  Entropy regularized algorithms such as Soft Q-learning and Soft Actor-Critic,
recently showed state-of-the-art performance on a number of challenging
reinforcement learning (RL) tasks. The regularized formulation modifies the
standard RL objective and thus generally converges to a policy different from
the optimal greedy policy of the original RL problem. Practically, it is
important to control the sub-optimality of the regularized optimal policy. In
this paper, we establish sufficient conditions for convergence of a large class
of regularized dynamic programming algorithms, unified under regularized
modified policy iteration (MPI) and conservative value iteration (VI) schemes.
We provide explicit convergence rates to the optimality depending on the
decrease rate of the regularization parameter. Our experiments show that the
empirical error closely follows the established theoretical convergence rates.
In addition to optimality, we demonstrate two desirable behaviours of the
regularized algorithms even in the absence of approximations: robustness to
stochasticity of environment and safety of trajectories induced by the policy
iterates.
