Variational Inference

We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernelized Stein Discrepancy ($\mathsf{KSD}$) and Wasserstein-2 metrics. Our key insight is that the time derivative of the relative entropy between the joint density of $N$ particle locations and the $N$-fold product target measure, starting from a regular initial distribution, splits into a dominant negative part' proportional to $N$ times the expected $\mathsf{KSD}^2$ and a smaller positive part'. This observation leads to $\mathsf{KSD}$ rates of order $1/\sqrt{N}$, in both continuous and discrete time, providing a near optimal (in the sense of matching the corresponding i.i.d. rates) double exponential improvement over the recent result by Shi and Mackey (2024). Under mild assumptions on the kernel and potential, these bounds also grow polynomially in the dimension $d$. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence in continuous time. For the case of `bilinear + Matérn' kernels, we derive Wasserstein-2 rates that exhibit a curse-of-dimensionality similar to the i.i.d. setting. We also obtain marginal convergence and long-time propagation of chaos results for the time-averaged particle laws.

We propose denoising diffusion variational inference (DDVI), a black-box variational inference algorithm for latent variable models which relies on diffusion models as flexible approximate posteriors. Specifically, our method introduces an expressive class of diffusion-based variational posteriors that perform iterative refinement in latent space; we train these posteriors with a novel regularized evidence lower bound (ELBO) on the marginal likelihood inspired by the wake-sleep algorithm. Our method is easy to implement (it fits a regularized extension of the ELBO), is compatible with black-box variational inference, and generally outperforms alternative classes of approximate posteriors based on normalizing flows or adversarial networks. We find that DDVI improves inference and learning in deep latent variable models across common benchmarks as well as on a motivating task in biology---inferring latent ancestry from human genomes---where it outperforms strong baselines on the 1000 Genomes dataset.

This paper builds bridges between two families of probabilistic algorithms: (hierarchical) variational inference (VI), which is typically used to model distributions over continuous spaces, and generative flow networks (GFlowNets), which have been used for distributions over discrete structures such as graphs. We demonstrate that, in certain cases, VI algorithms are equivalent to special cases of GFlowNets in the sense of equality of expected gradients of their learning objectives. We then point out the differences between the two families and show how these differences emerge experimentally. Notably, GFlowNets, which borrow ideas from reinforcement learning, are more amenable than VI to off-policy training without the cost of high gradient variance induced by importance sampling. We argue that this property of GFlowNets can provide advantages for capturing diversity in multimodal target distributions.