A Universal Approximation Result for Difference of log-sum-exp Neural Networks
This work addresses the need for optimizable surrogate models in design problems, such as medical applications, though it is incremental as it builds on existing LSE networks and difference-of-convex algorithms.
The paper tackled the problem of approximating continuous functions over convex compact sets using a new neural network architecture called Difference-LSE networks, which are smooth universal approximators and can be optimized efficiently due to their difference-of-convex-functions form. The result was demonstrated by applying the method to data-driven diet design for type-2 diabetes, showing practical effectiveness.
We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node (LSE networks) is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of networks maps to a family of subtraction-free ratios of generalized posynomials, which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of Difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for design that possess a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing effective optimization-based design. We illustrate the proposed approach by applying it to data-driven design of a diet for a patient with type-2 diabetes.