  We design a new provably efficient algorithm for episodic reinforcement
learning with generalized linear function approximation. We analyze the
algorithm under a new expressivity assumption that we call "optimistic
closure," which is strictly weaker than assumptions from prior analyses for the
linear setting. With optimistic closure, we prove that our algorithm enjoys a
regret bound of $\tilde{O}(\sqrt{d^3 T})$ where $d$ is the dimensionality of
the state-action features and $T$ is the number of episodes. This is the first
statistically and computationally efficient algorithm for reinforcement
learning with generalized linear functions.
