The softmax representation of probabilities for categorical variables plays a prominent role in modern machine learning with numerous applications in areas such as large scale classification, neural language modeling and recommendation systems. However, softmax estimation is very expensive for large scale inference because of the high cost associated with computing the normalizing constant. Here, we introduce an efficient approximation to softmax probabilities which takes the form of a rigorous lower bound on the exact probability. This bound is expressed as a product over pairwise probabilities and it leads to scalable estimation based on stochastic optimization. It allows us to perform doubly stochastic estimation by subsampling both training instances and class labels. We show that the new bound has interesting theoretical properties and we demonstrate its use in classification problems.

Neural Information Processing Systems http://nips.cc/

Further, we emphasize the significance of principled regularization of the network parameters and prediction.

Neural Information Processing Systems http://nips.cc/

Contributions: In this work, we develop a projected Stein variational Newton method (pSVN) to tackle the challenge of high-dimensional Bayesian inference by exploiting the intrinsic lowdimensional geometric structure of the posterior distribution (where it departs from the prior), as characterized by the dominant spectrum of the prior-preconditioned Hessian of the negative log likelihood.