Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities
This work provides theoretical motivation for architectural choices in neural networks, particularly for researchers in machine learning theory, but it is incremental as it builds on existing concepts like Banach spaces and representer theorems.
The authors tackled the problem of understanding the function-space optimality of shallow neural architectures with multivariate nonlinearities by constructing new Banach spaces and proving a representer theorem that characterizes solutions as such architectures, linking them to skip connections, orthogonal weight normalization, and multi-index models.
We investigate the function-space optimality (specifically, the Banach-space optimality) of a large class of shallow neural architectures with multivariate nonlinearities/activation functions. To that end, we construct a new family of Banach spaces defined via a regularization operator, the $k$-plane transform, and a sparsity-promoting norm. We prove a representer theorem that states that the solution sets to learning problems posed over these Banach spaces are completely characterized by neural architectures with multivariate nonlinearities. These optimal architectures have skip connections and are tightly connected to orthogonal weight normalization and multi-index models, both of which have received recent interest in the neural network community. Our framework is compatible with a number of classical nonlinearities including the rectified linear unit (ReLU) activation function, the norm activation function, and the radial basis functions found in the theory of thin-plate/polyharmonic splines. We also show that the underlying spaces are special instances of reproducing kernel Banach spaces and variation spaces. Our results shed light on the regularity of functions learned by neural networks trained on data, particularly with multivariate nonlinearities, and provide new theoretical motivation for several architectural choices found in practice.