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We construct pairs of distributions $\mu_d, \nu_d$ on $\mathbb{R}^d$ suchthat the quantity $|\mathbb{E}_{x \sim \mu_d} [F(x)] - \mathbb{E}_{x \sim\nu_d} [F(x)]|$ decreases as $\Omega(1/d^2)$ for some three-layer ReLU network$F$ with polynomial width and weights, while declining exponentially in $d$ if$F$ is any two-layer network with polynomial weights. This shows that deep GANdiscriminators are able to distinguish distributions that shallowdiscriminators cannot. Analogously, we build pairs of distributions $\mu_d,\nu_d$ on $\mathbb{R}^d$ such that $|\mathbb{E}_{x \sim \mu_d} [F(x)] -\mathbb{E}_{x \sim \nu_d} [F(x)]|$ decreases as $\Omega(1/(d\log d))$ fortwo-layer ReLU networks with polynomial weights, while declining exponentiallyfor bounded-norm functions in the associated RKHS. This confirms that featurelearning is beneficial for discriminators. Our bounds are based on Fouriertransforms.

Depth and Feature Learning are Provably Beneficial for Neural Network Discriminators