We investigate proximal descent methods, inspired by the minimizing movement scheme introduced by Jordan, Kinderlehrer and Otto, for optimizing entropy-regularized functionals on the Wasserstein space. We establish linear convergence under flat convexity assumptions, thereby relaxing the common reliance on geodesic convexity. Our analysis circumvents the need for discrete-time adaptations of the Evolution Variational Inequality (EVI). Instead, we leverage a uniform logarithmic Sobolev inequality (LSI) and the entropy "sandwich" lemma, extending the analysis from arXiv:2201.10469 and arXiv:2202.01009. The major challenge in the proof via LSI is to show that the relative Fisher information $I(\cdot|\pi)$ is well-defined at every step of the scheme. Since the relative entropy is not Wasserstein differentiable, we prove that along the scheme the iterates belong to a certain class of Sobolev regularity, and hence the relative entropy $\operatorname{KL}(\cdot|\pi)$ has a unique Wasserstein sub-gradient, and that the relative Fisher information is indeed finite.

翻译：本文研究了受Jordan、Kinderlehrer和Otto提出的最小化运动方案启发的邻近下降方法，用于优化Wasserstein空间上的熵正则化泛函。我们在平坦凸性假设下建立了线性收敛性，从而放宽了对测地凸性的常见依赖。我们的分析规避了对演化变分不等式离散时间适应的需求，转而利用一致的对数Sobolev不等式和熵“三明治”引理，扩展了arXiv:2201.10469和arXiv:2202.01009中的分析。通过LSI证明的主要挑战在于表明相对Fisher信息$I(\cdot|\pi)$在方案的每一步都有良好定义。由于相对熵在Wasserstein意义下不可微，我们证明了在方案迭代过程中，迭代点属于特定的Sobolev正则类，因此相对熵$\operatorname{KL}(\cdot|\pi)$具有唯一的Wasserstein次梯度，且相对Fisher信息确实有限。