Population loss in shallow ReLU networks: Bias&families of critical points

Abstract

The main result presented is a formula for the population loss in the student-teacher kernel model that is applicable to shallow ReLU networks with bias. This extends previous work of Choo and Saul (2009) and Brutzkus and Globerson (2017). The formula makes essential use of Owen's T-function. The necessary theory of the T-function is given and a high precision coding using MPFR for the T-function, based on an algorithm of Komelj (2023), is available on request. It is shown that various families of spurious minima described in past papers of Arjevani and the author extend to biased networks and that the loss is always strictly decreased when bias is added. The change in landscape geometry caused by adding bias appears to be relatively mild. Only the simplest examples are described in this paper where it is assumed that the number of inputs is equal to the number of neurons (this restriction is for reasons of length). A review of relevant previous results on unbiased networks is included. Aside from Gaussian statistics, the main mathematical tools and ideas come from analytic geometry (analytic and subanalytic sets, the Curve Selection Lemma).