Summary of Contributions

In this chapter, we focus on the predictive coder where nodes utilize only their previous state values. Similar to Chapter 2, we utilize a dithered quantizer and thus model the quantization effect as temporally and spatially uncorrelated ad-ditive analog noise which is also uncorrelated with the message. Under these assumptions, we prove that our scheme converges to a consensus inL². Since

characterization of the MSE performance, where MSE is defined as:

is a challenging problem for general graphs, we focus on a particular communi-cation patterns, i.e., regular graphs with symmetric connectivity. In this setting, we provide explicit expressions and scaling laws for the MSE. In particular, we show that MSE is inversely proportional with the network connectivity and ini-tial observation correlation. In addition, we show that MSE is a function of the network size and scales as O(N⁻¹)when other parameters are fixed. We further study the characteristics of the average consensus algorithm under variable rate quantization scheme. In a special case where the quantization rates are chosen such that quantization noise variance decreases like a geometric series, we de-termine the rate regions where asymptotic quantization rate approaches zero while convergence inL² is guaranteed. In addition, we show that the rate re-gions achieving zero asymptotic rate and L² convergence with bounded MSE, also achieve finite-sum rates. Therefore, the sum of the quantization rates over the iterations, is indeed finite. We conclude that transmitting more bits in early iterations and decreasing them gradually results in a better performance than using a predictive encoder with a fixed rate.

In the rest of the chapter, the quantization operation is denoted as Q[·] and variables with a tilde mark are quantized quantities ˜x = Q[x]. Throughout our analysis, we assume that each node encodes and transmits a long block (length K) of state variables where the block entries are i.i.d. random variables. i.i.d.

assumption on the entries of the message blocks results in utilization of optimal lattice quantizers inR^K. Nice properties regarding quantization noise for opti-mal lattice quantizers are discussed in Section 3.2. Therefore, the quantization

rate per iteration, indicated by R(t), is per symbol rate where 2^2KR(t) is the total number of lattice cells in R^K, in general changing with the iteration index (t) and vector quantizer dimension (K). While our notation considers one symbol at a time, the use of vector quantization allows us to consider rates per symbol which are not integers. Finally, Assumptions A(1)− A(4) of Chapter 2 are still in order throughout this chapter.

3.1.2 Chapter Organization

In Section 3.2, we introduce a predictive coding scheme for quantized average consensus problem. Section 3.3 summarizes some properties of the regular net-works, discussed convergence proof and characterizes MSE performance of the scheme using fixed rate quantization under the discussed regular networks. We derive asymptotic rate behavior and optimum rate allocation for the variable rate quantization scenario in Section 3.4. Finally, we conclude the chapter with Section 3.5.

3.2 Quantized Average Consensus

In the rest of the chapter, we assume that initial sensor observations are zero mean random variables with identical distributions (not necessarily indepen-dent). As we have described in Chapter 2 in detail, sensors will exchange their values with their neighbors at each iteration and update their state values as a linear combination of these quantities. We are interested in the case where sensors quantize their values before exchanging their state values.

- Quantizer

Figure 3.1: Differential encoder/decoder diagram with dithering.

Although, nodes encode a long block of data, state and noise values and their statistics mentioned in the chapter are per dimension quantities. As the source coding scheme, we will be using a simple first order differential en-coder/decoder depicted in Fig. 3.1 to explore increasing temporal correlation.

This coding scheme has been first introduce in Section 2.4. Moreover, we will utilize a dithered quantization to whiten the quantization noise. Our channel model is based on protocol model where there are no channel errors between neighboring nodes [42].

We denote di(t)as the innovation to be transmitted from node i to its neigh-bor at time t, zi(t) as the dither to be added (which is uniformly distributed in K dimensional lattice and independent from the message di(t)), ˜d_i(t)as the quantized message, ˜x_i(t)as the noisy state reconstruction and wi(t)as the

quan-tization error. Then, the noisy state reconstruction is follows:

d_i(t) = x_i(t)− ˜xi(t− 1) (3.2)

d˜_i(t) = Q[d_i(t) + z_i(t)] = Q[x_i(t)− ˜xi(t− 1) + zi(t)]

= x_i(t)− ˜xi(t− 1) + zi(t) + w_i(t) (3.3)

˜

x_i(t) = d˜_i(t) + ˜x_i(t− 1) − zi(t)

= x_i(t) + w_i(t) (3.4)

where (3.2) is due to the fact that at each iteration the difference between the current state and the previous quantized state is transmitted, (3.3) follows from utilizing a uniform dither and modeling the quantization error as additive noise, and (3.4) follows since the previous quantized state value and dither are also known at the neighbors’ decoders.

Once the prediction is transmitted and the noisy states are reconstructed at the nodes, network update is performed through:

x(t + 1) = W ˜x(t) = W (x(t) + w(t)) (3.5) where w(t) = [w1(t), w₂(t), . . . , w_N(t)]^T and W is a doubly stochastic matrix. In the rest of the chapter, we denote our algorithm defined in (3.2)-(3.5) as quan-tized consensus algorithm with predictive coding (QCPC). We note that (3.5) is slightly different than our formulation in (2.9). In Chapter 2, we have uti-lized the fact that each node i has access to the unquantized version of its own value xi(t), therefore it should utilize this particular value in the updates to further reduce the quantization error. In this chapter, for the sake of mathe-matical brevity, we focus on the case where nodes always use quantized state values in the updates. We note that w(t) is assumed to be uncorrelated with the messages and is also spatially and temporally uncorrelated, zero mean ran-dom vector, uniformly distributed on the K dimensional lattice. While such

assumptions require strict conditions on the distribution of the message and quantization rates [86, 100], since we have utilized uniform subtractive dither-ing, this approximation is accurate for all quantization levels [105]. Moreover, since we utilize the K dimensional optimal lattice quantizers and assume that input message data is i.i.d., second order statistics of the noise variances is given by [99, 105]:

E{w²i(t)} = CKE{di2(t)}2^−2R(t) (3.6) whereE{·} denotes the statistical expectation and CKis a constant whose value is a function of the quantizer dimension K. We note that state and noise statis-tics, and quantization rates are per symbol (dimension).

At this point we would like to remind our readers Lemma 1 of Chapter 2.

Lemma 7 The nodes converge to a consensus in mean squared sense, if and only if the noise variance at each sensor converges to 0, i.e..E{w²i(t)} → 0 as t → ∞ ∀i ∈ V.

We will use the above lemma to prove the convergence of both constant rate and variable rate schemes in Sections 3.3 and 3.4.