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arxiv: 2604.26420 · v1 · submitted 2026-04-29 · 🧮 math.OC

Accelerated Backward Forward Method for Convex Optimization

Zepeng Wang , Juan Peypouquet This is my paper

Pith reviewed 2026-05-07 13:30 UTC · model grok-4.3

classification 🧮 math.OC
keywords convex optimizationcomposite optimizationproximal methodsaccelerated algorithmsconvergence ratesbackward-forward splittingweak convergence
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The pith

The accelerated backward-forward method achieves O(1/k²) convergence for function values in convex composite optimization.

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

This work examines the convergence properties of an accelerated backward-forward method for convex composite optimization problems, where the objective combines a smooth convex function with a nonsmooth convex one. The method places the proximal operator before the gradient step, setting it apart from standard accelerated proximal gradient algorithms like FISTA. Under convexity of the smooth part, the analysis establishes an O(1/k²) rate for the decrease in function values and proves that the iterates converge weakly to an optimal solution. The authors also develop a variant for the strongly convex case that attains accelerated linear convergence rates.

Core claim

When the smooth component is convex, the accelerated backward-forward method attains a convergence rate of O(1/k²) for the objective function values and ensures weak convergence of the generated sequence to a minimizer. For strongly convex smooth parts, a modified version of the method is introduced that achieves an accelerated linear convergence rate for the function values.

What carries the argument

The accelerated backward-forward splitting method with inertial extrapolation terms, which applies the proximal mapping to the nonsmooth term prior to the gradient evaluation on the smooth term.

If this is right

  • Function values converge at a rate of O(1/k²) when the smooth function is convex.
  • The sequence of iterates converges weakly to a solution of the optimization problem.
  • A variant of the method achieves accelerated linear convergence when the smooth function is strongly convex.
  • The approach offers an alternative splitting scheme for problems where the backward-forward order simplifies computation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The proof techniques may extend to related inertial proximal algorithms with different step orderings.
  • Such rates could inform parameter choices in practical implementations for large-scale convex problems.
  • Connections to other operator splitting methods suggest broader applicability in monotone inclusion problems.

Load-bearing premise

The objective function must be convex and composite, with the smooth part convex or strongly convex, and the algorithm steps must follow the exact backward-forward proximal-gradient ordering.

What would settle it

A convex optimization problem where running the method produces function values that decrease slower than 1/k², such as on a quadratic objective with a nonsmooth term, would disprove the claimed convergence rate.

read the original abstract

We analyze the convergence rate of an accelerated backward forward method for solving convex composite optimization problems. The method was developed by Taylor, Hendrickx and Glineur, and is different from the FISTA algorithm in its placement of the proximal operator. When the smooth part of the objective function is convex, we establish a fast convergence rate of $\mathcal{O}\left( \frac{1}{k^2} \right)$ for the function values, and prove the weak convergence of the iterates. When the smooth part is strongly convex, we propose a variant of the method, and establish an accelerated linear convergence rate for the function values.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes the convergence of the accelerated backward-forward proximal-gradient method (introduced by Taylor, Hendrickx and Glineur) for composite convex optimization min f(x) + g(x), where f is L-smooth convex (or μ-strongly convex) and g is convex and proximable. It claims an O(1/k²) rate on function values together with weak convergence of the iterates when f is convex, and introduces a variant achieving accelerated linear convergence when f is strongly convex.

Significance. If the proofs hold, the work supplies the first rigorous rate analysis for this specific proximal ordering, which can be preferable in some applications. The O(1/k²) claim and the strongly-convex variant would be useful additions to the literature on accelerated first-order methods, provided the estimate-sequence argument is shown to survive the reversal of the proximal and gradient steps.

major comments (2)
  1. [§3] §3, proof of the O(1/k²) rate (Theorem 3.1 and the estimate-sequence construction): the standard three-point identity and descent lemma are applied after a forward gradient step on the extrapolated vector in FISTA-type analyses; the backward-forward ordering reverses which point receives the smoothness inequality. The manuscript must derive an explicit new identity or adjusted momentum condition that absorbs this reversal without tightening the step-size restriction beyond the stated L-smoothness assumption. Without that derivation the claimed rate does not follow from the given hypotheses.
  2. [§4] §4, strongly-convex variant and linear-rate proof: the acceleration parameter and the modified Lyapunov function must be shown to remain valid under the reversed proximal ordering; the current sketch appears to reuse the convex-case momentum sequence without re-deriving the contraction factor.
minor comments (2)
  1. [§2] Notation for the momentum sequence {t_k} and the extrapolation parameter is introduced without an explicit comparison table to the corresponding quantities in FISTA; adding such a table would improve readability.
  2. [Theorem 3.2] The weak-convergence argument for the iterates (Theorem 3.2) invokes Opial’s lemma but does not explicitly verify that the sequence is Fejér monotone with respect to the solution set under the backward-forward ordering.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each major comment below and will make the necessary revisions to strengthen the proofs.

read point-by-point responses

  1. Referee: [§3] §3, proof of the O(1/k²) rate (Theorem 3.1 and the estimate-sequence construction): the standard three-point identity and descent lemma are applied after a forward gradient step on the extrapolated vector in FISTA-type analyses; the backward-forward ordering reverses which point receives the smoothness inequality. The manuscript must derive an explicit new identity or adjusted momentum condition that absorbs this reversal without tightening the step-size restriction beyond the stated L-smoothness assumption. Without that derivation the claimed rate does not follow from the given hypotheses.

    Authors: We agree that the backward-forward ordering requires a specific treatment in the estimate sequence argument. In the revised version, we will add an explicit derivation of a modified three-point identity that accounts for applying the proximal operator first. This identity, combined with the descent lemma applied to the appropriate point, allows us to maintain the same step-size restriction and achieve the O(1/k²) rate. The momentum conditions are adjusted minimally to absorb the reversal, and we provide the full proof details. revision: yes

  2. Referee: [§4] §4, strongly-convex variant and linear-rate proof: the acceleration parameter and the modified Lyapunov function must be shown to remain valid under the reversed proximal ordering; the current sketch appears to reuse the convex-case momentum sequence without re-deriving the contraction factor.

    Authors: For the strongly convex variant, we will re-derive the contraction factor in the revised manuscript. The acceleration parameter is selected based on the strong convexity modulus μ and smoothness L, and we will demonstrate that the Lyapunov function decreases by a factor involving sqrt(μ/L) under the backward-forward ordering. This involves a new inequality that incorporates the proximal step's effect on the strongly convex function. The details will clarify that the linear rate holds. revision: yes

Circularity Check

0 steps flagged

No circularity: analysis of externally defined method

full rationale

The paper takes the backward-forward accelerated method as given from Taylor, Hendrickx and Glineur (external reference) and derives convergence rates via standard estimate-sequence or Lyapunov arguments applied to the stated algorithm. No quantity is defined in terms of the claimed rate, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation chain. The strong-convexity variant is likewise constructed from the base method without circular redefinition. The derivation chain is therefore self-contained against the external algorithm and the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard domain assumptions of convexity and strong convexity for the smooth component plus the specific algorithmic structure from the cited reference; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The objective function is convex composite with the smooth part convex or strongly convex.
    Invoked to obtain the stated rates; appears in the opening sentences of the abstract.

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Reference graph

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