Temperature is All You Need for Generalization in Langevin Dynamics and other Markov Processes

We analyze the generalization gap (gap between the training and test errors) when training a potentially over-parametrized model using a Markovian stochastic training algorithm, initialized from some distribution θ0 p0. We focus on Langevin dynamics with a positive temperature β 1, i.e. gradient descent on a training loss Lwith infinitesimal step size, perturbed with β 1-variances Gaussian noise, and lightly regularized or bounded. There, we bound the generalization gap, at any time during training, by p (βEL(θ0)+ln(1/δ))/N with probability 1 δ over the dataset, where N is the sample size, and EL(θ0) = O(1)with standard initialization scaling. In contrast to previous guarantees, we have no dependence on either training time or reliance on mixing, nor a dependence on dimensionality, gradient norms, or any other properties of the loss or model. This guarantee follows from a general analysis of any Markov process-based training that has a Gibbs-style stationary distribution. The proof is surprisingly simple, once we observe that the marginal distribution divergence from initialization remains bounded, as implied by a generalized second law of thermodynamics.