Bias and Extrapolation in Markovian Linear Stochastic Approximation with Constant Stepsizes
This addresses bias issues in reinforcement learning algorithms like Temporal Difference learning for practitioners, though it is incremental as it builds on existing LSA theory.
The paper tackles bias in Linear Stochastic Approximation with constant stepsizes and Markovian data, proving convergence to a stationary distribution with geometric rates and showing that bias is proportional to stepsize, unlike in i.i.d. cases. It demonstrates that Richardson-Romberg extrapolation reduces bias exponentially, improving mean squared error theoretically and empirically.
We consider Linear Stochastic Approximation (LSA) with a constant stepsize and Markovian data. Viewing the joint process of the data and LSA iterate as a time-homogeneous Markov chain, we prove its convergence to a unique limiting and stationary distribution in Wasserstein distance and establish non-asymptotic, geometric convergence rates. Furthermore, we show that the bias vector of this limit admits an infinite series expansion with respect to the stepsize. Consequently, the bias is proportional to the stepsize up to higher order terms. This result stands in contrast with LSA under i.i.d. data, for which the bias vanishes. In the reversible chain setting, we provide a general characterization of the relationship between the bias and the mixing time of the Markovian data, establishing that they are roughly proportional to each other. While Polyak-Ruppert tail-averaging reduces the variance of the LSA iterates, it does not affect the bias. The above characterization allows us to show that the bias can be reduced using Richardson-Romberg extrapolation with $m\ge 2$ stepsizes, which eliminates the $m-1$ leading terms in the bias expansion. This extrapolation scheme leads to an exponentially smaller bias and an improved mean squared error, both in theory and empirically. Our results immediately apply to the Temporal Difference learning algorithm with linear function approximation, Markovian data, and constant stepsizes.