Order Matters: Sequence to sequence for sets

TLDR

Recurrent neural networks have revived sequence modeling, enabling many tasks to be framed as sequence‑to‑sequence problems, yet variable‑size inputs or outputs that are not naturally ordered, such as sets of numbers or random variables, remain challenging. This work demonstrates that the ordering of input and output data critically affects learning and introduces a seq2seq extension that handles unordered input sets and a loss that searches over possible output orders during training. The authors extend the seq2seq architecture by treating input sets as unordered collections and employ a loss that iteratively explores different output permutations to mitigate the lack of inherent structure. Experiments on benchmark language modeling, parsing, sorting numbers, and joint probability estimation confirm that the proposed ordering‑aware seq2seq framework improves performance over baseline models.

Abstract

Sequences have become first class citizens in supervised learning thanks to the resurgence of recurrent neural networks. Many complex tasks that require mapping from or to a sequence of observations can now be formulated with the sequence-to-sequence (seq2seq) framework which employs the chain rule to efficiently represent the joint probability of sequences. In many cases, however, variable sized inputs and/or outputs might not be naturally expressed as sequences. For instance, it is not clear how to input a set of numbers into a model where the task is to sort them; similarly, we do not know how to organize outputs when they correspond to random variables and the task is to model their unknown joint probability. In this paper, we first show using various examples that the order in which we organize input and/or output data matters significantly when learning an underlying model. We then discuss an extension of the seq2seq framework that goes beyond sequences and handles input sets in a principled way. In addition, we propose a loss which, by searching over possible orders during training, deals with the lack of structure of output sets. We show empirical evidence of our claims regarding ordering, and on the modifications to the seq2seq framework on benchmark language modeling and parsing tasks, as well as two artificial tasks -- sorting numbers and estimating the joint probability of unknown graphical models.