Differentiable Optimization of Generalized Nondecomposable Functions using Linear Programs
Authors: Zihang Meng, Lopamudra Mukherjee, Vikas Singh, Sathya N. Ravi
Keywords: linear programming, nondecomposable functions, differentiable, AUC, Fscore
Abstract: We propose a framework which makes it feasible to directly train deep neural networks with respect to popular families of task-specific non-decomposable per- formance measures such as AUC, multi-class AUC, F -measure and others, as well as models such as non-negative matrix factorization. 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 influence the constraints. The constraint matrix is not only large but the constraints are also modified at each iteration. We show how adopting a set of influential ideas proposed by Mangasarian for 1-norm SVMs – which advocates for solving LPs with a generalized Newton method – provides a simple and effective solution. In particular, this strategy needs little unrolling, which makes it more efficient during backward pass. While a number of specialized algorithms have been proposed for the models that we de- scribe 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.