  This chapter studies emerging cyber-attacks on reinforcement learning (RL)
and introduces a quantitative approach to analyze the vulnerabilities of RL.
Focusing on adversarial manipulation on the cost signals, we analyze the
performance degradation of TD($\lambda$) and $Q$-learning algorithms under the
manipulation. For TD($\lambda$), the approximation learned from the manipulated
costs has an approximation error bound proportional to the magnitude of the
attack. The effect of the adversarial attacks on the bound does not depend on
the choice of $\lambda$. In $Q$-learning, we show that $Q$-learning algorithms
converge under stealthy attacks and bounded falsifications on cost signals. We
characterize the relation between the falsified cost and the $Q$-factors as
well as the policy learned by the learning agent which provides fundamental
limits for feasible offensive and defensive moves. We propose a robust region
in terms of the cost within which the adversary can never achieve the targeted
policy. We provide conditions on the falsified cost which can mislead the agent
to learn an adversary's favored policy. A case study of TD($\lambda$) learning
is provided to corroborate the results.
