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We develop a corrective mechanism for neural network approximation: the totalavailable non-linear units are divided into multiple groups and the first groupapproximates the function under consideration, the second group approximatesthe error in approximation produced by the first group and corrects it, thethird group approximates the error produced by the first and second groupstogether and so on. This technique yields several new representation andlearning results for neural networks. First, we show that two-layer neuralnetworks in the random features regime (RF) can memorize arbitrary labels forarbitrary points under under Euclidean distance separation condition using$\tilde{O}(n)$ ReLUs which is optimal in $n$ up to logarithmic factors. Next,we give a powerful representation result for two-layer neural networks withReLUs and smoothed ReLUs which can achieve a squared error of at most$\epsilon$ with $O(C(a,d)\epsilon^{-1/(a+1)})$ for $a \in \mathbb{N}\cup\{0\}$when the function is smooth enough (roughly when it has $\Theta(ad)$ boundedderivatives). In certain cases $d$ can be replaced with effective dimension $q\ll d$. Previous results of this type implement Taylor series approximationusing deep architectures. We also consider three-layer neural networks and showthat the corrective mechanism yields faster representation rates for smoothradial functions. Lastly, we obtain the first $O(\mathrm{subpoly}(1/\epsilon))$upper bound on the number of neurons required for a two layer network to learnlow degree polynomials up to squared error $\epsilon$ via gradient descent.Even though deep networks can express these polynomials with$O(\mathrm{polylog}(1/\epsilon))$ neurons, the best learning bounds on thisproblem require $\mathrm{poly}(1/\epsilon)$ neurons.

A Corrective View of Neural Networks: Representation, Memorization and Learning