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arxiv: 2605.28134 · v1 · pith:BPHKZV6Nnew · submitted 2026-05-27 · 🧮 math.OC · stat.ML

Convergence of empirical subgradients for optimal transport-based objectives

Tam Le (LPSM , UPCit\'e) This is my paper

Pith reviewed 2026-06-29 11:16 UTC · model grok-4.3

classification 🧮 math.OC stat.ML
keywords optimal transportsubdifferentialsgraphical convergenceempirical convergencesubgradient methodsparameterized objectivessliced Wasserstein
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The pith

Sampled optimal transport objectives have subdifferentials that converge graphically to the population subdifferential.

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

The paper proves that objectives built from finite samples of optimal transport costs have subdifferentials that converge graphically to those of the corresponding population objectives. This convergence implies that subgradient methods run on the sampled problem will approach stationary points of the full population problem. The result relies on smooth parameterizations to maintain stability between statistical consistency and optimization. Illustrations cover risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems. Nonsmooth costs or models can produce derivatives that destabilize as the sample size grows.

Core claim

We study parameterized objectives defined by sampled transport costs and prove graphical convergence of their subdifferentials to the subdifferential of the population objective. In particular, this ensures that standard subgradient methods consistently approach stationary points of the population-level problem. The analysis is illustrated in risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems, with smooth parameterizations providing a stable interface between sampling and optimization.

What carries the argument

Graphical convergence of subdifferentials between empirical and population optimal transport-based objectives

If this is right

  • Subgradient methods applied to the sampled problem approach stationary points of the population objective.
  • Smooth parameterizations ensure stable derivatives in the large-sample limit.
  • The convergence result applies directly to risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems.
  • Nonsmooth costs and models can produce unstable derivatives as sample size increases.

Where Pith is reading between the lines

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

  • Empirical optimal transport losses can be treated as reliable proxies for population-level optimization when parameters remain smooth.
  • The same graphical-convergence approach might apply to other sampling-based losses if analogous technical conditions hold.
  • Training pipelines using transport costs may benefit from enforcing smoothness on the model class to avoid limit instability.

Load-bearing premise

Smooth parameterizations are needed to translate statistical consistency into stable optimization behavior without unstable derivatives in the large-sample limit.

What would settle it

An explicit example of a smooth parameterization and transport cost where the empirical subdifferential fails to converge graphically to the population subdifferential, or where subgradient iterates on growing samples diverge from the population stationary points.

Figures

Figures reproduced from arXiv: 2605.28134 by Tam Le (LPSM, UPCit\'e).

Figure 1
Figure 1. Figure 1: Sample-induced local minimum One-dimensional transport costs illustrate this phenomenon well as they admit a quantile representation [61, Chapter 2], which reduces to an explicit sorting-based formula for discrete measures. This links transport costs with ranks and quantiles and makes them particularly convenient in learning pipelines. Such tractability has been exploited for instance in his￾togram matchin… view at source ↗
Figure 2
Figure 2. Figure 2: The population objective is increasing, with subgradients in [ 1 4 , 1] while the range of empirical derivatives contains zero. Experiment. For the numerical illustrations, we take w = 3/4 and M = 6. In [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗
read the original abstract

Optimal transport is widely used to learn distributions, enforce distributional constraints, and model uncertainty. In applications, transport losses are often computed from samples through tractable representations, such as one-dimensional sorting formulas or sliced Wasserstein costs, making them practical components in training pipelines. We study parameterized objectives defined by sampled transport costs and prove graphical convergence of their subdifferentials to the subdifferential of the population objective. In particular, this ensures that standard subgradient methods consistently approach stationary points of the population-level problem. We illustrate the results in several settings, including risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems. Our analysis highlights that smooth parameterizations provide a favorable interface between statistical consistency and optimization. By contrast, transport objectives with nonsmooth costs and models may exhibit unstable derivatives in the large-sample limit.

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 proves graphical convergence of the subdifferentials of empirical optimal transport (OT) objectives—defined via sampled transport costs such as one-dimensional sorting or sliced Wasserstein—to the subdifferential of the corresponding population objective. This convergence is shown to ensure that standard subgradient methods applied to the empirical problems consistently approach stationary points of the population problem. The results are illustrated in risk-averse optimization, fairness-constrained learning, and sliced Wasserstein settings, with emphasis on the favorable role of smooth parameterizations versus potential instability in nonsmooth cases.

Significance. If the graphical convergence result holds under the stated conditions, the work supplies a useful theoretical bridge between statistical consistency of empirical OT losses and the reliability of first-order optimization methods. This is relevant for machine learning pipelines that incorporate transport-based objectives, and the explicit contrast between smooth and nonsmooth regimes offers practical guidance on when subgradient consistency can be expected.

major comments (2)
  1. [Main theorem / assumptions paragraph] The central graphical convergence claim (abstract and main theorem) relies on technical conditions on the transport cost and parameterization class that are invoked but whose precise statement and necessity are not fully detailed in the provided abstract; the main result section should explicitly list all assumptions (e.g., on smoothness, compactness, or measurability) and verify they are minimal for the conclusion.
  2. [Section on illustrations] The illustrations (risk-averse optimization, fairness, sliced Wasserstein) are presented as supporting examples, but without quantitative verification that the empirical subdifferentials indeed converge in the reported regimes, it is unclear whether the examples confirm the rate or only the qualitative behavior; a numerical check or explicit error bound would strengthen the claim.
minor comments (2)
  1. Notation for the empirical versus population subdifferentials should be introduced once and used consistently; occasional shifts between ∂ and ∂_emp notation reduce readability.
  2. [Abstract / conclusion] The abstract states that nonsmooth costs 'may exhibit unstable derivatives in the large-sample limit,' but this is not accompanied by a counter-example or reference; adding a brief remark or citation would clarify the contrast.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will incorporate the suggested clarifications in a revised version of the manuscript.

read point-by-point responses

  1. Referee: [Main theorem / assumptions paragraph] The central graphical convergence claim (abstract and main theorem) relies on technical conditions on the transport cost and parameterization class that are invoked but whose precise statement and necessity are not fully detailed in the provided abstract; the main result section should explicitly list all assumptions (e.g., on smoothness, compactness, or measurability) and verify they are minimal for the conclusion.

    Authors: We agree that the assumptions should be stated more explicitly for clarity. In the revised manuscript we will insert a dedicated 'Assumptions' paragraph immediately preceding the statement of the main graphical convergence theorem. This paragraph will enumerate all conditions on the transport cost (continuity, growth, and measurability requirements) and on the parameterization class (compactness of the parameter domain and appropriate measurability of the maps). We will also add a short remark discussing the role of each assumption in the proof and note which ones are standard versus those that are tailored to the OT setting. revision: yes

  2. Referee: [Section on illustrations] The illustrations (risk-averse optimization, fairness, sliced Wasserstein) are presented as supporting examples, but without quantitative verification that the empirical subdifferentials indeed converge in the reported regimes, it is unclear whether the examples confirm the rate or only the qualitative behavior; a numerical check or explicit error bound would strengthen the claim.

    Authors: The illustrations are designed to highlight qualitative distinctions between smooth and nonsmooth regimes that follow from the theory, rather than to provide rate information. We acknowledge that a quantitative check would make the examples more convincing. In the revision we will add, in the sliced Wasserstein subsection, a small numerical study that tracks the distance between empirical and population subdifferentials (or a proxy such as the norm of the difference in subgradient evaluations) across increasing sample sizes, thereby supplying concrete evidence of the convergence behavior in at least one setting. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a mathematical proof of graphical convergence of subdifferentials for empirical OT-based objectives to the population subdifferential. The derivation relies on standard variational analysis tools and assumptions on smooth parameterizations, without reducing any central claim to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The result is framed as an independent convergence theorem that applies to the stated regimes (risk-averse optimization, fairness, sliced Wasserstein) and explicitly contrasts with nonsmooth cases, remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, invented entities, or ad-hoc axioms; the result rests on standard background from optimal transport and variational analysis.

axioms (1)
  • standard math Standard properties of subdifferentials and graphical convergence from variational analysis
    Invoked to establish the main convergence statement.

pith-pipeline@v0.9.1-grok · 5664 in / 1254 out tokens · 31139 ms · 2026-06-29T11:16:17.714422+00:00 · methodology

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