Sample Complexity Bounds for RNNs with Application to Combinatorial Graph Problems

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 ϵ can be realized with at most Õ(a4b/ϵ2) samples.1 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 Õ(n6/ϵ2), i. e., with only a polynomial number of samples. For combinatorial graph problems, this provides a theoretical foundation that renders RNNs competitive.