Automatic Optimisation of Normalised Neural Networks

We propose automatic optimisation methods considering the geometry of matrix manifold for the normalised parameters of neural networks. Layerwise weight normalisation with respect to Frobenius norm is utilised to bound the Lipschitz constant and to enhance gradient reliability so that the trained networks are suitable for control applications. Our approach first initialises the network and normalises the data with respect to the $\ell^{2}$-$\ell^{2}$ gain of the initialised network. Then, the proposed algorithms take the update structure based on the exponential map on high-dimensional spheres. Given an update direction such as that of the negative Riemannian gradient, we propose two different ways to determine the stepsize for descent. The first algorithm utilises automatic differentiation of the objective function along the update curve defined on the combined manifold of spheres. The directional second-order derivative information can be utilised without requiring explicit construction of the Hessian. The second algorithm utilises the majorisation-minimisation framework via architecture-aware majorisation for neural networks. With these new developments, the proposed methods avoid manual tuning and scheduling of the learning rate, thus providing an automated pipeline for optimizing normalised neural networks.