Sample Complexity Bounds for Recurrent Neural Networks with Application to Combinatorial Graph Problems

TL;DR

This paper establishes the first theoretical sample complexity bounds for real-valued recurrent neural networks, demonstrating their efficiency and competitiveness in solving combinatorial graph problems.

Contribution

It provides the first upper bounds on sample complexity for learning real-valued RNNs, extending theoretical understanding from feed-forward networks to recurrent architectures.

Findings

Sample complexity for single-layer RNNs is (a^4b/
^2)

Multi-layer RNNs have comparable sample complexity bounds

RNNs can learn graph problems with polynomial sample size

Abstract

Learning to predict solutions to real-valued combinatorial graph problems promises efficient approximations. As demonstrated based on the NP-hard edge clique cover number, recurrent neural networks (RNNs) are particularly suited for this task and can even outperform state-of-the-art heuristics. However, the theoretical framework for estimating real-valued RNNs is understood only poorly. As our primary contribution, this is the first work that upper bounds the sample complexity for learning real-valued RNNs. While such derivations have been made earlier for feed-forward and convolutional neural networks, our work presents the first such attempt for recurrent neural networks. Given a single-layer RNN with $a$ rectified linear units and input of length $b$, we show that a population prediction error of $\varepsilon$ can be realized with at most $\tilde{\mathcal{O}}(a^4b/\varepsilon^2)$ samples. We further derive comparable results for multi-layer RNNs. Accordingly, a size-adaptive RNN fed with graphs of at most $n$ vertices can be learned in $\tilde{\mathcal{O}}(n^6/\varepsilon^2)$, i.e., with only a polynomial number of samples. For combinatorial graph problems, this provides a theoretical foundation that renders RNNs competitive.