Stochastic Gradient Descent as Approximate Bayesian Inference

[I will be at ICLR next week, let's grab some coffee if you are there]

The recent distill pub on Why Momentum Really Works by Gabriel Goh does provide a some insight on why Gradient Descent might work. Overviews such as the ones listed below also do help:

• Optimization techniques comparison in Julia: SGD, Momentum, Adagrad, Adadelta, Adam
• An overview of gradient descent optimization algorithms by Sebastian Ruder

But today, we have an addtional insight in the mapping of SGD to Bayesian inference: Stochastic Gradient Descent as Approximate Bayesian Inference by Stephan Mandt, Matthew D. Hoffman, David M. Blei

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.

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