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The paper proves that the adaptive golden-ratio primal–dual algorithm needs no step-size cap—its own adaptive rule keeps step-sizes bounded—and provides two accelerated variants with ergodic 1/N² rates under strong convexity.

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2026-08-04 04:23 UTC pith:SZLT62NC

load-bearing objection The no-cap step-size result is a genuinely useful fix, and Algorithm 3's acceleration is well proved, but the second accelerated-rate theorem (Algorithm 4) is stated without its proof and relies on a trial-and-error threshold, so the paper needs revision before the abstract's two-variant claim is credible. the 3 major comments →

arxiv 2607.08174 v2 pith:SZLT62NC submitted 2026-07-09 math.OC

Two Adaptive Accelerated Golden Ratio Primal--Dual Algorithms With an Application to Poisson Imaging Problem

Santanu Soe , V. Vetrivel This is my paper
classification math.OC MSC 90C2565K1049M2765J10
keywords primal–dual algorithmsgolden ratioadaptive step-sizelocal smoothnessstrong convexityaccelerated convergencePoisson imagingergodic rate
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Structured convex problems in imaging often involve a differentiable term that is only locally smooth, making global Lipschitz constants unavailable. The paper shows that the adaptive golden-ratio primal–dual algorithm does not need an artificial cap on primal step-sizes: its step-size rule has a growth branch and a local-smoothness branch, and the elementary inequality min{a,b} ≤ √(ab) forces a uniform bound automatically. With the cap removed, the method retains a parameter-free ergodic O(1/N) rate and, when both nonsmooth primal and dual functions are strongly convex, converges linearly. Two accelerated variants—one using strong convexity of the nonsmooth component, one using strong convexity of the differentiable component—achieve ergodic O(1/N²) rates without a linesearch. Experiments on Poisson image reconstruction show faster residual decay for the accelerated methods.

Core claim

At the center of the paper is a single observation: the artificial upper bound τ_max placed on primal step-sizes in the earlier aEGRPDA is redundant. Writing the update as τ_n = min{ρτ_{n-1}, c_n/τ_{n-1}} with c_n bounded above, the inequality min{a,b} ≤ √(ab) yields τ_n ≤ √(ρ c_n) ≤ U, so the adaptive rule is self-bounding. This leads to Algorithm 2 (PF-GRPDA), a parameter-free method for locally smooth h, with ergodic O(1/N) guarantees for objective residual and feasibility violation and linear convergence when f and g* are strongly convex. The paper then derives two accelerated variants: Algorithm 3, which grows the dual scaling β_n via β_n = β_{n-1}(1 + μζ_n τ_{n-1}) when f is strongly c

What carries the argument

The machinery is the adaptive primal step-size update τ_n = min{ρτ_{n-1}, α_n α_{n-1} ψ θ_{n-1} / ((L_n² + βψ‖K‖²)τ_{n-1})}, where L_n is the local estimate of the Lipschitz constant of ∇h, combined with the golden-ratio convex averaging z_n = (ψ−1)/ψ x_{n-1} + 1/ψ z_{n-1} with ψ∈(1,φ]. The critical handle is Lemma 2.5: min{a,b} ≤ √(ab), applied to the two branches of the update, which yields the uniform bound τ_n ≤ U independent of all iterates. For the accelerated variants, the dual-primal ratio β_n is made to grow (β_n = β_{n-1}(1+μζ_n τ_{n-1}) for strong convexity of f, β_n = β_{n-1}(1+γτ_{n-1}) for strong convexity of h); the proofs show β_n grows quadratically, so the weighted ergodic

Load-bearing premise

The accelerated 1/N² rate for the h-strongly-convex variant (Algorithm 4) is only proved if the acceleration parameter γ stays below a threshold built from the global strong-convexity constant µ_h of h; if µ_h is unknown or misestimated, or γ is chosen above the threshold, the telescoping inequality that yields the rate does not follow.

What would settle it

Run Algorithm 2 on the quadratic test in Section 3 with no τ_max and record the primal step-sizes: for every n the bound τ_n ≤ √(ρ c_max) from Remark 4.2 must hold, and the iterates must converge; any τ_n exceeding that bound, or any divergence, refutes the parameter-free claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Users can drop the τ_max hyperparameter: the ergodic O(1/N) rates for objective residual and feasibility violation hold with no step-size cap.
  • When f and g* are both strongly convex, Algorithm 2 converges linearly, giving a linearly convergent, locally-smooth primal–dual method with no external tuning except β, ψ, α.
  • With f strongly convex, Algorithm 3 obtains an ergodic O(1/N²) rate while keeping h only locally smooth and needing no linesearch or global Lipschitz constant.
  • With h globally strongly convex, Algorithm 4 obtains the same O(1/N²) ergodic rate under a step-size rule that remains adaptive.
  • On Poisson imaging problems, the accelerated variants reduce feasibility violation and objective residual faster than the base method in the reported runs.

Where Pith is reading between the lines

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

  • The min-of-two-branches bounding trick likely extends to other adaptive first-order methods: any step-size update that combines a multiplicative growth branch with a reciprocal local-curvature branch is automatically bounded without a cap.
  • The two acceleration mechanisms are complementary and depend on how the problem is split: placing strong convexity in f or in h determines which accelerated variant applies, so the splitting choice itself becomes an algorithmic lever.
  • The threshold γ̄ for Algorithm 4 depends on the global strong-convexity constant µ_h; estimating µ_h adaptively along the iterates would complete the parameter-free story, and the paper explicitly lists this as an open direction.
  • The heuristic adaptive-β rule used in the experiments is outside the proved theory; a convergence proof for it would extend the parameter-free guarantee to the primal–dual scaling ratio.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies a class of adaptive golden-ratio primal–dual methods for composite convex optimization (1), where the differentiable term h is only locally smooth. It first revisits the aEGRPDA algorithm of Soe et al. [26] and removes the artificial step-size cap τ_max. The key structural argument is Remark 4.2, which uses Lemma 2.5 (min{a,b} ≤ √(ab)) to show that the adaptive rule (13) itself yields a uniform upper bound on τ_n, so no linesearch, no global Lipschitz constant, and no cap are needed. Algorithms 2, 3, and 4 are then analyzed: Algorithm 2 is shown to have ergodic O(1/N) objective and feasibility rates (Theorem 4.2), and linear convergence under strong convexity of f and g* (Theorem 5.1). Algorithm 3, which exploits strong convexity of f, is proved to have ergodic O(1/N²) rates (Theorem 6.1). Algorithm 4, which exploits global strong convexity of h, is claimed to have the same O(1/N²) rate (Theorem 6.2), but the proof is omitted. Numerical experiments on Poisson imaging problems illustrate the methods in three settings.

Significance. If the results are correct, the paper makes a useful contribution: it provides a principled way to run a golden-ratio primal–dual method without a step-size cap or linesearch under only local smoothness, and it extends accelerated O(1/N²) guarantees to a locally smooth setting, which is uncommon in the literature. The proof that the adaptive rule self-bounds τ_n (Remark 4.2) is simple and elegant, and the resulting O(1/N) estimates that are independent of τ_max genuinely improve on prior work [26]. The linear-convergence section (Section 5) is carefully written, with explicit constants and a clear contraction argument. The main weakness is that the second accelerated result (Theorem 6.2 for Algorithm 4) is stated without proof, and the numerical experiments choose γ via trial and error after computing γ̄ from (63), which depends on the global strong-convexity constant μ_h. This leaves the paper's second central accelerated-rate claim under-supported.

major comments (3)
  1. [Section 6.1, Theorem 6.2] Theorem 6.2, the O(1/N²) rate for Algorithm 4, is stated without proof: 'The proof follows an analogous argument as in Remark 6.3, Lemma 6.4, Lemma 6.5 and Theorem 6.1. Because of its simplicity, we omit the details.' This is a load-bearing omission. The analogous argument must establish the lower step-size bound τ_n ≥ M̄/√(L²+Cβ_n) with M̄ independent of n, the quadratic growth β_n ≥ c₀n², and β_nτ_n ≥ c₁n for the fixed-γ update (64). None of these steps appears in the text for Algorithm 4, and the presence of γ in (64) changes the recursion substantially from the μ-driven update of Algorithm 3. The rate (56) therefore is not demonstrated for Algorithm 4 as stated.
  2. [Section 6.1, Eq. (63) and Section 7.1.3] The threshold γ̄ in (63) depends on μ_h, the global strong-convexity constant of h. In the experiments, γ = 0.99 γ̄ is selected only after trial and error (Section 7.1.3: 'after a few trial and error, we are satisfied with β₀ = 5, and after computing γ̄ from (63), select γ = 0.99 ¯γ'). If μ_h is unknown or overestimated, Lemma 6.8's telescoping argument (the inequalities following (72)) may fail: the γτ_n terms would not be absorbed by μ_hτ_n and θ_n terms, so (69) would not telescope and (56) would not follow. The paper does not provide a strategy for estimating μ_h or a robustness analysis for misspecified μ_h. This is connected to the omission of the proof of Theorem 6.2 and should be addressed explicitly.
  3. [Section 4, Lemma 4.2] Lemma 4.2's lower bound on τ_n and θ_n relies on [37, Lemma 4.2] without reproducing the statement or verifying its hypotheses for Algorithm 2. Since this lower bound is used in Theorem 4.1 and Theorem 4.2 to obtain nonzero Δ and the O(1/N) rate, the dependence should be made transparent: state the lemma or at least explicitly verify the required conditions. The reader should not have to consult a separate paper to see whether the assumptions (e.g., the specific form of the step-size update) match.
minor comments (6)
  1. [Section 7.1.3, paragraph after Fig. 13] The text refers to 'Algorithm 6' in 'the accelerated method (Algorithm 6) yields a lower overall objective residual'. The algorithm is Algorithm 4. Please correct the numbering.
  2. [Abstract and Introduction] The phrase 'parameter-free' is used for Algorithm 2, but the method still depends on β, ψ, α_n, and τ₀. What the paper actually shows is that the rate estimates do not depend on an artificial cap τ_max. Please qualify the term, e.g., 'cap-free' or 'parameter-free in the convergence-rate bound'.
  3. [Section 4, Remark 4.1] The convention '0/0 = +∞' is stated, but the update (13) when x_n = x_{n-1} and L_n = 0 gives a denominator of zero. The convention makes the fraction equal to +∞, so τ_n = min{ρτ_{n-1}, +∞} = ρτ_{n-1}, which is consistent with the remark, but this should be stated more explicitly because the usual computer arithmetic would produce NaN.
  4. [Section 6, Algorithm 3 and Lemma 6.4] The definition of M̄ in Lemma 6.4 contains τ₁ and τ₂, which are defined by the algorithm. The induction proof then claims the bound for n=1,2 by definition. This is fine, but the notation M̄ is used before τ₁ and τ₂ are introduced as deterministic functions of the input. Suggest moving the induction start to n=3 or clarifying that τ₁,τ₂ are fixed by the initial data.
  5. [Section 7, general] Figure captions and text frequently write 'PFGRPDA' instead of 'PF-GRPDA'. Also, the caption of Figure 3 says 'Several heuristic adaptive-β n for PFGRPDA' and 'Thin faded curves show individual heuristic choices' – this is a bit confusing because the figure seems to show only one method with several β-update variants. Please clarify.
  6. [Section 2, Lemma 2.5] Lemma 2.5 is stated as 'min{a,b} ≤ √(ab)' for a,b>0. This is true but unnecessarily weak: the sharpest bound is min{a,b} ≤ √(ab), which is exactly the geometric mean. The proof is fine, but the lemma could be stated as an equality-like bound without overclaiming.

Circularity Check

0 steps flagged

No significant circularity: the no-cap step-size bound is derived in-line from the adaptive update, not assumed. The only caveats are a non-load-bearing self-citation to [26] and an omitted proof for the second accelerated rate, which is a completeness risk rather than a circular step.

full rationale

The central no-cap claim is self-contained. In Remark 4.2 the paper rewrites update (13) as tau_n = min(rho tau_{n-1}, c_n / tau_{n-1}), applies Lemma 2.5 (min{a,b} <= sqrt(ab), proved in the paper) to obtain tau_n <= sqrt(rho c_n), and then bounds c_n using theta_n <= 1+1/psi and D_n >= beta psi ||K||^2, giving tau_n <= U = sqrt(rho c_max). This is a direct algebraic consequence of the step-size rule, not an input equivalent to the conclusion. The lower bound in Lemma 4.2 cites [37, Lemma 4.2], an external lemma by Tam--Uteda rather than the present authors, and the boundedness needed for it is established in the same lemma from inequality (17); even if that citation were questioned, it would not make the upper-bound result circular. The self-citation to [26] supplies the base aEGRPDA and the previous constant C1, but the new proof does not use [26] to justify removal of tau_max, so it is not load-bearing. The O(1/N^2) rate for Algorithm 3 is fully proved, while Theorem 6.2 for Algorithm 4 is asserted with "we omit the details" (Section 6.1), and the gamma threshold (63) requires the strong-convexity constant mu_h, with gamma = 0.99 gamma_bar selected by trial and error in Section 7.1.3. These are completeness and parameter-sensitivity risks, not circular steps: no rate constant is fitted to data, and the theorem is conditional on the stated gamma bound. No prediction reduces by construction to a fitted value.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The algorithms require user-chosen hyperparameters (beta, psi, alpha, tau0, beta0, gamma), but the convergence theory states results for all admissible choices within the stated ranges, so no constants are fitted to data. The main external assumptions are local smoothness of h and the cited step-size lower-bound lemma. No new entities are postulated.

free parameters (5)
  • beta (Algorithm 2) / beta0 (Algorithms 3,4)
    User-chosen positive ratio sigma_n/tau_n; convergence holds for any beta>0, so not fitted to data, but it scales the step sizes.
  • psi = 1.61 in Algorithm 4 experiments
    Golden-ratio relaxation; must lie in (1,phi], (psi0,phi], or (1,phi) depending on the variant. Chosen by hand.
  • alpha_n / alpha
    Momentum/adaptive coefficient in the step-size update, required in (epsilon,1/3] or (0,1/3]. Chosen by user; no data fit.
  • tau0 = 0.30U in Algorithm 4 experiments
    Initial primal step size; positive, with tau0<=U in Algorithms 3 and 4. User-chosen.
  • gamma (Algorithm 4) = 0.99 gamma-bar
    Acceleration growth parameter for beta_n; must satisfy 0<=gamma<=gamma-bar in (63). In experiments chosen by trial and error; if too large the proof collapses.
axioms (6)
  • domain assumption Assumption 2.1: saddle point of (3) exists and prox of f,g are computable
    Used throughout; standard constraint qualification cited.
  • domain assumption Assumption 4.1: f,g proper convex lsc; h convex and locally smooth
    Standing assumption for Algorithms 2, 3, 4 and their rates.
  • domain assumption Assumption 5.1: f is mu_f-strongly convex and g* is mu_g*-strongly convex
    Enables linear convergence in Section 5.
  • domain assumption Assumption 6.1: f is mu-strongly convex
    Enables Algorithm 3 O(1/N^2).
  • domain assumption Assumption 6.2: h is globally mu_h-strongly convex and locally smooth
    Enables Algorithm 4 O(1/N^2).
  • standard math [37, Lemma 4.2] lower bound on tau_n for adaptive step sizes
    Used in Lemma 4.2 and Remark 6.3 to establish tau_n bounded away from zero.

pith-pipeline@v1.3.0-alltime-deepseek · 31622 in / 15619 out tokens · 137887 ms · 2026-08-04T04:23:02.153777+00:00 · methodology

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read the original abstract

This paper revisits the adaptive extended golden-ratio primal--dual algorithm (aEGRPDA) proposed by Soe et al. (2026) for structured convex optimisation problems involving a differentiable term that is only locally smooth. We prove that the artificial upper bound imposed on the primal step-size in aEGRPDA is redundant, since the adaptive rule itself keeps the step-sizes bounded above. As a consequence, the ergodic $\mathcal O(1/N)$ estimates for the objective residual and feasibility violation, where $N\ge1$ denotes the number of iterations, are independent of this hyperparameter. Consequently, the resulting adaptive golden-ratio primal--dual method, therefore, requires neither a step-size cap, nor a linesearch procedure, nor a known global Lipschitz constant. We establish linear convergence of the algorithm when both the primal and dual functions are strongly convex. Furthermore, we develop two accelerated variants, in addition to the local smoothness assumption: one for the case where the nonsmooth primal component is strongly convex, and another for the case where the differentiable term is globally strongly convex. For these accelerated methods, we prove an ergodic $\mathcal O(1/N^2)$ convergence rate. Preliminary numerical experiments on a Poisson imaging problem illustrate the efficiency and robustness of the proposed approaches.

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