Generalization and Estimation Error Bounds for Model-based Neural Networks
This work addresses a theoretical gap for practitioners using model-based networks in tasks like sparse coding, though it is incremental as it builds on existing complexity theory.
The paper tackles the lack of theoretical understanding for why model-based neural networks generalize better than ReLU networks, deriving upper bounds on generalization and estimation errors using complexity measures like Rademacher complexities. It shows that model-based networks outperform ReLU networks in sparse recovery, with experiments confirming higher generalization abilities, especially with small training samples.
Model-based neural networks provide unparalleled performance for various tasks, such as sparse coding and compressed sensing problems. Due to the strong connection with the sensing model, these networks are interpretable and inherit prior structure of the problem. In practice, model-based neural networks exhibit higher generalization capability compared to ReLU neural networks. However, this phenomenon was not addressed theoretically. Here, we leverage complexity measures including the global and local Rademacher complexities, in order to provide upper bounds on the generalization and estimation errors of model-based networks. We show that the generalization abilities of model-based networks for sparse recovery outperform those of regular ReLU networks, and derive practical design rules that allow to construct model-based networks with guaranteed high generalization. We demonstrate through a series of experiments that our theoretical insights shed light on a few behaviours experienced in practice, including the fact that ISTA and ADMM networks exhibit higher generalization abilities (especially for small number of training samples), compared to ReLU networks.