  We consider the problem of learning stabilizable systems governed by
nonlinear state equation $h_{t+1}=\phi(h_t,u_t;\theta)+w_t$. Here $\theta$ is
the unknown system dynamics, $h_t $ is the state, $u_t$ is the input and $w_t$
is the additive noise vector. We study gradient based algorithms to learn the
system dynamics $\theta$ from samples obtained from a single finite trajectory.
If the system is run by a stabilizing input policy, we show that
temporally-dependent samples can be approximated by i.i.d. samples via a
truncation argument by using mixing-time arguments. We then develop new
guarantees for the uniform convergence of the gradients of empirical loss.
Unlike existing work, our bounds are noise sensitive which allows for learning
ground-truth dynamics with high accuracy and small sample complexity. Together,
our results facilitate efficient learning of the general nonlinear system under
stabilizing policy. We specialize our guarantees to entry-wise nonlinear
activations and verify our theory in various numerical experiments
