Distributed cooperative learning algorithms using wavelet neural network

Abstract

This paper investigates the distributed cooperative learning (DCL) problems over networks, where each node only has access to its own data generated by the unknown pattern (map or function) uniformly, and all nodes cooperatively learn the pattern by exchanging local information with their neighboring nodes. These problems cannot be solved by using traditional centralized algorithms. To solve these problems, two novel DCL algorithms using wavelet neural networks are proposed, including continuous-time DCL (CT-DCL) algorithm and discrete-time DCL (DT-DCL) algorithm. Combining the characteristics of neural networks with the properties of the wavelet approximation, the wavelet series are used to approximate the unknown pattern. The DCL algorithms are used to train the optimal weight coefficient matrix of wavelet series. Moreover, the convergence of the proposed algorithms is guaranteed by using the Lyapunov method. Compared with existing distributed optimization strategies such as distributed average consensus (DAC) and alternating direction method of multipliers (ADMM), our DT-DCL algorithm requires less information communications and training time than ADMM strategy. In addition, it achieves higher accuracy than DAC strategy when the network consists of large amounts of nodes. Moreover, the proposed CT-DCL algorithm using a proper step size is more accurate than the DT-DCL algorithm if the training time is not considered. Several illustrative examples are presented to show the efficiencies and advantages of the proposed algorithms.

Appendices

Appendix A

Proof of Theorem 1

For convenience of analyzing the convergence of the CT-DCL algorithm, we rewrite algorithm (20) in a matrix form as

$$\begin{aligned}{l} {\left\{ \begin{array}{ll} \dot{W}(t)=-\gamma (H^TH+\sigma \otimes I_{ml})^{-1}(\mathcal {L}\otimes I_{ml})W(t)\\ {W}(0)=(H^TH+\sigma \otimes I_{ml})^{-1}H^TY \end{array}\right. }, \end{aligned}$$

(33)

where $W(t)=[W_1^T(t),W_2^T(t),...,W_N^T(t)]^T\in {\mathbf{R}}^{mlN\times 1}$, $H=diag \{H_1,H_2,\ldots,H_N\}\in {\mathbf{R}}^{mN_iN\times mlN}$, $\sigma =diag\{\sigma _1,\sigma _2,\ldots,\sigma _N\}\in {\mathbf{R}}^{N\times N}$, $Y=[Y_1^T,Y_2^T,\ldots,Y_N^T]^T\in {\mathbf{R}}^{mN_iN\times 1}$, ${\mathcal {L}}$ is the Laplacian matrix of the graph ${\mathcal {G}}$.

Proof

Consider CT-DCL algorithm (20), the following Lyapunov function candidate is constructed:

$$\begin{aligned} V(W(t))=\frac{1}{2}\sum _{i=1}^{N}\left( (W^*-W_i)^T(H_i^TH_i+\sigma _iI_{ml})(W^*-W_i)\right) , \end{aligned}$$

(34)

where $W^*,W_i \in {\mathbf{R}}^{ml\times 1}$, $V:{\mathbf{R}}^{ml}\rightarrow {\mathbf{R}}$.

It is easy to verify that

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

(35)

In addition, the following inequality holds (Proof see “Appendix C''):

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

(36)

Now, we are in the position to give the main result on the convergence of algorithm (20). Consider Lyapunov function candidate (34). Then, along the solution of (20), we have

$$\begin{aligned} \frac{\mathrm{d}V(W(t))}{\mathrm{d}t}=&-\sum _{i=1}^{N}\dot{W_i}^T\left( H_i^TH_i+\sigma _iI_{ml}\right) ^T(W^*-\!W_i)\nonumber \\ =&-W^{*T}\sum _{i=1}^{N}\left( H_i^TH_i+\sigma _iI_{ml}\right) \dot{W_i}(t)\nonumber \\&+\sum _{i=1}^{N}W_i^T\left( H_i^TH_i+\sigma _iI_{ml}\right) \dot{W_i}(t)\nonumber \\ =&-\gamma W(t)^T\left( {\mathcal {L}}\otimes I_{ml}\right) W(t). \end{aligned}$$

(37)

Since graph ${\mathcal {G}}$ is undirected and connected, we deduce that $\sum _{i=1}^{N}(H_i^TH_i+\sigma _iI_{ml})\dot{W_i}(t)=\gamma \sum _{i=1}^{N}\sum \nolimits _{j\in {\mathcal {N}}_{i}}a_{ij}\left( W_j(t)-W_i(t)\right) =0$ according to CT-DCL algorithm (20). Then,

$$\begin{aligned} \frac{\mathrm{d}V(W(t))}{\mathrm{d}t}=-\gamma W(t)^T({\mathcal {L}}\otimes I_{ml})W(t). \end{aligned}$$

(38)

According to inequality (36), we have $W(t)^T({\mathcal {L}}\otimes I_{ml})W(t)\ge \frac{2\lambda _2}{\overline{\varTheta }} V(W)$. Therefore,

$$\begin{aligned} \frac{\mathrm{d}V(W(t))}{\mathrm{d}t}\le -\frac{2\gamma \lambda _2}{\overline{\varTheta }} V(W)=-\kappa V(W). \end{aligned}$$

(39)

Integrating both sides of (39) from 0 to t leads to

$$\begin{aligned} V\left( W(t)\right) \le V\left( W(0)\right) e^{-\kappa t}. \end{aligned}$$

(40)

According to inequality (35), we have

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

(41)

It will be seen from this that $W_i(t)$ exponential convergence to $W^*$ when $t\rightarrow \infty$. This implies that problem (18) $W^*=\arg \min \nolimits _{W_i}(G(W_i)$ is solved. Then, our goal $\lim \nolimits _{t\rightarrow \infty }W_i(t)=W^*,\forall i\in {\mathcal {V}}$ is achieved by CT-DCL algorithm (20). $\tilde{f_i}^*(x)=Q_i(x)W^*$ is the estimation function of original function $f_i(x)$.

The proof is completed. $\square$

Appendix B

Proof of Theorem 2

We also rewrite DT-DCL algorithm (23) in a matrix form as

$$\begin{aligned}{l} {\left\{ \begin{array}{ll} {W}(k+1)=-\gamma (H^T\!H+\sigma \!\otimes \!I_{ml})^{-1}\!({\mathcal {L}} \otimes I_{ml})\!W(k)+W(k)\\ {W}(0)=(H^T\!H+\sigma \otimes I_{ml})^{-1}H^TY \end{array}\right. }, \end{aligned}$$

(42)

where $W(k),H,\sigma$ and Y are similar to the form in CT-DCL algorithm (33).

Proof

Consider DT-DCL algorithm (23), the discrete form of Lyapunov function candidate (34) is as follows:

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

(43)

The following two inequalities are still established under discrete form:

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

(44)

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

(45)

Now, we are in the position to give the main result on the convergence of algorithm (23). Consider Lyapunov function candidate (43), whose difference is given by

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

(46)

Under the discrete form, $\sum\nolimits _{i=1}^{N}(H_i^TH_i+\sigma _iI_l)\left( {W_i}(k+1)-{W_i}(k)\right) =\gamma \sum\nolimits _{i=1}^{N}\sum \nolimits _{j\in {\mathcal {N}}_{i}}a_{ij}\left( W_j(k)-W_i(k)\right) =0$ according to DT-DCL algorithm (23). Then,

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

(47)

By increasing and decreasing function terms, we have

$$\begin{aligned}&\bigtriangleup {V\left( {W}(k+1)\right) }\nonumber \\&\quad =-\frac{1}{2}\sum _{i=1}^{N}\left( \left( {W_i}(k+1)-{W_i}(k)\right) ^T(H_i^TH_i+\sigma _iI_{ml})\right. \nonumber \\&\qquad \left. \times \left( {W_i}(k+1)-{W_i}(k)\right) \right) +\sum _{i=1}^{N}\left( {W_i}(k+1)-{W_i}(k)\right) ^T\nonumber \\&\qquad \times (H_i^TH_i+\sigma _iI_{ml})W_i(k+1)\nonumber \\&\quad \le \sum _{i=1}^{N}\left( {W_i}(k+1)-{W_i}(k)\right) ^T(H_i^TH_i+\sigma _iI_{ml})W_i(k+1)\nonumber \\&\quad =\left( {W}(k+1)-{W}(k)\right) ^T(H^TH+\sigma \otimes I_{ml})W(k+1)\nonumber \\&\quad =-\gamma W(k)^T({\mathcal {L}}\otimes I_{ml})W(k+1). \end{aligned}$$

(48)

Due to (42), we obtain

$$\begin{aligned}&\bigtriangleup {V\left( {W}(k+1)\right) }\nonumber \\&\quad \le -\gamma W(k)^T({\mathcal {L}}\otimes I_{ml})\left[ W(k)\right. \nonumber \\&\qquad \left. -\gamma (H^TH+\sigma \otimes I_{ml})^{-1}({\mathcal {L}}\otimes I_{ml})W(k)\right] \nonumber \\&\quad =-\gamma W(k)^T({\mathcal {L}}\otimes I_{ml})W(k)+\gamma ^2W(k)^T({\mathcal {L}}\otimes I_{ml})\nonumber \\&\qquad \times (H^TH+\sigma \otimes I_{ml})^{-1}({\mathcal {L}}\otimes I_{ml})W(k)\nonumber \\&\quad \le -\gamma W(k)^T\left[ ({\mathcal {L}}\otimes I_{ml})-\frac{1}{\bar{\lambda }}\gamma \left( {\mathcal {L}}^2\otimes I_{ml}\right) \right] W(k), \end{aligned}$$

(49)

where $\bar{\lambda }=\lambda _{min}(H^TH+\sigma \otimes I_{ml})$.

Because of ${\mathcal {L}}^2\le \eta {\mathcal {L}}$, where $\eta =\lambda _{max}({\mathcal {L}})$. Then, it follows that

$$\begin{aligned} \bigtriangleup&{V\left( {W}(k+1)\right) }\le -\gamma (1-\frac{\eta \gamma }{\bar{\lambda }})W(k)^T({\mathcal {L}}\otimes I_{ml})W(k). \end{aligned}$$

(50)

If $\gamma$ can be chosen such that $0<\gamma <\frac{\bar{\lambda }}{\eta }$, $\{V\left( {W}(k)\right) \}_{k=0}^{\infty }$ is nonnegative and non-increasing.

Due to (45), we obtain

$$\begin{aligned} V\left( {W}(k+1)\right) -V\left( {W}(k)\right) \le -\frac{ \gamma \lambda _2}{\overline{\varTheta }}(1-\frac{\eta \gamma }{\bar{\lambda }})V\left( {W}(k)\right) . \end{aligned}$$

(51)

Then,

$$\begin{aligned} V\left( {W}(k+1)\right) \le \varepsilon V\left( {W}(k)\right) , \end{aligned}$$

(52)

where $\varepsilon =1-\frac{2 \gamma \lambda _2}{\overline{\varTheta }}(1-\frac{\eta \gamma }{\bar{\lambda }})$.

Because of $0<\gamma <\frac{\bar{\lambda }}{\eta }$, then, $0<1-\frac{\eta \gamma }{\bar{\lambda }}<1$. And because $\{V\left( {W}(k)\right) \}_{k=0}^{\infty }$ is nonnegative and non-increasing, thus $0<\frac{2 \gamma \lambda _2}{\overline{\varTheta }}(1-\frac{\eta \gamma }{\bar{\lambda }})<1$, so $\gamma$ just need can be chosen such that $0<\gamma <\frac{\,\overline{\varTheta }\,}{\,2\lambda _2\,}$.

Therefore, based on the above analysis, if $\gamma$ can be chosen such that $0<\gamma <\min \{\frac{\,\overline{\varTheta }\,}{\,2\lambda _2\,},\frac{\bar{\lambda }}{\eta }\}$, then $0<\varepsilon <1$, and $\{V\left( {W}(k)\right) \}_{k=0}^{\infty }$ is nonnegative and non-increasing.

Due to (52), we obtain

$$\begin{aligned} V\left( {W}(k)\right) \le \varepsilon V\left( {W}(k-1)\right) \le \cdots \le \varepsilon ^k V\left( {W}(0)\right) . \end{aligned}$$

(53)

According to inequality (44), we have

$$\begin{aligned} \sum \limits _{i\in {\mathcal {V}}}\parallel W^*-W_i(k)\parallel ^2&\le \frac{2}{\underline{\theta }}V\left( {W}(k)\right) \le \frac{2}{\underline{\theta }}V\left( W(0)\right) \varepsilon ^{k}. \end{aligned}$$

(54)

Because of $0<\varepsilon <1$, then, our goal $\lim \nolimits _{k\rightarrow \infty }W_i(k)=W^*$, $\forall i\in {\mathcal {V}}$ is achieved by DT-DCL algorithm (23). $\tilde{f_i}^*(x)=Q_i(x)W^*$ is the estimation function of original function $f_i(x)$.

The proof is completed. $\square$

Appendix C

Proof of the inequality (36)

Proof

From optimization problem (18), we have $g_i(W_i)=\frac{1}{2}\left( \parallel Y_i-H_iW_i\parallel ^2 +\sigma _i \parallel W_i\parallel ^2\right)$, then

$$\begin{aligned}&\sum _{i=1}^{N}\left( g_i(W^*)-g_i(W_i(t))-\bigtriangledown {g_i\left( W_i(t)\right) }^T(W^*-W_i(t))\right) \nonumber \\&\quad =\sum _{i=1}^{N}\left( \frac{1}{2}\left( \parallel Y_i-H_iW^*\parallel ^2 +\sigma _i \parallel W^*\parallel ^2\right) \right. \nonumber \\&\qquad -\frac{1}{2}\left( \parallel Y_i-H_iW_i\parallel ^2 +\sigma _i \parallel W_i\parallel ^2\right) \nonumber \\&\qquad \left. (-H_i^TY_i+H_i^TH_iW_i+\sigma _iW_i)^T(W^*-W_i(t))\right) \nonumber \\&\qquad \vdots \nonumber \\&\quad =\sum _{i=1}^{N}\left( \frac{1}{2}(W^*-W_i)^T(H_i^TH_i+\sigma _iI_{ml})(W^*\!\!-\!\!W_i)\right) \nonumber \\&\quad =V(W(t)). \end{aligned}$$

(55)

Thus, the following inequality holds [44]:

$$\begin{aligned} V(W(t))&\le \frac{\overline{\varTheta }}{2}\sum _{i=1}^{N} \parallel W_i(t)-\frac{1}{N}\sum _{j=1}^{N}W_j(t)\parallel ^2\nonumber \\&= \frac{\overline{\varTheta }}{2N}W(t)^T(\tilde{\mathcal {L}}\otimes I_{ml})W(t), \end{aligned}$$

(56)

where $\tilde{\mathcal {L}}\in {\mathbf{R}}^{N\times N}$ is the Laplacian matrix of the complete graph $\tilde{{\mathcal {G}}}$.

For the undirected and connected graph ${\mathcal {G}}$, the following inequality holds [44]:

$$\begin{aligned} \lambda _2\tilde{\mathcal {L}}\le N\mathcal {L}. \end{aligned}$$

(57)

Substituting (57) into (56), we have

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

(58)

The proof is completed. $\square$