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REVIEW 3 major objections 5 minor 16 references

Nonlinearly Preconditioned Gradient Methods under Generalized Smoothness

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A single generalized smoothness condition unifies the convergence analysis of gradient clipping, Adam, Adagrad, and (L0,L1)-smooth methods.

desk verdict Solid two-constant smoothness framework with correct conditional convergence rates, but the injectivity proof for nonsmooth reference functions has a gap that should be fixed. read the letter →

arxiv 2502.08532 v2 pith:4JWMPV33 submitted 2025-02-12 math.OC

Konstantinos Oikonomidis , Jan Quan , Emanuel Laude , Panagiotis Patrinos This is my paper · ORCID
classification math.OC MSC 90C2590C3049J52
keywords nonlinearpreconditioninggeneralizedsmoothnessanisotropicgradientclippingadaptivemethodsconvexoptimizationnonconvex(L0L1)-smoothness
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 claims that many widely used gradient preconditioning schemes—gradient clipping, Adam and Adagrad without momentum, and newer methods for (L0,L1)-smooth objectives—are all instances of one iteration, $x^{k+1}=x^k-\gamma\nabla\phi^*(\lambda\nabla f(x^k))$, governed by a single generalized descent inequality called $(L,\bar L)$-anisotropic smoothness. This property is broader than ordinary Lipschitz smoothness, so the same analysis covers objectives whose Hessians grow with the gradient norm. The paper proves an $O(1/K)$ rate for the stationarity measure in the nonconvex case and $O(1/K)$ rates for the function-value gap in two convex settings, including anisotropic reference functions via an envelope representation. If correct, one convergence proof explains and extends guarantees for clipping, Adam, Adagrad, and $(L_0,L_1)$-smooth optimization.

What carries the argument

The load-bearing object is the pair (reference function $\phi$, dual $\phi^*$) through the forward operator $T_{\gamma,\lambda}(x)=x-\gamma\nabla\phi^*(\lambda\nabla f(x))$, together with the $(L,\bar L)$-anisotropic smoothness inequality. The argument's engine is that convexity and evenness of $\phi$ make the descent gap $\phi((1-\alpha)v)-\phi(v)$ bounded by $(|1-\alpha|-1)\phi(v)$, which gives the factor $\beta=1-|1-\alpha|$ in the rates. A second engine is the envelope identity $f=\inf_y \bar L(L^{-1}\star\phi)(\cdot-y)+\xi(y)$, which turns the algorithm into a nonlinear proximal point method and yields the convex anisotropic rate.

What would settle it

Choose a twice differentiable function such as a variant of $\frac14\|x\|^4$ with an added non-quartic term and a reference function from Table 1, then scan a fine grid of pairs $(x,\bar x)$ for a violation of inequality (4) with the constants claimed by the paper's sufficient conditions; finding any pair that violates (4) while satisfying the second-order condition and forward-operator coercivity would disprove Proposition 2.9. Conversely, running iteration (2) on an $(L_0,L_1)$-smooth function and checking that the stationarity measure decays by the predicted $O(1/K)$ bound would confirm the mechanism.

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Extended reading notes

Core claim

The central discovery is that replacing the quadratic upper bound of Lipschitz smoothness with a reference-function upper bound $f(x)\le f(\bar x)+\bar L[(L^{-1}\star\phi)(x-\bar y)-(L^{-1}\star\phi)(\bar x-\bar y)]$, where $\bar y=\bar x-L^{-1}\nabla\phi^*(\bar L^{-1}\nabla f(\bar x))$, yields a strictly larger class of smooth functions still amenable to first-order methods. The iteration that minimizes this bound is gradient descent with a nonlinear, coordinate-wise or radial step determined by the conjugate of $\phi$. On this class the paper proves the nonconvex stationarity bound $\min_{0\le k\le K}\phi(\nabla\phi^*(\bar L^{-1}\nabla f(x^k)))\le L(f(x^0)-f^*)/(\bar L \beta(K+1))$, and convex $O(1/K)$ suboptimality-gap rates for isotropic and $2$-subhomogeneous anisotropic reference functions.

Load-bearing premise

The whole convergence theory stands on the assumption that at every point the objective $f$ satisfies the anisotropic descent inequality (4) with the particular point $\bar y$ produced by the algorithm; if that inequality is violated for even one iterate, the descent step used in every rate proof breaks down.

Editorial extensions

If this is right

  • Stepsizes up to $\gamma=2/L$ are allowed in the nonconvex case, with rate depending on $\beta=1-|1-\alpha|$ (Theorem 3.2).
  • Under isotropic reference functions, iterates are Fej\'er monotone, gradient norms decrease monotonically, and the function-value gap decays as $O(1/K)$ (Theorem 3.6).
  • For convex objectives with $2$-subhomogeneous anisotropic reference functions, the gap decays as $O(1/K)$ without an initial-gradient-dependent constant (Theorem 3.7).
  • Every $(L_0,L_1)$-smooth $C^2$ function satisfies the generalized smoothness property for a specific kernel, giving new guarantees and characterizations for that class (Corollary 2.11).
  • Gradient clipping, Adam and Adagrad without memory are recovered as special cases, so their convergence is covered by the same proof.

Reading between the lines

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

  • The kernel view suggests a way to design new preconditioners by choosing $\phi$, with the second-order condition in Theorem 2.4 as a checkable design criterion; one could generate a family of sigmoid-step methods beyond the listed examples.
  • The framework's weakest point is the need to verify the anisotropic descent inequality itself; the second-order sufficient condition relies on a coercivity/injectivity assumption, and a useful extension would be to characterize the class of functions satisfying (4) for non-smooth kernels such as the clipping kernel without the extra monotonicity assumption.
  • The convex results might extend to composite objectives with an additive nonsmooth term, which the paper explicitly leaves as future work; a proof alongside the anisotropic proximal gradient would likely carry the same rates.
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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 / 5 minor

Summary. The paper introduces (L,\bar L)-anisotropic smoothness, a two-constant, possibly non-full-domain generalization of the anisotropic descent property of Laude and Patrinos, and studies the nonlinearly preconditioned gradient iteration x^{k+1}=x^k-\gamma\nabla\phi^*(\bar L^{-1}\nabla f(x^k)). The main theoretical results are a nonconvex stationarity rate (Theorem 3.2), convex O(1/K) suboptimality rates for isotropic and 2-subhomogeneous anisotropic reference functions (Theorems 3.6 and 3.7), and a second-order sufficient condition route (Definition 2.4, Propositions 2.6 and 2.9) intended to certify the new smoothness condition for classes of functions including clipping, Adagrad-like, Adam-like, and (L_0,L_1)-smooth objectives. The paper also contains numerical experiments on a quartic problem, phase retrieval, and a small neural network, with public code.

Significance. If the breadth claims are fully established, this is a valuable unifying framework: it places gradient clipping, signed/adaptive preconditioners, and recent (L_0,L_1)-smooth methods under one majorization-minimization umbrella and gives explicit convergence rates that are not conditional on a Euclidean Lipschitz gradient. The core convergence argument is refreshingly simple and appears correct: Theorem 3.2 derives the stationarity rate directly from the defining descent inequality, and the convex rates in Theorems 3.6 and 3.7 are nontrivial extensions of existing results. The paper ships reproducible code and states its constants explicitly, which I regard as strengths. The main caveat is that the convergence theory is conditional on Definition 2.1, whose breadth is supposed to be certified by the second-order sufficiency theory; as detailed below, that certification has a gap for nonsmooth reference functions, and the worked example in Appendix D.1 contains a false inequality. These issues do not invalidate the conditional convergence theorems, but they do affect the advertised scope of the framework.

major comments (3)
  1. [Appendix B.4, Proposition 2.6] The proof that T_{L^{-1},\bar L^{-1}} is a local homeomorphism is incomplete. From the chain-rule inclusion, each element A of the Clarke Jacobian satisfies that A(x-\bar x) lies in the convex hull of matrices of the form M_H = I - L^{-1}\bar L^{-1} H \nabla^2 f(\bar x) applied to (x-\bar x). Condition (5) only shows that each individual M_H has eigenvalues in (0,\infty) (when the product has real eigenvalues). It does not imply that every matrix in the convex hull is nonsingular: for example, the invertible matrices [[1,2],[0,1]] and [[1,0],[2,1]] both have all eigenvalues equal to 1, yet their 1/2-convex combination is the singular matrix [[1,1],[1,1]]. Thus the application of Clarke's inverse function theorem as stated is not justified. Since Proposition 2.9 requires injectivity of T, the sufficient route to (L,\bar L)-anisotropic smoothness is left unproved exactly for nonsmooth reference functions such as the clipping kernel in Example 1.7 and Section D.2, where the generalized Jacobian of \nabla\phi^* is multi-valued and H\nabla^2 f is generally nonsymmetric. Proposition D.2 does not cover this case because it assumes H\nabla^2 f is symmetric. The convergence theorems in Section 3 are conditional on Definition 2.1 and are not affected; what needs repair is the sufficiency/breadth claim. A fix could impose a condition on the entire convex hull or prove strong monotonicity of the forward operator under additional structure.
  2. [Appendix D.1, Example D.3] The proof of the constants in Table 2 asserts that (p-1)a_i(\bar L^{-1}\|x\|^{p-1}) \ge b_i(\bar L^{-1}\|x\|^{p-1}) for all reference functions considered. For \phi_3, one has a_3(t)=1/(1+t)^2 and b_3(t)=1/(1+t), so the inequality fails as soon as t>p-2. Consequently the eigenvalue formula (37) is not maximized by (p-1)a_i in that regime, and the subsequent bound is not justified as written. The final value of L_{\rm norm} appears to remain valid for p=4 (the maximum of the second branch at the switch point t=2 is smaller than the maximum of the first branch at t=1/2), but the argument should be replaced by a genuine case analysis rather than the stated universal inequality.
  3. [Definition 2.1 and Proposition 2.9] The main circularity risk of the framework deserves an explicit statement: the defining inequality (4) is almost literally the descent condition used in the proof of Theorem 3.2, so the convergence results are conditional on a condition that is not algorithm-independent. The paper does provide an independent sufficiency theory, which is the right way to address this concern, but because that theory currently hinges on the unproved injectivity step in Proposition 2.6, the reader cannot yet conclude that the advertised classes (clipping, Adam, Adagrad, (L_0,L_1)-smooth) satisfy Definition 2.1 in the nonsmooth cases. I would ask the authors to either repair Proposition 2.6 or explicitly state the injectivity condition as an additional hypothesis in Proposition 2.9 for the nonsmooth-reference-function applications.
minor comments (5)
  1. [Throughout, numbering] The text repeatedly cites numbered display items as 'Theorem' even when they are Assumptions, Definitions, Examples, or Propositions (for instance, 'Theorem 1.1' for Assumption 1.1, 'Theorem 1.7' for Example 1.7, and 'Theorem 2.2' for Proposition 2.2). This is confusing and should be normalized to the correct labels.
  2. [Section 1.5] The sentence 'Theorem 1.1 is considered valid throughout the paper' should read 'Assumption 1.1 is assumed throughout', since the item is an assumption, not a theorem.
  3. [Appendix C.4, proof of Theorem 3.6] In the display around equation (24), the quotient h^{*\prime}(\|\bar L^{-1}\nabla f(x^k)\|)/\|\bar L^{-1}\nabla f(x^k)\| is undefined when \nabla f(x^k)=0. The stationary case should be handled separately, or the argument should be stated for the active regime \nabla f(x^k)\ne 0.
  4. [Table 2 / Section 4.1] The experiment in Section 4.1 says that L is computed 'according to the rules established in section D', but the precise values of L used for each \bar L are not listed in the text or caption. Since Table 2 gives formulas, adding the numerical constants would improve reproducibility; the public code presumably contains them, but a short sentence would help the reader.
  5. [Example D.4] The bounds leading to the L_{\rm logistic} entries are stated as a chain of inequalities without indicating which bound corresponds to which \phi_i. A short annotation would make the case analysis much easier to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence rates are conditional theorems under the stated anisotropic smoothness assumption, which is independently supported by second-order and (L0,L1) sufficiency results.

full rationale

The paper's main convergence results, Theorems 3.2, 3.6, and 3.7, are valid conditional statements: they assume (L,bar-L)-anisotropic smoothness (Definition 2.1) and derive rates by summing the descent inequality (4). This is a standard majorization-minimization structure, not a circular reduction, because the smoothness condition is stated as an a priori property of f and is not fitted to the target rate. The paper provides independent routes for verifying the condition: Proposition 2.3 shows Lipschitz-smooth functions satisfy it, Corollary 2.11 connects it to (L0,L1)-smoothness, and Section D verifies it for concrete examples using second-order conditions. The self-citations to Laude and Patrinos, Laude et al., and Laude are used for context, for the definitional ancestry of the framework, and for a technical proximal-equivalence lemma in the proof of Theorem 3.7; none of these import an unverified uniqueness claim, and the cited lemma does not assume the target convergence rate. The gap in Proposition 2.6 noted by the skeptic is a soundness concern about one sufficient route to injectivity, not a circularity: even if that route fails, Theorem 3.2's proof follows directly from the assumed inequality, and alternative sufficiency conditions such as Proposition D.2 remain available. The paper is self-contained in its main derivation chain and does not rename a known result as a prediction or fit a parameter and call it a prediction.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper does not fit parameters to data; the only parameters are those defining the smoothness class and the algorithm. It introduces a mathematical definition rather than a physical or conceptual entity, so no invented entities appear. The listed axioms are the explicit assumptions of the theorems.

free parameters (1)
  • Smoothness constants L and \bar L and stepsizes \gamma, \lambda = not fitted to data; chosen per problem, e.g. L > L_norm in Appendix D
    These parameters define the generalized smoothness condition and the algorithm (2). They are not estimated from data; the theory provides sufficient conditions for their existence.
assumptions (6)
  • domain assumption Reference function \phi is proper, lsc, strongly convex, even, with \phi(0)=0 (Assumption 1.1).
    Used throughout to ensure \phi \ge 0, \nabla \phi^*(0)=0, and the evenness-based step-size extension in Theorem 3.2.
  • domain assumption \phi \in C^2(int dom \phi) and is essentially smooth (Assumption 1.2).
    Yields \phi^* \in C^2 with positive-definite Hessian, used in the second-order conditions (Lemma 2.5) and in the main convex proofs.
  • domain assumption f is C^1, (L,\bar L)-anisotropically smooth relative to \phi, and bounded below (Assumption 3.1).
    This is the central smoothness hypothesis; all convergence theorems are conditional on it.
  • domain assumption arg min f is nonempty and f is convex in the convex rate theorems (Assumption 3.4).
    Needed for the dual convexity argument in Prop 3.5 and for defining x^* in Theorems 3.6 and 3.7.
  • domain assumption In Theorem 3.7, dom \phi = R^n and \phi is 2-subhomogeneous, i.e. \phi(\theta x) \le \theta^2 \phi(x) for \theta \in [0,1].
    Used in the nonlinear proximal point / envelope proof of the anisotropic convex sublinear rate.
  • standard math Standard tools from convex analysis and Clarke generalized calculus (Rockafellar and Wets, Clarke, Dontchev and Rockafellar).
    Background for conjugate duality, generalized Jacobians, and global homeomorphism arguments in the proofs.

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Pith. "Pith review of Nonlinearly Preconditioned Gradient Methods under Generalized Smoothness." pith.science (2026). https://pith.science/paper/4JWMPV33

@misc{pith2026250208532,
  author       = {Pith},
  title        = {Pith review of: Nonlinearly Preconditioned Gradient Methods under Generalized Smoothness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4JWMPV33}},
  note         = {Machine review of arXiv:2502.08532}
}
abstract

We analyze nonlinearly preconditioned gradient methods for solving smooth minimization problems. We introduce a generalized smoothness property, based on the notion of abstract convexity, that is broader than Lipschitz smoothness and provide sufficient first- and second-order conditions. Notably, our framework encapsulates algorithms associated with the gradient clipping method and brings out novel insights for the class of $(L_0,L_1)$-smooth functions that has received widespread interest recently, thus allowing us to extend beyond already established methods. We investigate the convergence of the proposed method in both the convex and nonconvex setting.

Figures

Figures reproduced from arXiv: 2502.08532 by the authors.

Figure 1
Figure 1. Visualization of the quadratic upper bounds of the func￾tion f(x) at various points. By flipping the figure it can be seen that −f is a Φ-convex function: it is the pointwise supremum over concave quadratics of the form −ϕ(x−y) +β, with ϕ = L 2 ∥ · ∥2 and y, β ∈ R. Note that this function is not convex in the classical sense, as there are no linear functions supporting it. 1.2. Our contribution Our approach departs … view at source ↗
Figure 2
Figure 2. Minimizing 1 4 ∥x∥ 4 using (2). The figure on the left corresponds to ϕ1(x) = cosh(∥x∥) − 1, the middle one to ϕ2(x) = exp(∥x∥) − ∥x∥ − 1 and the one on the right to ϕ3(x) = −∥x∥ − ln(1 − ∥x∥). We choose values of L¯, set λ = L¯−1 and then compute γ = L −1 with L as in section D. Theorem 3.2. Let Theorem 3.1 hold and {x k}k∈N0 be the sequence of iterates generated by (2) with γ = αL−1 , α ∈ (0, 2), λ = L¯−1 and let … view at source ↗
Figure 3
Figure 3. Nonconvex phase retrieval. ϕ1 corresponds to the isotropic reference function and ϕ2 to the anisotropic one, both of which are generated by cosh(·) − 1. The two figures on the left compare the algorithms for one instance of the problem. The figure on the right displays the results of gradient clipping and the isotropic version of (2) averaged across 100 random instances. 0 50 100 150 200 250 Number of iterations 10−… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Simple NN training. (left) results for (2) with ϕ1(x) = cosh(∥x∥) − 1; (middle) ϕ2(x) = −∥x∥ − ln(1 − ∥x∥); (right) gradient clipping method as presented in Theorem 1.7. from (Vankov et al., 2024) or (Gorbunov et al., 2024) to achieve a better complexity when specifyin…

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