  Gradient-based Meta-RL (GMRL) refers to methods that maintain two-level
optimisation procedures wherein the outer-loop meta-learner guides the
inner-loop gradient-based reinforcement learner to achieve fast adaptations. In
this paper, we develop a unified framework that describes variations of GMRL
algorithms and points out that existing stochastic meta-gradient estimators
adopted by GMRL are actually \textbf{biased}. Such meta-gradient bias comes
from two sources: 1) the compositional bias incurred by the two-level problem
structure, which has an upper bound of
$\mathcal{O}\big(K\alpha^{K}\hat{\sigma}_{\text{In}}|\tau|^{-0.5}\big)$
\emph{w.r.t.} inner-loop update step $K$, learning rate $\alpha$, estimate
variance $\hat{\sigma}^{2}_{\text{In}}$ and sample size $|\tau|$, and 2) the
multi-step Hessian estimation bias $\hat{\Delta}_{H}$ due to the use of
autodiff, which has a polynomial impact
$\mathcal{O}\big((K-1)(\hat{\Delta}_{H})^{K-1}\big)$ on the meta-gradient bias.
We study tabular MDPs empirically and offer quantitative evidence that
testifies our theoretical findings on existing stochastic meta-gradient
estimators. Furthermore, we conduct experiments on Iterated Prisoner's Dilemma
and Atari games to show how other methods such as off-policy learning and
low-bias estimator can help fix the gradient bias for GMRL algorithms in
general.
