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 iterations for entropic optimal transport in the continuous setting correspond to a mirror descent, and we obtain a new proof of its (sub)linear convergence. We also show that Expectation Maximization (EM) can always formally be written as a mirror descent, and, when optimizing on the latent distribution while fixing the mixtures, we derive sublinear rates of convergence.

翻译：机器学习中的许多问题可以被描述为优化一个连接功能, 而不是一个测量空间。 本文研究了镜像下沉算法在这一无限维度环境中的趋同性。 定义布雷格曼通过定向衍生物的差异, 我们从中得出相对顺畅和强烈的曲线功能对子的组合。 将我们的结果应用到联合分布和 Kullback- Leiber (KL) 差异中, 我们显示辛克霍恩在连续环境中对正向最佳迁移的原始迭代与镜影下沉相对应, 我们获得了其( 子线性) 趋同性的新证据。 我们还表明, 期望最大化( EM) 总是可以正式写成一个反向下沉, 当在修正混合物时优化潜在分布时, 我们得出子线性趋同率 。