On bounds for norms of reparameterized ReLU artificial neural network parameters: sums of fractional powers of the Lipschitz norm control the network parameter vector
This work provides theoretical insights into the relationship between network parameters and Lipschitz norms, which is incremental for understanding neural network stability and optimization in machine learning.
The authors tackled the problem of bounding the norm of ReLU neural network parameters in terms of the Lipschitz norm of the network's realization function, proving that for shallow networks, the parameter norm is bounded above by sums of fractional powers of the Lipschitz norm, specifically with exponents 1/2 and 1, and showing this does not hold for other norms like Hölder or Sobolev-Slobodeckij.
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 Hölder 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.