Natural Langevin Dynamics for Neural Networks

One way to avoid overfitting in machine learning is to use model parameters distributed according to a Bayesian posterior given the data, rather than the maximum likelihood estimator. Stochastic gradient Langevin dynamics (SGLD) is one algorithm to approximate such Bayesian posteriors for large models and datasets. SGLD is a standard stochastic gradient descent to which is added a controlled amount of noise, specifically scaled so that the parameter converges in law to the posterior distribution [WT11, TTV16]. The posterior predictive distribution can be approximated by an ensemble of samples from the trajectory. Choice of the variance of the noise is known to impact the practical behavior of SGLD: for instance, noise should be smaller for sensitive parameter directions. Theoretically, it has been suggested to use the inverse Fisher information matrix of the model as the variance of the noise, since it is also the variance of the Bayesian posterior [PT13, AKW12, GC11]. But the Fisher matrix is costly to compute for largedimensional models. Here we use the easily computed Fisher matrix approximations for deep neural networks from [MO16, Oll15]. The resulting natural Langevin dynamics combines the advantages of Amari’s natural gradient descent and Fisher-preconditioned Langevin dynamics for large neural networks. Small-scale experiments on MNIST show that Fisher matrix preconditioning brings SGLD close to dropout as a regularizing technique. Consider a supervised learning problem with a dataset D = {(x1, y1), . . . , (xN , yN )} of N input-output pairs, to be modelled by a parametric probabilistic distribution yi ∼ pθ(y|xi) (x = ∅ amounts to unsupervised learning of y). Defining the log-loss lθ(yi|xi) := − ln pθ(yi|xi), the maximum likelihood estimator is the value θ that minimizes E(x,y)∈Dlθ(y|x), where E(x,y)∈D denotes averaging over the dataset. Stochastic gradient descent is often used to tackle this minimization problem for large-scale datasets [BL03, Bot10]. This consists in iterating θ ← θ − η Ê(x,y)∈D ∂θlθ(y|x), (1) MILA, Université de Montréal, Canada CNRS, TAU, Université Paris-Saclay, France