Bregman Divergence for Stochastic Variance Reduction: Saddle-Point and Adversarial Prediction
Adversarial machines, where a learner competes against an adversary, have regained much recent interest in machine learning. They are naturally in the form of saddle-point optimization, often with separable structure but sometimes also with unmanageably large dimension. In this work we show that adversarial prediction under multivariate losses can be solved much faster than they used to be. We first reduce the problem size exponentially by using appropriate sufficient statistics, and then we adapt the new stochastic variance-reduced algorithm of Balamurugan & Bach (2016) to allow any Bregman divergence. We prove that the same linear rate of convergence is retained and we show that for adversarial prediction using KL-divergence we can further achieve a speedup of #example times compared with the Euclidean alternative. We verify the theoretical findings through extensive experiments on two example applications: adversarial prediction and LPboosting.
This paper shows that "certain" adversarial prediction problems under multivariate losses can be solved "much faster than they used to be". The paper stands on two main ideas: (1) that the general saddle function optimization problem stated in eq. (9) can be simplified from an exponential to a quadratic complexity (on the sample size), and (2) that the simplified optimization problem, with some regularization, can be solved using some extension of the SVRG (stochastic variance reduction gradient) method to Bregman divergences.
The paper is quite focused on the idea of obtaining a faster solution of the adversarial problem. However, the key simplification is applied to a specific loss, the F-score, so one may wonder if the benefits of the proposed method could be extended to other losses. The extension of the SVRG is a more general result, it seems that the paper could have been focused on proposing Breg-SVRG, showing the adversarial optimization with the F-score as a particular application.
In any case, I think the paper is technically correct and interesting enough to be accepted.