Sampling from a high-dimensional distribution serves as one of the key components in statistics, machine learning, and scientific computing, and constitutes the foundation of the fields including Bayesian statistics and generative models [Liu and Liu, 2001, Brooks et al., 2011, Song et al.,

Sampling from a high-dimensional distribution serves as one of the key components in statistics, machine learning, and scientific computing, and constitutes the foundation of the fields including Bayesian statistics and generative models [Liu and Liu, 2001, Brooks et al., 2011, Song et al.,

Takentogether, our results provide theoretical justification for the observedempirical success of sketch-DL.

Sampling from a high-dimensional distribution serves as one of the key components in statistics, machine learning, and scientific computing, and constitutes the foundation of the fields including Bayesian statistics and generative models [Liu and Liu, 2001, Brooks et al., 2011, Song et al.,

Sampling from a high-dimensional distribution serves as one of the key components in statistics, machine learning, and scientific computing, and constitutes the foundation of the fields including Bayesian statistics and generative models [Liu and Liu, 2001, Brooks et al., 2011, Song et al.,

We prove that, given a mean-field location-scale variational family, black-box variational inference (BBVI) with the reparametrization gradient converges at an almost dimension-independent rate. Specifically, for strongly log-concave and log-smooth targets, the number of iterations for BBVI with a sub-Gaussian family to achieve an objective $ε$-close to the global optimum is $\mathrm{O}(\log d)$, which improves over the $\mathrm{O}(d)$ dependence of full-rank location-scale families. For heavy-tailed families, we provide a weaker $\mathrm{O}(d^{2/k})$ dimension dependence, where $k$ is the number of finite moments. Additionally, if the Hessian of the target log-density is constant, the complexity is free of any explicit dimension dependence. We also prove that our bound on the gradient variance, which is key to our result, cannot be improved using only spectral bounds on the Hessian of the target log-density.