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arxiv: 2605.14345 · v1 · submitted 2026-05-14 · 🧮 math.OC

Recognition: 3 theorem links

· Lean Theorem

Convergence of difference inclusions via a diameter criterion

Lexiao Lai , Mingzhi Song
Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:20 UTC · model grok-4.3

classification 🧮 math.OC
keywords diameterconvergencecriterionboundeddifferencediscreteframeworklimit
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The pith

A diameter criterion tied to a potential function certifies convergence of difference inclusions, enabling discrete proofs for first-order optimization methods with diminishing steps.

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

Consider a sequence of points updated by picking a direction from a set-valued map and adding noise. The work proves that if the gap between one point and the next stays small enough relative to how much a potential function varies, the sequence must approach a point where no improving direction exists in the limit map. To establish the needed diameter bound, the space is divided into layers called a stratification, and the potential is shown to decrease except for errors that sum to a finite total. This discrete argument covers inexact and noisy subgradient steps as well as momentum, provided the objective is locally Lipschitz and belongs to a broad class of functions definable in o-minimal structures. No continuous-time limit is used.

Core claim

Combining the diameter criterion with a diameter estimate obtained from this framework yields convergence of common first-order optimization methods under step sizes of order 1/k. The guarantees cover inexact and stochastic subgradient methods, as well as the momentum method, for locally Lipschitz objectives definable in polynomially bounded o-minimal structures.

Load-bearing premise

The objectives must be locally Lipschitz and definable in polynomially bounded o-minimal structures so that the stratified descent framework produces a summable error term in the potential decrease.

read the original abstract

We study discrete dynamics governed by a difference inclusion whose increment is the sum of a selection from a set-valued map and a noise term. For any bounded realization, convergence follows once the inter-iterate diameter is controlled by the variation of a continuous potential. The limit point is then critical for a scaled outer limit of the update map. To certify this diameter criterion, we develop a stratified descent framework: we project iterates onto a suitable stratification and track a potential that decreases up to a summable error. Combining the diameter criterion with a diameter estimate obtained from this framework yields convergence of common first-order optimization methods under step sizes of order $1/k$. The guarantees cover inexact and stochastic subgradient methods, as well as the momentum method, for locally Lipschitz objectives definable in polynomially bounded o-minimal structures. Our arguments are entirely discrete, with no appeal to continuous-time approximations.

Editorial analysis

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Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on domain assumptions about the function class and the existence of a suitable stratification; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption Objectives are locally Lipschitz and definable in polynomially bounded o-minimal structures
    Invoked to guarantee that the stratified descent produces a summable error in potential decrease.

pith-pipeline@v0.9.0 · 5440 in / 1170 out tokens · 109455 ms · 2026-05-15T02:20:51.712683+00:00 · methodology

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Reference graph

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