Real-valued average consensus over noisy quantized channels Andrea - PowerPoint PPT Presentation

Real-valued average consensus over noisy quantized channels Andrea Censi Richard Murray Control & Dynamical Systems, California Institute of Technology

Consensus problems Consensus: reach the agreement of agent beliefs or agent states, ■ respecting the given communication constraints. Basic average consensus problem: x i ( k ) → 1 n ∑ x i ( 0 ) . ■ Interesting to me because it is an example of distributed computation ■ done by a network of simple units. Example success story of control-theory + computation: ■ R. W. Brockett, "Dynamical Systems That Sort Lists, Diagonalize Matrices and Solve Linear Programming Problems," – magic formula: ˙ H = [ H , [ N , H ]] . Computation/control on distributed/noisy substrates will be an ■ important topic: neuronal networks (neuroscience) ◆ noisy electronic components (precision vs. efficiency) ◆ chemical reaction networks ◆ 2 / 11

Ideas from neuroscience The brain is the only instance of intelligence we know. We are very ■ very far from understanding how it works. How about neurons ? ■ Asynchronous distributed computation using spikes. ◆ They are slow with respect to the dynamics they control (e.g. fruit ◆ fly). They are noisy. ◆ Lots of models (we don’t have a clue of what is important) ◆ Simplest non-trivial: linear sum of inputs + noisy nonlinearity. ■ Can a control theorist tell something interesting? ■ Useless things to prove: Interesting things to prove: “stability” computational properties ■ ■ “synchronization” adaptation/learning ■ ■ Can a noisy spiking network solve the consensus problem? ■ 3 / 11

Some related work Real-valued consensus over quantized channels is a two-part strategy: Communication strategy: decide the value y j ( k ) ∈ Z to send. 1. Update strategy: update the node’s state x i ( k ) based on received y j ( k ) 2. [Aysal et al ’07]: Given P stochastic, let ■ x i ( k ) = ∑ j P i , j y j ( k ) y j ( k ) = q p ( x j ( k )) � ⌈ x ⌉ with probability x − ⌊ x ⌋ Uses “probabilistic quantization” q p ( x ) = ⌊ x ⌋ otherwise Results: consensus is reached to a value τ ∈ Z ; E { τ } = average. [Carli et al. ’08]: Given P doubly stochastic, let ■ x i ( k + 1 ) = x i ( k ) − y i ( k ) + ∑ j P i , j y j ( k ) y j ( k ) = round ( x j ( k )) Results: the average is conserved; the consensus is not reached. 4 / 11

Model/approach Update strategy : We adapt from [Olfati-Saber ’07]: x i ( k + 1 ) = x i ( k ) + η � � ∆ ∑ j a ij y j ( k ) − x i ( k ) a ij = a ji is an element of the adjacency matrix; ∆ is the degree of the ■ graph; η ∈ ( 0, 1 ) a parameter. Communication strategy Assume y ( k ) = ψ ( x ( k )) , with ψ arbitrary function: ■ | ψ ( x ) − x | ≤ β y i y j ψ x j 5 / 11

Model/approach Update strategy : We adapt from [Olfati-Saber ’07]: x i ( k + 1 ) = x i ( k ) + η � � ∆ ∑ j a ij y j ( k ) − x i ( k ) a ij = a ji is an element of the adjacency matrix; ∆ is the degree of the ■ graph; η ∈ ( 0, 1 ) a parameter. Communication strategy Assume y ( k ) = ψ ( x ( k )) , with ψ arbitrary function: ■ | ψ ( x ) − x | ≤ β + � y i − y j − ψ x j 5 / 11

Model/approach Update strategy : We adapt from [Olfati-Saber ’07]: x i ( k + 1 ) = x i ( k ) + η � � ∆ ∑ j a ij y j ( k ) − x i ( k ) a ij = a ji is an element of the adjacency matrix; ∆ is the degree of the ■ graph; η ∈ ( 0, 1 ) a parameter. Communication strategy Assume y ( k ) = ψ ( x ( k )) , with ψ arbitrary function: ■ | ψ ( x ) − x | ≤ β � � y j ( k ) = x j ( k ) − c j ( k ) ψ � � c j ( k + 1 ) = c j ( k ) + y j ( k ) − x j ( k ) � �� � transmission error Has the flavor of a self-inhibitory action potential. 5 / 11

Behavior of the drift Define the drift d ( k ) as ■ � � 1 � � d ( k ) � n ∑ i x i ( k ) − α � � � � α � 1 n ∑ i x i ( 0 ) is the goal state. ◆ Proposition: The drift is bounded: ■ d ( k ) ≤ ηβ β is the bound on the quantization error ◆ η is the speed of the update strategy ◆ By choosing η , we can make the drift as small as desired. ■ 6 / 11

Behavior of the disagreement error Take as an error measure the average disagreement: ■ � � 1/2 1 � � 2 ϕ ( k ) � n ∆ ∑ x i ( k ) − x j ( k ) a ij i , j ∆ is the degree of the graph ( n ∆ ≃ number of edges) ◆ Proposition: Eventually, the disagreement is bounded by: ■ √ 6 · ηβ · λ n { L } | ϕ ( k ) | ≤ λ 2 { L } λ 2 { L } is the second smallest eigenvalue, ̸ = 0 if graph connected. ◆ β is the bound on the quantization error ◆ η is the speed of the update strategy ◆ By choosing η , we can make the disagreement as small as desired. ■ 7 / 11

Comparison Method Drift Disagreement d ( k ) = 0 ϕ ( k ) → 0 No quanti- zation d ( k ) = 0 ϕ ( k ) → c > 0 Carli et al. d ( k ) ̸ = 0 ϕ ( k ) → 0 Aysal et al. k → ∞ ϕ ( k ) ≤ c · ηβλ n { L } Proposed d ( k ) ≤ ηβ lim λ 2 { L } strategy Therefore, consensus can be reached with arbitrary precision. ■ But small η implies slow convergence. ■ 8 / 11

Characterization of the bound For some graphs, λ n L / λ 2 L depends on the number of nodes n . ■ yet the performance appear to be largely independent of n ◆ graph λ n L λ 2 L λ n L / λ 2 L star n 1 n complete n n 1 � 2 π � n 2 2 − 2 cos ring 4 n � π � π � � n 2 2 + 2 cos 2 − 2 cos path n n 9 / 11

Examples ψ = round; ring graph with n = 10 nodes. η = 0.1, overall behavior States 8 6 x 4 2 Disagreement log(x T Lx) 10 0 −10 Drift 0.05 d(k) 0 −0.05 100 200 300 400 500 600 700 800 900 1000 time steps 10 / 11

Examples ψ = round; ring graph with n = 10 nodes. η = 0.1, last 100 steps States 5.35 x 5.3 Disagreement log(x T Lx) −4 −6 −8 Drift 0.05 d(k) 0 −0.05 900 920 940 960 980 1000 time steps 10 / 11

Examples ψ = round; ring graph with n = 10 nodes. η = 0.05, overall behavior States 8 6 x 4 2 Disagreement log(x T Lx) 10 0 −10 Drift 0.02 d(k) 0 −0.02 100 200 300 400 500 600 700 800 900 1000 time steps 10 / 11

Examples ψ = round; ring graph with n = 10 nodes. η = 0.05, last 100 steps States 5.32 x 5.3 Disagreement log(x T Lx) −5 −6 −7 Drift 0.01 d(k) 0 −0.01 900 920 940 960 980 1000 time steps 10 / 11

Conclusions Consensus can be reached with arbitrary precision regardless of ■ quantization and noise. Possible improvements: ■ Characterization of convergence speed / precision tradeoffs with ◆ choosing η . Find better bounds ◆ In practice, the error appears independent of the number of ■ nodes. However, λ n { L } / λ 2 { L } ≃ O ( n 2 ) , for ring graphs. Consider with specific quantization functions ψ or topologies. ■ Prove that, if ψ deterministic, it converges to a periodic orbit ◆ 11 / 11