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arxiv: 2511.06868 · v3 · submitted 2025-11-10 · 🧮 math.OC · math.DS

On the diameter of subgradient sequences in o-minimal structures

Lexiao Lai , Mingzhi Song This is my paper

Pith reviewed 2026-05-18 00:09 UTC · model grok-4.3

classification 🧮 math.OC math.DS
keywords subgradient sequenceso-minimal structuresconvergence analysisdiameter boundstep sizesLipschitz stratificationsnonsmooth optimization
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The pith

Subgradient sequences for locally Lipschitz o-minimal functions have diameters controlled by function value variation plus double sums of step sizes.

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

The paper examines subgradient sequences generated by locally Lipschitz functions that are definable in polynomially bounded o-minimal structures. It derives a bound on the diameter of any such sequence in terms of the total variation of the function values, where the remaining discrepancy is governed by a double summation of the chosen step sizes. From this diameter relation the authors conclude that any bounded sequence converges whenever the step sizes are of order 1/k. The argument proceeds by invoking Lipschitz L-regular stratifications to decompose the domain and project the sequence onto successive strata where the subgradient behavior simplifies. Readers working on nonsmooth optimization may value the result because it supplies a convergence guarantee under definability assumptions rather than classical smoothness or convexity.

Core claim

We show that the diameter of any subgradient sequence is related to the variation in function values, with error terms dominated by a double summation of step sizes. Consequently, we prove that bounded subgradient sequences converge if the step sizes are of order 1/k. The proof uses Lipschitz L-regular stratifications in o-minimal structures to analyze subgradient sequences via their projections onto different strata.

What carries the argument

Lipschitz L-regular stratifications that decompose the domain and allow projection of the sequence onto individual strata for local analysis.

If this is right

  • Bounded subgradient sequences converge when the step sizes decrease as order 1/k.
  • The diameter of the sequence remains controlled by the net change in function values along the sequence.
  • Error contributions to the diameter bound are governed by double summations of the step sizes.

Where Pith is reading between the lines

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

  • The diameter bound may supply a template for proving convergence of related first-order methods under the same definability hypothesis.
  • Relaxing the polynomially bounded condition while retaining o-minimality could be tested by constructing explicit examples outside the current setting.

Load-bearing premise

The functions are locally Lipschitz and definable in a polynomially bounded o-minimal structure so that the required stratifications exist.

What would settle it

A locally Lipschitz function definable in a polynomially bounded o-minimal structure together with a bounded subgradient sequence using steps of size 1/k that fails to converge would refute the convergence claim.

Figures

Figures reproduced from arXiv: 2511.06868 by Lexiao Lai, Mingzhi Song.

Figure 1
Figure 1. Figure 1: Constructed regions N0(i, α) for three adjacent strata. Proof. We will first construct the constants ci , βi , γi recursively for each stratum Mi so that Mi ⊂ Mx for all x ∈ ∪α∈(0,1]N0(i, α) where the projection PMi is L-Lipschitz continuous and C 2 , and then establish the desired estimates of Lipschitz constants (i.e., inequalities (9)-(11)) towards the end of the proof. We start with the strata of the l… view at source ↗
read the original abstract

We study subgradient sequences of locally Lipschitz functions definable in a polynomially bounded o-minimal structure. We show that the diameter of any subgradient sequence is related to the variation in function values, with error terms dominated by a double summation of step sizes. Consequently, we prove that bounded subgradient sequences converge if the step sizes are of order $1/k$. The proof uses Lipschitz $L$-regular stratifications in o-minimal structures to analyze subgradient sequences via their projections onto different strata.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript studies subgradient sequences for locally Lipschitz functions definable in polynomially bounded o-minimal structures. It claims that the diameter of any such sequence is controlled by the variation in function values plus error terms dominated by a double summation of step sizes. The proof proceeds by invoking Lipschitz L-regular stratifications to project the sequence onto strata and transfer the bound; as a consequence, bounded sequences are shown to converge when the step sizes are of order 1/k.

Significance. If the diameter bound and its transfer via projections hold, the work supplies a geometric convergence tool for nonsmooth subgradient methods in the definable setting, extending classical results beyond convex or smooth cases. The explicit reliance on o-minimal stratifications is a methodological strength that could support further results on tame optimization.

major comments (2)
  1. [Proof of the main theorem (stratification decomposition)] The projection argument (abstract and proof of the main diameter theorem): the original g_k lies in ∂f(x_k), but after projection onto a stratum S the sequence satisfies a subgradient relation only for the restriction f|S. It is not shown that the normal components are absorbed into the double-sum error term without additional constants that may accumulate across the finite number of strata transitions; this step is load-bearing for the claimed diameter bound.
  2. [Derivation of the diameter bound] Error-term domination (statement following the diameter relation): the claim that projection-induced errors are dominated by the double summation of step sizes requires an explicit estimate showing uniformity with respect to the Lipschitz constants of the strata and the number of transitions; without this, the subsequent convergence statement for 1/k steps may fail to hold uniformly.
minor comments (2)
  1. [Abstract] The abstract and introduction could state the precise polynomial boundedness assumption on the o-minimal structure earlier, as it is used to guarantee the existence of the Lipschitz L-regular stratification.
  2. [Main theorem statement] Notation for the double summation of step sizes is introduced late; defining it explicitly in the statement of the main theorem would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each major comment below and have made revisions to strengthen the proof as suggested.

read point-by-point responses

  1. Referee: [Proof of the main theorem (stratification decomposition)] The projection argument (abstract and proof of the main diameter theorem): the original g_k lies in ∂f(x_k), but after projection onto a stratum S the sequence satisfies a subgradient relation only for the restriction f|S. It is not shown that the normal components are absorbed into the double-sum error term without additional constants that may accumulate across the finite number of strata transitions; this step is load-bearing for the claimed diameter bound.

    Authors: We appreciate this observation. The proof relies on the properties of Lipschitz L-regular stratifications in polynomially bounded o-minimal structures, where the finite stratification ensures only finitely many transitions occur. The normal components are controlled by the local Lipschitz constant of f, and due to the definability, these constants are uniform across the strata. We have added a new lemma in the revised manuscript that explicitly bounds the accumulation of these constants by a factor depending only on the number of strata and the global Lipschitz constant, which is then absorbed into the double-sum error term without affecting the overall bound. revision: yes

  2. Referee: [Derivation of the diameter bound] Error-term domination (statement following the diameter relation): the claim that projection-induced errors are dominated by the double summation of step sizes requires an explicit estimate showing uniformity with respect to the Lipschitz constants of the strata and the number of transitions; without this, the subsequent convergence statement for 1/k steps may fail to hold uniformly.

    Authors: The referee correctly identifies the need for uniformity in the error estimate. In the original manuscript, the domination is argued using the o-minimal tameness, but we agree that an explicit estimate enhances clarity. We have included in the revision a detailed calculation showing that the projection errors are bounded by K * sum_{i=1}^k sum_{j=1}^i alpha_i alpha_j, where K is a constant depending on the maximum Lipschitz constant over the finite strata and the maximum number of transitions (which is bounded by the number of strata). This ensures the convergence for step sizes of order 1/k holds uniformly. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external o-minimal stratifications

full rationale

The paper derives the diameter bound for subgradient sequences and the consequent convergence for 1/k step sizes by invoking Lipschitz L-regular stratifications of definable functions in polynomially bounded o-minimal structures. These stratifications and their projection properties are standard external results from o-minimal geometry, not defined, fitted, or assumed within the paper. The analysis decomposes the sequence across strata, relates diameter to function-value variation plus a double sum of step sizes, and transfers the subgradient relation via the restriction to each stratum; none of these steps reduce by the paper's own equations to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation. The central claims therefore remain independent of the paper's inputs and are supported by external mathematical facts.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the definability in o-minimal structures and the applicability of stratification techniques, which are standard in the field but specific assumptions here.

axioms (2)
  • domain assumption The functions under study are locally Lipschitz and definable in a polynomially bounded o-minimal structure.
    This is the primary setting stated in the abstract for which the results hold.
  • domain assumption Lipschitz L-regular stratifications exist and can be used to analyze the sequences via projections onto strata.
    The proof relies on this technique from o-minimal geometry.

pith-pipeline@v0.9.0 · 5370 in / 1284 out tokens · 57349 ms · 2026-05-18T00:09:51.316603+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On convergence rates of subgradient descent on semialgebraic functions

    math.OC 2026-04 unverdicted novelty 8.0

    Under Lipschitz stratification assumptions that hold automatically for semialgebraic functions, constant-step subgradient descent achieves explicit rates that improve with fewer strata and recover smooth-case rates up...

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Works this paper leans on

3 extracted references · 3 canonical work pages · cited by 1 Pith paper

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