On bounds for norms of reparameterized ReLU artificial neural network parameters: sums of fractional powers of the Lipschitz norm control the network parameter vector

TL;DR

This paper establishes a two-sided bound linking the Lipschitz norm of shallow ReLU neural networks to sums of fractional powers of their parameter norms, revealing new insights into their functional and parameter space relationship.

Contribution

It proves that for shallow ReLU networks, the Lipschitz norm and sums of fractional powers of the parameter norm are equivalent up to constants, clarifying the bounds and limitations of such relationships.

Findings

The Lipschitz norm of the network can be bounded by sums of fractional powers of the parameter norm.

This bound only holds for the Lipschitz norm, not for Hölder or Sobolev-Slobodeckij norms.

The bounds are valid specifically for sums with exponents 1/2 and 1, not for the Lipschitz norm alone.

Abstract

It is an elementary fact in the scientific literature that the Lipschitz norm of the realization function of a feedforward fully-connected rectified linear unit (ReLU) artificial neural network (ANN) can, up to a multiplicative constant, be bounded from above by sums of powers of the norm of the ANN parameter vector. Roughly speaking, in this work we reveal in the case of shallow ANNs that the converse inequality is also true. More formally, we prove that the norm of the equivalence class of ANN parameter vectors with the same realization function is, up to a multiplicative constant, bounded from above by the sum of powers of the Lipschitz norm of the ANN realization function (with the exponents $ 1/2 $ and $ 1 $). Moreover, we prove that this upper bound only holds when employing the Lipschitz norm but does neither hold for H\"older norms nor for Sobolev-Slobodeckij norms. Furthermore, we prove that this upper bound only holds for sums of powers of the Lipschitz norm with the exponents $ 1/2 $ and $ 1 $ but does not hold for the Lipschitz norm alone.