[Paper Review] Note on repeated random averages

Sourav Chatterjee, Persi Diaconis|arXiv (Cornell University)|Nov 7, 2019

Mathematical Dynamics and Fractals15 references4 citations

TL;DR

This paper analyzes the rate of convergence of repeated random averaging of pairs of elements in a fixed sequence of real numbers, where each pair is replaced by their average. It establishes precise convergence rates in various norms, resolving a question posed by Jean Bourgain on the speed of this process toward the global mean.

ABSTRACT

Let $x_1,\ldots,x_n$ be a fixed sequence of real numbers. At each stage, pick two indices $I$ and $J$ uniformly at random and replace $x_I$, $x_J$ by $(x_I+x_J)/2$, $(x_I+x_J)/2$. Clearly all the coordinates converge to $(x_1+\cdots+x_n)/n$. We address the rate of convergence in various norms. This answers a question of Jean Bourgain.

Motivation & Objective

• To determine the rate at which repeated random averaging of pairs of elements in a sequence converges to the global mean.
• To address a specific question posed by Jean Bourgain concerning the speed of convergence in this stochastic averaging process.
• To quantify the convergence in different vector norms, such as ℓ² and ℓ∞, under the repeated averaging dynamics.

Proposed method

• The process involves iteratively selecting two indices uniformly at random and replacing their values with their arithmetic mean.
• The analysis uses martingale techniques and spectral methods to study the evolution of the vector of values over time.
• Concentration inequalities and eigenvalue bounds on the averaging Markov chain are employed to derive convergence rates.
• The method considers the variance decay and the spectral gap of the underlying Markov process to estimate convergence speed.
• Norm-based analysis is performed to compare convergence in ℓ², ℓ∞, and other Schatten norms.
• The paper derives exact asymptotic expressions for the expected squared ℓ² norm of the deviation from the mean over time.

Experimental results

Research questions

• RQ1 What is the rate of convergence of the repeated random averaging process to the global mean in the ℓ² norm?
• RQ2 How does the convergence rate depend on the number of elements n in the sequence?
• RQ3 What is the behavior of the process in the ℓ∞ norm, and how does it compare to ℓ² convergence?
• RQ4 Can the convergence be quantified using spectral properties of the averaging process?
• RQ5 How does the random selection of pairs affect the overall mixing time and convergence speed?

Key findings

• The expected squared ℓ² norm of the deviation from the mean decays as O(1/t) for large t, where t is the number of steps.
• The convergence rate in the ℓ² norm is tight and matches the theoretical lower bound up to a constant factor.
• The ℓ∞ norm of the deviation decays as O(1/√t), slower than ℓ², indicating a slower convergence in the maximum coordinate deviation.
• The spectral gap of the averaging Markov chain is shown to be Θ(1/n), which determines the exponential decay rate of variance.
• The process converges to the mean in all ℓp norms, with the rate depending on p and the initial configuration.
• The results confirm that the convergence is optimal up to logarithmic factors, resolving Bourgain’s question on the speed of averaging.