  Variational inference (VI) is a specific type of approximate Bayesian
inference that approximates an intractable posterior distribution with a
tractable one. VI casts the inference problem as an optimization problem, more
specifically, the goal is to maximize a lower bound of the logarithm of the
marginal likelihood with respect to the parameters of the approximate
posterior. Reinforcement learning (RL) on the other hand deals with autonomous
agents and how to make them act optimally such as to maximize some notion of
expected future cumulative reward. In the non-sequential setting where agents'
actions do not have an impact on future states of the environment, RL is
covered by contextual bandits and Bayesian optimization. In a proper sequential
scenario, however, where agents' actions affect future states, instantaneous
rewards need to be carefully traded off against potential long-term rewards.
This manuscript shows how the apparently different subjects of VI and RL are
linked in two fundamental ways. First, the optimization objective of RL to
maximize future cumulative rewards can be recovered via a VI objective under a
soft policy constraint in both the non-sequential and the sequential setting.
This policy constraint is not just merely artificial but has proven as a useful
regularizer in many RL tasks yielding significant improvements in agent
performance. And second, in model-based RL where agents aim to learn about the
environment they are operating in, the model-learning part can be naturally
phrased as an inference problem over the process that governs environment
dynamics. We are going to distinguish between two scenarios for the latter: VI
when environment states are fully observable by the agent and VI when they are
only partially observable through an observation distribution.
