We provide the first convergence guarantee for full black-box variational inference (BBVI), also known as Monte Carlo variational inference. While preliminary investigations worked on simplified versions of BBVI (e.g., bounded domain, bounded support, only optimizing for the scale, and such), our setup does not need any such algorithmic modifications. Our results hold for log-smooth posterior densities with and without strong log-concavity and the location-scale variational family. Also, our analysis reveals that certain algorithm design choices commonly employed in practice, particularly, nonlinear parameterizations of the scale of the variational approximation, can result in suboptimal convergence rates. Fortunately, running BBVI with proximal stochastic gradient descent fixes these limitations, and thus achieves the strongest known convergence rate guarantees. We evaluate this theoretical insight by comparing proximal SGD against other standard implementations of BBVI on large-scale Bayesian inference problems.

翻译：我们首次为全黑箱变分推断（亦称蒙特卡洛变分推断）提供了收敛性保证。以往研究仅针对简化版BBVI（如有界定义域、有界支撑、仅优化尺度参数等特殊情况），而本文的框架无需任何此类算法改进。我们的结论适用于对数光滑后验密度（无论是否具有强对数凹性）及位置-尺度变分族。此外，分析表明实践中某些常见算法设计选择（尤其是变分近似尺度的非线性参数化）可能导致次优收敛速率。幸运的是，采用近端随机梯度下降法运行BBVI可克服这些限制，从而获得已知最强的收敛速率保证。我们通过在大规模贝叶斯推断问题中对比近端SGD与其他标准BBVI实现，验证了这一理论洞见。