Depth and Feature Learning are Provably Beneficial for Neural Network Discriminators

• URL: http://arxiv.org/abs/2112.13867v1
• Date: Mon, 27 Dec 2021 19:03:22 GMT
• Title: Depth and Feature Learning are Provably Beneficial for Neural Network Discriminators
• Authors: Carles Domingo-Enrich
• Abstract summary: We show that deep GAN discriminators are able to distinguish distributions that shallow discriminators cannot distinguish. This confirms that feature learning is beneficial for discriminators.
• Score: 3.04585143845864
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We construct pairs of distributions $\mu_d, \nu_d$ on $\mathbb{R}^d$ such that 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 GAN discriminators are able to distinguish distributions that shallow discriminators 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 $\Omega(1/(d\log d))$ for two-layer ReLU networks with polynomial weights, while declining exponentially for bounded-norm functions in the associated RKHS. This confirms that feature learning is beneficial for discriminators. Our bounds are based on Fourier transforms.

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