We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods. Our main application is to the problem of mean-field variational inference, which seeks to approximate a distribution $\pi$ over $\mathbb{R}^d$ by a product measure $\pi^\star$. When $\pi$ is strongly log-concave and log-smooth, we provide (1) approximation rates certifying that $\pi^\star$ is close to the minimizer $\pi^\star_\diamond$ of the KL divergence over a \emph{polyhedral} set $\mathcal{P}_\diamond$, and (2) an algorithm for minimizing $\text{KL}(\cdot\|\pi)$ over $\mathcal{P}_\diamond$ with accelerated complexity $O(\sqrt \kappa \log(\kappa d/\varepsilon^2))$, where $\kappa$ is the condition number of $\pi$.

翻译：我们发展了Wasserstein空间上有限维多面体子集的理论，并研究了通过一阶方法对其上的泛函进行优化。主要应用针对平均场变分推断问题，即通过乘积测度$\pi^\star$近似$\mathbb{R}^d$上的分布$\pi$。当$\pi$具有强对数凹性和对数光滑性时，我们提供了：(1) 逼近速率，证明了$\pi^\star$接近于多面体集合$\mathcal{P}_\diamond$上KL散度极小化器$\pi^\star_\diamond$的程度；(2) 在$\mathcal{P}_\diamond$上最小化$\text{KL}(\cdot\|\pi)$的算法，其加速复杂度为$O(\sqrt \kappa \log(\kappa d/\varepsilon^2))$，其中$\kappa$为$\pi$的条件数。