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.

In my view, the main reason the proposed lower bound is interesting is that it offers a potential way to speed up training for multi-class models with a very large number of classes. While it is useful to understand other properties of the lower bound, the paper could be improved by emphasizing this primary use case in machine learning. Figure 1c and Figure 3 need a more clear explanation of what is being displayed, and why it is important. In particular, what value is being plotted on the y-axis, and at what setting of the parameters w. Here is how I understand it, for Figure 1c say: Blue Line - value of Eq. (13) at the setting of parameters w that maximize 13 Red Line - value of Eq. (13) at the setting of parameters w that maximize 14 Green Line - value of Eq. (13)? at the setting of parameters w that maximize the Bouchard lower bound (?) Red dashed line - value of Eq. (13)? at parameters w based on the given iterations of training?