  This paper studies model-based bandit and reinforcement learning (RL) with
nonlinear function approximations. We propose to study convergence to
approximate local maxima because we show that global convergence is
statistically intractable even for one-layer neural net bandit with a
deterministic reward. For both nonlinear bandit and RL, the paper presents a
model-based algorithm, Virtual Ascent with Online Model Learner (ViOlin), which
provably converges to a local maximum with sample complexity that only depends
on the sequential Rademacher complexity of the model class. Our results imply
novel global or local regret bounds on several concrete settings such as linear
bandit with finite or sparse model class, and two-layer neural net bandit. A
key algorithmic insight is that optimism may lead to over-exploration even for
two-layer neural net model class. On the other hand, for convergence to local
maxima, it suffices to maximize the virtual return if the model can also
reasonably predict the size of the gradient and Hessian of the real return.
