Sharper Exponential Convergence Rates for Sinkhorn’s Algorithm in Continuous Settings
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, µ, admits a
density over R
d
. For a semi-concave cost function bounded by c∞ and a regularization parameter λ > 0, we obtain exponential convergence guarantees on the dual sub-optimality
gap with contraction rate polynomial in λ/c∞. This represents an exponential improvement over the known contraction rate 1 − Θ(exp(−c∞/λ)) achievable via Hilbert’s projective metric. Specifically, we prove a contraction rate value of 1 − Θ(λ
2/c2
∞) when µ
has a bounded log-density. In some cases, such as when µ is log-concave and the cost
function is c(x, y) = −hx, yi, this rate improves to 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 λ/c∞. Our results are fully non-asymptotic and explicit
in all the parameters of the problem.
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