Convergence of empirical subgradients for optimal transport-based objectives

• URL: http://arxiv.org/abs/2605.28134v1
• Date: Wed, 27 May 2026 08:20:24 GMT
• Title: Convergence of empirical subgradients for optimal transport-based objectives
• Abstract summary: We study parameterized objectives defined by sampled transport costs and prove graphical convergence of their subdifferentials to the subdifferential of the population objective.<n>We illustrate the results in several settings, including risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems.
• Score: 13.295141528871966
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Coupled Optimal Transport with Landmark Constraints [64.98367856457412]
  An optimal transport (OT) model seeks a map or plan between distributions by minimizing a prescribed transport cost or distortion.<n>This paper proposes a novel coupled OT framework that leverages a small number of annotated landmarks to guide the recovery of an underlying deformation.<n>The proposed framework provides a principled connection between landmark-based registration and transport-based distribution matching.
  arXiv  Detail & Related papers  (2026-08-20T08:28:13Z)
• Variational Entropic Optimal Transport [67.76725267984578]
  We propose Variational Entropic Optimal Transport (VarEOT) for domain translation problems.<n>VarEOT is based on an exact variational reformulation of the log-partition $log mathbbE[exp(cdot)$ as a tractable generalization over an auxiliary positive normalizer.<n> Experiments on synthetic data and unpaired image-to-image translation demonstrate competitive or improved translation quality.
  arXiv  Detail & Related papers  (2026-02-02T15:48:44Z)
• HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization [4.249024052507976]
  We introduce a novel framework that reformulates hard-constrained sampling as a trajectory optimization problem.<n>Our key insight is to leverage numerical optimal control to steer the sampling trajectory so that constraints are satisfied precisely at the terminal time.<n>Our algorithm, which we name $textitHardFlow$, substantially outperforms existing methods in both constraint satisfaction and sample quality.
  arXiv  Detail & Related papers  (2025-11-11T16:33:57Z)
• On the Optimal Construction of Unbiased Gradient Estimators for Zeroth-Order Optimization [57.179679246370114]
  A potential limitation of existing methods is the bias inherent in most perturbation estimators unless a stepsize is proposed.<n>We propose a novel family of unbiased gradient scaling estimators that eliminate bias while maintaining favorable construction.
  arXiv  Detail & Related papers  (2025-10-22T18:25:43Z)
• Stochastic Optimization with Optimal Importance Sampling [49.484190237840714]
  We propose an iterative-based algorithm that jointly updates the decision and the IS distribution without requiring time-scale separation between the two.<n>Our method achieves the lowest possible variable variance and guarantees global convergence under convexity of the objective and mild assumptions on the IS distribution family.
  arXiv  Detail & Related papers  (2025-04-04T16:10:18Z)
• Generalization Bounds of Surrogate Policies for Combinatorial Optimization Problems [53.03951222945921]
  We analyze smoothed (perturbed) policies, adding controlled random perturbations to the direction used by the linear oracle.<n>Our main contribution is a generalization bound that decomposes the excess risk into perturbation bias, statistical estimation error, and optimization error.<n>We illustrate the scope of the results on applications such as vehicle scheduling, highlighting how smoothing enables both tractable training and controlled generalization.
  arXiv  Detail & Related papers  (2024-07-24T12:00:30Z)
• Dynamical Measure Transport and Neural PDE Solvers for Sampling [77.38204731939273]
  We tackle the task of sampling from a probability density as transporting a tractable density function to the target. We employ physics-informed neural networks (PINNs) to approximate the respective partial differential equations (PDEs) solutions. PINNs allow for simulation- and discretization-free optimization and can be trained very efficiently.
  arXiv  Detail & Related papers  (2024-07-10T17:39:50Z)
• Diffusion Models as Constrained Samplers for Optimization with Unknown Constraints [55.39203337683045]
  We propose to perform optimization within the data manifold using diffusion models.<n>Depending on the differentiability of the objective function, we propose two different sampling methods.<n>Our method achieves better or comparable performance with previous state-of-the-art baselines.
  arXiv  Detail & Related papers  (2024-02-28T03:09:12Z)
• Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization [7.841376286262178]
  We study optimal transport-based distributionally robust optimization problems where a fictitious adversary, often envisioned as nature, can choose the distribution of the uncertain problem parameters by a prescribed reference distribution at a finite transportation cost.
  arXiv  Detail & Related papers  (2023-03-07T13:52:32Z)
• On the Existence of Optimal Transport Gradient for Learning Generative Models [8.602553195689513]
  Training of Wasserstein Generative Adversarial Networks (WGAN) relies on the calculation of the gradient of the optimal transport cost. We first demonstrate that such gradient may not be defined, which can result in numerical instabilities during gradient-based optimization. By exploiting the discrete nature of empirical data, we formulate the gradient in a semi-discrete setting and propose an algorithm for the optimization of the generative model parameters.
  arXiv  Detail & Related papers  (2021-02-10T16:28:20Z)
• Statistical Optimal Transport posed as Learning Kernel Embedding [0.0]
  This work takes the novel approach of posing statistical Optimal Transport (OT) as that of learning the transport plan's kernel mean embedding from sample based estimates of marginal embeddings. A key result is that, under very mild conditions, $epsilon$-optimal recovery of the transport plan as well as the Barycentric-projection based transport map is possible with a sample complexity that is completely dimension-free.
  arXiv  Detail & Related papers  (2020-02-08T14:58:53Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.