  The remarkable success of reinforcement learning (RL) heavily relies on
observing the reward of every visited state-action pair. In many real world
applications, however, an agent can observe only a score that represents the
quality of the whole trajectory, which is referred to as the {\em
trajectory-wise reward}. In such a situation, it is difficult for standard RL
methods to well utilize trajectory-wise reward, and large bias and variance
errors can be incurred in policy evaluation. In this work, we propose a novel
offline RL algorithm, called Pessimistic vAlue iteRaTion with rEward
Decomposition (PARTED), which decomposes the trajectory return into per-step
proxy rewards via least-squares-based reward redistribution, and then performs
pessimistic value iteration based on the learned proxy reward. To ensure the
value functions constructed by PARTED are always pessimistic with respect to
the optimal ones, we design a new penalty term to offset the uncertainty of the
proxy reward. For general episodic MDPs with large state space, we show that
PARTED with overparameterized neural network function approximation achieves an
$\tilde{\mathcal{O}}(D_{\text{eff}}H^2/\sqrt{N})$ suboptimality, where $H$ is
the length of episode, $N$ is the total number of samples, and $D_{\text{eff}}$
is the effective dimension of the neural tangent kernel matrix. To further
illustrate the result, we show that PARTED achieves an
$\tilde{\mathcal{O}}(dH^3/\sqrt{N})$ suboptimality with linear MDPs, where $d$
is the feature dimension, which matches with that with neural network function
approximation, when $D_{\text{eff}}=dH$. To the best of our knowledge, PARTED
is the first offline RL algorithm that is provably efficient in general MDP
with trajectory-wise reward.
