Soft Mode in the Dynamics of Over-realizable On-line Learning for Soft Committee Machines

TL;DR

This paper investigates the dynamics of over-realizable two-layer soft committee machines trained online, revealing a power-law convergence to perfect learning and the replication of teacher nodes by student nodes.

Contribution

It demonstrates that in over-realizable scenarios, the learning approach follows a power-law rather than exponential decay, with all student nodes learning and replicating teacher nodes.

Findings

Power-law approach to perfect learning in over-realizable case

Student nodes replicate teacher nodes when outputs are rescaled

Convergence dynamics differ from the realizable case

Abstract

Over-parametrized deep neural networks trained by stochastic gradient descent are successful in performing many tasks of practical relevance. One aspect of over-parametrization is the possibility that the student network has a larger expressivity than the data generating process. In the context of a student-teacher scenario, this corresponds to the so-called over-realizable case, where the student network has a larger number of hidden units than the teacher. For on-line learning of a two-layer soft committee machine in the over-realizable case, we find that the approach to perfect learning occurs in a power-law fashion rather than exponentially as in the realizable case. All student nodes learn and replicate one of the teacher nodes if teacher and student outputs are suitably rescaled.