In this talk, we study fully connected Gaussian deep neural networks, focusing on their covariance process. In particular, we establish a large deviation principle (LDP) for this process in a functional framework, viewing it as a trajectory in the space of continuous functions. As key applications of our main results, we derive posterior LDPs under Gaussian likelihoods in both the infinite-width and mean-field regimes. We will outline the main ideas of the proof, emphasizing that it relies on an LDP for the covariance process regarded as a Markov process taking values in the space of non-negative, symmetric, trace-class operators equipped with the trace norm.
This talk is based on joint work with F. Bassetti and C. Hirsch.

We consider fully connected and feedforward deep neural networks with dependent and possibly heavy-tailed weights. These networks have been introduced to alleviate the drawbacks due to a Gaussian initialization. In a Bayesian framework, when the likelihood is Gaussian, we identify the posterior distribution of the output in the sequential wide-width limit and, in the shallow case, we compute explicitly the posterior distribution of these models.
The talk is based on a joint work with Nicola Apollonio and Giovanni Franzina.