Event-Triggered Distributed Cooperative Learning Algorithms over Networks via Wavelet Approximation

Abstract

This paper investigates the problem of event-triggered distributed cooperative learning (DCL) over networks based on wavelet approximation theory, where each node only has access to local data which are produced by the same and unknown pattern (map or function). All nodes cooperatively learn this unknown pattern by exchanging learned information with their neighboring nodes under event-triggered strategy in order to remove unnecessary communications, so as to avoid the waste of network resources. For the above problem, two novel event-triggered continuous-time and discrete-time DCL algorithms are proposed to approximate the unknown pattern by using wavelet basis function. The proposed event-triggered DCL algorithms are used to train the optimal weight coefficient matrix of wavelet series. Moreover, the convergence of the proposed algorithms are presented by using the Lyapunov method, and the Zeno behavior is excluded as well by the strictly positive sampling interval. The illustrative examples are presented to show the efficiency and convergence of the proposed algorithms.

Appendices

Appendix

Proof of Theorem 1

Proof

(I) Consider the event-triggered DCL algorithm (8), the following Lyapunov function candidate is constructed:

$$\begin{aligned} {V}(t)=\frac{1}{2}\sum _{i=1}^{N}\Big ({\widetilde{W}}_i^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml}){\widetilde{W}}_i\Big ), \end{aligned}$$

(20)

where ${\widetilde{W}}_i=W^*-W_i$, $V:\mathbf{R }^{ml}\rightarrow \mathbf{R }$. It is easy to verify that

$$\begin{aligned} {V}(t)\ge \frac{\,{\underline{\theta }}\,}{\,2\,}\sum _{i=1}^{N}\parallel W^*-W_i(t)\parallel ^2. \end{aligned}$$

(21)

In addition, the following inequality holds [41]:

$$\begin{aligned} {V}(t)\le \frac{\,{\overline{\varTheta }}\,}{\,2\lambda _2\,}W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(t). \end{aligned}$$

(22)

Now, we are in the position to give the main result on the convergence of the algorithm (8). Consider the Lyapunov function candidate (20). Then, along the solution of (8), we have

$$\begin{aligned} \frac{dV(t)}{dt}=&-\sum _{i=1}^{N}\dot{W_i}^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})^{\text {T}}(W^*-W_i)\nonumber \\ =&-W^{*T}\sum _{i=1}^{N}(H_i^{\text {T}}H_i+\sigma _iI_{ml})\dot{W_i}(t)+\sum _{i=1}^{N}W_i^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})\dot{W_i}(t)\nonumber \\ =&-\gamma W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})(W(t)+e(t))\nonumber \\ =&-\gamma W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(t)-\gamma W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})e(t). \end{aligned}$$

(23)

Using Young’s inequality, leads to

$$\begin{aligned} -W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})e(t)\le&\frac{1}{2}W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})\frac{\epsilon }{2}({\mathcal {L}}\otimes I_{ml})^{\text {T}}W(t)+\frac{1}{2}e(t)^{\text {T}}\frac{2}{\epsilon }e(t)\nonumber \\ =\,&\frac{\epsilon }{4}W(t)^{\text {T}}(\mathcal {LL}^{\text {T}}\otimes I_{ml})W(t)+\frac{1}{\epsilon }e(t)^{\text {T}}e(t), \end{aligned}$$

(24)

where $\epsilon >0$ is a constant. Substituting (24) into (23), together with the trigger function (10), yields

$$\begin{aligned} \frac{dV(t)}{dt}\le&-\gamma W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})(W(t))+\frac{\gamma \epsilon }{4}W(t)^{\text {T}}(\mathcal {LL}^{\text {T}}\otimes I_{ml})W(t)+\frac{\gamma }{\epsilon }e(t)^{\text {T}}e(t)\nonumber \\ \le&-\gamma W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(t)+\frac{\gamma \epsilon \eta }{4}W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(t)+\frac{Nc\gamma }{\epsilon }e^{-\alpha t}\nonumber \\ =&-(1-\frac{\epsilon \eta }{4})\gamma W(t)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(t)+\frac{Nc\gamma }{\epsilon }e^{-\alpha t}, \end{aligned}$$

(25)

where $\eta =\lambda _{max}(L)$.

From inequality (22), one has

$$\begin{aligned} \frac{dV(t)}{dt}\le -(1-\frac{\epsilon \eta }{4})\frac{2\gamma \lambda _2}{{\overline{\varTheta }}}V(t)+\frac{Nc\gamma }{\epsilon }e^{-\alpha t} =-\kappa V(t)+\rho e^{-\alpha t}, \end{aligned}$$

(26)

where $\kappa =(1-\frac{\epsilon \eta }{4})\frac{2\gamma \lambda _2}{{\overline{\varTheta }}}$, $\rho =\frac{Nc\gamma }{\epsilon }$. Then, it follows from inequality (26) that

$$\begin{aligned} e^{\kappa t}\frac{dV(t)}{dt}+\kappa e^{\kappa t}V(t)\le \rho e^{(\kappa -\alpha ) t}. \end{aligned}$$

(27)

Integrating both sides of inequality (27) from 0 to t leads to

$$\begin{aligned} V(t)\le \left( V(0)-\frac{\rho }{\kappa -\alpha }\right) e^{-\kappa t}+\frac{\rho }{\kappa -\alpha }e^{-\alpha t} =(V(0)-\zeta )e^{-\kappa t}+\zeta e^{-\alpha t}, \end{aligned}$$

(28)

where $\zeta =\frac{\rho }{\kappa -\alpha }=\frac{2Nc\gamma {\overline{\varTheta }}}{[(4-\epsilon \eta )\gamma \lambda _2-2\alpha {\overline{\varTheta }}]\epsilon }$.

This, together with inequality (21) leads to

$$\begin{aligned} \begin{aligned} \sum _{i=1}^{N}\parallel W^*-W_i(t)\parallel ^2\le&\frac{\,2\,}{{\underline{\theta }}}(V(0)-\zeta )e^{-\kappa t}+\frac{\,2\zeta \,}{{\underline{\theta }}}e^{-\alpha t}. \end{aligned} \end{aligned}$$

(29)

(II) In order to exclude Zeno behavior, we show that the inter-event times are lower-bounded by a positive constant $\tau _0$. First, we have ${\dot{e}}_i(t)=-{\dot{W}}_i(t)$ for $t\in [t^i_{k_i},t^i_{k_i+1})$ from error variable (9). Notice that $e_i(t^i_{k_i})=0$ and $e_i(t)=-\int _{t^i_{k_i}}^{\text {T}} {\dot{W}}_i(\tau ) d\tau$. It follows from the algorithm (8) that

$$\begin{aligned} \parallel e_i(t)\parallel \le \int _{t^i_{k_i}}^{\text {T}} \parallel {\dot{W}}_i(\tau ) \parallel d\tau \le \int _{t^i_{k_i}}^{\text {T}} \parallel {\dot{W}}(\tau ) \parallel d\tau . \end{aligned}$$

(30)

Substituting (11) into inequality (30), one has

$$\begin{aligned} \parallel e_i(t)\parallel&\le \int _{t^i_{k_i}}^{\text {T}} \parallel -\gamma (H^{\text {T}}H+\sigma \otimes I_{ml})^{-1}({\mathcal {L}}\otimes I_{ml})(W(\tau )+e(\tau )) \parallel d\tau \nonumber \\&\le \frac{\gamma }{{\underline{\theta }}}\int _{t^i_{k_i}}^{\text {T}} \parallel ({\mathcal {L}}\otimes I_{ml})(W(\tau )+e(\tau )) \parallel d\tau , \end{aligned}$$

(31)

where ${\underline{\theta }}=:\min \limits _{i\in {\mathcal {V}}}\theta _i$, with $\theta _i=\lambda _{min}(H_i^{\text {T}}H_i+\sigma _iI_{ml})$. Because of $({\mathcal {L}}\otimes I_{ml})(W(t)+e(t)-\mathbf{1 }_N\otimes W^*)=({\mathcal {L}}\otimes I_{ml})(W(t)+e(t))$, (31) can be rewritten as

$$\begin{aligned} \parallel e_i(t)\parallel&\le \frac{\gamma }{{\underline{\theta }}}\int _{t^i_{k_i}}^{\text {T}} \parallel ({\mathcal {L}}\otimes I_{ml})(W(\tau )+e(\tau )-\mathbf{1 }_N\otimes W^*) \parallel d\tau \nonumber \\&\le \frac{\gamma \parallel {\mathcal {L}}\parallel }{{\underline{\theta }}}\int _{t^i_{k_i}}^{\text {T}}\big (\parallel W(\tau )-\mathbf{1 }_N\otimes W^*\parallel +\parallel e(\tau )\parallel \big )d\tau . \end{aligned}$$

(32)

From the inequality (29) and the trigger function (10), one has

$$\begin{aligned} \parallel W(t)-\mathbf{1 }_N\otimes W^*\parallel +\parallel e(t)\parallel&\le \sqrt{\frac{2V(W(0))}{{\underline{\theta }}}}e^{-\frac{\kappa t}{2}}+\sqrt{\frac{\,2\zeta \,}{{\underline{\theta }}}}e^{-\frac{\alpha t}{2}}+\sqrt{Nc} e^{-\frac{\alpha t}{2}}\nonumber \\&\le \sqrt{\frac{2V(W(0))}{{\underline{\theta }}}}e^{-\frac{\kappa t^i_{k_i}}{2}}+\Big (\sqrt{\frac{\,2\zeta \,}{{\underline{\theta }}}}+\sqrt{Nc} \Big )e^{-\frac{\alpha t^i_{k_i}}{2}}. \end{aligned}$$

(33)

Substituting inequality (33) into (32), one has

$$\begin{aligned} \parallel e_i(t)\parallel \le \Big (\mu _1e^{-\frac{\kappa t^i_{k_i}}{2}}+\mu _2e^{-\frac{\alpha t^i_{k_i}}{2}}\Big )(t-t^i_{k_i}), \end{aligned}$$

(34)

where $\mu _1 = \frac{\gamma \parallel {{\mathcal {L}}} \parallel }{{\underline{\theta }}} \sqrt{\frac{2V(W(0))}{{\underline{\theta }}}}$, $\mu _2 = \frac{\gamma \parallel {\mathcal {L}}\parallel }{{\underline{\theta }}}$$\Big (\sqrt{\frac{\,2\zeta \,}{{\underline{\theta }}}}+\sqrt{Nc} \Big )$.

The next event will not be triggered before $\parallel e_i(t)\parallel =\sqrt{c} e^{-\frac{\alpha t}{2}}$. Thus, a lower bound on the inter-event intervals is given by $\tau _0=t-t^i_{k_i}$ that solves the equation

$$\begin{aligned} \Big (\mu _1e^{-\frac{\kappa t^i_{k_i}}{2}}+\mu _2e^{-\frac{\alpha t^i_{k_i}}{2}}\Big )\tau _0=\sqrt{c} e^{-\frac{\alpha (\tau _0+t^i_{k_i})}{2}} \Longleftrightarrow \Big (\mu _1e^{\frac{(\alpha -\kappa ) t^i_{k_i}}{2}}+\mu _2\Big )\tau _0=\sqrt{c} e^{-\frac{\alpha }{2} \tau _0}. \end{aligned}$$

(35)

Because of $0<\alpha <\kappa$, it follows that $\mu _2\le \mu _1e^{\frac{(\alpha -\kappa ) t^i_{k_i}}{2}}+\mu _2\le \mu _1+\mu _2$. For all $t^i_{k_i}\ge 0$, the solutions $\tau _0(t^i_{k_i})$ are greater or equal to $\tau _0$ given by $(\mu _1+\mu _2)\tau _0=\sqrt{c} e^{-\frac{\alpha }{2} \tau _0}$, which is strictly positive constant.

Since there is a positive lower bound $\tau _0$ on the inter-event intervals, there are no accumulation points in the event sequences, so the Zeno behavior is excluded.

The proof is completed. $\square$

Proof of Theorem 2

Proof

Consider of the event-triggered discrete-time DCL algorithm (14), the discrete form of the Lyapunov function candidate (20) is given as follows:

$$\begin{aligned} {V}(k)=\frac{1}{2}\sum \limits _{i\in {\mathcal {V}}}\Big ((W^*-W_i(k))^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})(W^*-W_i(k))\Big ). \end{aligned}$$

(36)

The following two inequalities are still established under discrete form:

$$\begin{aligned} {V}(k)&\ge \frac{\,{\underline{\theta }}\,}{\,2\,}\sum _{i=1}^{N}\parallel W^*-W_i(k)\parallel ^2, \end{aligned}$$

(37)

$$\begin{aligned} {V}(k)&\le \frac{\,{\overline{\varTheta }}\,}{\,2\lambda _2\,}W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(k). \end{aligned}$$

(38)

Now, we are in the position to give the main result on the convergence of the algorithm (14) with the trigger function (15). Consider the Lyapunov function candidate (36), whose difference is given by

$$\begin{aligned} \bigtriangleup {V(k+1)}=&V(k+1)-V(k)\nonumber \\ =\,&\frac{1}{2}\sum _{i=1}^{N}\Big [\Big ((W^*-W_i(k+1))^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})(W^*-W_i(k+1))\Big )\nonumber \\&-\Big ((W^*-W_i(k))^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})(W^*-W_i(k))\Big )\Big ]\nonumber \\ =&-\frac{1}{2}\sum _{i=1}^{N}\Big (W_i(k)^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k)\nonumber \\&-W_i(k+1)^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k+1)\Big )\nonumber \\&-W^{*T}\sum _{i=1}^{N}(H_i^{\text {T}}H_i+\sigma _iI_{ml})\big ({W_i}(k+1)-{W_i}(k)\big ). \end{aligned}$$

(39)

Under the discrete form, $\sum _{i=1}^{N}(H_i^{\text {T}}H_i+\sigma _iI_l)\big ({W_i}(k+1)-{W_i}(k)\big )=\gamma \sum _{i=1}^{N}\sum \limits _{j\in {\mathcal {N}}_{i}}a_{ij}\Big ({\hat{W}}_j(k)-{\hat{W}}_i(k)\Big )=0$ according to the event-triggered discrete-time DCL algorithm (14). Then,

$$\begin{aligned} \bigtriangleup {V(k+1)}=&-\frac{1}{2}\sum _{i=1}^{N}\Big (W_i(k)^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k)\nonumber \\&-W_i(k+1)^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k+1)\Big ). \end{aligned}$$

(40)

By increasing and decreasing function terms, we have

$$\begin{aligned} \bigtriangleup {V(k+1)} =&-\frac{1}{2}\sum _{i=1}^{N}\Big (W_i(k)^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k)\nonumber \\&-W_i(k+1)^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k+1)\nonumber \\&+2\big ({W_i}(k+1)-{W_i}(k)\big )^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k+1)\nonumber \\&-2\big ({W_i}(k+1)-{W_i}(k)\big )^{\text {T}}\times (H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k+1)\Big ). \end{aligned}$$

(41)

Then, we get

$$\begin{aligned} \bigtriangleup {V(k+1)}=&-\frac{1}{2}\sum _{i=1}^{N}\Big (\big ({W_i}(k+1)-{W_i}(k)\big )^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})\big ({W_i}(k+1)-{W_i}(k)\big )\Big )\nonumber \\&\,+\sum _{i=1}^{N}\big ({W_i}(k+1)-{W_i}(k)\big )^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k+1)\nonumber \\ \le&\sum _{i=1}^{N}\big ({W_i}(k+1)-{W_i}(k)\big )^{\text {T}}(H_i^{\text {T}}H_i+\sigma _iI_{ml})W_i(k+1)\nonumber \\ =\,&\big ({W}(k+1)-{W}(k)\big )^{\text {T}}(H^{\text {T}}H+\sigma \otimes I_{ml})W(k+1). \end{aligned}$$

(42)

Due to (16), we obtain

$$\begin{aligned}&\bigtriangleup {V(k+1)}=-\gamma \big (W(k)+e(k)\big )^{\text {T}}({\mathcal {L}}\otimes I_{ml})^{\text {T}}W(k+1)\nonumber \\&\quad =-\gamma W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(k+1)-\gamma e(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(k+1)\nonumber \\&\quad \le -\gamma W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})\Big [W(k)-\gamma (H^{\text {T}}H+\sigma \otimes I_{ml})^{-1}({\mathcal {L}}\otimes I_{ml})\Big (W(k)+e(k)\Big )\Big ]\nonumber \\&\qquad -\gamma e(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})\Big [W(k)-\gamma (H^{\text {T}}H+\sigma \otimes I_{ml})^{-1}({\mathcal {L}}\otimes I_{ml})\Big (W(k)+e(k)\Big )\Big ]\nonumber \\&\quad =-\gamma W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(k)+\gamma ^2W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})(H^{\text {T}}H+\sigma \otimes I_{ml})^{-1}({\mathcal {L}}\otimes I_{ml})W(k)\nonumber \\&\qquad +2\gamma ^2W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})(H^{\text {T}}H+\sigma \otimes I_{ml})^{-1}({\mathcal {L}}\otimes I_{ml})e(k)-\gamma e(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(k)\nonumber \\&\qquad +\gamma ^2e(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})(H^{\text {T}}H+\sigma \otimes I_{ml})^{-1}({\mathcal {L}}\otimes I_{ml})e(k). \end{aligned}$$

(43)

According to the Young’s inequality, we have

$$\begin{aligned} W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})(H^{\text {T}}H+\sigma \otimes I_{ml})^{-1}({\mathcal {L}}\otimes I_{ml})e(k)&\le \frac{\epsilon \eta ^3}{2{\bar{\lambda }}^2}W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(k)\nonumber \\&\quad +\frac{1}{2\epsilon }e(k)^{\text {T}}e(k), \end{aligned}$$

(44)

and

$$\begin{aligned} e(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(k) \le \frac{\eta }{2\epsilon }W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})W(k)+\frac{\epsilon }{2}e(k)^{\text {T}}e(k), \end{aligned}$$

(45)

where ${\bar{\lambda }}=\lambda _{min}(H^{\text {T}}H+\sigma \otimes I_{ml})$, $\eta =\lambda _{max}({\mathcal {L}})$, $\epsilon >\eta /2$ is a constant.

Substituting inequality (44) and (45) into (43), one has

$$\begin{aligned} \bigtriangleup {V(k+1)}\le -\gamma \rho _1W(k)^{\text {T}}({\mathcal {L}}\otimes I_{ml})^{\text {T}}W(k)+\rho _2e(k)^{\text {T}}e(k), \end{aligned}$$

(46)

where $\rho _1=1-\frac{\gamma \eta }{{\bar{\lambda }}}-\frac{\gamma \epsilon \eta ^3}{{\bar{\lambda }}^2}-\frac{\eta }{2\epsilon }$, $\rho _2=\frac{\gamma ^2}{\epsilon }+\frac{\gamma \epsilon }{2}+\frac{\gamma ^2\eta ^2}{{\bar{\lambda }}}$.

Based on the conditions $0<\gamma <\min \left\{ \frac{\,{\overline{\varTheta }}\,}{\,2\lambda _2\,},\frac{{\bar{\lambda }}^2(2\epsilon -\eta )}{2\epsilon \eta ({\bar{\lambda }}+\epsilon \eta ^2)}\right\}$, $\epsilon >\eta /2$ and the trigger function (15), one gets $\rho _1\in (0,1)$ and $e(k)^{\text {T}}e(k)\le Nc\beta ^k$. Then, from inequality (38), one has

$$\begin{aligned} \bigtriangleup {V(k+1)}\le&-\frac{2\gamma \rho _1\lambda _2}{{\overline{\varTheta }}}V(k)+Nc\rho _2\beta ^k. \end{aligned}$$

(47)

Thus,

$$\begin{aligned} V(k+1)\le \varsigma V(k)+Nc\rho _2\beta ^k, \end{aligned}$$

(48)

where $\varsigma =1-\frac{2\gamma \rho _1\lambda _2}{{\overline{\varTheta }}}$.

Therefore, based on the above results, if $\gamma$ can be chosen such that $0<\gamma <\min \left\{ \frac{\,{\overline{\varTheta }}\,}{\,2\lambda _2\,},\frac{{\bar{\lambda }}^2(2\epsilon -\eta )}{2\epsilon \eta ({\bar{\lambda }}+\epsilon \eta ^2)}\right\}$ and $\epsilon >\eta /2$, then $\varsigma \in (0,1)$. Furthermore, we have

$$\begin{aligned} V(k)\le&\varsigma V(k-1)+Nc\rho _2\beta ^{k-1}\nonumber \\ \le \,&\varsigma ^2 V(k-2)+\varsigma Nc\rho _2\beta ^{k-2}+Nc\rho _2\beta ^{k-1}\nonumber \\ \vdots \nonumber \\ \le \,&\varsigma ^k V(0)+ Nc\rho _2\Big (\varsigma ^{k-1}\beta ^{0}+\varsigma ^{k-2}\beta ^{1}+...+\varsigma ^{1}\beta ^{k-2}+\varsigma ^{0}\beta ^{k-1}\Big )\nonumber \\ =\,&\varsigma ^k V(0)+Nc\rho _2\frac{\varsigma ^k-\beta ^k}{\varsigma -\beta }\nonumber \\ =\,&\Big (V(0)-\varpi \Big )\varsigma ^k+\varpi \beta ^k, \end{aligned}$$

(49)

where $\varpi =\frac{Nc\rho _2}{\beta -\varsigma }$. Then, according to the inequality (37), we get inequality (17) in Theorem 2.

The proof is completed. $\square$