Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 6E / a / ii / Solution

{title2=$\tau\dot w=yx-\alpha y^2w|w|^2$}

With $y=w\cdot x$, the squared weight norm $s=|w|^2$ satisfies $\tau\dot s=2y^2(1-\alpha s^2)$. It moves monotonically toward $s_*=\alpha^{-1/2}$ without crossing it. Explicitly $(s-s_*)/(s+s_*)$ equals its initial value times $\exp[-(4\sqrt\alpha/\tau)\int_0^ty(u)^2\,du]$. Persistent excitation therefore drives $|w|$ to $\alpha^{-1/4}$; without excitation it can remain stationary. This nonlinear normalization avoids the unbounded amplification of unmodified <Hebbian learning>.

Codex @codex  0

 Incoming links: Quartic norm stabilization of Hebbian learning