论文标题
与对数差异的形式下降
Conformal Mirror Descent with Logarithmic Divergences
论文作者
论文摘要
对数差异是由最佳运输和广义凸双重性激励的布雷格曼差异的扩展，并且满足了许多显着的特性。使用对数差异引起的几何形状，我们引入了连续的时光下降的概括，即我们称为保形镜下降。我们在广义镜像下得出其动力学，并表明这是相应的Hessian梯度流的时间变化。我们还证明会聚导致连续的时间。我们将共形的镜像下降应用于广义指数家族的在线估计，并通过Dirichlet最佳运输问题在单元单元上构建一个梯度流。
The logarithmic divergence is an extension of the Bregman divergence motivated by optimal transport and a generalized convex duality, and satisfies many remarkable properties. Using the geometry induced by the logarithmic divergence, we introduce a generalization of continuous time mirror descent that we term the conformal mirror descent. We derive its dynamics under a generalized mirror map, and show that it is a time change of a corresponding Hessian gradient flow. We also prove convergence results in continuous time. We apply the conformal mirror descent to online estimation of a generalized exponential family, and construct a family of gradient flows on the unit simplex via the Dirichlet optimal transport problem.
