Expected Values Estimated via Mean-Field Approximation are 1/N-Accurate

Abstract

Mean-field approximation is a powerful tool to study large-scale stochastic systems such as data-centers-one example being the famous power of two-choice paradigm. It is shown in the literature that under quite general conditions, the empirical measure of a system of N interacting objects converges at rate O (1/ √ N) to a deterministic dynamical system, called its mean-field approximation. In this paper, we revisit the accuracy of mean-field approximation by focusing on expected values. We show that, under almost the same general conditions, the expectation of any performance functional converges at rate O (1/N) to its mean-field approximation. Our result applies for finite and infinite-dimensional mean-field models. We also develop a new perturbation theory argument that shows that the result holds for the stationary regime if the dynamical system is asymptotically exponentially stable. We provide numerical experiments that demonstrate that this rate of convergence is tight and that illustrate the necessity of our conditions. As an example, we apply our result to the classical two-choice model. By combining our theory with numerical experiments, we claim that, as the load ρ goes to 1, the average queue length of a two-choice system with N servers is log 2 1 1−ρ + 1 2N (1−ρ) + O 1 N 2. CCS Concepts: • General and reference → Performance; • Mathematics of computing → Stochastic processes; Queueing theory; Ordinary differential equations;

A Refined Mean Field Approximation

“…The use of Stein's method to compute bounds for stationary distributions has been recently popularized in [4,12]. This methodology has then been used in [31] and further extended in [9,32] to establish the rate of convergence of stochastic processes to their mean eld approximation, in light or heavy trac. Our paper strongly relies on the methodology developed in those papers.…”

“…Our paper strongly relies on the methodology developed in those papers. In particular, it is shown in [9] that the mean eld approximation is 1/N -accurate. We improve this result as we are able, for a large class of models, to express the constant V h by a simple formula that can be easily evaluated numerically.…”

“…We improve this result as we are able, for a large class of models, to express the constant V h by a simple formula that can be easily evaluated numerically. Note that the idea of using h( ) + V h /N as an improved approximation was already proposed in [9] for the two-choice model, where the constant V h was estimated by simulation. The idea of improving classical approximations via Stein's method was also presented in [4] but, according to the authors, their computations seem hard to generalize to high dimensional settings.…”

“…Let B be the bounded set of Assumption (A5). We proceed as for the proof of [9,Theorem 3.2]. First, it should be noted that one may assume without loss of generality that h equals h( ) outside a bounded set.…”

“…The essential dierence lies in the fact that the h-th derivative of function x 7 ! R 1 0 t xdt is bounded on any bounded set, but not necessarily on the whole state-space E (see [9,Lemma 6.3]). This is why we need (A5).…”