Learning Activation Functions in Deep (Spline) Neural Networks

We develop an efficient computational solution to train deep neural networks (DNN) with free-form activation functions. To make the problem well-posed, we augment the cost functional of the DNN by adding an appropriate shape regularization: the sum of the second-order total-variations of the trainable nonlinearities. The representer theorem for DNNs tells us that the optimal activation functions are adaptive piecewise-linear splines, which allows us to recast the problem as a parametric optimization. The challenging point is that the corresponding basis functions (ReLUs) are poorly conditioned and that the determination of their number and positioning is also part of the problem. We circumvent the difficulty by using an equivalent B-spline basis to encode the activation functions and by expressing the regularization as an $\ell _1$-penalty. This results in the specification of parametric activation function modules that can be implemented and optimized efficiently on standard development platforms. We present experimental results that demonstrate the benefit of our approach.