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Using a sparsity inducing penalty in artificial neural networks (ANNs) avoidsover-fitting, especially in situations where noise is high and the training setis small in comparison to the number of features. For linear models, such anapproach provably also recovers the important features with high probability inregimes for a well-chosen penalty parameter. The typical way of setting thepenalty parameter is by splitting the data set and performing thecross-validation, which is (1) computationally expensive and (2) not desirablewhen the data set is already small to be further split (for example,whole-genome sequence data). In this study, we establish the theoreticalfoundation to select the penalty parameter without cross-validation based onbounding with a high probability the infinite norm of the gradient of the lossfunction at zero under the zero-feature assumption. Our approach is ageneralization of the universal threshold of Donoho and Johnstone (1994) tononlinear ANN learning. We perform a set of comprehensive Monte Carlosimulations on a simple model, and the numerical results show the effectivenessof the proposed approach.

What needles do sparse neural networks find in nonlinear haystacks