uses V(x0,x*)=d(x*)≤R0, but this equality holds only when x0 minimizes d and d(x0)=0; the assumption should be stated as V(x0,x*)≤R0 or d(x*)≤R0.
  5. [Eq. (52) and Theorem 5.4 proof] The assumption 0<α_k≤B_k≤A_k is stated for all k but is not verified in the paper; please add a short proof or a reference showing that the updates of Algorithm 4 maintain these inequalities.
  6. [General] The manuscript contains numerous typographical and grammatical errors (e.g., 'we concluded its convergence rate', 'Itakuro–Saito', 'with constant γ >0'); a thorough language revision is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence proofs are self-contained and reduce to the stated assumptions, not to fitted constants or self-citation chains.

full rationale

The derivation chain is self-contained. Theorems 3.1, 4.1, 4.5, and 5.4 each prove their rates by explicit telescoping arguments (e.g., summing inequalities (24), (35), and (42)) from the standing assumptions: relative smoothness (10), the inexact-oracle inequality (16), and the triangular scaling property (17). No fitted parameter is renamed as a prediction: the parameters L, delta, and gamma are inputs or structural assumptions, not quantities tuned to the target bound. The cited works ([6] for the triangular scaling property and the APBG analogy, [12] for the similar-triangles framework, [28] for the relatively-Lipschitz inequality) are external lemmas with stated assumptions and are used as tools rather than as substitutes for the target convergence proofs. The numerical experiments' use of gamma = 2, 1.4, and 1.1 for a problem whose Bregman divergence is the Itakura-Saito divergence (gamma < 1 by Example 2.9) is an application-consistency issue, not a circular one. Thus no step reduces the paper's conclusions to its own inputs by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. The central results rest on existing constructs (Bregman divergence, triangular scaling, inexact oracle) plus a few technical assumptions, two of which (the gamma range and the scale of the adaptive condition) are load-bearing and problematic.

free parameters (1)
  • gamma (triangular scaling factor) in numerical experiments = 2, 1.4, 1.1
    The experiments in Figure 1 run the same Poisson inverse problem with gamma=2, 1.4, and 1.1. Since gamma is a property of the Bregman divergence (Example 2.9 gives gamma<1 for IS divergence), hand-tuning it as a free parameter affects the reported convergence comparison and is not justified.
assumptions (4)
  • domain assumption Bregman divergence satisfies the triangular scaling property with exponent gamma in (1,2].
    Assumed in Section 3 (gamma=2), Section 4, and Theorem 5.4. Examples 2.8 and 2.9 state that KL divergence has gamma=1 and IS divergence has gamma<1, so the accelerated results exclude the motivating divergences.
  • domain assumption The pair (f_delta, grad f_delta) is a (delta,L)-oracle satisfying Definition 2.5.
    Used throughout the proofs. For relatively Lipschitz functions, this oracle inequality is not shown to hold, so the universality remarks do not provide convergence guarantees.
  • ad hoc to paper The relatively Lipschitz inequality from [28] (f(x) <= f(y)+<grad f(y),x-y>+L V(x,y)+L V(y,x)+delta) supports the inexact-oracle line search.
    Invoked in Remarks 1 and 2 to show the line-search condition terminates, but no convergence rate is proven in the Lipschitz setting. This supports an overclaim in the abstract and contributions.
  • ad hoc to paper The adaptive condition L_{k+1} >= (theta_{k+1}/theta_k)^gamma in Algorithm 3 is scale-invariant.
    As written, the condition is dimensionless while L has units. If the true L is less than 1, the condition can force L_{k+1} above 2L, invalidating the assumption L_N < 2L in Theorem 4.5.

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Cite this review

Pith. "Pith review of Accelerated Bregman gradient methods for relatively smooth and relatively Lipschitz continuous minimization problems." pith.science (2026). https://pith.science/paper/V3FS5UXV

@misc{pith2026241116743,
  author       = {Pith},
  title        = {Pith review of: Accelerated Bregman gradient methods for relatively smooth and relatively Lipschitz continuous minimization problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3FS5UXV}},
  note         = {Machine review of arXiv:2411.16743}
}
read the original abstract

In this paper, we propose some accelerated methods for solving optimization problems under the condition of relatively smooth and relatively Lipschitz continuous functions with an inexact oracle. We consider the problem of minimizing the convex differentiable and relatively smooth function concerning a reference convex function. The first proposed method is based on a similar triangles method with an inexact oracle, which uses a special triangular scaling property for the used Bregman divergence. The other proposed methods are non-adaptive and adaptive (tuning to the relative smoothness parameter) accelerated Bregman proximal gradient methods with an inexact oracle. These methods are universal in the sense that they are applicable not only to relatively smooth but also to relatively Lipschitz continuous optimization problems. We also introduced an adaptive intermediate Bregman method which interpolates between slower but more robust algorithms non-accelerated and faster, but less robust accelerated algorithms. We conclude the paper with the results of numerical experiments demonstrating the advantages of the proposed algorithms for the Poisson inverse problem.

Figures

Figures reproduced from arXiv: 2411.16743 by the authors.

Figure 1
Figure 1. The results of Algorithms 1 (AdapFGM), 2 (AccBPGM-1), 3 (AccBPGM-2), 4 AIBM, and BPG￾Adapt [31], for Poisson inverse problem with m = 150, n = 100 and and different values of γ. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. The results of the compared algorithms, with noise, for the Poisson inverse problem. All other problem parameters are still the same as in the previous experiment. 7. Conclusion In this paper, we proposed some accelerated methods for solving the problem of min￾imizing the convex differentiable and relatively smooth function. The first proposed method is an adaptive fast gradient method based on a similar triangles m… view at source ↗
Figure 3
Figure 3. The results of the AIBM, with different δ expectancy, uniform distribution, for the Poisson inverse problem. All other problem parameters are still the same as in the previous experiment. compared to other approaches was shown by the adaptive version of the accelerated proximal gradient Bregman method with an inexact oracle. With increasing the value of the triangular scaling parameter, the accuracy and convergence … view at source ↗

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