Adding a single memory per agent gives the fastest average consensus

Previous papers have proposed to add memory registers to the individual dynamics of discrete-time linear agents to move faster towards average consensus under interactions dictated by a given but unknown graph. They have proved that adding one memory slot per agent allows faster convergence. We here prove that this situation cannot be improved by adding more memory slots. We conclude by discussing a more general framework for our result in an algorithmic context. 1 Problem definition The basic consensus algorithm in discrete time [1] writes x(t+ 1) = x(t)− αLx(t) , x = [x1; x2; ...xN ] ∈ R , (1) where xi ∈ R is the state of agent i, α > 0 is a gain, and L is the Laplacian matrix characterizing interactions among agents: component j of Lx equals ∑N i=1 wj,i (xj − xi) with weights wj,i ≥ 0. Usually L contains many zeros, as each agent interacts with a limited number of fellows. For the algorithm to compute the average of the initial values xi(0), we assume wj,i = wi,j for all i, j. The Laplacian is then symmetric nonnegative definite, and if the interactions form a connected graph it has a single eigenvalue λ1 = 0 with eigenvector v1 = [1; 1; ...1], i.e. consensus. Correspondingly, the average 1 N ∑N i=1 xi(t) is invariant under (1). When (IN − αL) has nonnegative entries, with IN the N × N identity matrix, it can equivalently be viewed as the transition matrix of a Markov chain; symmetric L implies that the limiting distribution is uniform. For time-invariant L, the convergence speed of (1) is dictated by the largest eigenvalue, in modulus, of (IN − αL) excluding the trivial λ1 = 0. In the orthonormal basis corresponding to the eigenvectors of L (so-called “modes”), the system decouples into x̃i(t+ 1) = (1− αλi)x̃i(t) , i = 1, 2, ..., N, (2) with x̃i the coefficient of mode i. If we only know that the eigenvalues of L belong to an interval λi ∈ [λ, λ̄] ⊂ R>0 for i = 2, 3, ..., N , then the convergence speed — in terms of eigenvalues of (2) guaranteed over all λi ∈ [λ, λ̄] — is optimal when α is selected to satisfy (1− αλ) = −(1− αλ̄). This gives α = 2 λ̄+λ and worst eigenvalue μ = λ̄−λ λ̄+λ .