Stochastic Gradient Descent with a constant learning rate (constant SGD) simulates a Markov chain with a stationary distribution. With this perspective, we derive several new results. (1) We show that constant SGD can be used as an approximate Bayesian posterior inference algorithm. Specifically, we show how to adjust the tuning parameters of constant SGD to best match the stationary distribution to a posterior, minimizing the Kullback-Leibler divergence between these two distributions. (2) We demonstrate that constant SGD gives rise to a new variational EM algorithm that optimizes hyperparameters in complex probabilistic models. (3) We also propose SGD with momentum for sampling and show how to adjust the damping coefficient accordingly. (4) We analyze MCMC algorithms. For Langevin Dynamics and Stochastic Gradient Fisher Scoring, we quantify the approximation errors due to finite learning rates. Finally (5), we use the stochastic process perspective to give a short proof of why Polyak averaging is optimal. Based on this idea, we propose a scalable approximate MCMC algorithm, the Averaged Stochastic Gradient Sampler. Recently, stochastic gradients (SG) have also been used in the service of scalable Bayesian Markov Chain Monte-Carlo (MCMC) methods, where the goal is to generate samples from a conditional distribution of latent variables given a data set. In Bayesian inference, we assume a probabilistic model p(θ , x) with 1 arXiv:1704.04289v1 [stat.ML]
The authors of REF approximate SGD by a different SDE and show that SGD can be used as an approximate Bayesian posterior inference algorithm.
Stochastic Gradient Descent as Approximate Bayesian Inference
Journal of Machine Learning Research 18 (2017) 1-35
Mathematics Computer Science