论文标题
多尺度平均场模型的偏差和完善
Bias and Refinement of Multiscale Mean Field Models
论文作者
论文摘要
平均场近似是一种强大的技术，已在许多设置中用于研究大规模随机系统。在两次计算系统的情况下，近似是通过比例参数和平均原理的使用结合而获得的。本文分析了此“平均”平均字段模型的近似错误$（\ boldsymbol {\ boldsymbol {x}，\ boldsymbol {y}）$，其中慢的组件$ \ boldsymbol {x} $描述了与快速更改环境$ $ bolds $ bold $ bolds的相互作用的粒子的人群。该模型通过缩放系数$ n $进行参数化，例如与快速组件的不变动力相比，随着$ n $的增长，人口大小会大大减少慢组分的跳跃大小。我们表明，在相对温和的条件下，“平均”平均字段近似具有$ o（1/n）$的偏差，而$ \ mathbb {e} [\ boldsymbol {x x}] $。如果模型呈指数稳定，则在瞬态策略中的任何连续性能指标以及稳态下，这是正确的。为了更进一步，我们得出了稳态的偏置校正项，从中定义了一个新的近似值，称为精制的“平均”平均场近似值，其偏差为$ o（1/n^2）$。这种精制的“平均”平均场近似允许计算精确的近似值，即使对于小缩放因子，即$ n \ of $ n \ of 10 -50 $。我们通过应用于随机访问CSMA模型来说明开发的框架和准确性结果。
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, the `average' 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 refined `average' 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.
