Nonmonotone subgradient methods based on a local descent lemma

David Torregrosa-Bel\'en; Francisco J. Arag\'on-Artacho; Pedro P\'erez-Aros; Rub\'en Campoy

arxiv: 2510.19341 · v2 · submitted 2025-10-22 · 🧮 math.OC · cs.LG

Nonmonotone subgradient methods based on a local descent lemma

Francisco J. Arag\'on-Artacho , Rub\'en Campoy , Pedro P\'erez-Aros , David Torregrosa-Bel\'en This is my paper

Pith reviewed 2026-05-18 05:04 UTC · model grok-4.3

classification 🧮 math.OC cs.LG

keywords nonmonotone line searchsubgradient methodsupper-C2 functionsconvergence analysisnonsmooth optimizationNewton methodsclustering problem

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The pith

A nonmonotone line search based on a local descent lemma makes subgradient methods converge on upper-C² functions.

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

This paper develops a nonmonotone line search for subgradient algorithms that works with upper-C² functions, a class of nonsmooth nonconvex functions satisfying a local descent property. The key result is subsequential convergence of the iterates to a stationary point. The framework is general enough to include Newton and quasi-Newton methods as instances. A self-adaptive version is proposed and tested on a clustering problem, where it shows practical benefits.

Core claim

We present a nonmonotone line search subgradient algorithm tailored to upper-C² functions. This family of functions satisfies a nonsmooth and local version of the descent lemma, making them suitable for line searches. We prove subsequential convergence of the proposed algorithm to a stationary point of the optimization problem. Our approach allows us to cover the setting of various subgradient algorithms, including Newton and quasi-Newton methods.

What carries the argument

The nonsmooth local descent lemma satisfied by upper-C² functions, which justifies the nonmonotone line search step.

If this is right

The convergence guarantee extends to Newton and quasi-Newton methods as special cases of the general scheme.
The self-adaptive SNSM version automatically updates the line search parameters without manual tuning.
The framework produces a concrete algorithm for the minimum sum-of-squares clustering problem.
Numerical tests indicate practical advantages over existing subgradient algorithms on clustering instances.

Where Pith is reading between the lines

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

The same local descent idea could support line searches for other nonsmooth function classes that obey similar inequalities.
The method might improve performance in machine learning tasks that reduce to nonconvex nonsmooth optimization.
Further work could test whether the subsequential convergence strengthens to full convergence under additional assumptions.

Load-bearing premise

The objective function must belong to the upper-C² class so that it satisfies the nonsmooth local descent lemma that enables the line search.

What would settle it

A concrete run of the algorithm on an upper-C² function that fails to approach any stationary point, or where the line search step cannot be completed, would disprove the convergence result.

read the original abstract

In this paper we present a nonmonotone line search subgradient algorithm tailored to upper-$\mathcal{C}^2$ functions. This is a family of nonsmooth and nonconvex functions that satisfies a nonsmooth and local version of the descent lemma, making them suitable for line searches. We prove subsequential convergence of the proposed algorithm to a stationary point of the optimization problem. Our approach allows us to cover the setting of various subgradient algorithms, including Newton and quasi-Newton methods. In addition, we propose a specification of the general scheme, named Self-adaptive Nonmonotone Subgradient Method (SNSM), which automatically updates the parameters of the line search. Particular attention is paid to the minimum sum-of-squares clustering problem, for which we provide a concrete implementation of SNSM. We conclude with some numerical experiments where we exhibit the advantages of SNSM in comparison with some known algorithms.

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.

Desk Editor's Note private letter to a colleague

The paper gives a unified nonmonotone line-search subgradient scheme for upper-C² functions with subsequential convergence and a self-adaptive version tested on clustering.

read the letter

The paper's main contribution is a nonmonotone line-search subgradient method built on a local descent lemma for upper-C² functions. It establishes subsequential convergence to stationary points and incorporates various subgradient algorithms, including Newton and quasi-Newton ones, under the same framework. A self-adaptive variant, SNSM, is proposed and applied to minimum sum-of-squares clustering with numerical tests. What works well is the adaptation of the line search to the nonsmooth local descent property. This allows the analysis to go through for a class of functions that are neither smooth nor convex in the usual sense. Unifying existing methods this way provides a cleaner picture than treating them separately. The automatic parameter tuning in SNSM is a practical addition, and the clustering example shows how to implement it specifically. The experiments demonstrate advantages over some standard algorithms on clustering problems. They use typical datasets, which is fine for validation but could be expanded to more challenging cases in a revision. A minor soft spot is that convergence is only subsequential. This is standard in the area, but it leaves the door open for further work on full sequence convergence or rates. The assumption that the function is upper-C² is necessary for the lemma, and the paper handles it directly without circularity. This is aimed at optimization specialists dealing with nonsmooth nonconvex problems. Readers interested in subgradient methods or clustering would find the unified analysis and implementation details valuable. The math and logic hold up based on the stated conditions. I recommend putting it through peer review. It is focused and the evidence supports the claims without major issues.

Referee Report

1 major / 3 minor

Summary. The manuscript introduces a nonmonotone line-search subgradient algorithm designed for upper-C² functions, which satisfy a local nonsmooth version of the descent lemma. It proves subsequential convergence of the generated sequence to a stationary point and shows that the framework encompasses several subgradient schemes, including Newton and quasi-Newton methods. A self-adaptive specialization called SNSM is proposed, with a concrete implementation for the minimum sum-of-squares clustering problem, followed by numerical comparisons against existing algorithms.

Significance. If the convergence analysis holds, the work supplies a coherent line-search framework for a practically relevant class of nonsmooth nonconvex functions and unifies several algorithmic families under a single set of assumptions. The self-adaptive parameter rule and the explicit clustering instantiation add immediate applicability. The numerical section, if the experimental protocol is reproducible, provides concrete evidence of practical advantage.

major comments (1)

§4, Theorem 4.1: the subsequential convergence argument invokes the local descent lemma to guarantee that accepted steps produce sufficient decrease; the proof sketch does not explicitly verify that the upper-C² modulus remains controlled along the sequence when the subgradient is replaced by a Newton or quasi-Newton direction, which is required for the claimed generality.

minor comments (3)

§2.1: the precise statement of the local descent lemma (inequality (2.3)) should be restated immediately before the line-search acceptance condition to improve readability.
Table 2: the clustering results report only final objective values; adding iteration counts and CPU times with standard deviations over 10 random initializations would strengthen the comparison.
§5.3: the update rule for the self-adaptive parameter β_k is given without a reference to the earlier general scheme; a short cross-reference would clarify the specialization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the proof of Theorem 4.1. We address the point below.

read point-by-point responses

Referee: [—] §4, Theorem 4.1: the subsequential convergence argument invokes the local descent lemma to guarantee that accepted steps produce sufficient decrease; the proof sketch does not explicitly verify that the upper-C² modulus remains controlled along the sequence when the subgradient is replaced by a Newton or quasi-Newton direction, which is required for the claimed generality.

Authors: We agree that an explicit verification would strengthen the presentation. The upper-C² property and its associated local descent lemma are intrinsic to the objective function f and hold in a neighborhood of any point, with the modulus depending only on local properties of f (independent of the search direction). In the proof of Theorem 4.1 the line search accepts a step only when the sufficient-decrease condition furnished by the lemma is satisfied; the argument then proceeds from the resulting decrease to show that every accumulation point is stationary. Because subsequential convergence is established, any accumulation point x* lies in a compact set on which f is upper-C², so the modulus can be taken uniform in a neighborhood of x*. The same reasoning applies verbatim when the direction is a Newton or quasi-Newton vector, provided only that the vector is admissible for the line search (i.e., the step-size selection remains well-defined). We will revise the proof to insert a short paragraph making this uniformity explicit and confirming that the argument does not rely on the direction being a subgradient beyond the decrease condition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under upper-C² hypothesis

full rationale

The paper introduces a nonmonotone line-search subgradient scheme for upper-C² functions, which by definition satisfy a local nonsmooth descent lemma. Subsequential convergence to a stationary point is proved directly from this property and standard line-search acceptance conditions, without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations whose validity reduces to the present work. The SNSM specialization and clustering instantiation are explicit constructions from the general framework. The argument remains internally consistent and independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central construction rests on the upper-C² property as the enabling assumption for the line search; no explicit free parameters or new invented entities are described in the abstract.

axioms (1)

domain assumption The objective function is upper-C² and therefore satisfies a nonsmooth local descent lemma.
This property is invoked to justify the line-search acceptance condition and the convergence analysis.

pith-pipeline@v0.9.0 · 5701 in / 1179 out tokens · 32604 ms · 2026-05-18T05:04:48.051894+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 3.2(iii): for each x̄ there exist κ≥0 and neighborhood V such that φ(y)≤φ(x)+⟨w,y-x⟩+κ‖y-x‖² for all y∈V, w∈∂φ(x)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4.5: subsequential convergence to stationary point under Assumptions 4.1 for upper-C² objective

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