  Reinforcement learning is a powerful paradigm for learning optimal policies
from experimental data. However, to find optimal policies, most reinforcement
learning algorithms explore all possible actions, which may be harmful for
real-world systems. As a consequence, learning algorithms are rarely applied on
safety-critical systems in the real world. In this paper, we present a learning
algorithm that explicitly considers safety, defined in terms of stability
guarantees. Specifically, we extend control-theoretic results on Lyapunov
stability verification and show how to use statistical models of the dynamics
to obtain high-performance control policies with provable stability
certificates. Moreover, under additional regularity assumptions in terms of a
Gaussian process prior, we prove that one can effectively and safely collect
data in order to learn about the dynamics and thus both improve control
performance and expand the safe region of the state space. In our experiments,
we show how the resulting algorithm can safely optimize a neural network policy
on a simulated inverted pendulum, without the pendulum ever falling down.
