Semi-discrete optimization through semi-discrete optimal transport: a framework for neural architecture search

In this paper we introduce a theoretical framework for semi-discrete optimization using ideas from optimal transport. Our primary motivation is in the field of deep learning, and specifically in the task of neural architecture search. With this aim in mind, we discuss the geometric and theoretical motivation for new techniques for neural architecture search (in the companion work \cite{practical}; we show that algorithms inspired by our framework are competitive with contemporaneous methods). We introduce a Riemannian like metric on the space of probability measures over a semi-discrete space $\mathbb{R}^d \times \mathcal{G}$ where $\mathcal{G}$ is a finite weighted graph. With such Riemmanian structure in hand, we derive formal expressions for the gradient flow of a relative entropy functional, as well as second order dynamics for the optimization of said energy. Then, with the aim of providing a rigorous motivation for the gradient flow equations derived formally we also consider an iterative procedure known as minimizing movement scheme (i.e., Implicit Euler scheme, or JKO scheme) and apply it to the relative entropy with respect to a suitable cost function. For some specific choices of metric and cost, we rigorously show that the minimizing movement scheme of the relative entropy functional converges to the gradient flow process provided by the formal Riemannian structure. This flow coincides with a system of reaction-diffusion equations on $\mathbb{R}^d$.