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Relaxed and Inertial Nonlinear Forward-Backward with Momentum

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Pith's one-line read Two inertial variants of the nonlinear forward-backward algorithm with momentum are proved to converge weakly to solutions of monotone inclusions under explicit step-size and momentum conditions.

desk verdict Solid inertial/relaxed extension of NFBM; the stress-test's Young-bound objection is a false alarm. read the letter →

arxiv 2412.02045 v1 pith:F6AMAYVM submitted 2024-12-03 math.OC

Fernando Roldán , Cristian Vega This is my paper · ORCID
classification math.OC MSC 47H0565K0565K1590C25
keywords monotoneinclusionnonlinearforward-backwardinertialalgorithmmomentumcorrectionrelaxationparameterweakconvergenceforward-half-reflect-backwardimagerestoration
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 proposes two momentum-accelerated versions of the nonlinear forward-backward with momentum (NFBM) algorithm for solving monotone inclusions of the form $0 \in (A+C)x$, where $A$ is maximally monotone and $C$ is cocoercive in a real Hilbert space. The first version adds a relaxation parameter $\lambda$ to an inertial step, and the second adds a double-inertial step with an extra momentum term. The authors prove that, under a strict Lipschitz condition on the warped operators and a bounded step-size regime, the iterates converge weakly to a solution point. Because NFBM specializes to forward-backward, forward-half-reflect-backward, Chambolle\u2013Pock, and Condat\u2013V\u0169, the results yield inertial and relaxed versions of those algorithms and recover known step-size conditions from the literature. Numerical experiments on image restoration show that the inertial variants reduce iteration counts for fixed step sizes, and a restart heuristic gives further gains in CPU time.

What carries the argument

The engine of the proof is the Lyapunov function $C_n(x)=\|x_n-x\|_S^2 - \alpha_n\|x_{n-1}-x\|_S^2 + 2\lambda\langle u_n \mid x_n-x\rangle + \lambda(1+|1-\lambda|)\zeta_n\|p_n-y_{n-1}\|_S^2 + \xi_n\|x_n-x_{n-1}\|^2$ for Algorithm 2.2, with an analogous function for Algorithm 2.3. The warped resolvent step $(M_n+A)^{-1}$ with momentum correction $u_n$ is the object that generalizes forward-backward, and the coefficients $\rho_n,\eta_n,\xi_n$ encode the step-size, cocoercivity constant, and Lipschitz constants. The central descent inequality $C_{n+1}(x)\le C_n(x)-(\eta_n-\xi_{n+1})\|x_{n+1}-x_n\|^2$ is what forces summability of successive differences and eventual weak convergence.

What would settle it

Run Algorithm 2.2 in $\mathbb{R}^2$ with $A$ the subdifferential of the indicator of the unit square, $C(x)=0.5x$, $S=\mathrm{Id}$, constant $\gamma$, $\alpha_n\equiv 0.2$, $\lambda=1.5$, choosing $\gamma$ so that (4.22) holds, and compute $C_{n+1}(x)-C_n(x)+(\eta_n-\xi_{n+1})\|x_{n+1}-x_n\|^2$ at each iteration. If this quantity is ever positive, the Lyapunov descent inequality (4.8), which is the core of the proof, is false and the theorem collapses.

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Extended reading notes

Core claim

The central claim is that inertia and relaxation can be added to the nonlinear forward-backward with momentum algorithm without sacrificing convergence. Theorems 4.4 and 4.7 assert that, under Assumption 4.1 and the parameter conditions (4.22) or (4.41), the iterates of Algorithms 2.2 and 2.3 converge weakly to a point in the solution set $Z=\mathrm{zer}(A+C)$ of the monotone inclusion $0\in (A+C)x$. The proof constructs a non-increasing Lyapunov function $C_n(x)$ in the $S$-norm that includes momentum-correction terms, derives a descent inequality whose coefficient $\eta_n-\xi_{n+1}$ is uniformly positive, and then uses Opial's lemma after showing every weak cluster point lies in $Z$. By choosing the warping operators and metric in specific ways, the same theorems recover inertial and relaxed versions of forward-backward, forward-half-reflect-backward, Chambolle\u2013Pock, and Condat\u2013V\u0169, together with their known step-size restrictions.

Load-bearing premise

The whole convergence proof rests on Assumption 4.1: at every iteration the warped operator $\gamma_n M_n - S$ must be Lipschitz with constant $\zeta_n \le 1-\varepsilon$ with respect to the fixed strongly monotone linear metric $S$, and the step sizes $\gamma_n$ must stay in a fixed bounded interval; if this strict contractivity is violated, the Lyapunov descent inequality fails.

Editorial extensions

If this is right

  • If the theorems hold, every algorithm that is a special case of Algorithm 2.2 or 2.3 inherits a weak-convergence guarantee under explicit parameter inequalities, covering inertial and relaxed FB, FHRB, Chambolle\u2013Pock, and Condat\u2013V\u0169.
  • The parameter conditions (4.22) and (4.41) give a user-ready recipe: the inertial coefficients must be non-decreasing, and the admissible step-size interval shrinks as the inertial parameters grow for fixed cocoercivity and Lipschitz data.
  • A direct corollary is that when relaxation is removed ($\lambda=1$), the inertial FB condition reduces to $1-3\alpha-\gamma(1-\alpha)^2/(2\mu)>0$ in the constant-coefficient case, matching known inertial FB thresholds.
  • The second algorithm with double inertia allows more flexibility in the choice of the two inertial parameters, since only the sum $\tilde{\alpha}_n=\alpha_n+\theta_n$ and the combination $\alpha_n-\gamma_n\beta_n/(2\mu)$ are constrained by the conditions.
  • Numerically, for image restoration, inertial FHRB variants cut the number of iterations for fixed step-sizes, and the double-inertial variant performed best in iteration count while a semi-double-inertial variant was fastest in CPU time.

Reading between the lines

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

  • Because the Lyapunov analysis uses only $S$-norm geometry and the Lipschitz bound on the warped operator, the same descent-inequality template should transfer to fixed-point iterations of averaged operators in other metric spaces, though the authors do not explore that direction.
  • The restart strategy tested numerically has no convergence proof; a natural next step would be to prove weak convergence for the restarted sequence, which currently holds only because the inertial parameter is eventually zero, making the acceleration heuristic rather than guaranteed.
  • The numerical observation that larger step-sizes shrink admissible inertia suggests a trade-off: acceleration from momentum and acceleration from step-size compete, and an adaptive scheme that modulates inertia as the step-size approaches its limit might achieve better worst-case rates.
  • One can test the sharpness of (4.22) on a two-dimensional example where $A$ is a normal cone and $C$ is a linear cocoercive map: if the inequality is necessary as well as sufficient, iterates should fail to converge as soon as $\eta_n-\xi_{n+1}$ turns negative.
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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

1 major / 6 minor

Summary. This manuscript studies inertial and relaxed versions of the nonlinear forward-backward with momentum (NFBM) of Morin, Banert, and Giselsson. It proposes two algorithms: Algorithm 2.2, an inertial NFBM with a relaxation step, and Algorithm 2.3, a double-inertial variant with an additional momentum term. For both, it proves weak convergence to a zero of A+C under Assumption 4.1 and the parameter conditions (4.22) and (4.41). It then derives inertial and relaxed versions of forward-backward, forward-half-reflected-backward, Chambolle-Pock, Condat-Vu, and related primal-dual methods, recovering previously known convergence conditions as special cases. Numerical experiments on image restoration compare the variants, and a restart strategy is proposed to preserve acceleration for large step sizes.

Significance. If the convergence theorems are fully correct, the paper gives a valuable unified framework: the same Lyapunov analysis covers inertia, relaxation, and momentum for NFBM and its many special cases. A notable strength is that the special-case reductions are independently checkable and match known results, for example (5.1) recovers the condition in [45] and (5.7) recovers the conditions in [46] and [58]. The algebraic derivations in Section 4 are detailed, and the numerical code is made available. The main caveat is that the proof of the under-relaxed case lambda in (0,1) in Theorem 4.4 contains a false inequality, so the claim as stated is not fully established; the argument for lambda in [1,2) appears sound.

major comments (1)
  1. [§4.1, Proposition 4.2(4), Eq. (4.21)] The displayed lower bound in the proof of Proposition 4.2(4) is false for lambda in (0,1). From u_n = T_{n-1}p_n - T_{n-1}y_{n-1} and the zeta_{n-1}-Lipschitz property of T_{n-1}, together with p_n - y_{n-1} = (x_n - y_{n-1})/lambda, one obtains ||u_n||_{S^{-1}} <= (zeta_{n-1}/lambda)||x_n - y_{n-1}||_S, hence 2lambda <u_n | x_n - x> >= -2 zeta_{n-1} ||x_n - y_{n-1}||_S ||x_n - x||_S. Applying Young's inequality with coefficient c = lambda(1+|1-lambda|) gives the term -zeta_{n-1}/(lambda(1+|1-lambda|)) ||x_n - x||_S^2, not the claimed -lambda zeta_{n-1}/(1+|1-lambda|) ||x_n - x||_S^2. For lambda in (0,1) the claimed coefficient is strictly smaller than any coefficient obtainable by this argument, and the concrete example lambda=1/2, zeta=1, x_n=0, x=1, y_{n-1}=1, T=-Id gives LHS=-2 and RHS=-13/12. Therefore the proof that C_n(x) >= 0 via (4.21) fails in the under-relaxed case. Since Theorem 4.4 explicitly allows lambda in (0,1) with (xi_n)_{n>=N0} non-decreasing, the weak-convergence claim in that case is not proved as written. The argument for lambda in [1,2) is unaffected by this particular inequality.
minor comments (6)
  1. [§4.1, proof of Proposition 4.2] The sentence 'it follows from (2.2)' just before (4.12) should refer to (2.3), since the displayed identity concerns Algorithm 2.2.
  2. [Table 3] The header repeats 'FHRBDI' twice instead of listing FHRBDI and FHRBSDI separately, making the table difficult to read.
  3. [Figures 3-5] The captions and text use 'PNSR' where the standard abbreviation is 'PSNR'.
  4. [§5.2] The phrase 'alpha_n -> alpha in (0,+infinity)' should be 'alpha in [0,1]' in accordance with Algorithm 2.2; the same applies to beta and theta.
  5. [§6, restart strategy] The claim that the restart choice alpha_n = alpha for n <= N0 and alpha_n = 0 afterwards satisfies the assumptions of Theorem 4.4 should state explicitly that the theorem's N0 is taken after the restart, since the sequence is non-increasing across the whole index set.
  6. [Captions of Figures 4 and 5] The labels 'FHRBR' appear to be typos for 'FHRBIR'; the notation should be unified with the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence proofs are genuine derivations from stated assumptions, and the special-case conditions are algebraic substitutions, not fitted predictions.

full rationale

The paper's central claims, Theorem 4.4 and Theorem 4.7, are proved directly from Problem 2.1, Assumption 4.1, and the Lyapunov inequalities in Propositions 4.2 and 4.6. Assumption 4.1, requiring gamma_n M_n - S to be zeta_n-Lipschitz with respect to S for a self-adjoint strongly monotone S, is explicitly stated as a hypothesis and is not a restatement of the weak-convergence conclusion; importing it from [48] does not make the argument circular because the present convergence result is not assumed inside that assumption. The particular cases in Section 5 are obtained by substituting concrete choices of S, M_n, gamma_n, and zeta_n into (4.22) and (4.41); for example, the FHRB condition (5.5) and the double-inertial condition (5.7) follow by algebra from the master conditions, and the paper explicitly verifies reduction to known conditions such as [5, Corollary 3.12], [45], and [46, Theorem 4.3]. No parameter is fitted to numerical output and then renamed as a prediction: Section 6 merely runs the algorithms with parameters chosen to satisfy the displayed sufficient conditions and reports iteration counts and CPU times. The only self-citations, [17] and [53], appear in background lists and do not carry the load-bearing convergence argument. Section 7's admission that the restart strategy lacks a theoretical framework is an honest limitation and is not presented as a derived prediction. The alleged defect in Proposition 4.2(4) identified by the skeptic concerns the validity of a Young-inequality bound; if substantiated, it would be a correctness gap in the proof, not a circularity, and does not change the circularity assessment. The derivation chain is therefore self-contained with respect to circularity.

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

The central theorems introduce no fitted constants. The proof uses standard Hilbert space monotone operator theory, the problem hypotheses on A and C, an assumption imported from [48] on the metric/warping operators, and new sufficient Lyapunov conditions. No invented physical or mathematical entities are introduced.

assumptions (4)
  • standard math Standard real Hilbert space monotone operator theory, including resolvents, maximal monotonicity, cocoercivity, weak-strong graph closedness, and Opial's lemma.
    Invoked throughout Section 4; results from Bauschke-Combettes [8] are used without proof.
  • domain assumption The solution set Z of 0 in (A+C)x is nonempty.
    Problem 2.1 explicitly assumes Z is nonempty; this is necessary for convergence to a solution to make sense.
  • domain assumption Assumption 4.1: uniform step bounds gamma_n in [gamma, gamma_bar] and gamma_n M_n - S is zeta_n-Lipschitz with zeta_n in [0, 1-epsilon].
    Imported from [48, Assumption 2.2]; all Lyapunov estimates in Section 4 depend on this property.
  • ad hoc to paper The sufficient Lyapunov conditions (4.22)/(4.41): rho_n >= 0, eta_n - xi_{n+1} >= epsilon, and monotonicity of alpha_n or tilde alpha_n.
    These are new conditions introduced for the proof; the paper does not show they are necessary for convergence.

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Pith. "Pith review of Relaxed and Inertial Nonlinear Forward-Backward with Momentum." pith.science (2026). https://pith.science/paper/F6AMAYVM

@misc{pith2026241202045,
  author       = {Pith},
  title        = {Pith review of: Relaxed and Inertial Nonlinear Forward-Backward with Momentum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F6AMAYVM}},
  note         = {Machine review of arXiv:2412.02045}
}
read the original abstract

In this article, we study inertial algorithms for numerically solving monotone inclusions involving the sum of a maximally monotone and a cocoercive operator. In particular, we analyze the convergence of inertial and relaxed versions of the nonlinear forward-backward with momentum (NFBM). We propose an inertial version including a relaxation step, and a second version considering a double-inertial step with additional momentum. By applying NFBM to specific monotone inclusions, we derive inertial and relaxed versions of algorithms such as forward-backward, forward-half-reflect-backward (FHRB), Chambolle-Pock, Condat-V\~u, among others, thereby recovering and extending previous results from the literature for solving monotone inclusions involving maximally monotone, cocoercive, monotone and Lipschitz, and linear bounded operators. We also present numerical experiments on image restoration, comparing the proposed inertial and relaxation algorithms. In particular, we compare the inertial FHRB with its non-inertial and momentum versions. Additionally, we compare the numerical convergence for larger step-sizes versus relaxation parameters and introduce a restart strategy that incorporates larger step-sizes and inertial steps to further enhance numerical convergence.

Figures

Figures reproduced from arXiv: 2412.02045 by the authors.

Figure 1
Figure 1. Relative error along iteration number for the random observation 10. In this case, the inertial parameters do not satisfy the hypothesis guaranteeing convergence. See [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Relative error along iteration number for the random observation 10. See [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Original image, blur and noisy observation 10, and recovered images for FHRB, FHRBSI, FHRBI, FHRBDI, and FHRBIR (α = 0.2) with their respective PNSR (dB), blur of kernel 3 × 3 and N = 256. See [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Original image, blur and noisy observation 10, and recovered images for FHRB and FHRBIR (N0 = 3000) with their respective PNSR (dB), blur of kernel 9 × 9 and N = 256. See [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Original image, blur and noisy observation 10, and recovered images for FHRB and FHRBIR (N0 = 3000) with their respective PNSR (dB), blur of kernel 9 × 9 and N = 512. See [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]

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Pith tools

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