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The task of using machine learning to approximate the mapping$\mathbf{x}\mapsto\sum_{i=1}^d x_i^2$ with $x_i\in[-1,1]$ seems to be a trivialone. Given the knowledge of the separable structure of the function, one candesign a sparse network to represent the function very accurately, or evenexactly. When such structural information is not available, and we may only usea dense neural network, the optimization procedure to find the sparse networkembedded in the dense network is similar to finding the needle in a haystack,using a given number of samples of the function. We demonstrate that the cost(measured by sample complexity) of finding the needle is directly related tothe Barron norm of the function. While only a small number of samples is neededto train a sparse network, the dense network trained with the same number ofsamples exhibits large test loss and a large generalization gap. In order tocontrol the size of the generalization gap, we find that the use of explicitregularization becomes increasingly more important as $d$ increases. Thenumerically observed sample complexity with explicit regularization scales as$\mathcal{O}(d^{2.5})$, which is in fact better than the theoreticallypredicted sample complexity that scales as $\mathcal{O}(d^{4})$. Withoutexplicit regularization (also called implicit regularization), the numericallyobserved sample complexity is significantly higher and is close to$\mathcal{O}(d^{4.5})$.

Learning the mapping $\mathbf{x}\mapsto \sum_{i=1}^d x_i^2$: the cost of finding the needle in a haystack