SCIENCE CHINA Information Sciences, Volume 61, Issue 9: 092202(2018) https://doi.org/10.1007/s11432-016-9173-8

## Distributed regression estimation with incomplete data in multi-agent networks

• AcceptedJun 21, 2017
• PublishedJan 4, 2018
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### Abstract

In this paper, distributed regression estimation problem with incomplete data in a time-varying multi-agent network isinvestigated. Regression estimation is carried out based on local agentinformation with incomplete in the non-ignorable mechanism. By virtue of gradient-based design and adaptive filter,a distributed algorithm is proposed to deal with aregression estimation problem with incomplete data. With the help ofconvex analysis and stochastic approximation techniques, the exactconvergence is obtained for the proposed algorithm with incomplete dataand a jointly-connected multi-agent topology. Moreover, online regretanalysis is also given for real-time learning. Then, simulationsfor the proposed algorithm are also given to demonstrate how it cansolve the estimation problem in a distributed way, even when thenetwork configuration is time-varying.

### Acknowledgment

This work was supported by National Key Research and Development Program of China (Grant No. 2016YFB0901902) and National Natural Science Foundation of China (Grant Nos. 61573344, 61333001, 61374168).

### Supplement

Appendix

Proof of Lemma Lm3

With the observation noise of (14), we obtain

\begin{align} \mathbb{E}\|\epsilon^{i}_k\|=\mathbb{E}\|R^{\bar{A},i}_k\xi^{i}_{k}-R^{\bar{A},i}\xi^{i}_{k}+r^{y\bar{A},i}-y^{i}_{k}\bar{A}^{i}_{k}\|\leqslant \mathbb{E}\|R^{\bar{A},i}_k-R^{\bar{A},i}\|\|\xi^{i}_{k}\|+\mathbb{E}\|r^{y\bar{A},i}-y^{i}_{k}\bar{A}^{i}_{k}\|, \forall i \in \mathcal{N}. \tag{28} \end{align}

Therefore, for all $\epsilon>0$, there exists a $k_{1}$ ($k_{1}$ is an integer) for $k>~k_{1}$, such that $\|R^{\bar{A},i}_k-R^{\bar{A},i}\|<\epsilon$. Define $M_{1}=\max\{\|R^{\bar{A},i}_{1}-R^{\bar{A},i}\|,\|R^{\bar{A},i}_{2}-R^{\bar{A},i}\|,\ldots,\|R^{\bar{A},i}_{k_{1}}-R^{\bar{A},i}\|,\epsilon\}$. Then $\|R^{\bar{A},i}_k-R^{\bar{A},i}\|\leqslant~M_{1}$ for $k\geqslant~0$. Analogously, $\|r^{y\bar{A},i}_{k}-R^{\bar{A},i}\|\leqslant~M_{2}$ for $k\geqslant~0$. From Remark Rem3, we have $\|\xi^{i}_{k}\|<C_{x}$. Hence, $\mathbb{E}\|\epsilon_i(k)\|\leqslant~M_1C_x+M_2=M_{\epsilon},~\forall k\geqslant~0$. By (14), $d^{i}_{k}=\nabla~g_i(k)+\epsilon_i(k)$. Thus, $\mathbb{E}\|d^{i}_{k}\|\leqslant~\mathbb{E}\|\nabla~g_i(k)\|+\mathbb{E}\|\epsilon_i(k)\|\leqslant~C_g+M_{\epsilon}=M_d$, which is bounded.

Proof of Lemma Lm4

For all $i\in~\mathcal{N},\;k\geqslant~0$, define $p^{i}_{k+1}=\xi^{i}_{k+1}-\sum_{j=1}^{N}w_{ij}(k)\xi^j_{k}$. We rewrite (8) compactly in terms of $\Psi(k,s)$ as follows: $\xi^{i}_{k+1}=\sum_{j=1}^{N}~[\Psi(k,0)]_{ij}\xi^{j}_{0}+p^{i}_{k+1}+\sum_{s=1}^{k}\sum_{j=1}^{N}~[\Psi(k,s)]_{ij}p^j_{s}$, for $k\geqslant~s$. Moreover, with Assumption Ass1 and by induction, the following equality holds: $\bar{\xi}_{k+1}=\frac{1}{N}\sum_{i=1}^{N}\xi^{i}_{0}+\frac{1}{N}\sum_{s=1}^{k+1}\sum_{j=1}^{N}p^j_{s}$. Consequently, we obtain that, for $i\in~\mathcal{N}$, $\xi^{i}_{k+1}-\bar{\xi}_{k+1}=\sum_{j=1}^{N}( [\Psi(k,0)]_{ij}-\frac{1}{N})\xi^{j}_{0}+(p^{i}_{k+1} -\frac{1}{N}\sum_{j=1}^{N}p^j_{k+1})+\sum_{s=1}^{k}\sum_{j=1}^{N} (~[\Psi(k,0)]_{ij}-\frac{1}{N})p^j_{s}$. Therefore, $\forall~i\in~\mathcal{N}$,

\begin{align} \|\xi^{i}_{k+1}-\bar{\xi}_{k+1}\|\leqslant\sum_{j=1}^{N} | [\Psi(k,0)]_{ij}-\frac{1}{N}\||\xi^{i}_{0}\|+\|p^{i}_{k+1}\| +\left\|\frac{1}{N}\sum_{j=1}^{N}p^j_{k+1}\right\|+\sum_{s=1}^{k}\sum_{j=1}^{N} | [\Psi(k,0)]_{ij}-\frac{1}{N}\||p^j_{s}\|. \tag{29} \end{align}

Plugging in the estimate of $\Psi(k,s)$ in Lemma Lm1 and $\|\xi^{i}_{0}\|\leqslant~\max_{1\leqslant~i\leqslant~N}\|\xi^{i}_{0}\|$, we have

\begin{align} \|\xi^{i}_{k+1}-\bar{\xi}_{k+1}\|\leqslant N\lambda \beta^{k}\max_{1\leqslant i\leqslant N}\|\xi^{i}_{0}\|+\|p^{i}_{k+1}\|+\frac{1}{N}\sum_{j=1}^{N}\|p^{j}_{k+1}\|+\lambda\sum_{s=1}^{k}\beta^{k-s}\sum_{j=1}^{N}\|p^{j}_{s}\|. \tag{30} \end{align}

Next, from the definition of $p_i(k)$, we get

\begin{align} \|p^i_{k+1}\|=\left\|P_{X}\left(\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\iota_{k}d^{i}_{k}\right)-\sum_{j=1}^{N}w_{ij}(k)\xi^j_{k}\right\|\leqslant\iota_{k}\|d^{i}_{k}\|. \tag{31} \end{align}

With (30) and (31), the proof is completed.

Proof of Theorem Thm2

From Theorem Thm1, $\|\xi^i_{k+1}-\bar{\xi}_{k+1}\|$ converges in mean. Then, on the base of Fatou's Lemma 1), the following relation holds $0\leqslant\mathbb{E}[\underset{k\rightarrow\infty}{\liminf}\|\xi^i_{k+1}-\bar{\xi}_{k+1}\|]\leqslant\underset{k\rightarrow\infty}{\liminf}\mathbb{E}[\|\xi^i_{k+1} -\bar{\xi}_{k+1}\|]=0$, which yields $\mathbb{E}[\underset{k\rightarrow\infty}{\liminf}\|\xi^i_{k+1}-\bar{\xi}_{k+1}\|]=0$. Therefore,$\underset{k\rightarrow\infty}{\liminf}\|\xi^i_{k+1}-\bar{\xi}_{k+1}\|=0$ holds almost surely. Since $~\|\xi^i_{k+1}-\bar{\xi}_{k}\|^2\leqslant\|\hat{\xi}^{i}_{k+1}-\bar{\xi}_{k}\|^2$,

\begin{align} \|\xi^i_{k+1}-\bar{\xi}_{k}\|^2\leqslant \|\hat{\xi}^{i}_{k+1}-\bar{\xi}_{k}\|^2\leqslant\sum_{j=1}^{N}w_{ij}(k)\|\xi^j_{k}-\bar{\xi}_{k}\|^2+\iota^2_k\|d^{i}_{k}\|^2+2\iota_{k}\|d^{i}_{k}\|\sum_{j=1}^{N}w_{ij}(k)\|\xi^j_{k}-\bar{\xi}_{k}\|. \tag{32} \end{align}

Note that $\sum_{i=1}^{N}~\|\xi^i_{k+1}-\bar{\xi}_{k}\|^2\leqslant\sum_{i=1}^{N}\sum_{j=1}^{N}w_{ij}(k)\|\xi^{j}_{k}-\bar{\xi}_{k}\|^2+\sum_{i=1}^{N}\iota^2_k\|d^{i}_{k}\|^2+2\sum_{i=1}^{N}\iota_{k}\|d^{i}_{k}\|\sum_{j=1}^{N}w_{ij}(k)\|\xi^j_{k}-\bar{\xi}_{k}\|$, $~i\in\mathcal{N}$, which implies $\sum_{i=1}^{N}\sum_{j=1}^{N}w_{ij}(k)\|\xi^{j}_{k}-\bar{\xi}_{k}\|^2=\sum_{i=1}^{N}\|\xi^{i}_{k}-\bar{\xi}_{k}\|^2$. Therefore,

\begin{align} \sum_{i=1}^{N}\|\xi^i_{k+1}-\bar{\xi}_{k}\|^2\leqslant\sum_{i=1}^{N}\|\xi^{i}_{k}-\bar{\xi}_{k}\|^2+\sum_{i=1}^{N}\iota^2_k\|d^{i}_{k}\|^2 +2\sum_{i=1}^{N}\iota_{k}\|d^{i}_{k}\|\sum_{j=1}^{N}w_{ij}(k)\|\xi^j_{k}-\bar{\xi}_{k}\|. \tag{33} \end{align}

Taking the conditional expectation of both side of (33) yields

\begin{align} \sum_{i=1}^{N}\mathbb{E}[\|\xi^i_{k+1}-\bar{\xi}_{k+1}\|^2|F_k]&\leqslant \sum_{i=1}^{N}\|\xi^{i}_{k}-\bar{\xi}_{k}\|^2+2M_{d}\sum_{j=1}^{N}\iota_{k}\|\xi^j_{k}-\bar{\xi}_{k}\|+N\iota^2_kM_{d}^2. \tag{34} \end{align}

According to Theorem 6.2 of [12], $\sum_{k=1}^{\infty}\iota_{k}\|\xi^j_{k}-\bar{\xi}_{k}\|<\infty$ with probability $1$. Therefore, together with $\sum_{k=1}^{\infty}N\iota^2_kM_{d}^2<\infty$, $\|\xi^i_{k+1}-\bar{\xi}_{k}\|^2$ converges almost surely by Lemma Lm2. Hence, the conclusion follows.

Rudin W. Real and Complex Analysis. New York: McGraw-Hill Book Company, 1986. 5–71.

Proof of Theorem Thm3

Clearly, $\|\xi^i_{k+1}-\xi^*\|^2\leqslant~\|\hat{\xi}^{i}_{k+1}-\xi^{*}\|^2$, and then $\|\xi^i_{k+1}-\xi^*\|^2\leqslant~\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*~\|^2~+ \iota^2_k \|d^{i}_{k}\|^2-2\iota_{k}(d^{i}_{k})^\text{T}~(\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*)$, which follows from [21] that, $\forall~x_{1},x_{2}$, $g(x_{2})\geqslant~g(x_{1})+\nabla~g(x_{1})^\text{T}(x_{2}-x_{1})$. Recalling that, $\mathbb{E}\|d^{i}_{k}\|\leqslant~M_{d}$ in Lemma Lm3 and $\|\nabla g^{i}(\xi)\|\leqslant~C_{g}$ in Remark Rem5, we have

\begin{align} (\nabla g^{i}_{k})^\text{T}\left(\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*\right)\geqslant g^{i}(\bar{\xi}_{k})-g_{i}(\xi^*)-C_{g}\left\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\bar{\xi}_{k}\right\|, \tag{35} \end{align}

and $\mathbb{E}[\epsilon_{i}^\text{T}(k)(\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*)]\leqslant \mathbb{E}\|\epsilon^{i}_k\|\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*\|$ for all $k=0,1,2,\ldots$. Therefore,

\begin{align} \mathbb{E}[\|\xi^i_{k+1}-\xi^*\|^2|F_{k}] \leqslant & \mathbb{E}\left[\left\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^* \right\|^2\Bigg|F_{k}\right]+\iota^2_k\mathbb{E}\|d^{i}_{k}\|^2+2\iota_{k}C_{g}\mathbb{E} \left[\left\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\bar{\xi}_{k}\right\|\Bigg|F_{k}\right] \\ &-2\iota_{k}\mathbb{E}\|\epsilon^{i}_k\|\left\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*\right\| -2\iota_{k}(g_{i}(\bar{\xi}_{k})-g_{i}(\xi^*)). \tag{36} \end{align}

By the double stochasticity of matrix $W(k)$,

\begin{align} \begin{cases}\displaystyle \sum_{i=1}^n \mathbb{E}\left[\left\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*\right\|^2 \Bigg|F_{k}\right]\leqslant \sum_{i=1}^n \|\xi^i_{k}-\xi^*\|^2, \tag{37} \\ \displaystyle \sum_{i=1}^n \mathbb{E}\left[\left\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\bar{\xi}_{k}\right\| \Bigg|F_{k}\right]\leqslant \sum_{i=1}^n \|\xi^i_{k}-\bar{\xi}_{k}\|. \end{cases} \tag{38} \end{align}

Then, with probability $1$, for $i\in~\mathcal{N}$, it holds

\begin{align} \sum_{i=1}^N \mathbb{E}[\|\xi^i_{k+1}-\xi^* \|^2|F_{k}]&\leqslant\sum_{i=1}^{N} \|\xi^i_{k}-\xi^*\|^2+w_{k}-v_{k}, \tag{39} \end{align}

where

\begin{align} \begin{cases}\displaystyle w_{k}=\sum_{i=1}^N\iota^2_k\mathbb{E}\|d^{i}_{k}\|^2+2\iota_{k}C_{g}\sum_{i=1}^N\mathbb{E}\|\xi^{i}_{k}-\bar{\xi}_{k}\|, \tag{40} \\ \displaystyle v_{k}=2\sum_{i=1}^N\iota_{k}\mathbb{E}[\|\epsilon^{i}_k\|]\left\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*\right\|+2\iota_{k}g(\bar{\xi}_{k}-g(\xi^*)). \end{cases} \tag{41} \end{align}

By Theorem 6.2 in [12], $\sum_{k=1}^{\infty}2\iota_{k}C_{g}\sum_{i=1}^N\mathbb{E}\|\xi^{i}_{k}-\bar{\xi}_{k}\|<\infty$. Since $\sum_{k=1}^{\infty}\iota^2_k<\infty$, $\sum_{k=1}^{\infty} \iota^2_k\mathbb{E}\|d^{i}_{k}\|^2~\leqslant \sum_{k=1}^{\infty}\iota^2_kNM_{d}^{2}<\infty$. Therefore, $\sum_{k=1}^{\infty}w_{k}<\infty$.

From Lemma Lm2, the sequence $\sum_{i=1}^{N}~\|\xi^i_{k}-\xi^*\|^2$ converges with probability 1 and $\sum_{k=1}^{\infty}v_{k}<\infty$.

As for $v_{k}$, according to the boundedness of $\xi^{i}_{k}$ and the ergodicity of $\bar{A}^{i}_{k}$, we conclude that $\lim_{k\rightarrow\infty}R^{\bar{A},i}_k\xi^{i}_{k}-R^{\bar{A},i}\xi^{i}_{k}=0$. Moreover, $\lim_{k\rightarrow\infty}r^{y\bar{A},i}-y^{i}_{k}\bar{A}^{i}_{k}=0$ by the stationary property of $y^{i}_{k},~a^{i}_{k}$.

Therefore

$$\lim_{k\rightarrow\infty}2\sum_{i=1}^N\mathbb{E}[\|\epsilon^{i}_{k}\|]\left\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*\right\|=0. \tag{42}$$

Similar to the demonstration of Theorem 6.2 in [12], we get $\sum_{k=1}^{\infty}2\sum_{i=1}^N\iota_{k}\mathbb{E}[\|\epsilon^{i}_k\|]\|\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\xi^*\|<\infty$, which implies $\sum_{k=1}^{\infty}2\iota_{k}(g(\bar{\xi}_{k})-g(\xi^*))<\infty$. Since $\sum_{k=1}^{\infty}2\iota_{k}(g(\bar{\xi}_{k})-g(\xi^*))<\infty$ and $\sum_{i=1}^{\infty}\iota_{k}=\infty$, $\liminf_{k\rightarrow \infty}~g(\bar{\xi}_{k})=g(\xi^*)$ holds almost surely. Therefore, $\lim_{k\rightarrow~\infty}\|\xi^i_{k}-\bar{\xi}_{k}~\|=0$ holds almost surely for all $i$, which yields the conclusion.

Proof of Lemma Lm5

Define $r^{i}_{k}=\xi^{i}_{k}-\hat{\xi}^{i}_{k}=P_{X}(\hat{\xi}^{i}_{k})-\hat{\xi}^{i}_{k}$. Since $X$ is convex and $W(k)$ is doubly stochastic, we have $\sum_{j=1}^{n}w_{ij}(k)x^{j}_k\in~X$, which leads to $\|r^{i}_{k+1}\||\leqslant~\|P_{X}(\hat{\xi}^{i}_{k+1})-\sum_{j=1}^{n}w_{ij}(k)\xi^{j}_{k}\|+\iota_{k}\|d^{i}_{k}\|\leqslant~2\iota_{k}\|d^{i}_{k}\|$. By Algorithm 1, we obtain $\bar{\xi}_{k+1}=\bar{\xi}_{k}-\frac{\iota_{k}}{N}\sum_{i=1}^{N}(\triangledown g^{i}_k+\epsilon^{i}_k)+\frac{1}{N}\sum_{i=1}^{N}r^{i}_{k+1}$. As a result, we can decompose $\|\bar{\xi}_{k+1}-\xi\|^{2}$ by

\begin{align} \|\bar{\xi}_{k+1}-\xi\|^{2}=\|\bar{\xi}_{k}-\xi\|^{2}+\frac{1}{N^{2}} \left\|\sum_{i=1}^{N}(r^{i}_{k+1}+\iota_{k}d^{i}_{k})\right\|^{2} +\frac{2}{N}\sum_{i=1}^{N}\langle r^{i}_{k+1},\bar{\xi}_{k}-\xi\rangle-\frac{2\iota_{k}}{N}\sum_{i=1}^{N}\langle\triangledown g^{i}_{k},\bar{\xi}_{k}-\xi\rangle-\frac{2\iota_{k}}{N}\sum_{i=1}^{N}\langle\epsilon^{i}_k, \bar{\xi}_{k}-\xi\rangle. \tag{43} \end{align}

Let us check $-\sum_{i=1}^{N}\langle~\triangledown g^{i}_{k},\bar{\xi}_{k}-\xi\rangle$. Based on Lemma (Lm1), we obtain

\begin{align} -\langle \triangledown g^{i}_{k},\bar{\xi}_{k}-\xi\rangle =& -\langle \triangledown g^{i}_{k},\bar{\xi}_{k}-\xi^{i}_{k}\rangle-\langle \triangledown g^{i}_{k},\xi^{i}_{k}-\xi\rangle \leqslant \|\triangledown g^{i}_{k}\|\|\bar{\xi}_{k}-\xi^{i}_{k}\|+g^{i}(\bar{\xi}_{k})-g^{i}_{k}-\frac{\mu}{2}\|\xi^{i}_{k}-x\|^{2}+g^{i}(\xi)-g^{i}(\bar{\xi}_{k}) \\ \leqslant & \|\triangledown g^{i}_{k}\|\|\bar{\xi}_{k}-\xi^{i}_{k}\|+\langle \triangledown \bar{g}^{i}_{k},\bar{\xi}_{k}-\xi^{i}_{k}\rangle -\frac{\mu}{2}\|\xi^{i}_{k}-\xi\|^{2}-\frac{\mu}{2}\|\xi^{i}_{k}-\bar{\xi}_{k}\|^{2}+g^{i}(\xi)-g^{i}(\bar{\xi}_{k}). \tag{44} \end{align}

Since $\langle~\triangledown~\bar{g}^{i}_{k},\bar{\xi}_{k}-\xi^{i}_{k}\rangle\leqslant~\|\triangledown~\bar{g}^{i}_{k}\|\|\bar{\xi}_{k}-\xi^{i}_{k}\|$ and $\|\xi^{i}_{k}-\xi\|^{2}+\|\xi^{i}_{k}-\bar{\xi}_{k}\|^{2}~\geqslant~\frac{1}{2}\|\bar{\xi}_{k}-\xi\|^{2}$, we can estimate $-\langle~\triangledown~g^{i}_{k},\bar{\xi}_{k}-\xi\rangle$ as follows: $-\langle~\triangledown~g^{i}_{k},\bar{\xi}_{k}-\xi\rangle\leqslant~(\|\triangledown~g^{i}_{k}\|+\|\triangledown~\bar{g}^{i}_{k}\|)\|\bar{\xi}_{k}-\xi^{i}_{k}\| +g^{i}(\xi)-g^{i}(\bar{\xi}_{k})-\frac{\mu}{4}\|\bar{\xi}_{k}-\xi\|^{2}$.

Summing up over $i=1,2,\ldots,N$, the following inequality holds:

\begin{align} -\sum_{i=1}^{N}\langle \triangledown g^{i}_{k},\bar{\xi}_{k}-\xi\rangle\leqslant \sum_{i=1}^{N} (\|\triangledown g^{i}_{k}\|+\|\triangledown \bar{g}^{i}_{k}\|) \|\bar{\xi}_{k}-\xi^{i}_{k}\|+g(\xi)-g(\bar{\xi}_{k})-\frac{\mu N}{4}\|\bar{\xi}_{k}-\xi\|^{2}. \tag{45} \end{align}

Next, it is not hard to get that, for $k=0,1,\ldots$,

\begin{align} -\sum_{i=1}^{N}\langle \epsilon^{i}_k,\bar{\xi}_{k}-\xi\rangle\leqslant \sum_{i=1}^{N}\|\epsilon^{i}_k\|\|\bar{\xi}_{k}-\xi^{i}_{k}\|+\sum_{i=1}^{N}\| \epsilon^{i}_k\|\|\xi^{i}_{k}-\xi\|. \tag{46} \end{align}

Then

\begin{align} \langle r^{i}_{k+1},\bar{\xi}_{k}-\xi\rangle\leqslant \langle r^{i}_{k+1},\bar{\xi}_{k}-\hat{\xi}^{i}_{k+1}\rangle+\langle P_{X}(\hat{\xi}^{i}_{k+1}) -\hat{\xi}^{i}_{k+1},\hat{\xi}^{i}_{k+1}-\xi\rangle. \tag{47} \end{align}

Because the projection operator satisfies the following inequality

\begin{align} \langle P_{X}(\hat{\xi})-\hat{\xi},\hat{\xi}-\xi\rangle\leqslant -\|P_{X}(\hat{\xi})-\hat{\xi}\|^{2}\leqslant 0,\; \forall \xi\in X, \tag{48} \end{align}

it follows from (48) with (47) that

\begin{align} \langle r^{i}_{k+1},\bar{\xi}_{k}-\xi\rangle\leqslant \langle r^{i}_{k+1},\bar{\xi}_{k}-\hat{\xi}^{i}_{k+1}\rangle\leqslant 2\iota_{k}\|d^{i}_{k}\|\|\bar{\xi}_{k} -\hat{\xi}^{i}_{k+1}\|. \tag{49} \end{align}

Moreover,

\begin{align} \frac{1}{N^{2}}\left\|\sum_{i=1}^{N}(r^{i}_{k+1}+\iota_{k}d^{i}_{k})\right\|^{2}= \frac{1}{N^{2}}\left(\sum_{i=1}^{N}(r^{i}_{k+1}+\iota_{k}d^{i}_{k})\right)^{2}\leqslant \frac{9\iota_{k}^{2}}{N^{2}}\left(\sum_{i=1}^{N}\|d^{i}_{k}\|\right)^{2}. \tag{50} \end{align}

Combining (45), (46), (49), and (50) with (43) yields the conclusion.

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• Figure 1

The topology of the networks.

• Figure 2

(Color online) The learning curves. (a) Missing data with threshold $[-10,-4,-1,-2,-10]^\top$; (b) missing data with threshold $[-10,-10,-10,-10,-10]^\top$.

• Figure 3

(Color online) The performances of $R(T)$ for all agents.

•

$R^{\bar{A},i}_k=(1-\rho_{k})R^{\bar{A},i}_{k-1}+\rho_{k}\bar{A}^{i}_{k}(\bar{A}^{i}_{k})^\text{T}$;

$r^{y\bar{A},i}_{k}=(1-\rho_{k})r^{y~\bar{A},i}_{k-1}+\rho_{k}y^{i}_{k}\bar{A}^{i}_{k}$;

$d^{i}_{k}=R^{\bar{A},i}_k\xi^{i}_{k}-r^{y\bar{A},i}_{k}$;

$\hat{\xi}^{i}_{k+1}=\sum_{j=1}^{N}w_{ij}(k)\xi^{i}_{k}-\iota_{k}d^{i}_{k}$;

$\xi^{i}_{k+1}=P_{X}(\hat{\xi}^{i}_{k+1})$.

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