Training Neural Networks with Property-Preserving Parameter Perturbations

Many types of neural network layers rely on matrix properties such as invertibility or orthogonality. Retaining such properties during optimization with gradient-based stochastic optimizers is a challenging task, which is usually addressed by either reparameterization of the affected parameters or by directly optimizing on the manifold. In contrast, this work presents a novel, general approach of preserving matrix properties by using parameterized perturbations. In lieu of directly optimizing the network parameters, the introduced P$^{4}$ update optimizes perturbations and merges them into the actual parameters infrequently such that the desired property is preserved. As a demonstration, we use this concept to preserve invertibility of linear layers during training. This P$^{4}$Inv update allows keeping track of inverses and determinants via rank-one updates and without ever explicitly computing them. We show how such invertible blocks improve the mixing of coupling layers and thus the mode separation of the resulting normalizing flows.