Many problems in machine learning can be formulated as optimizing a convex functional over a space of measures. This paper studies the convergence of the mirror descent algorithm in this infinite-dimensional setting. Defining Bregman divergences through directional derivatives, we derive the convergence of the scheme for relatively smooth and strongly convex pairs of functionals. Applying our result to joint distributions and the Kullback-Leibler (KL) divergence, we show that Sinkhorn's primal i…