A Theoretical Understanding of Gradient Bias in Meta-Reinforcement Learning

Abstract

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 extbf{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 (KαKσ̂ In|τ|−0.5) mph{w.r.t.} inner-loop update step K, learning rate α, estimate variance σ̂ 2In and sample size |τ|, and 2) the multi-step Hessian estimation bias Δ̂ H due to the use of autodiff, which has a polynomial impact ((K−1)(Δ̂ H)K−1) 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.