变分传输：一种用于分布优化的收敛粒子算法
Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization
AI总结：
本文提出变分传输算法，通过推动粒子近似执行 Wasserstein 梯度下降来求解分布优化问题，并在 PL 条件下证明其线性收敛且统计误差随粒子数增加次线性衰减。
摘要：
我们考虑在一族概率分布上最小化一个泛函的优化问题，其中目标泛函被假定具有变分形式。这类分布优化问题广泛出现在机器学习和统计学中，例如蒙特卡洛采样、变分推断、策略优化和生成对抗网络。针对该问题，我们提出一种新的基于粒子的算法，称为变分传输（variational transport），它通过迭代地推动一组粒子，在概率分布流形上近似执行 Wasserstein 梯度下降。具体而言，我们证明沿着关于二阶 Wasserstein 距离的泛函梯度方向沿测地线移动，等价于对概率分布应用一个前推映射，而这种映射可以通过推动一组粒子来准确近似。具体来说，在变分传输的每次迭代中，我们首先利用粒子求解与目标泛函相关的变分问题，其解给出 Wasserstein 梯度方向。然后，我们通过沿着该解所指定的方向推动每个粒子来更新当前分布。通过刻画估计 Wasserstein 梯度时产生的统计误差以及优化算法的进展，我们证明当目标函数满足 Polyak-Łojasiewicz (PL)（Polyak, 1963）条件和光滑性条件的泛函版本时，变分传输线性收敛到目标泛函的全局最小值，且存在一定的统计误差，该误差随着粒子数趋于无穷而以次线性速度衰减到零。
英文摘要：
We consider the optimization problem of minimizing a functional defined over a family of probability distributions, where the objective functional is assumed to possess a variational form. Such a distributional optimization problem arises widely in machine learning and statistics, with Monte-Carlo sampling, variational inference, policy optimization, and generative adversarial network as examples. For this problem, we propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent over the manifold of probability distributions via iteratively pushing a set of particles. Specifically, we prove that moving along the geodesic in the direction of functional gradient with respect to the second-order Wasserstein distance is equivalent to applying a pushforward mapping to a probability distribution, which can be approximated accurately by pushing a set of particles. Specifically, in each iteration of variational transport, we first solve the variational problem associated with the objective functional using the particles, whose solution yields the Wasserstein gradient direction. Then we update the current distribution by pushing each particle along the direction specified by such a solution. By characterizing both the statistical error incurred in estimating the Wasserstein gradient and the progress of the optimization algorithm, we prove that when the objective function satisfies a functional version of the Polyak-Łojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly to the global minimum of the objective functional up to a certain statistical error, which decays to zero sublinearly as the number of particles goes to infinity.
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