Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions

Nikolaos Karalias, Joshua David Robinson, Andreas Loukas, Stefanie Jegelka

Keywords: deep learning, unsupervised learning, combinatorial optimization, algorithmic reasoning

Abstract: Integrating functions on discrete domains into neural networks is key to developing their capability to reason about discrete objects. But, discrete domains are (1) not naturally amenable to gradient-based optimization, and (2) incompatible with deep learning architectures that rely on representations in high-dimensional vector spaces. In this work, we address both difficulties for set functions, which capture many important discrete problems. First, we develop a framework for extending set functions onto low-dimensional continuous domains, where many extensions are naturally defined. Our framework subsumes many well-known extensions as special cases. Second, to avoid undesirable low-dimensional neural network bottlenecks, we convert low-dimensional extensions into representations in high-dimensional spaces, taking inspiration from the success of semidefinite programs for combinatorial optimization. Empirically, we observe benefits of our extensions for unsupervised neural combinatorial optimization, in particular with high-dimensional representations.

TL;DR: We present a framework that extends discrete set functions onto continuous and high-dimensional domains with the purpose of using them in neural network architectures.