  Stability is one of the most fundamental requirements for systems synthesis.
In this paper, we address the stabilization problem for unknown linear systems
via policy gradient (PG) methods. We leverage a key feature of PG for Linear
Quadratic Regulator (LQR), i.e., it drives the policy away from the boundary of
the unstabilizing region along the descent direction, provided with an initial
policy with finite cost. To this end, we discount the LQR cost with a factor,
by adaptively increasing which gradient leads the policy to the stabilizing set
while maintaining a finite cost. Based on the Lyapunov theory, we design an
update rule for the discount factor which can be directly computed from data,
rendering our method purely model-free. Compared to recent work
\citep{perdomo2021stabilizing}, our algorithm allows the policy to be updated
only once for each discount factor. Moreover, the number of sampled
trajectories and simulation time for gradient descent is significantly reduced
to $\mathcal{O}(\log(1/\epsilon))$ for the desired accuracy $\epsilon$.
Finally, we conduct simulations on both small-scale and large-scale examples to
show the efficiency of our discount PG method.
