Stability of Sequential and Parallel Coordinate Ascent Variational Inference

We study the non-asymptotic contraction in Wasserstein distance of the sequential, parallel, and random-scan coordinate ascent variational inference algorithms. This is shown to hold under a functional smoothness condition of the optimality maps and a transportation-information inequality at their fixed points. Our results are sharp and general, and as opposed to those based on global strong log-concavity assumptions, they allow for local convergence on smooth, non-smooth, and discrete manifolds, including within the context of data augmentation. We consider many applications in statistical physics and Bayesian statistics. These include pairwise Markov Random field models such as Ising and Curie-Weiss, unbalanced Bayesian Gaussian Mixture Models, high-dimensional Bayesian Probit Regression, and high-dimensional Logistic Regression with Pólya--Gamma random variables (i.e. Jaakkola-Jordan's algorithm). In many of these models, these represent the first available convergence results of their kind.

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.

Cyclic block coordinate methods are a fundamental class of first-order algorithms, widely used in practice for their simplicity and strong empirical performance. Yet, their theoretical behavior remains challenging to explain, and setting their step sizes -- beyond classical coordinate descent for minimization -- typically requires careful tuning or line-search machinery. In this work, we develop $\texttt{ADUCA}$ (Adaptive Delayed-Update Cyclic Algorithm), a cyclic algorithm addressing a broad class of Minty variational inequalities with monotone Lipschitz operators. $\texttt{ADUCA}$ is parameter-free: it requires no global or block-wise Lipschitz constants and uses no per-epoch line search, except at initialization. A key feature of the algorithm is using operator information delayed by a full cycle, which makes the algorithm compatible with parallel and distributed implementations, and attractive due to weakened synchronization requirements across blocks. We prove that $\texttt{ADUCA}$ attains (near) optimal global oracle complexity as a function of target error $ε>0,$ scaling with $1/ε$ for monotone operators, or with $\log^2(1/ε)$ for operators that are strongly monotone.