Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy
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| Format: | Preprint |
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2023
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| author | Malamut, Hugo Sylvestre, Maxime |
| author_facet | Malamut, Hugo Sylvestre, Maxime |
| contents | We study the convergence of the transport plans $γ_ε$ towards $γ_0$ as well as the cost of the entropy-regularized optimal transport $(c,γ_ε)$ towards $(c,γ_0)$ as the regularization parameter $ε$ vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance $W_2(γ_ε,γ_0)$ is asymptotically greater than $C\sqrtε$ and the suboptimality $(c,γ_ε)-(c,γ_0)$ is of order $ε$. In the quadratic cost case the compactness assumption is relaxed into a moment of order $2+δ$ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance $W_2(γ_ε,γ_0)$ converges to $0$ at rate $\sqrtε$. Finally, if in addition the marginals have finite Fisher information, we prove $(c,γ_ε)-(c,γ_0) \sim dε/2$ and we provide a companion expansion of $H(γ_ε)$. These results are achieved by disentangling the role of the cost and the entropy in the regularized problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_06940 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy Malamut, Hugo Sylvestre, Maxime Optimization and Control 49Q22, 94A17, 49K40 We study the convergence of the transport plans $γ_ε$ towards $γ_0$ as well as the cost of the entropy-regularized optimal transport $(c,γ_ε)$ towards $(c,γ_0)$ as the regularization parameter $ε$ vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance $W_2(γ_ε,γ_0)$ is asymptotically greater than $C\sqrtε$ and the suboptimality $(c,γ_ε)-(c,γ_0)$ is of order $ε$. In the quadratic cost case the compactness assumption is relaxed into a moment of order $2+δ$ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance $W_2(γ_ε,γ_0)$ converges to $0$ at rate $\sqrtε$. Finally, if in addition the marginals have finite Fisher information, we prove $(c,γ_ε)-(c,γ_0) \sim dε/2$ and we provide a companion expansion of $H(γ_ε)$. These results are achieved by disentangling the role of the cost and the entropy in the regularized problem. |
| title | Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy |
| topic | Optimization and Control 49Q22, 94A17, 49K40 |
| url | https://arxiv.org/abs/2306.06940 |