Lower Complexity Adaptation for Empirical Entropic Optimal Transport

• URL: http://arxiv.org/abs/2306.13580v4
• Date: Fri, 18 Oct 2024 14:30:19 GMT
• Title: Lower Complexity Adaptation for Empirical Entropic Optimal Transport
• Authors: Michel Groppe, Shayan Hundrieser,
• Abstract summary: Entropic optimal transport (EOT) presents an effective and computationally viable alternative to unregularized optimal transport (OT) We derive novel statistical bounds for empirical plug-in estimators of the EOT cost. Our techniques employ empirical process theory and rely on a dual formulation of EOT over a single function class.
• Score: 0.0
• License:
• Abstract: Entropic optimal transport (EOT) presents an effective and computationally viable alternative to unregularized optimal transport (OT), offering diverse applications for large-scale data analysis. In this work, we derive novel statistical bounds for empirical plug-in estimators of the EOT cost and show that their statistical performance in the entropy regularization parameter $\epsilon$ and the sample size $n$ only depends on the simpler of the two probability measures. For instance, under sufficiently smooth costs this yields the parametric rate $n^{-1/2}$ with factor $\epsilon^{-d/2}$, where $d$ is the minimum dimension of the two population measures. This confirms that empirical EOT also adheres to the lower complexity adaptation principle, a hallmark feature only recently identified for unregularized OT. As a consequence of our theory, we show that the empirical entropic Gromov-Wasserstein distance and its unregularized version for measures on Euclidean spaces also obey this principle. Additionally, we comment on computational aspects and complement our findings with Monte Carlo simulations. Our techniques employ empirical process theory and rely on a dual formulation of EOT over a single function class. Crucial to our analysis is the observation that the entropic cost-transformation of a function class does not increase its uniform metric entropy by much.

Related papers

• Online non-parametric likelihood-ratio estimation by Pearson-divergence functional minimization [55.98760097296213]
  We introduce a new framework for online non-parametric LRE (OLRE) for the setting where pairs of iid observations $(x_t sim p, x'_t sim q)$ are observed over time. We provide theoretical guarantees for the performance of the OLRE method along with empirical validation in synthetic experiments.
  arXiv  Detail & Related papers  (2023-11-03T13:20:11Z)
• A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport [92.96250725599958]
  Kernel-based optimal transport (OT) estimators offer an alternative, functional estimation procedure to address OT problems from samples. We show that our SSN method achieves a global convergence rate of $O (1/sqrtk)$, and a local quadratic convergence rate under standard regularity conditions.
  arXiv  Detail & Related papers  (2023-10-21T18:48:45Z)
• Projected Langevin dynamics and a gradient flow for entropic optimal transport [0.8057006406834466]
  We introduce analogous diffusion dynamics that sample from an entropy-regularized optimal transport. By studying the induced Wasserstein geometry of the submanifold $Pi(mu,nu)$, we argue that the SDE can be viewed as a Wasserstein gradient flow on this space of couplings.
  arXiv  Detail & Related papers  (2023-09-15T17:55:56Z)
• Fast Rates for Maximum Entropy Exploration [52.946307632704645]
  We address the challenge of exploration in reinforcement learning (RL) when the agent operates in an unknown environment with sparse or no rewards. We study the maximum entropy exploration problem two different types. For visitation entropy, we propose a game-theoretic algorithm that has $widetildemathcalO(H3S2A/varepsilon2)$ sample complexity. For the trajectory entropy, we propose a simple algorithm that has a sample of complexity of order $widetildemathcalO(mathrmpoly(S,
  arXiv  Detail & Related papers  (2023-03-14T16:51:14Z)
• Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency [53.90687548731265]
  We study optimal procedures for estimating a linear functional based on observational data. For any convex and symmetric function class $mathcalF$, we derive a non-asymptotic local minimax bound on the mean-squared error.
  arXiv  Detail & Related papers  (2023-01-16T02:57:37Z)
• KL-Entropy-Regularized RL with a Generative Model is Minimax Optimal [70.15267479220691]
  We consider and analyze the sample complexity of model reinforcement learning with a generative variance-free model. Our analysis shows that it is nearly minimax-optimal for finding an $varepsilon$-optimal policy when $varepsilon$ is sufficiently small.
  arXiv  Detail & Related papers  (2022-05-27T19:39:24Z)
• Single Trajectory Nonparametric Learning of Nonlinear Dynamics [8.438421942654292]
  Given a single trajectory of a dynamical system, we analyze the performance of the nonparametric least squares estimator (LSE) We leverage recently developed information-theoretic methods to establish the optimality of the LSE for non hypotheses classes. We specialize our results to a number of scenarios of practical interest, such as Lipschitz dynamics, generalized linear models, and dynamics described by functions in certain classes of Reproducing Kernel Hilbert Spaces (RKHS)
  arXiv  Detail & Related papers  (2022-02-16T19:38:54Z)
• Probabilistic Inverse Optimal Transport [11.425633112192521]
  Optimal transport (OT) formalizes the problem of finding an optimal coupling between probability measures given a cost matrix. The inverse problem of inferring the cost given a coupling is Inverse Optimal Transport (IOT) We formalize and systematically analyze the properties of IOT using tools from the study of entropy-regularized OT.
  arXiv  Detail & Related papers  (2021-12-17T20:33:27Z)
• On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification [101.0377583883137]
  Projection robust (PR) OT seeks to maximize the OT cost between two measures by choosing a $k$-dimensional subspace onto which they can be projected. Our first contribution is to establish several fundamental statistical properties of PR Wasserstein distances. Next, we propose the integral PR Wasserstein (IPRW) distance as an alternative to the PRW distance, by averaging rather than optimizing on subspaces.
  arXiv  Detail & Related papers  (2020-06-22T14:35:33Z)
• A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-$\ell_1$-Norm Interpolated Classifiers [3.167685495996986]
  This paper establishes a precise high-dimensional theory for boosting on separable data. Under a class of statistical models, we provide an exact analysis of the universality error of boosting. We also explicitly pin down the relation between the boosting test error and the optimal Bayes error.
  arXiv  Detail & Related papers  (2020-02-05T00:24:53Z)

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.