用价格梯度提升变分推断，让参数空间方法也能媲美测度空间效果
Stochastic Gradient Variational Inference with Price's Gradient Estimator from Bures-Wasserstein to Parameter Space
- 引入价格梯度（基于目标密度的海森矩阵）改进变分推断
- 在高斯变分族下实现与测度空间方法相当的收敛速度
- 实验证明性能提升主要来自二阶信息利用，适合追求高效推断的研究者
在仅能获取未归一化对数密度的情况下，基于随机梯度的变分推断（VI）是常用方法。例如，水恩斯坦变分推断（WVI）在测度空间（布雷斯-瓦瑟斯坦空间）中进行梯度下降，而黑箱变分推断（BBVI）则在参数空间中操作。此前研究表明，对于高斯变分族，WVI 的收敛性优于使用重参数化梯度的现有 BBVI，暗示测度空间方法具有独特优势。本文通过分析发现，该优势实际源于其使用的特定梯度估计器——即与价格定理相关的、利用目标对数密度二阶信息（海森矩阵）的估计器。我们称其为价格梯度。通过简单修改，BBVI 亦可采用该梯度，从而获得相同的状态最优迭代复杂度。反之，若使用仅需梯度的重参数化梯度，可使 WVI 更广泛适用。实验表明，价格梯度是性能提升的主要来源。
原文摘要 · Abstract (English)
For approximating a target distribution given only its unnormalized log-density, stochastic gradient-based variational inference (VI) algorithms are a popular approach. For example, Wasserstein VI (WVI) and black-box VI (BBVI) perform gradient descent in measure space (Bures-Wasserstein space) and parameter space, respectively. Previously, for the Gaussian variational family, convergence guarantees for WVI have shown superiority over existing results for black-box VI with the reparametrization gradient, suggesting the measure space approach might provide some unique benefits. In this work, however, we close this gap by obtaining identical state-of-the-art iteration complexity guarantees for both. In particular, we identify that WVI's superiority stems from the specific gradient estimator it uses, which BBVI can also leverage with minor modifications. The estimator in question is usually associated with Price's theorem and utilizes second-order information (Hessians) of the target log-density. We will refer to this as Price's gradient. On the flip side, WVI can be made more widely applicable by using the reparametrization gradient, which requires only gradients of the log-density. We empirically demonstrate that the use of Price's gradient is the major source of performance improvement.
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