FedBat: a self-adapting bat algorithm-based federated learning approach

Abstract

Federated learning (FL) is an advanced distributed machine learning (ML) framework designed to address issues related to data silos and data privacy. In real-world applications, common problems like non-convex optimization and nonindependent and identically distributed (Non-IID) client data reduce training efficiency, cause local optima, and degrade performance. Therefore, we propose a FL scheme based on the bat algorithm (FedBat), which leverages the echolocation mechanism of bats to effectively balance global and local search capabilities, enabling the algorithm to escape local optima with a certain probability. By combining global optimal model weight optimization with dynamically adjusted search strategies, FedBat guides weaker client models toward the global optimum, thereby accelerating convergence. Additionally, FedBat allows for adaptive parameter adjustments across various datasets. To mitigate client drift, we extend FedBat with Jensen–Shannon (JS) divergence to quantify differences between local and global models. Clients decide whether to upload their local models based on this divergence, to enhance the global model’s generalization and minimize communication overhead. Experimental results demonstrate that FedBat converges 5 times faster and enhances test accuracy by more than 40$\%$ compared to FedAvg. The extended FedBat effectively mitigates the decrease in the generalization performance of the global model and reduces communication costs by approximately 20$\%$. Comparing FedPso, FedGwo, and FedProx shows that FedBat demonstrates superior performance in terms of convergence speed and test accuracy. Furthermore, we derive the formula for the expected convergence rate of FedBat, analyze the impact of various parameters on FL performance, and establish the upper bound of FedBat to evaluate its model divergence.

Appendix 1

In this section, we perform a convergence analysis of the proposed FedBat algorithm. Each participating client uses the bat algorithm combined with SGD to update its local model. We will repeatedly use the following facts and illustrate some assumptions throughout the text.

• Fact 1: $\langle \textbf{p}, \textbf{q} \rangle = \Vert \textbf{p} \Vert \Vert \textbf{p} \Vert \cos \theta \ge -\Vert \textbf{p} \Vert \Vert \textbf{q} \Vert$

• Fact 2: $\Vert \textbf{p}+\textbf{q}\Vert \le \Vert \textbf{p}\Vert +\Vert \textbf{q}\Vert$

• Fact 3: $\Vert \textbf{pq}\Vert \le \Vert \textbf{p}\Vert \Vert \textbf{q}\Vert$

• Fact 4: For any set of vectors$\left\{ \textbf{p}_i\right\} _{i=1}^n$ and nonnegative scalars $\left\{ q_i\right\} _{i=1}^n$, define the weighted sum $s=\sum _{i=1}^nq_i$. Then, according to Jensen’s inequality, we have: $\left| \left| \sum _{i=1}^n\frac{q_i}{s}\textbf{p}_i\right| \right| ^2\le \sum _{i=1}^n\frac{q_i}{s}\Vert \textbf{p}_i\Vert ^2$

• Fact 5: For any set of vectors $\left\{ \textbf{p}_i\right\} _{i=1}^n$, we have $\left\| \sum _{i=1}^n\textbf{p}_i\right\| ^2\le n\sum _{i=1}^n\Vert \textbf{p}_i\Vert ^2$.

Proof of Theorem 1:

According to Assumption 1, where $F_i(\cdot )$ is L-smooth, we have:

$$F_{i} (\user2{u}) \le F_{i} (\user2{v}) + <\user2{u} - \user2{v},\nabla F_{i} (\user2{v})>+ \frac{L}{2}\left\| {\user2{u} - \user2{v}} \right\|^{2}$$

(38)

$$F_{i} ({\mathbf{w}}_{{t + 1}} ) \le F_{i} ({\mathbf{w}}_{t} ) +<{\mathbf{w}}_{{i,t + 1}} - {\mathbf{w}}_{{i,t}} ,\nabla F({\mathbf{w}}_{{i,t}} )> + \frac{L}{2}\left\| {{\mathbf{w}}_{{i,t + 1}} - {\mathbf{w}}_{{i,t}} } \right\|^{2}$$

(39)

$$F_{i} (\mathbf{w}_{{t + 1}} ) - F_{i} (\mathbf{w}_{t} )\underbrace {{ - \langle \mathbf{v}{\mathbf{R}}_{{i,t + 1}} + p\epsilon {\mathbf{AR}}_{{t + 1}} - \eta \nabla F_{i} (\mathbf{w}_{{i,t}} ),\nabla F_{i} (\mathbf{w}_{t} )\rangle }}_{A} \le \underbrace {{\frac{L}{2}\left\| {\mathbf{w}_{{i,t + 1}} - \mathbf{w}_{{i,t}} } \right\|^{2} }}_{B}$$

(40)

We next focus on bounding A. According to the Fact 1 and eqn. (6), we have:

$$\begin{aligned}&A \Leftrightarrow \langle -\mathbf{vR}_{i,t+1} - p\epsilon\mathbf{AR}_{t+1} + \eta\nabla F_{i}(\mathbf{w}_{i,t}) , \nabla F_{i}(\mathbf{w}_{t}) \rangle \\&\Leftrightarrow (\|-\mathbf{vR}_{i,t+1} - p\epsilon\mathbf{AR}_{t+1}+\eta\nabla F_{i}(\mathbf{w}_{i,t})\|\|\nabla F_{i}(\mathbf{w}_{t})\|\cos\theta) \\&\Leftrightarrow (\|-\mathbf{vR}_{i,t} - (\mathbf{w}_{i,t}-\mathbf{w}_{*,t})f_{i} - p\epsilon\mathbf{AR}_{t+1}+\eta\nabla F_{i}(\mathbf{w}_{i,t})\|\|\nabla F_{i}(\mathbf{w}_{t})\|\cos\theta)\end{aligned}$$

(41)

By combining eqn. (14) and eqn. (15), it follows that:

$$\begin{aligned} \underbrace{(\Vert -\textbf{vR}_{i,t} - (\textbf{vR}_{i,t}-\textbf{vR}_{*,t})f_{i} - p\epsilon \textbf{AR}_{t+1} + \eta \nabla F_{i}(\textbf{w}_{i,t})\Vert \Vert \nabla F_{i}(\textbf{w}_{t})\Vert \cos \theta )}_{\text {A1}} \end{aligned}$$

(42)

By combining the Fact 2 and eqn. (10), we have:

$$\begin{aligned} &A1\le [(\Vert \textbf{vR}_{i,t}\Vert +\Vert (\textbf{vR}_{i,t}-\textbf{vR}_{*,t})f_{i}\Vert +p\epsilon \varphi \Vert \textbf{AR}_{t}\Vert +\eta \Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert )\Vert \nabla F_{i}(\textbf{w}_{t})\Vert \cos \theta ] \\&{\mathop {\le }\limits ^{(a)}}\underbrace{[(\Vert \textbf{vR}_{i,t}\Vert +\Vert \textbf{vR}_{i,t}\Vert f_{i}+\Vert \textbf{vR}_{*,t}\Vert f_{i}+p\epsilon \varphi \Vert \textbf{AR}_{t}\Vert +\eta \Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert )\Vert \nabla F_{i}(\textbf{w}_{t})\Vert \cos \theta ]}_{\text {A2}} \end{aligned}$$

(43)

where (a) uses Face 2 again.

According to the eqn. (20)-(25), we have:

$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q} \left\| {\nabla F_{i} ({\mathbf{w}}_{{i,t}} )} \right\|^{2} \le \left\| {{\mathbf{vR}}_{{i,t}} } \right\|\left\| {\nabla F_{i} ({\mathbf{w}}_{{i,t}} )} \right\|\cos \theta _{{i,t}} \le \bar{u}\bar{q}\left\| {\nabla F_{i} ({\mathbf{w}}_{{i,t}} )} \right\|^{2}$$

(44)

$${\underline{u}}_{*}{\underline{q}}_{*}\left\| {\nabla F_{i} ({\mathbf{w}}_{{i,t}} )} \right\|^{2} \le \left\| {{\mathbf{vR}}_{{*,t}} } \right\|\left\| {\nabla F_{i} ({\mathbf{w}}_{{i,t}} )} \right\|\cos \theta _{{*,t}} \le \bar{u}_{*} \bar{q}_{*} \left\| {\nabla F_{i} ({\mathbf{w}}_{{i,t}} )} \right\|^{2}$$

(45)

$$\begin{aligned} &{\underline{u}}_A{\underline{q}}_A\Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2\le \Vert \textbf{A}\textbf{R}_{t}\Vert \Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert \cos \theta _{A,t} \le {\overline{u}}_A{\overline{q}}_A\Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2 \end{aligned}$$

(46)

Substituting eqn. (44)-(46) to A2, we have:

$$\begin{aligned} &A2 \Leftrightarrow [(\Vert \textbf{vR}_{i,t}\Vert (1+f_{i})+\Vert \textbf{vR}_{*,t}\Vert f_{i}+p\epsilon \varphi \Vert \textbf{AR}_{t}\Vert +\eta \Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert )\Vert \nabla F_{i}(\textbf{w}_{t})\Vert \cos \theta ] \\&\le [({\underline{u}}{\underline{q}}(1+f_{i})+{\underline{u}}_{*}{\underline{q}}_{*}+ \eta +p\epsilon \varphi {\underline{u}}_{A}{\underline{q}}_{A})\Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2] \end{aligned}$$

(47)

We next aim to bound B.

$$\begin{aligned} &B \Leftrightarrow \frac{L}{2}\Vert \textbf{w}_{i,t+1}-\textbf{w}_{i,t}\Vert ^2 \\&\Leftrightarrow \frac{L}{2}\Vert \textbf{vR}_{i,t}+(\textbf{vR}_{i,t}-\textbf{v}_{*,t})f_{i}+p\epsilon \varphi \textbf{AR}_{t}-\eta \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2 \end{aligned}$$

(48)

By combining the Fact 2 and eqn. (11), we have:

$$\begin{aligned} &B\le \frac{L}{2}(\Vert \textbf{vR}_{i,t}\Vert +\Vert \textbf{vR}_{i,t}\Vert f_{i}+\Vert \textbf{vR}_{*,t}\Vert f_{i}+p\epsilon \varphi \Vert \textbf{AR}_{t}\Vert +\eta \Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert )^2 \\&{\mathop {\le }\limits ^{(b)}}\underbrace{\frac{5L}{2}\left( \left\| \textbf{vR}_{i,t}\right\| ^2 + {f_i}^2 \Vert \textbf{vR}_{i,t}^2\Vert + {f_i}^2\Vert \textbf{vR}_{*,t}\Vert ^2 + \eta ^2 \left\| \nabla F_i(\textbf{w}_{i,t})\right\| ^2 + p^2 \epsilon ^2 \varphi ^2 \left\| \textbf{AR}_t\right\| ^2\right) }_{\text {B1}} \end{aligned}$$

(49)

where (b) uses Jensen’s inequality in Fact 5, the quadratic term is expanded.

According to the eqn. (23)-(25), we have:

$$\left\| {{\mathbf{vR}}_{{i,t - 1}} } \right\| \le \bar{u}\left\| {\nabla F_{i} ({\mathbf{w}}_{{i,t - 1}} )} \right\|$$

(50)

$$\left\| {{\mathbf{vR}}_{{*,t - 1}} } \right\| \le \bar{u}_{*} \left\| {\nabla F_{i} ({\mathbf{w}}_{{i,t - 1}} )} \right\|$$

(51)

$$\begin{aligned} \left\| \textbf{A}\textbf{R}_{t-1}\right\| \le {\overline{u}}_A\left\| \nabla F_{i}(\textbf{w}_{i,t-1})\right\| \end{aligned}$$

(52)

Substituting eqn. (50)-(52) to B1, we have:

$$\begin{aligned} B1\le \frac{5L}{2} \left( {\overline{u}}^2(1 + {f_i}^2) + {f_i}^2{\overline{u}}_*^2 + \eta ^2 + p^2 \epsilon ^2 \varphi ^2 {\overline{u}}_A^2 \right) \Vert \nabla F_i(\textbf{w}_{i,t})\Vert ^2 \end{aligned}$$

(53)

Using eqn. (53) and (47), we complete the proof.

$$\begin{aligned} &F_{i}(\textbf{w}_{t+1}) - F_{i}(\textbf{w}_{t}) +[{\underline{u}}{\underline{q}}(1+f_{i})+{\underline{u}}_{*}{\underline{q}}_{*}+ \eta +p\epsilon \varphi {\underline{u}}_{A}{\underline{q}}_{A}]\Vert \nabla F_i(\textbf{w}_{i,t})\Vert ^2- \\&[\frac{5L}{2} \left( {\overline{u}}^2(1 + {f_i}^2) + {f_i}^2{\overline{u}}_*^2 + \eta ^2 + p^2 \epsilon ^2 \varphi ^2 {\overline{u}}_A^2 \right) ]\Vert \nabla F_i(\textbf{w}_{i,t})\Vert ^2\le 0 \end{aligned}$$

(54)

Let $\Phi ={\underline{u}}{\underline{q}}(1+f_{i})+{\underline{u}}_{*}{\underline{q}}_{*}+\eta +p\epsilon \varphi {\underline{u}}_{A}{\underline{q}}_{A}-\frac{5\,L}{2}[{\overline{u}}^2(1+{f_i}^2)+{f_i}^2{\overline{u}}_*^2 + \eta ^2 + p^2 \epsilon ^2 \varphi ^2{\overline{u}}_A^2]$, we have:

$$\begin{aligned} &F_{i}(\textbf{w}_{t+1})-F_{i}(\textbf{w}_{t})+\Phi \Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2\le 0 \\&\Leftrightarrow \Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2 \le \frac{[F_{i}(\textbf{w}_{t})-F_{i}(\textbf{w}_{t+1})]}{\Phi } \end{aligned}$$

(55)

Take expected values on both sides:

$$\begin{aligned} {\mathbb {E}}\Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2\le \frac{{\mathbb {E}}[F_{i}(\textbf{w}_{t})-F_{i}(\textbf{w}_{t+1})]}{\Phi } \end{aligned}$$

(56)

Now we apply eqn. (56) over and over again:

$$\begin{aligned} &{\mathbb {E}}\Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2\le \frac{{\mathbb {E}}[F_{i}(\textbf{w}_{1})-F_{i}(\textbf{w}_{2})]}{\Phi }\\&{\mathbb {E}}\Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2\le \frac{{\mathbb {E}}[F_{i}(\textbf{w}_{2})-F_{i}(\textbf{w}_{3})]}{\Phi }\\&\hspace{7em} \vdots \nonumber \end{aligned}$$

Sum the above up, we have:

$$\begin{aligned} \sum _{t=1}^T{\mathbb {E}}\left\| \nabla F_i(\textbf{w}_{t})\right\| ^2\quad \le \frac{F_{1}-F_{*}}{\Phi } \end{aligned}$$

(57)

Divide both sides by T, we have:

$$\begin{aligned} \min _{t=1:T}{\mathbb {E}}\left\| \nabla F_i(\textbf{w}_t)\right\| ^2\le \frac{\sum _{t=1}^T{\mathbb {E}}\left\| \nabla F_i(\textbf{w}_t)\right\| ^2}{T}&\le \frac{F_{1}-F_{*}}{\Phi T} \end{aligned}$$

(58)

Let us assume that $F_{1}-F_{*}\le D$. Then if we set $\eta \le \frac{1}{5\,L}$, we can get:

$$\begin{aligned} \min _{t=1:T}{\mathbb {E}}\left\| \nabla F_i(\textbf{w}_t)\right\| ^2\le \frac{5LD}{T} \end{aligned}$$

(59)

We can see that the convergence rate of FedBat satisfies ${\mathcal {O}}(\frac{1}{T})$.

Proof of Theorem 2:

In the proof process, we derived eqn. (59), which provides the convergence bound for FedBat. To enhance these guarantees, we introduce the Polyak-Lojasiewicz (PL) condition. Starting from equ. (56), we have:

$$\begin{aligned} &{\mathbb {E}}[F(\textbf{w}_{t+1})-F(\textbf{w}_{t})]+\Phi {\mathbb {E}}[\Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2]\le 0 \\&\Leftrightarrow {\mathbb {E}}[F(\textbf{w}_{t+1})-F(\textbf{w}_{*})-(F(\textbf{w}_{t})-F(\textbf{w}_{*}))]+\Phi {\mathbb {E}}[\Vert \nabla F_{i}(\textbf{w}_{i,t})\Vert ^2]\le 0 \end{aligned}$$

(60)

By combining the Assumption 2:

$$\begin{aligned} &{\mathbb {E}}[F(\textbf{w}_{t+1})-F(\textbf{w}_{*})]\le {\mathbb {E}}[F(\textbf{w}_{t})-F(\textbf{w}_{*})]-\Phi \mu {\mathbb {E}}[F(\textbf{w}_t)-F(\textbf{w}_{*})]\\&\hspace{9em}\le (1-\Phi \mu ){\mathbb {E}}[F(\textbf{w}_t)-F(\textbf{w}_{*})]\\&\hspace{9em}\le (1-\Phi \mu )^2{\mathbb {E}}[F(\textbf{w}_{t-1})-F(\textbf{w}_{*})]\\&\hspace{12em} \vdots \nonumber \\&\hspace{9em}\le (1-\Phi \mu )^{k+1}{\mathbb {E}}[F(\textbf{w}_0)-F(\textbf{w}_{*})] \end{aligned}$$

To make this value less than or equal to $\Lambda$, we have:

$$\begin{aligned}&(1-\Phi \mu )^{k+1}{\mathbb {E}}[F(\textbf{w}_0)-F(\textbf{w}_{*})] \le \Lambda \\&T_{\Lambda }={\mathcal {O}}(\log {\frac{{\mathbb {E}}[F(\textbf{w}_0)-F(\textbf{w}_{*})]}{\Lambda }}\frac{1}{\Phi \mu }) \end{aligned}$$

Proof of Theorem 3:

$$\begin{aligned} \Vert \textbf{w}_{i,t+1}-\textbf{w}_{*,t+1}\Vert&=\Vert \textbf{w}_{i,t}-\textbf{w}_{*,t}+\textbf{vR}_{i,t+1}-\textbf{vR}_{*,t+1}-\eta \nabla F_{i}(\textbf{w}_{i,t})+\eta \nabla F(\textbf{w}_{*,t})\Vert \\&\le \Vert \textbf{w}_{i,t}-\textbf{w}_{*,t}\Vert +\Vert \textbf{vR}_{i,t+1}-\textbf{vR}_{*,t+1}\Vert + \left\| \eta \nabla F_i\big (\textbf{w}_{i,t}\big )-\eta \nabla F\big (\textbf{w}_{*,t}\big )\right\| \end{aligned}$$

(61)

Based on the definitions of gradients at both the local clients and the virtual client, the gradients are given by $\nabla F_i(\textbf{w}_{i,t})=\sum _{s=1}^Sm_i(b{=}s)\nabla {\mathbb {E}}_{\textbf{a}|b{=}s}[I_s(\textbf{a},\textbf{w}_{i,t})]$ and $\nabla F_i(\textbf{w}_{*,t})=\sum _{s=1}^Sm_i(b{=}s)\nabla {\mathbb {E}}_{\textbf{a}|b{=}s}[I_s(\textbf{a},\textbf{w}_{*,t})]$.

$$\begin{aligned} &\left\| \nabla F_i\big (\textbf{w}_{i,t}\big )-\nabla F\big (\textbf{w}_{*,t}\big )\right\| \\&=\left\| \sum _{s=1}^Sm_i(b=s)\nabla {\mathbb {E}}_{\textbf{a}|b=s}[I_s(\textbf{a},\textbf{w}_{i,t})]-\sum _{s=1}^Sm(b=s)\nabla {\mathbb {E}}_{\textbf{a}|b=s}[I_s(\textbf{a},\textbf{w}_{*,t})]\right\| \\&=\left\| \sum _{s=1}^Sm_i(b=s)\nabla {\mathbb {E}}_{\textbf{a}|b=s}[I_s(\textbf{a},\textbf{w}_{i,t})]-\sum _{s=1}^Sm_i(b=s)\nabla {\mathbb {E}}_{\textbf{a}|b=s}[I_s(\textbf{a},\textbf{w}_{*,t})] \right. \\&\quad \left. +\sum _{s=1}^Sm_i(b=s)\nabla {\mathbb {E}}_{\textbf{a}|b=s}\left[ I_s\left( \textbf{a},\textbf{w}_{*,t}\right) \right] -\sum _{s=1}^Sm(b=s)\nabla {\mathbb {E}}_{\textbf{a}|b=s}\left[ I_s\left( \textbf{a},\textbf{w}_{*,t}\right) \right] \right\| \\&\le \left\| \sum _{s=1}^Sm_i(b{=}s)(\nabla {\mathbb {E}}_{\textbf{a}|b{=}s}[I_s(\textbf{a},\textbf{w}_{i,t})]-\nabla {\mathbb {E}}_{\textbf{a}|b=s}[I_s(\textbf{a},\textbf{w}_{*,t})])\right\| \\&\quad +\left\| \sum _{s=1}^S(m_i(b{=}s){-}m(b{=}s))\nabla {\mathbb {E}}_{\textbf{a}|b{=}s}[I_s(\textbf{a},\textbf{w}_{*,t})]\right\| \end{aligned}$$

(62)

Letting $I_{\text {max}}(\textbf{w}_{*,t}) = \max \{\nabla {\mathbb {E}}_{\textbf{a}|b=s}[I_s(\textbf{a}, \textbf{w}_{*,t})]\}_{s=1}^S$. Using the Lipschitz continuity$\Vert \nabla {\mathbb {E}}_{\textbf{a}|b=s}[I_s(\textbf{a},\mathbf {w_1})]-\nabla {\mathbb {E}}_{\textbf{a}|b=s}[I_s(\textbf{a},\mathbf {w_2})]\Vert \le L_s\Vert \mathbf {w_1}-\mathbf {w_2}\Vert$, the equality of eqn. (62) can be further rewritten as:

$$\begin{aligned} &\Vert \nabla F_i\left( \textbf{w}_{i,t}\right) -\nabla F(\textbf{w}_{*,t})\Vert \\&\le \sum _{s=1}^Sm_i(b=s)L_s\Vert \textbf{w}_{i,t}-\textbf{w}_t\Vert + I_{max}\big (\textbf{w}_{*,t}\big )\sum _{s=1}^S|(m_i(b=s)-m(b=s))| \end{aligned}$$

(63)

Then, we have:

$$\begin{aligned}&\left\| \textbf{w}_{i,t+1}-\textbf{w}_{*,t+1}\right\| \\&\le \left\| \textbf{w}_{i,t}-\textbf{w}_{*,t}\right\| +\left\| \textbf{vR}_{i,t+1}-\textbf{vR}_{*,t+1}\right\| \\&\le \Vert \textbf{w}_{i,t}-\textbf{w}_{*,t}\Vert +|\left( 1+f_{i}\right) \textbf{vR}_{i,t}-\left( f_{i}+1\right) \textbf{vR}_{*,t}\Vert +\eta \Vert \nabla F_i\big (\textbf{w}_{i,t}\big )-\nabla F\big (\textbf{w}_{*,t}\big )\Vert \\&\le \Vert \textbf{w}_{i,t}-\textbf{w}_{*,t}\Vert +|\left( 1+f_{i}\right) \textbf{vR}_{i,t}-\left( f_{i}+1\right) \textbf{vR}_{*,t}\Vert \\ & +\eta \sum _{s=1}^Sm_i(b = s)L_s\Vert \textbf{w}_{i,t}-\textbf{w}_{*,t}\Vert +\eta I_{max}\big(\textbf{w}_{*,t}\big)\sum _{s=1}^S|m_i(b = s)-m(b=s)| \\ &=\left( 1+\eta \sum _{s=1}^Sm_i(b=s)L_s\right) \Vert \textbf{w}_{i,t}-\textbf{w}_{*,t}\Vert +\Vert \left( 1+f_{i}\right) \textbf{vR}_{i,t}-\left( f_{i}+1\right) \textbf{vR}_{*,t}\Vert \\&\quad+\eta I_{max}\big (\textbf{w}_{*,t}\big )\sum _{s=1}^S|m_i(b=s)-m(b=s)| \end{aligned}$$

(64)

Letting $\beta = 1 + \eta \sum _{s=1}^S m_i(b = s)L_s$, we rewrite eqn. (64) as:

$$\begin{aligned} &\left\| \textbf{w}_{i,t+1}-\textbf{w}_{*,t+1}\right\| \le \beta \Vert \textbf{w}_{i,t}-\textbf{w}_{*,t}\Vert +\left( 1+f_{i}\right) (\Vert \textbf{vR}_{i,t}-\textbf{vR}_{*,t}\Vert ) \\&\quad+\eta I_{max}\left( \textbf{w}_{*,t}\right) \sum _{s=1}^S|m_i(b=s)-m(b=s)| \\&\le \beta ^2\Vert \textbf{w}_{i,t-1}-\textbf{w}_{*,t-1}\Vert +\beta \left( 1+f_{i}\right) (\Vert \textbf{vR}_{i,t}-\textbf{vR}_{*,t}\Vert ) \\&\quad+\beta \eta I_{max}\left( \textbf{w}_{*,t}\right) \sum _{s=1}^S|m_i(b=s)-m(b=s)|+\left( 1+f_{i}\right) (\Vert \textbf{vR}_{i,t}-\textbf{vR}_{*,t}\Vert ) \\&\quad+\eta I_{max}\left( \textbf{w}_{*,t}\right) \sum _{s=1}^S|m_i(b=s)-m(b=s)| \\&\le \beta ^{t+1}\Vert \textbf{w}_{i,0}-\textbf{w}_{*,0}\Vert +\left( 1+f_{i}\right) \sum _{j=0}^t\beta ^{t-j}\Vert \textbf{vR}_{i,j}-\textbf{vR}_{*,j}\Vert \\&\quad+\eta \sum _{s=1}^S|m_i(b=s)-m(b=s)|\sum _{j=0}^tI_{max}\left( \textbf{w}_{*,j}\right) \end{aligned}$$

(65)

In conclusion, the proof is complete.