More is less: inducing sparsity via overparameterization

Information and Inference A Journal of the IMA, vol. 12, pp. 1437–1460

Abstract

Abstract In deep learning, it is common to overparameterize neural networks, that is, to use more parameters than training samples. Quite surprisingly training the neural network via (stochastic) gradient descent leads to models that generalize very well, while classical statistics would suggest overfitting. In order to gain understanding of this implicit bias phenomenon, we study the special case of sparse recovery (compressed sensing) which is of interest on its own. More precisely, in order to reconstruct a vector from underdetermined linear measurements, we introduce a corresponding overparameterized square loss functional, where the vector to be reconstructed is deeply factorized into several vectors. We show that, if there exists an exact solution, vanilla gradient flow for the overparameterized loss functional converges to a good approximation of the solution of minimal $\ell _1$-norm. The latter is well-known to promote sparse solutions. As a by-product, our results significantly improve the sample complexity for compressed sensing via gradient flow/descent on overparameterized models derived in previous works. The theory accurately predicts the recovery rate in numerical experiments. Our proof relies on analyzing a certain Bregman divergence of the flow. This bypasses the obstacles caused by non-convexity and should be of independent interest.

Authors 3

1. Hung-Hsu Chou corresponding

   Ludwig-Maximilians-Universität München

   Affiliation as printed

   Mathematics Institute, Ludwig Maximilian University of Munich , Theresienstr. 39, 80333 Munich , Germany

2. Johannes Maly

   Ludwig-Maximilians-Universität München

   Affiliation as printed

   Mathematics Institute, Ludwig Maximilian University of Munich , Theresienstr. 39, 80333 Munich , Germany

3. Holger Rauhut Aachen Chair of Mathematics of Information Processing

   RWTH Aachen University

   Affiliation as printed

   Chair of Mathematics of Information Processing, RWTH Aachen University , Pontdriesch 12-14, 52062 Aachen , Germany