On the Convergence of Gradient Extrapolation Methods for Unbalanced Optimal Transport

We study the Unbalanced Optimal Transport (UOT) between two measures of possibly different masses with at most $n$ components, where marginal constraints of the standard Optimal Transport (OT) are relaxed via Kullback-Leibler divergence with regularization factor $\tau$. We propose a novel algorithm based on Gradient Extrapolation Method (GEM-UOT) to find an $\varepsilon$-approximate solution to the UOT problem in $O\big( \kappa n^2 \log\big(\frac{\tau n}{\varepsilon}\big) \big)$, where $\kappa$ is the condition number depending on only the two input measures. Compared to the only known complexity ${O}\big(\tfrac{\tau n^2 \log(n)}{\varepsilon} \log\big(\tfrac{\log(n)}{{\varepsilon}}\big)\big)$ for solving the UOT problem via the Sinkhorn algorithm, ours is better in $\varepsilon$ and lifts Sinkhorn's linear dependence on $\tau$, which hindered its practicality to approximate the standard OT via UOT. Our proof technique is based on a novel dual formulation of the squared $\ell_2$-norm regularized UOT objective, which is of independent interest and also leads to a new characterization of approximation error between UOT and OT in terms of both the transportation plan and transport distance. To this end, we further present an algorithm, based on GEM-UOT with fine tuned $\tau$ and a post-process projection step, to find an $\varepsilon$-approximate solution to the standard OT problem in $O\big( \kappa n^2 \log\big(\frac{ n}{\varepsilon}\big) \big)$, which is a new complexity in the literature of OT. Extensive experiments on synthetic and real datasets validate our theories and demonstrate the favorable performance of our methods in practice.