While black-box variational inference is widely used, there is no proof that its stochastic optimization succeeds. We suggest this is due to a theoretical gap in existing stochastic optimization proofs-namely the challenge of gradient estimators with unusual noise bounds, and a composite non-smooth objective. For dense Gaussian variational families, we observe that existing gradient estimators based on reparameterization satisfy a quadratic noise bound and give novel convergence guarantees for proximal and projected stochastic gradient descent using this bound. This provides the first rigorous guarantee that black-box variational inference converges for realistic inference problems.

翻译：尽管黑箱变分推断被广泛使用，但其随机优化方法是否成功尚无理论证明。我们认为这是由于现有随机优化证明中存在理论空白——即具有特殊噪声界的梯度估计器以及复合非光滑目标函数带来的挑战。针对稠密高斯变分族，我们观察到基于重参数化的现有梯度估计器满足二次噪声界，并利用该界给出了近端和投影随机梯度下降方法的新颖收敛性保证。这首次为黑箱变分推断在真实推断问题中的收敛性提供了严格保证。