Bias and Refinement of Multiscale Mean Field Models

Abstract: Mean field approximation is a powerful technique which has been used in many settings to study large-scale stochastic systems. In the case of two-timescale systems, the approximation is obtained by a combination of scaling arguments and the use of the averaging principle. This paper analyzes the approximation error of this average' mean field model for a two-timescale model $(\boldsymbol{X}, \boldsymbol{Y})$, where the slow component $\boldsymbol{X}$ describes a population of interacting particles which is fully coupled with a rapidly changing environment $\boldsymbol{Y}$. The model is parametrized by a scaling factor $N$, e.g. the population size, which as $N$ gets large decreases the jump size of the slow component in contrast to the unchanged dynamics of the fast component. We show that under relatively mild conditions, theaverage' mean field approximation has a bias of order $O(1/N)$ compared to $\mathbb{E}[\boldsymbol{X}]$. This holds true under any continuous performance metric in the transient regime, as well as for the steady-state if the model is exponentially stable. To go one step further, we derive a bias correction term for the steady-state, from which we define a new approximation called the refined average' mean field approximation whose bias is of order $O(1/N^2)$. This refinedaverage' mean field approximation allows computing an accurate approximation even for small scaling factors, i.e., $N\approx 10 -50$. We illustrate the developed framework and accuracy results through an application to a random access CSMA model.