Motivated by learning of correlated equilibria in non-cooperative games, we perform a large deviations analysis of a regret minimizing stochastic approximation algorithm. The regret minimization algorithm we consider comprises multiple agents that communicate over a graph to coordinate their decisions. We derive an exponential decay rate towards the algorithm's stable point using large deviations theory. Our analysis leverages the variational representation of the Laplace functionals and weak convergence methods to characterize the exponential decay rate.