Input Invex Neural Network

In this paper, we present a novel method to constrain invexity on Neural Networks (NN). Invex functions ensure every stationary point is global minima. Hence, gradient descent commenced from any point will lead to the global minima. Another advantage of invexity on NN is to divide data space locally into two connected sets with a highly non-linear decision boundary by simply thresholding the output. To this end, we formulate a universal invex function approximator and employ it to enforce invexity in NN. We call it Input Invex Neural Networks (II-NN). We first fit data with a known invex function, followed by modification with a NN, compare the direction of the gradient and penalize the direction of gradient on NN if it contradicts with the direction of reference invex function. In order to penalize the direction of the gradient we perform Gradient Clipped Gradient Penalty (GC-GP). We applied our method to the existing NNs for both image classification and regression tasks. From the extensive empirical and qualitative experiments, we observe that our method gives the performance similar to ordinary NN yet having invexity. Our method outperforms linear NN and Input Convex Neural Network (ICNN) with a large margin. We publish our code and implementation details at github.