  We establish an equivalence between the $\ell_2$-regularized solution path
for a convex loss function, and the solution of an ordinary differentiable
equation (ODE). Importantly, this equivalence reveals that the solution path
can be viewed as the flow of a hybrid of gradient descent and Newton method
applying to the empirical loss, which is similar to a widely used optimization
technique called trust region method. This provides an interesting algorithmic
view of $\ell_2$ regularization, and is in contrast to the conventional view
that the $\ell_2$ regularization solution path is similar to the gradient flow
of the empirical loss.New path-following algorithms based on homotopy methods
and numerical ODE solvers are proposed to numerically approximate the solution
path. In particular, we consider respectively Newton method and gradient
descent method as the basis algorithm for the homotopy method, and establish
their approximation error rates over the solution path. Importantly, our theory
suggests novel schemes to choose grid points that guarantee an arbitrarily
small suboptimality for the solution path. In terms of computational cost, we
prove that in order to achieve an $\epsilon$-suboptimality for the entire
solution path, the number of Newton steps required for the Newton method is
$\mathcal O(\epsilon^{-1/2})$, while the number of gradient steps required for
the gradient descent method is $\mathcal O\left(\epsilon^{-1}
\ln(\epsilon^{-1})\right)$. Finally, we use $\ell_2$-regularized logistic
regression as an illustrating example to demonstrate the effectiveness of the
proposed path-following algorithms.
