  Provably sample-efficient Reinforcement Learning (RL) with rich observations
and function approximation has witnessed tremendous recent progress,
particularly when the underlying function approximators are linear. In this
linear regime, computationally and statistically efficient methods exist where
the potentially infinite state and action spaces can be captured through a
known feature embedding, with the sample complexity scaling with the
(intrinsic) dimension of these features. When the action space is finite,
significantly more sophisticated results allow non-linear function
approximation under appropriate structural constraints on the underlying RL
problem, permitting for instance, the learning of good features instead of
assuming access to them. In this work, we present the first result for
non-linear function approximation which holds for general action spaces under a
linear embeddability condition, which generalizes all linear and finite action
settings. We design a novel optimistic posterior sampling strategy, TS^3 for
such problems, and show worst case sample complexity guarantees that scale with
a rank parameter of the RL problem, the linear embedding dimension introduced
in this work and standard measures of the function class complexity.
