## 4 Decentralized Data Valuation
### Decentralized Calculation of Data Valuation  
One approach to approximate \(q\) is via decentralized cross-validation in distributed data networks. We define the utility of a global model as the validation accuracy based on private local data. Let \(\mathbf{d}_{n}^{\text{val}}\) be the validation dataset of smartphone \(n\). The validation accuracy of any model \(\mathbf{w}\), denoted by \(\mathcal{A}(\mathbf{w},\mathbf{d}_{n}^{\text{val}})\), is calculated by the ratio of matching predictions to the total number of validation samples. More formally, for any smartphone \(n\), the utility of the global model in round \(r\) is calculated by  
\[U_{n}(\mathbf{w}_{r})=\mathcal{A}(\mathbf{w}_{r},\mathbf{d}_{n}^{\text{val}}). \tag{13}\]  
Let \(\Phi_{i,n,r}(\mathbf{d}_{n,r})=U_{i}(\mathbf{w}_{n,r})-U_{i}(\mathbf{w}_{r-1})\) be the contribution of \(\mathbf{d}_{n,r}\) to smartphone \(i\), \(i,n\in\mathcal{N}\). Then, \(\Phi_{i,n,r}(\mathbf{d}_{n,r})\) is defined by  
\[\Phi_{i,n,r}(\mathbf{d}_{n,r})=\left\{ \begin{array}{ll}U_{i}(\mathbf{w}_{n,r})-U_{i} (\mathbf{w}_{r-1}),&\text{if }U_{i}(\mathbf{w}_{n,r})>U_{i}(\mathbf{w}_{r-1}),\\ 0,&\text{if }U_{i}(\mathbf{w}_{n,r})\leq U_{i}(\mathbf{w}_{r-1}).\end{array} \right. \tag{14}\]  
Note that \(\Phi_{i,n,r}(\mathbf{d}_{j,r}+\mathbf{d}_{k,r})=U_{i}(\mathbf{w}_{j,r})+U_{i}(\mathbf{w}_{k,r} )-2U_{i}(\mathbf{w}_{r-1}),\forall i,j,k\in\mathcal{N},i\
eq j\
eq k\) because datasets from individual smartphones remain localized and are never transmitted beyond their respective devices. Then, the valuation of data contribution from any smartphone \(n\) is determined by all smartphones except \(n\). That is,  
\[V(\mathbf{d}_{n,r})=\frac{1}{N-1}\sum_{i\in\mathcal{N},i\
eq n}\Phi_{i,n,r}(\mathbf{d} _{n,r}),\forall n\in\mathcal{N}. \tag{15}\]  
Equation (15) describes a cross-validation of contributions. Every smartphone's data valuation is truly based on its uploaded model weights and is determined by other smartphones. Therefore, the calculation of \(V_{n,r}\) (i.e., \(V(\mathbf{d}_{n,r})\)) is fully decentralized: relying solely on the blockchain records of model weights. To solve (8) and maximize \(U_{n}\), we argue that decentralized calculation of \(V_{n,r}\) is a critical step to minimize \(D_{KL}(p_{n,r}\|q)\) at the network edge, especially when \(q\) is unpredictable. By examining (14) and (15), our data valuation function \(V\) has the following properties:  
1. Group rationality: The valuation of the per-round data contribution is completely distributed among all data contributors, i.e., \(V_{r}=\sum_{n\in\mathcal{N}}V(\mathbf{d}_{n,r})\). 2. Fairness: Two data contributors with identical data contributions should have the same valuation, i.e., \(V(\mathbf{d}_{i,r})=V(\mathbf{d}_{j,r})\) if datasets \(\mathbf{d}_{i,r}\) and \(\mathbf{d}_{j,r}\) are identical; a free-rider \(n\) with zero \(\Phi_{i,n,r}\) for all other \(N-1\) smartphones has zero valuation, i.e., \(V(\mathbf{d}_{n,r})=V(\emptyset)=0\). 3. Additivity: In any round \(r\), the data valuation of multiple data contributors equals the sum of the data valuations of individual data contributors, i.e., \(V(\mathbf{d}_{1,r})+V(\mathbf{d}_{2,r})=V(\mathbf{d}_{1,r}+\mathbf{d}_{2,r})\). Furthermore, we present the following theorem to elucidate how decentralized data valuation benefits collaborative edge intelligence. **Theorem 1**. : _Cross-validation of data contributions using (15) can prevent free-riders._  
Proof of Theorem 1.: We will prove this by contradiction. Suppose in any round \(r\), \(\forall r\in\mathcal{R}\), a free-rider \(j\), \(j\in\mathcal{N}\), has data valuation of \(V(\mathbf{d}_{j,r})\). An honest data contributor \(k\), \(k\in\mathcal{N},k\
eq j\), has \(V(\mathbf{d}_{k,r})\) as the lowest data contribution among other honest contributors. Suppose for the sake of contradiction proof, that the free-rider, smartphone \(j\), can mimic the honest data contributor, smartphone \(k\).