  In this paper, we made an extension to the convergence analysis of the
dynamics of two-layered bias-free networks with one $ReLU$ output. We took into
consideration two popular regularization terms: the $\ell_1$ and $\ell_2$ norm
of the parameter vector $w$, and added it to the square loss function with
coefficient $\lambda/2$. We proved that when $\lambda$ is small, the weight
vector $w$ converges to the optimal solution $\hat{w}$ (with respect to the new
loss function) with probability $\geq (1-\varepsilon)(1-A_d)/2$ under random
initiations in a sphere centered at the origin, where $\varepsilon$ is a small
value and $A_d$ is a constant. Numerical experiments including phase diagrams
and repeated simulations verified our theory.
