Year: 2025
Type: article
Abstract: We study the convergence rate of Sinkhorn’s algorithm for solving entropy-regularized optimal transport problems when at least one of the probability measures, $$\mu $$ μ , admits a density over $$\mathbb {R}^d$$ R d . For a semi-concave cost function bounded by $$c_{\infty }$$ c ∞ and a regularization parameter $$\lambda > 0$$ λ > 0 , we obtain exponential convergence guarantees on the dual sub-optimality gap with contraction rates that are polynomial in $$\lambda /c_{\infty }$$ λ / c ∞ . This represents an exponential improvement over the known contraction rate $$1 - \Theta (\exp (-c_{\infty }/\lambda ))$$ 1 - Θ ( exp ( - c ∞ / λ ) ) achievable via Hilbert’s projective metric. Specifically, we prove a contraction rate value of $$1-\Theta (\lambda ^2/c_\infty ^2)$$ 1 - Θ ( λ 2 / c ∞ 2 ) when $$\mu $$ μ has a bounded log-density. In some cases, such as when $$\mu $$ μ is log-concave and the cost function is $$c(x,y)=-\langle x, y\rangle $$ c ( x , y ) = - ⟨ x , y ⟩ , this rate improves to $$1-\Theta (\lambda /c_\infty )$$ 1 - Θ ( λ / c ∞ ) . The latter rate matches the one that we derive for the transport between isotropic Gaussian measures, indicating tightness in the dependency in $$\lambda /c_\infty $$ λ / c ∞ . Our results are fully non-asymptotic and explicit in all the parameters of the problem.
Source: Mathematical Programming
Institution: École Polytechnique Fédérale de Lausanne
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4411911684