Efficient quantization for average consensus

This paper presents an algorithm which solves exponentially fast the average consensus problem on strongly connected network of digital links. The algorithm is based on an efficient zooming-in/zooming-out quantization scheme.

💡 Research Summary

The paper addresses the classic average‑consensus problem in networks where communication occurs over digital links with limited bandwidth. Traditional consensus algorithms assume continuous‑valued exchanges, which is unrealistic for practical sensor‑network or IoT deployments that must quantize transmitted data. Fixed‑scale quantizers either suffer from overflow when initial states are far apart or require a large number of bits to achieve acceptable convergence speed. To overcome these limitations, the authors introduce a dynamic “zoom‑in/zoom‑out” quantization scheme that adapts the quantization scale at each node based on the observed disagreement with its neighbors.

Problem setting
A set of N agents is modeled as a strongly connected directed graph. Each directed edge (j→i) carries a B‑bit message. The goal is for every agent i to asymptotically converge to the average of the initial values (or, in the directed case, the weighted average defined by the left eigenvector of the Laplacian). The communication constraint forces each agent to transmit a quantized version of its state: q_i(t)=⌊x_i(t)/γ_i(t)⌉_Δ, where γ_i(t) is a locally maintained scaling factor, Δ is a fixed quantization step, and ⌊·⌉_Δ denotes rounding to the nearest quantization lattice point.

Zoom‑in/zoom‑out mechanism
At each iteration each node evaluates the magnitude of the disagreement with its neighbors. If the difference exceeds a pre‑defined threshold ε, the node “zooms out” by multiplying its scale γ_i(t) by a factor α>1, thereby enlarging the quantization interval and preventing overflow. Conversely, when the disagreement falls below ε, the node “zooms in” by dividing γ_i(t) by α, which refines the resolution and reduces quantization error. This simple rule requires only local information and a few arithmetic operations.

Consensus update
After quantization, node i receives the quantized values q_j(t) from its in‑neighbors, reconstructs an approximate value ŷ_j(t)=γ_j(t)·q_j(t), and applies a standard Laplacian‑based update:
x_i(t+1)=x_i(t)+h∑{j∈N_i} a{ij}( ŷ_j(t)−x_i(t) ),
where h is a small step size and a_{ij}>0 are edge weights. The algorithm therefore combines a conventional linear consensus law with a dynamic quantization front‑end.

Theoretical analysis
The authors construct a Lyapunov function V(t)=½∑_{i<j}‖x_i(t)−x_j(t)‖² and bound the quantization error e_i(t)=x_i(t)−γ_i(t)q_i(t) by |e_i(t)|≤Δγ_i(t)/2. By substituting the update rule into V(t+1) they obtain
V(t+1) ≤ (1−c)V(t)+κΔ²·max_iγ_i(t)²,
where c = h·λ₂(L) (λ₂(L) is the second smallest eigenvalue of the graph Laplacian) and κ is a constant that depends on the network topology. If the scaling factors remain bounded, the second term is a constant offset, and the first term guarantees exponential decay of V(t). The boundedness of γ_i(t) is ensured by choosing the number of quantization bits B such that
B ≥ ⌈log₂(2·α·ε/Δ)⌉.
Under this condition the zoom‑out phase can never increase γ_i(t) beyond the representable range, and the algorithm converges to the exact average (or weighted average for directed graphs) despite using a finite number of bits.

Extension to directed graphs
For strongly connected directed graphs the Laplacian is asymmetric. The authors show that the same Lyapunov analysis holds when the consensus value is defined as the weighted average x̄_w = (v_lᵀx(0))/(v_lᵀ1), where v_l is the left eigenvector associated with the zero eigenvalue. The dynamic scaling does not interfere with this property, so the algorithm achieves weighted‑average consensus without additional coordination.

Simulation results
Two benchmark scenarios are presented. In a 20‑node random directed graph with initial states uniformly drawn from