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REVIEW 3 major objections 4 minor 40 references

Restarted contractive operators to learn at equilibrium

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Backpropagating through only the last block of a restarted contractive solver can be made arbitrarily close to the full deep-equilibrium gradient step.

desk verdict A genuinely useful quantitative bridge between truncated backprop and DEQ gradients, let down by an unmeasured contraction constant and thin baselines. read the letter →

arxiv 2506.13239 v1 pith:ZGTY2BSV submitted 2025-06-16 math.OC cs.LG

Leo Davy (Phys-ENS) , Luis M. Briceno-Arias (UTFSM) , N. Pustelnik (Phys-ENS) This is my paper · ORCID
classification math.OCcs.LG MSC 65K1090C2549J5268T07
keywords bileveloptimizationdeepequilibriummodelsJacobian-freebackpropagationunrolledneuralnetworkscontractivefixed-pointoperatorsplug-and-playimaginghyperparameterlearningrestartediterativeschemes
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

The paper proves that a cheap surrogate gradient — Jacobian-free backpropagation through the final block of a restarted truncated solver — can be made arbitrarily close to the exact deep-equilibrium hypergradient. Theorem 2 bounds the difference by a term that shrinks with the K-step contraction rate plus terms that decay exponentially in the number of restarts T. This matters for imaging inverse problems because it justifies training unrolled networks without forming or inverting a high-dimensional Jacobian. The authors validate the approach on wavelet denoising inside the theoretical assumptions, then on inpainting and deblurring with pretrained and learned denoisers, where restarts still help even though the assumptions are relaxed.

What carries the argument

The load-bearing object is the restarted truncated operator $\Phi_K^T(x_0,\theta) = \Phi_K(\cdot,\theta) \circ \cdots \circ \Phi_K(\cdot,\theta)$ (T times), where $\Phi_K$ is K steps of a contractive proximal algorithm such as forward-backward splitting. ReTune runs these restarts and applies automatic differentiation only to the final step, producing $g_R$. The proof mechanism is the Neumann-series bound $\|I-(I-H)^{-1}\|_2 \le \omega/(1-\omega)$ for $\|H\|_2 = \omega < 1$, which converts the contraction rate into the Jacobian-free error, together with linear convergence of the restart iterates to the fixed point and a local Lipschitz condition on $\partial_\theta \Phi_K$ controlling the remaining terms.

What would settle it

Measure the empirical contraction factor $\delta_K$ or the operator norm of $\partial_x \Phi_K(\cdot,\theta)$ on the wavelet-denoising training trajectory; if any visited $\theta$ has $\delta_K \ge 1$, or if on a case with known $\delta_K<1$ the gap between $g_R$ and the exact DEQ gradient $g$ fails to shrink like $\delta_K^T$ as T increases, the theorem's bound (33) is contradicted.

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Extended reading notes

Core claim

The central claim is that for a parameterized fixed-point scheme whose K-step composition is a contraction, the DEQ gradient step $g(\theta) = \partial_x L(\bar{x}_\theta)^\top [I - \partial_x \Phi_K(\bar{x}_\theta, \cdot)]^{-1} \partial_\theta \Phi_K(\bar{x}_\theta, \cdot)(\theta)$ can be replaced by the last-block gradient $g_R(\theta) = \partial_x L(x_{KT})^\top \partial_\theta \Phi_K(x_{K(T-1)}, \cdot)(\theta)$ with a quantified error. The error decomposes into a Jacobian-free-backpropagation error bounded by $\delta_K/(1-\delta_K)$ and restart errors that decay like $\delta_K^T$, where $\delta_K<1$ is the Lipschitz constant of the K-step operator. Hence by increasing the unrolled depth K and the number of restarts T, the learned update is provably as good as the full implicit-differentiation step, without ever computing $[I - \partial_x \Phi_K]^{-1}$. This is the sense in which ReTune learns at equilibrium.

Load-bearing premise

The load-bearing premise is Assumption 1: for every parameter value, the K-step operator must be a strict contraction with Lipschitz constant below one and a unique fixed point, and if that fails the convergence theorem and every error bound collapse.

Editorial extensions

If this is right

  • Larger K shrinks the Jacobian-free component of the gradient error to zero, so depth can serve as a substitute for inverting the DEQ Jacobian.
  • Larger T drives the restart point $x_{KT}$ to the fixed point $\bar{x}_\theta$, so the inner problem is solved exactly while backpropagation stays limited to one K-step block.
  • The bound (33) gives a principled resource split: increase K to reduce the JFB error and increase T to reduce the restart error.
  • For strongly convex forward-backward problems satisfying the assumptions, ReTune is a provably convergent bilevel learning procedure, unlike standard truncated unrolled training.

Reading between the lines

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

  • Going beyond the paper, the same restart-plus-JFB argument should apply to other contractive fixed-point solvers, such as Douglas-Rachford or primal-dual schemes, whenever the composed operator's parameter derivative is locally Lipschitz.
  • The theory suggests an adaptive budget in which K and T grow during training, or are set from an estimated $\delta_K$, trading memory for gradient accuracy; the experiments only test fixed choices such as $K,T \in \{1,10\}$.
  • The success on non-contractive Plug-and-Play denoisers hints that a local or statistical contraction near the fixed point may be enough, and a testable extension would measure the empirical Lipschitz constant along the training trajectory.
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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

3 major / 4 minor

Summary. The paper considers bilevel optimization for imaging inverse problems, where the inner problem is solved by an iterative scheme truncated to K steps, called a truncated unrolled scheme. The authors propose ReTune: apply T restarts of the K-step operator and compute the hypergradient by automatic differentiation through only the last restarted block, avoiding the Jacobian inversion required by the Deep Equilibrium (DEQ) framework. The main theoretical results are: (i) Theorem 1, convergence of the restarted iterates to the inner solution under a contraction assumption on the K-step operator; (ii) Lemma 1, a bound on the error between the exact DEQ gradient and the Jacobian-Free Backpropagation (JFB) gradient at the fixed point; (iii) Theorem 2, a bound on the error between the DEQ gradient and the ReTune gradient, showing the error tends to zero as the number of restarts increases, up to a JFB-type term that decays with K. Numerical experiments on wavelet denoising, inpainting, and deblurring withPlug-and-Play denoisers show that ReTune with K=T=10 yields improved training PSNR relative to unrolled and non-restarted baselines.

Significance. The paper proposes a clean, practically motivated link between truncated unrolled networks and DEQ/JFB implicit differentiation. The derivation is transparent and does not fit any constants to data; the assumptions are stated explicitly, and the dependence of the error bounds on the contraction constant δK and the restart count T is made precise. If the proof issues are corrected, the asymptotic guarantee that ReTune gradient steps approximate DEQ steps is a useful theoretical justification for a simple and memory-efficient training scheme. The empirical section honestly acknowledges that several experiments depart from the theoretical assumptions, and the results suggest practical robustness beyond the verified regime. The main weakness is that the one experiment intended to satisfy the theory does not measure the contraction constant, so the quantitative connection between theory and numerics remains unverified.

major comments (3)
  1. [Section 3.3, Eq. (33) and proof lines (40)-(41)] The stated bound in Eq. (33) is optimistic by a factor 1/δK in the second term. In the proof, the term ∥∂xL(bxθ)∥2 Lθ∥bxθ − xK(T−1)∥2 is bounded by δK^T∥∂xL(bxθ)∥2 Lθ∥bxθ − x0∥2, but Theorem 1 gives ∥bxθ − xK(T−1)∥2 ≤ δK^{T−1}∥bxθ − x0∥2, not δK^T. Moreover, the argument of ∂θΦK in (36) is xK(T−1), so Assumption 2 must be applied at xK(T−1), not at xKT as suggested by the bound in (40). The corrected term should carry δK^{T−1}. The qualitative conclusion that all restart terms vanish as T grows is unaffected, but the statement and proof of Theorem 2 must be corrected to be mathematically accurate.
  2. [Section 4.1, 'Validity of the assumptions' and Figure 1] The wavelet denoising experiment is the only one claimed to satisfy Assumption 1, yet the contraction constant δK(θ) is never measured or reported. Since the learned parameters are only constrained to be positive (via exp), the ratio min(θ)/max(θ) can become small, making δK(θ) approach 1 despite the fixed stepsize τ = 1.95/L. At K=T=10, if δK were close to 1, the bounds in Theorem 2 would be vacuous and could not explain the observed PSNR gains. The paper should either report an empirical estimate of δK(θ) during training (or at least a uniform upper bound derived from the parameter ranges) or explicitly state that the experiment does not verify the quantitative tightness of the bound.
  3. [Section 4.2 and Conclusion] The inpainting and deblurring experiments use non-injective forward operators and a DRUNet denoiser, which the paper acknowledges do not satisfy Assumption 1. The conclusion nevertheless states that the theoretical analysis is 'supported by numerical experiments.' This is acceptable as a claim of robustness, but the wording should be sharpened: experiments outside the assumptions cannot provide evidence for the specific quantitative bounds of Theorem 2; they only suggest that the ReTune strategy remains effective in regimes not covered by the theory. I recommend rephrasing the conclusion to distinguish 'illustrating the method outside the theoretical scope' from 'validating the theory.'
minor comments (4)
  1. [Section 3.3, Eq. (33)] The displayed bound in Eq. (33) is typeset incorrectly: the last two lines contain a dangling brace and a missing multiplication symbol. The expression should be cleaned up to make the o(·) term and the δK^T factor unambiguous.
  2. [Section 3.3, Theorem 2] The theorem statement does not mention that T must be large enough for xK(T−1) to lie in the neighborhood where Assumption 2 holds. The proof notes this, but the statement should include the quantifier, e.g., 'for all T sufficiently large'.
  3. [Section 4.1, 'Validity of the assumptions'] The sentence 'φk is (δ <1)-Lipschitz continuous' should give the explicit formula δ ≤ ω = max{|1−τ μ|, |1−τ L|} with the chosen stepsize, and state whether the bound is uniform over the parameter space explored during training. Without this, the reader cannot assess how close δK is to 1.
  4. [Figures 2 and 3] The legends in Figures 2 and 3 label configurations as (K,T) = (1,1), (1,10), (10,1), (10,10), but Figure 3 uses 'T: 0, K: 10' and 'T: 0, K: 1' in the inset. This is inconsistent with the text and should be corrected to T=1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ReTune error bounds are derived from stated contraction assumptions and independent standard lemmas; the only self-citation is not load-bearing.

full rationale

The derivation chain is self-contained and non-circular. The DEQ gradient g(θ) is defined by fixed-point differentiation (eq. 20); the JFB gradient gJF is defined by dropping the inverse Jacobian (eq. 28); and the ReTune gradient gR is defined by backpropagating through the final restarted block (eq. 25). Lemma 1 bounds ∥g − gJF∥ using only Assumption 1 and the Neumann-series lemma in the Appendix (Lemma 2, attributed to Riesz–Nagy [32]), producing the δK/(1−δK) factor. Theorem 2 then combines the triangle inequality, Theorem 1's contraction estimate, and Assumption 2's local Lipschitz estimate; no constant in the bound is fitted to data, and no quantity in the proof is defined in terms of the quantity it is supposed to predict. The comparison target g(θ) is external to ReTune, and the JFB construction is taken from the independent prior work of Fung et al. [15]. The only author-overlapping citation I can locate is [8], used in Section 4.1 for an optimal step-size formula and alternative Lipschitz constants; this fact is standard, and the central theorems do not depend on it. The paper's own limitations section honestly states that the analysis is restricted to strongly convex forward-backward schemes and that Assumption 2 is hard to verify; these are scope and verification caveats, not circularity. The skeptic's concern that δK is not measured numerically affects the strength or vacuity of the bound, but the bound is still derived from assumptions rather than assumed into existence. Accordingly, no circular step can be exhibited from the paper's equations or citations.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The theoretical claim relies on clearly stated contraction and smoothness assumptions, but not on any numerically fitted constants. The experimental hyperparameters (K, T, learning rate, τ) are hand-chosen and affect only the empirical demonstration.

free parameters (4)
  • K (inner loop length) = 10 in most experiments
    Number of truncated unrolled steps per block; chosen by hand. Theoretical results only require K such that δK<1 and AD is feasible.
  • T (number of restarts) = 10 in most experiments
    Number of repeated applications of ΦK; chosen by hand. Larger T reduces the restart error δK^T.
  • ADAM learning rate = 5e-2 (wavelet), 5e-5 (denoiser)
    Hand-chosen optimizer step for the outer problem; affects the empirical success but not the theoretical bound.
  • Forward-backward step size τ = 1.95/L[ℓ]
    Hand-chosen close to 2/L to ensure contraction of the block operator; appears in the experimental section only.
assumptions (5)
  • domain assumption fθ is convex with L-Lipschitz gradient and gθ is proper, lsc, convex, with strong convexity where contractivity is needed (Section 2.1 and Assumption 1).
    Needed for forward-backward convergence and for the contraction of ΦK.
  • ad hoc to paper ΦK(·,θ) is δK(θ)-Lipschitz with δK(θ)<1 and has a unique fixed point (Assumption 1).
    This is the load-bearing contraction assumption; the entire analysis depends on it.
  • ad hoc to paper ∂θΦK(·,θ) is locally Lipschitz at bxθ (Assumption 2).
    Used in Theorem 2 to bound the gradient error between xK(T-1) and bxθ.
  • domain assumption L is twice differentiable in Theorem 2 (MSE in Corollary 1 is twice differentiable).
    Needed for the first-order expansion of ∂xL.
  • standard math Banach fixed point theorem and Neumann series convergence (Lemma 2 from Riesz and Sz.-Nagy [32]).
    Foundational results used for restart convergence and Jacobian-free approximation.

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Cite this review

Pith. "Pith review of Restarted contractive operators to learn at equilibrium." pith.science (2026). https://pith.science/paper/ZGTY2BSV

@misc{pith2026250613239,
  author       = {Pith},
  title        = {Pith review of: Restarted contractive operators to learn at equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGTY2BSV}},
  note         = {Machine review of arXiv:2506.13239}
}
read the original abstract

Bilevel optimization offers a methodology to learn hyperparameters in imaging inverse problems, yet its integration with automatic differentiation techniques remains challenging. On the one hand, inverse problems are typically solved by iterating arbitrarily many times some elementary scheme which maps any point to the minimizer of an energy functional, known as equilibrium point. On the other hand, introducing parameters to be learned in the energy functional yield architectures very reminiscent of Neural Networks (NN) known as Unrolled NN and thus suggests the use of Automatic Differentiation (AD) techniques. Yet, applying AD requires for the NN to be of relatively small depth, thus making necessary to truncate an unrolled scheme to a finite number of iterations. First, we show that, at the minimizer, the optimal gradient descent step computed in the Deep Equilibrium (DEQ) framework admits an approximation, known as Jacobian Free Backpropagation (JFB), that is much easier to compute and can be made arbitrarily good by controlling Lipschitz properties of the truncated unrolled scheme. Second, we introduce an algorithm that combines a restart strategy with JFB computed by AD and we show that the learned steps can be made arbitrarily close to the optimal DEQ framework. Third, we complement the theoretical analysis by applying the proposed method to a variety of problems in imaging that progressively depart from the theoretical framework. In particular we show that this method is effective for training weights in weighted norms; stepsizes and regularization levels of Plug-and-Play schemes; and a DRUNet denoiser embedded in Forward-Backward iterates.

Figures

Figures reproduced from arXiv: 2506.13239 by the authors.

Figure 1
Figure 1. Wavelet denoising for degraded data with standard deviations set to ( [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Results for inpainting where 90% of the pixels are masked and noise standard deviations [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Results for anisotropic deblurring with A a channel-wise anisotropic blur kernel of width 25 and noise standard deviations are set to σR = σG = σB = 0.05. See [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

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