  To ensure stability of learning, state-of-the-art generalized policy
iteration algorithms augment the policy improvement step with a trust region
constraint bounding the information loss. The size of the trust region is
commonly determined by the Kullback-Leibler (KL) divergence, which not only
captures the notion of distance well but also yields closed-form solutions. In
this paper, we consider a more general class of f-divergences and derive the
corresponding policy update rules. The generic solution is expressed through
the derivative of the convex conjugate function to f and includes the KL
solution as a special case. Within the class of f-divergences, we further focus
on a one-parameter family of $\alpha$-divergences to study effects of the
choice of divergence on policy improvement. Previously known as well as new
policy updates emerge for different values of $\alpha$. We show that every type
of policy update comes with a compatible policy evaluation resulting from the
chosen f-divergence. Interestingly, the mean-squared Bellman error minimization
is closely related to policy evaluation with the Pearson $\chi^2$-divergence
penalty, while the KL divergence results in the soft-max policy update and a
log-sum-exp critic. We carry out asymptotic analysis of the solutions for
different values of $\alpha$ and demonstrate the effects of using different
divergence functions on a multi-armed bandit problem and on common standard
reinforcement learning problems.
