This paper studies the behavior of the extragradient algorithm when applied to hypomonotone operators, a class of problems that extends beyond the classical monotone setting. While the extragradient method is widely known for its efficacy in solving variational inequalities with monotone and Lipschitz continuous operators, we demonstrate that its convergence is not guaranteed in the hypomonotone setting. We provide a characterization theorem that identifies the conditions under which the extragradient algorithm fails to converge. Our results highlight the necessity of stronger assumptions to guarantee convergence of extragradient and to further develop the existing VI methods for broader problems.

We provide a rigorous analysis of training by variational inference (VI) of Bayesian neural networks in the two-layer and infinite-width case. We consider a regression problem with a regularized evidence lower bound (ELBO) which is decomposed into the expected log-likelihood of the data and the Kullback-Leibler (KL) divergence between the a priori distribution and the variational posterior. With an appropriate weighting of the KL, we prove a law of large numbers for three different training schemes: (i) the idealized case with exact estimation of a multiple Gaussian integral from the reparametrization trick, (ii) a minibatch scheme using Monte Carlo sampling, commonly known as Bayes by Backprop, and (iii) a new and computationally cheaper algorithm which we introduce as Minimal VI. An important result is that all methods converge to the same mean-field limit. Finally, we illustrate our results numerically and discuss the need for the derivation of a central limit theorem.

Neural network based models for collaborative filtering have started to gain attention recently. One branch of research is based on using deep generative models to model user preferences where variational autoencoders were shown to produce state-of-the-art results. However, there are some potentially problematic characteristics of the current variational autoencoder for CF. The first is the too simplistic prior that VAEs incorporate for learning the latent representations of user preference. The other is the model's inability to learn deeper representations with more than one hidden layer for each network. Our goal is to incorporate appropriate techniques to mitigate the aforementioned problems of variational autoencoder CF and further improve the recommendation performance. Our work is the first to apply flexible priors to collaborative filtering and show that simple priors (in original VAEs) may be too restrictive to fully model user preferences and setting a more flexible prior gives significant gains. We experiment with the VampPrior, originally proposed for image generation, to examine the effect of flexible priors in CF. We also show that VampPriors coupled with gating mechanisms outperform SOTA results including the Variational Autoencoder for Collaborative Filtering by meaningful margins on 2 popular benchmark datasets (MovieLens & Netflix).