Differentiable Optimization of Generalized Nondecomposable Functions using Linear Programs
Authors: Zihang Meng, Lopamudra Mukherjee, Yichao Wu, Vikas Singh, Sathya Ravi

We propose a framework which makes it feasible to directly train deep neural networks with respect to popular families of task-specific non-decomposable performance measures such as AUC, multi-class AUC, $F$-measure and others. A common feature of the optimization model that emerges from these tasks is that it involves solving a Linear Programs (LP) during training where representations learned by upstream layers characterize the constraints or the feasible set. The constraint matrix is not only large but the constraints are also modified at each iteration. We show how adopting a set of ingenious ideas proposed by Mangasarian for 1-norm SVMs -- which advocates for solving LPs with a generalized Newton method -- provides a simple and effective solution that can be run on the GPU. In particular, this strategy needs little unrolling, which makes it more efficient during backward pass. Further, even when the constraint matrix is too large to fit on the GPU memory (say large minibatch settings), we show that running the Newton method in a lower dimensional space yields accurate gradients for training, by utilizing a statistical concept called {\em sufficient}  dimension reduction. While a number of specialized algorithms have been proposed for the models that we describe here, our module turns out to be applicable without any  specific adjustments or relaxations. We describe each use case, study its properties and demonstrate the efficacy of the approach over alternatives which use surrogate lower bounds and often, specialized optimization schemes. Frequently, we achieve superior computational behavior and performance improvements on common datasets used in the literature.