Bifurcations of Renormalization Dynamics in Self-organizing Neural Networks

Abstract

Self-organizing neural networks (SONN) driven by softmax weight renormalization are capable of finding high quality solutions of difficult assignment optimization problems. The renormalization is shaped by a temperature parameter - as the system cools down the assignment weights become increasingly crisp. It has been reported that SONN search process can exhibit complex adaptation patterns as the system cools down. Moreover, there exists a critical temperature setting at which SONN is capable of powerful intermittent search through a multitude of high quality solutions represented as meta-stable states. To shed light on such observed phenomena, we present a detailed bifurcation study of the renormalization process. As SONN cools down, new renormalization equilibria emerge in a strong structure leading to a complex skeleton of saddle type equilibria surrounding an unstable maximum entropy point, with decision enforcing “one-hot” stable equilibria. This, in synergy with the SONN input driving process, can lead to sensitivity to annealing schedules and adaptation dynamics exhibiting signatures of complex dynamical behavior. We also show that (as hypothesized in earlier studies) the intermittent search by SONN can occur only at temperatures close to the first (symmetry breaking) bifurcation temperature.