Log-sum-exp neural networks and posynomial models for convex and log-log-convex data

TL;DR

This paper introduces neural networks with exponential and logarithmic activations as universal approximators for convex and log-log-convex functions, enabling efficient optimization via convex programming and geometric programming.

Contribution

It demonstrates that a specific one-layer neural network architecture can universally approximate convex and log-log-convex functions, linking neural networks with posynomial models for optimization.

Findings

Neural networks with exponential and logarithmic activations are universal approximators.

LSET networks are convex and suitable for convex optimization.

GPOST models can be optimized using geometric programming.

Abstract

We show in this paper that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is an universal approximator of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named LSET. Under a suitable exponential transformation, the class of LSET functions maps to a family of generalized posynomials GPOST, which we similarly show to be universal approximators for log-log-convex functions. A key feature of an LSET network is that, once it is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a GPOST model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using geometric programming (GP). The proposed methodology is illustrated by two numerical examples, in which, first, models are constructed from simulation data of the two physical processes (namely, the level of vibration in a vehicle suspension system, and the peak power generated by the combustion of propane), and then optimization-based design is performed on these models.