Large deviations for stochastic approximations is a well-studied field that yields convergence properties for many useful algorithms in statistics, machine learning and statistical physics. In this article, we prove, under certain assumptions, a large deviation principle for a stochastic approximation with state-dependent Markovian noise and with decreasing step size. Common algorithms that satisfy these conditions include stochastic gradient descent, persistent contrastive divergence and the Wang-Landau algorithm. The proof is based don't he weak convergence approach to the theory of large deviations and uses a representation formula to rewrite the problem into a stochastic control problem. The resulting rate function is an action potential over a local rate function that is the Fenchel-Legendre transform of a limiting Hamiltonian.

Evaluating generative models have become an important task in modern machine learn-ing. Despite this, most existing evaluation metrics are constricted to specific types of data orrequire a supervised setting, limiting their usefulness in the general case. In this article, wetake inspiration from large deviation theory, to propose an evaluation score for variationalautoencoders. The metric evaluates the latent space of the variational autoencoder and istherefore independent of the type of data and works in a completely unsupervised setting.Experimental results on MNIST and Fashion-MNIST prove that this new large deviationscore has a high correlation with other well know evaluation metrics.

In this article, we look at the problem of minimizing an action potential that arises from large deviation theory for stochastic approximations. The solutions to the minimising problem satisfy, in the sense of a viscosity solution, a Hamilton-Jacobi equation. From weak KAM theory, we know that these viscosity solutions are characterised by the projected Aubryset. The main result of this paper is that, for a specific rate function corresponding to the astochastic approximation algorithm, we prove that the projected Aubry set is equal to the forward limit set to the limit ODE.