  The $Q$-function is a central quantity in many Reinforcement Learning (RL)
algorithms for which RL agents behave following a (soft)-greedy policy w.r.t.
to $Q$. It is a powerful tool that allows action selection without a model of
the environment and even without explicitly modeling the policy. Yet, this
scheme can only be used in discrete action tasks, with small numbers of
actions, as the softmax cannot be computed exactly otherwise. Especially the
usage of function approximation, to deal with continuous action spaces in
modern actor-critic architectures, intrinsically prevents the exact computation
of a softmax. We propose to alleviate this issue by parametrizing the
$Q$-function implicitly, as the sum of a log-policy and of a value function. We
use the resulting parametrization to derive a practical off-policy deep RL
algorithm, suitable for large action spaces, and that enforces the softmax
relation between the policy and the $Q$-value. We provide a theoretical
analysis of our algorithm: from an Approximate Dynamic Programming perspective,
we show its equivalence to a regularized version of value iteration, accounting
for both entropy and Kullback-Leibler regularization, and that enjoys
beneficial error propagation results. We then evaluate our algorithm on classic
control tasks, where its results compete with state-of-the-art methods.
