Stochastic natural gradient descent draws posterior samples in function space
This work addresses the challenge of efficient Bayesian inference in machine learning, offering a method to sample from posteriors without traditional MCMC, though it is incremental as it builds on prior stochastic gradient descent correspondence.
The paper tackles the problem of approximating Bayesian posterior uncertainty in function space using stochastic gradient methods, proving that minibatch natural gradient descent draws posterior samples near local minima under specific conditions and proposing a novel optimiser, stochastic NGD, to maintain parameterisation invariance and validity away from minima.
Recent work has argued that stochastic gradient descent can approximate the Bayesian uncertainty in model parameters near local minima. In this work we develop a similar correspondence for minibatch natural gradient descent (NGD). We prove that for sufficiently small learning rates, if the model predictions on the training set approach the true conditional distribution of labels given inputs, the stationary distribution of minibatch NGD approaches a Bayesian posterior near local minima. The temperature $T = εN / (2B)$ is controlled by the learning rate $ε$, training set size $N$ and batch size $B$. However minibatch NGD is not parameterisation invariant and it does not sample a valid posterior away from local minima. We therefore propose a novel optimiser, "stochastic NGD", which introduces the additional correction terms required to preserve both properties.