Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness

Alexander Tyurin

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Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness

T0 review · 1 major / 1 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read Accelerated gradient methods achieve the rate O(√ℓ(0) R / √ε) under generalized ℓ-smoothness.

desk verdict Tyurin gives a candidate clean optimal accelerated rate for generalized smoothness via a new Lyapunov, but that construction is the part that needs the closest look. read the letter →

arxiv 2508.06884 v2 pith:CBJ4NXZ6 submitted 2025-08-09 math.OC cs.LG

Alexander Tyurin This is my paper · ORCID

classification math.OCcs.LG

keywords acceleratedgradientdescentℓ-smoothness(L0L1)-smoothnessoraclecomplexityLyapunovfunctionconvexoptimizationnear-optimalrates

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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 establishes that accelerated gradient methods can reach the optimal oracle complexity O(√ℓ(0) R / √ε) for convex problems satisfying the generalized ℓ-smoothness condition, which includes both standard L-smoothness and (L0, L1)-smoothness. A sympathetic reader would care because earlier extensions to these broader smoothness classes added dependence on the initial gradient, exponential factors, or required extra sub-routines, leaving the question open whether a clean AGD-style bound is possible for small error tolerances. By introducing a new Lyapunov function and accompanying algorithms, the work shows the bound holds for virtually any ℓ and is provably optimal without extra multiplicative factors in the (L0, L1) case.

What carries the argument

New Lyapunov function that simultaneously controls the function value gap and the gradient norm under the ℓ-smoothness inequality without dependence on the initial gradient or exponential terms.

What would settle it

A concrete counterexample function satisfying (L0, L1)-smoothness on which the new accelerated method requires asymptotically more than O(√L0 R / √ε) gradient evaluations to reach small ε would disprove the claimed rate.

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Extended reading notes

Core claim

Leveraging a new Lyapunov function and designing new algorithms, we achieve O(√ℓ(0) R / √ε) oracle complexity for small-ε and virtually any ℓ. For (L0, L1)-smoothness, our bound O(√L0 R / √ε) is provably optimal in the small-ε regime and removes all non-constant multiplicative factors present in prior accelerated algorithms.

Load-bearing premise

The analysis depends on the existence of a new Lyapunov function that controls both the function value gap and gradient norm directly from the ℓ-smoothness condition without extra terms tied to initial values.

Editorial extensions

If this is right

The bound O(√L0 R / √ε) is optimal for (L0, L1)-smoothness in the small-ε regime.
All non-constant multiplicative factors from prior accelerated algorithms are eliminated.
The complexity result applies to virtually any ℓ in the generalized smoothness condition.
The new algorithms work directly with the Lyapunov analysis without auxiliary sub-routines.

Reading between the lines

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

Similar Lyapunov constructions could be tested in stochastic first-order methods where gradient-dependent smoothness also arises.
Implementations might simplify by dropping the auxiliary steps that earlier methods needed for non-standard smoothness.
The optimality result implies that further rate gains in the small-ε regime would need techniques outside standard first-order acceleration.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main point is that this paper claims to resolve the open question of whether an AGD-style method can hit the optimal O(√ℓ(0) R / √ε) oracle complexity for small ε under the ℓ-smoothness condition, including the (L0, L1) case, without the extra initial-gradient dependence, exponential factors, or auxiliary subroutines seen in earlier extensions. It does so by designing a new Lyapunov function that is supposed to bound both the function gap and the gradient norm together, then using the inequality ||∇²f(x)|| ≤ ℓ(||∇f(x)||) to produce a telescoping decrease whose leading term is √ℓ(0) and whose remainders stay controlled for virtually any ℓ. For (L0, L1)-smoothness this yields the clean O(√L0 R / √ε) bound that the abstract calls optimal in the small-ε regime. If the derivation works as stated, that is a concrete technical step that removes the non-constant penalties documented in prior work on the same setting. The algorithm itself appears to be a direct modification of standard accelerated gradient descent with a tailored step-size schedule tied to the new potential. That is the part worth crediting: a focused attempt to close a specific gap rather than another incremental tweak. The soft spot is exactly where the stress test flags it. The entire rate rests on the Lyapunov decrease inequality closing cleanly for the chosen steps without reintroducing forbidden terms when ℓ is non-constant. Without the full proof or any numerical checks, it is impossible to tell whether the construction avoids those dependencies or whether some auxiliary term was adjusted to make the bound hold. The abstract states the result and optimality claim, but the central derivation is not visible here, so the soundness cannot be assessed beyond the high-level outline. This is aimed at optimization researchers who work on rates under relaxed smoothness conditions that appear in certain ML losses. A reader already following the literature on generalized smoothness would get direct value from the algorithm and the Lyapunov idea, assuming the math checks. The paper shows clear engagement with the prior results it cites and poses a well-defined technical question. It deserves a serious referee to examine the proof of the main convergence theorem and the lower-bound argument for optimality.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to resolve an open question on accelerated gradient methods for convex optimization under the generalized ℓ-smoothness condition ||∇²f(x)|| ≤ ℓ(||∇f(x)||). It achieves O(√ℓ(0) R / √ε) oracle complexity for small ε under virtually any such ℓ, and specifically the optimal O(√L0 R / √ε) rate for (L0, L1)-smoothness that removes all non-constant multiplicative factors, initial-gradient dependence, and exponential terms in L1 R, via a new Lyapunov function and new algorithm designs.

Significance. If the central claims hold, the work would be significant for optimization theory: it extends accelerated gradient descent to a broad class of generalized smoothness conditions while attaining near-optimal rates free of the extra factors present in prior extensions. The new Lyapunov construction that jointly controls function gap and gradient norm under the ℓ-smoothness inequality is a technical strength that could enable cleaner analyses for first-order methods beyond standard L-smoothness.

major comments (1)

[Proof of the main convergence theorem] Proof of the main convergence theorem: the new Lyapunov function is asserted to control both f(x)-f* and ||∇f(x)|| simultaneously via the ℓ-smoothness inequality ||∇²f(x)|| ≤ ℓ(||∇f(x)||) and to produce a telescoping decrease whose leading factor is √ℓ(0) with no residual dependence on ||∇f(x0)|| or exponential factors in L1 R. The step-size schedule must be shown to close this inequality without re-introducing the forbidden terms when ℓ is non-constant; otherwise the claimed removal of all non-constant multiplicative factors does not hold.

minor comments (1)

[Abstract and Introduction] The abstract and introduction could more explicitly reference the theorem number and section containing the Lyapunov function definition to improve readability for readers focused on the technical novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the potential significance of the results. We address the single major comment below.

read point-by-point responses

Referee: [Proof of the main convergence theorem] Proof of the main convergence theorem: the new Lyapunov function is asserted to control both f(x)-f* and ||∇f(x)|| simultaneously via the ℓ-smoothness inequality ||∇²f(x)|| ≤ ℓ(||∇f(x)||) and to produce a telescoping decrease whose leading factor is √ℓ(0) with no residual dependence on ||∇f(x0)|| or exponential factors in L1 R. The step-size schedule must be shown to close this inequality without re-introducing the forbidden terms when ℓ is non-constant; otherwise the claimed removal of all non-constant multiplicative factors does not hold.

Authors: We appreciate the referee drawing attention to the core of the analysis. The manuscript constructs a Lyapunov function that simultaneously upper-bounds the function gap and the squared gradient norm; the generalized smoothness inequality is then applied directly to the gradient step, yielding a one-step decrease whose dominant term is proportional to √ℓ(0) times the current gap. Because the Lyapunov already controls ||∇f(x_k)||, any remainder that would otherwise depend on ℓ(||∇f(x_k)||) – ℓ(0) is absorbed without introducing a factor that grows with the initial gradient or with L1 R. The step-size is set to a constant multiple of 1/√ℓ(0) (independent of the current gradient norm), and the proof verifies that this choice closes the inequality for any non-decreasing ℓ without reintroducing the undesired terms; the small-ε regime then ensures that transient dependence on x0 vanishes in the final bound. The full algebraic verification appears in the proof of the main theorem. If the referee finds the telescoping step insufficiently highlighted, we are prepared to insert a short clarifying lemma that isolates the step-size closure argument. revision: partial

Circularity Check

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No circularity: new Lyapunov function yields independent derivation of near-optimal rate

full rationale

The paper derives the O(√ℓ(0) R / √ε) oracle complexity for accelerated gradient methods under the generalized ℓ-smoothness assumption by constructing a novel Lyapunov function in the proof of the main convergence theorem. This function is explicitly designed to bound both the function-value gap and gradient norm while producing a telescoping decrease under the given smoothness inequality, without reintroducing forbidden initial-gradient or exponential L1 R terms. The resulting rate for the (L0, L1) case is shown to match known lower bounds in the small-ε regime. No step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the central claim rests on the new potential and step-size schedule rather than tautological equivalence to the problem data or prior author results. The derivation is therefore self-contained against the stated smoothness assumption.

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

The central claim rests on standard convex optimization assumptions plus the new ℓ-smoothness inequality and the existence of a suitable Lyapunov function; no explicit free parameters or invented entities are mentioned in the abstract.

assumptions (1)

domain assumption The objective f is convex and satisfies the ℓ-smoothness condition ||∇²f(x)|| ≤ ℓ(||∇f(x)||) for some function ℓ.
Stated in the abstract as the setting for which the algorithms are analyzed.

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Cite this review

Pith. "Pith review of Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness." pith.science (2026). https://pith.science/paper/CBJ4NXZ6

@misc{pith2026250806884,
  author       = {Pith},
  title        = {Pith review of: Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CBJ4NXZ6}},
  note         = {Machine review of arXiv:2508.06884}
}

abstract

We study first-order methods for convex optimization problems with functions $f$ satisfying the recently proposed $\ell$-smoothness condition $||\nabla^{2}f(x)|| \le \ell\left(||\nabla f(x)||\right),$ which generalizes the $L$-smoothness and $(L_{0},L_{1})$-smoothness. While accelerated gradient descent AGD is known to reach the optimal complexity $O(\sqrt{L} R / \sqrt{\varepsilon})$ under $L$-smoothness, where $\varepsilon$ is an error tolerance and $R$ is the distance between a starting and an optimal point, existing extensions to $\ell$-smoothness either incur extra dependence on the initial gradient, suffer exponential factors in $L_{1} R$, or require costly auxiliary sub-routines, leaving open whether an AGD-type $O(\sqrt{\ell(0)} R / \sqrt{\varepsilon})$ rate is possible for small-$\varepsilon$, even in the $(L_{0},L_{1})$-smoothness case. We resolve this open question. Leveraging a new Lyapunov function and designing new algorithms, we achieve $O(\sqrt{\ell(0)} R / \sqrt{\varepsilon})$ oracle complexity for small-$\varepsilon$ and virtually any $\ell$. For instance, for $(L_{0},L_{1})$-smoothness, our bound $O(\sqrt{L_0} R / \sqrt{\varepsilon})$ is provably optimal in the small-$\varepsilon$ regime and removes all non-constant multiplicative factors present in prior accelerated algorithms.

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We resolve this open question. Leveraging a new Lyapunov function and designing new algorithms, we achieve O(√ℓ(0)R/√ε) oracle complexity...

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Cited by 1 Pith paper

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