  This paper considers an online (real-time) control problem that involves an
agent performing a discrete-time random walk over a finite state space. The
agent's action at each time step is to specify the probability distribution for
the next state given the current state. Following the set-up of Todorov, the
state-action cost at each time step is a sum of a state cost and a control cost
given by the Kullback-Leibler (KL) divergence between the agent's next-state
distribution and that determined by some fixed passive dynamics. The online
aspect of the problem is due to the fact that the state cost functions are
generated by a dynamic environment, and the agent learns the current state cost
only after selecting an action. An explicit construction of a computationally
efficient strategy with small regret (i.e., expected difference between its
actual total cost and the smallest cost attainable using noncausal knowledge of
the state costs) under mild regularity conditions is presented, along with a
demonstration of the performance of the proposed strategy on a simulated target
tracking problem. A number of new results on Markov decision processes with KL
control cost are also obtained.
