  Offline reinforcement learning (RL), where the agent aims to learn the
optimal policy based on the data collected by a behavior policy, has attracted
increasing attention in recent years. While offline RL with linear function
approximation has been extensively studied with optimal results achieved under
certain assumptions, many works shift their interest to offline RL with
non-linear function approximation. However, limited works on offline RL with
non-linear function approximation have instance-dependent regret guarantees. In
this paper, we propose an oracle-efficient algorithm, dubbed Pessimistic
Nonlinear Least-Square Value Iteration (PNLSVI), for offline RL with non-linear
function approximation. Our algorithmic design comprises three innovative
components: (1) a variance-based weighted regression scheme that can be applied
to a wide range of function classes, (2) a subroutine for variance estimation,
and (3) a planning phase that utilizes a pessimistic value iteration approach.
Our algorithm enjoys a regret bound that has a tight dependency on the function
class complexity and achieves minimax optimal instance-dependent regret when
specialized to linear function approximation. Our work extends the previous
instance-dependent results within simpler function classes, such as linear and
differentiable function to a more general framework.
