Variational inference (VI) is a popular approach in Bayesian inference, that looks for the best approximation of the posterior distribution within a parametric family, minimizing a loss that is (typically) the reverse Kullback-Leibler (KL) divergence. Despite its empirical success, the theoretical properties of VI have only recently received attention, and is restricted to the Gaussian case. This research paper aims to contribute to the theoretical study of VI in the non-Gaussian case by investigating the setting of Mixture of Gaussians with fixed covariance. In this view, VI over this specific family can be casted as the minimization of a Mollified relative entropy, i.e. the KL between the convolution (with respect to a Gaussian kernel) of an atomic measure supported on Diracs, where the support of the atomic measure correspond to the localization of the Gaussian components, and the target distribution. Hence, solving variational inference is equivalent to optimizing the positions of the Diracs (the particles), which can be done through gradient descent and takes the form of an interacting particle system. We study two sources of error in variational inference in this context. The first is an optimization result that is a descent lemma establishing that the algorithm decreases the objective at each iteration. The second is an approximation error that upper bounds the mollified relative entropy between an optimal finite mixture and the target distribution.

In this paper, we consider high dimensional contextual bandit problems. Within this setting, Thompson Sampling and its variants have been proposed and have been successfully applied to multiple machine learning problems. Existing theory on Thompson Sampling shows that it has suboptimal dimension dependency in contrast to upper confidence bound (UCB) algorithms. To circumvent this issue and obtain optimal regret bounds, (Zhang, 2021) recently proposed to modify Thompson Sampling by enforcing more exploration and hence is able to attain optimal regret bounds. Nonetheless, this analysis does not permit tractable implementation in high dimensions. The main challenge therein is the simulation of the posterior samples at each step given the available observations. To overcome this, we propose and analyze the use of Markov Chain Monte Carlo methods. As a corollary, we show that for contextual linear bandits, using Langevin Monte Carlo (LMC) or Metropolis Adjusted Langevin Algorithm (MALA), our algorithm attains optimal regret bounds of $\tilde{O}(d\sqrt{T})$. Furthermore, we show that this is obtained with $\tilde{O}(dT^4)$, $\tilde{O}(dT^2)$ data evaluations respectively for LMC and MALA. Finally, we validate our findings through numerical simulations and show that we outperform vanilla Thompson sampling in high dimensions.

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.