Polyak-Lojasiewicz Inequality for Quadratically Regularized Optimal Transport

Abstract: Quadratically regularized optimal transport (QOT) is an alternative to entropic regularization that yields sparse couplings and avoids numerical instabilities due to exponential scaling. From an optimization viewpoint, the dual QOT objective is concave but features a positive part function which prevents strong concavity and reduces smoothness of optimizers. Consequently, standard arguments for linear convergence of algorithms do not apply. In this paper, we nevertheless establish a quantitative curvature property for the QOT dual. Under mild assumptions covering both continuous and semi-discrete transport problems, we prove a local error bound and a Polyak-Lojasiewicz (PL) inequality, with explicit constants depending only on the problem primitives. These results are obtained by functional-analytic techniques exploiting that near the optimum, the argument of the positive part function is positive on the interior of the support of the optimal coupling. As applications, we derive linear convergence of the gradient ascent, coordinate ascent, and coordinate gradient ascent algorithms on the dual problem, with explicit contraction rates.