# Intelligent Radio Frequency Identification for URLLC in Industrial IoT Networks
## 3 RFI Based on Semisupervised Learning For URLLCs
### Novel Architecture Illustration of Semisupervised Learning Model
We can obtain the derivatives by chain rule  
\[\frac{\partial L_{l}(w_{s}^{\text{gsl}}(w_{t}))}{\partial w_{t}} =\frac{\partial}{\partial w_{t}}\mathcal{C}\Bigg{(}y_{t},\underbrace{ S\Big{(}x_{l};\mathbb{E}_{\widehat{y}_{u}\sim T(x_{u};w_{t})}[w_{s}-\eta_{s} \
abla_{\eta_{s}}\mathcal{C}(\widehat{y}_{u},S(x_{u};w_{s}))]\Big{)}}_{\text{ student prediction}}\Bigg{)} \tag{11}\] \[=\frac{\partial}{\partial w_{T}}\mathcal{C}\big{(}y_{t},S\big{(} x_{l};\widehat{w}_{s}^{\prime}\big{)}\big{)}\] \[=\underbrace{\frac{\partial\mathcal{C}(y_{l},S(x_{l};\widehat{w }_{s}^{\prime}))}{\partial w_{s}}}_{\text{PartI}}|_{w_{s}=\widehat{w}_{s}^{ \prime}}\underbrace{\frac{\partial\widehat{w}_{s}^{\prime}}{\partial w_{t}}}_ {\text{PartII}}\]  
where,  
\[\widehat{w}_{s}^{\prime}=\mathbb{E}_{\widehat{y}_{u}\sim T(x_{u};w_{T})}\big{[} w_{s}-\eta_{s}\
abla_{\eta_{s}}\mathcal{C}(\widehat{y}_{u},S(x_{u};w_{s})) \big{]} \tag{12}\]  
As shown in (11), the Part I factor can be calculated by backpropagation. Then, we can rewrite Part II as following  
\[ \begin{split}\frac{\partial\widehat{w}_{s}^{\prime}}{\partial w _{t}}&=\frac{\partial}{\partial w_{t}}\mathbb{E}_{\widehat{y}_{u} \sim T(x_{u};w_{t})}\big{[}w_{s}-\eta_{s}\
abla_{\eta_{s}}\mathcal{C}( \widehat{y}_{u},S(x_{u};w_{s}))\big{]}\\ &=\frac{\partial}{\partial w_{t}}\mathbb{E}_{\widehat{y}_{u} \sim T(x_{u};w_{t})}\Bigg{[}w_{s}-\eta_{s}\cdot\left(\frac{\partial\mathcal{C }(\widehat{y}_{u},S(x_{u};w_{s}))}{\partial w_{s}}\right)^{T}\Bigg{]}\end{split}  \tag{13}\]  
where \(w_{s}\) is independent with \(w_{t}\), and we can leave the first part out of subsequent derivations. Furthermore, we can obtain the simplified expression as  
\[ \begin{split}\frac{\partial\widehat{w}_{s}^{\prime}}{\partial w _{t}}&=-\eta_{s}\cdot\frac{\partial}{\partial w_{t}}\mathbb{E}_{ \widehat{y}_{u}\sim T(x_{u};w_{t})}\Bigg{[}\left(\frac{\partial\mathcal{C}( \widehat{y}_{u},S(x_{u};w_{s}))}{\partial w_{s}}\right)^{T}\Bigg{]}\\ &=-\eta_{s}\cdot\frac{\partial}{\partial w_{t}}\mathbb{E}_{ \widehat{y}_{u}\sim T(x_{u};\widehat{a}_{T})}[g_{s}(\widehat{y}_{u})]\end{split}  \tag{14}\]  
where  
\[g_{s}(\widehat{y}_{u})=\left(\frac{\partial\mathcal{C}(\widehat{y}_{u},S(x_{u };w_{s}))}{\partial w_{s}}\right)^{T} \tag{15}\]  
(15) shows that \(g_{s}(\widehat{y}_{u})\) has no dependency on \(w_{t}\), except for via \(\widehat{y}_{u}\). So, the REINFORCE equation [34] was applied to obtain  
\[ \begin{split}\frac{\partial\widehat{w}_{s}^{\prime}}{\partial w _{t}}&=-\eta_{s}\cdot\frac{\partial}{\partial w_{t}}\mathbb{E}_{ \widehat{y}_{u}\sim T(x_{u};w_{t})}[g_{s}(\widehat{y}_{u})]\\ &=\eta_{s}\cdot\mathbb{E}_{\widehat{y}_{u}\sim T(x_{u};w_{t})} \Bigg{[}g_{s}(\widehat{y}_{u})\cdot\underbrace{\frac{\partial\mathcal{C}( \widehat{y}_{u},T(x_{u};w_{t}))}{\partial w_{t}}}_{\text{gradient of cross-entropy loss}}\Bigg{]}\end{split}  \tag{16}\]Then, we can substitute (16) into (11) to acquire the final result:  
\[ \begin{split}\frac{\partial L_{l}(w_{s}^{ssl}(w_{t}))}{\partial w_{t}} &=\underbrace{\frac{\partial\mathcal{C}(y_{t},S(x_{l};\hat{w}_{s}^ {\prime}))}{\partial w_{s}}|_{w_{s}=\hat{w}_{s}^{\prime}}}_{\text{PartI}} \underbrace{\frac{\partial\hat{w}_{s}^{\prime}}{\partial w_{t}}}_{\text{PartI}} \\ &=\eta_{s}\cdot\frac{\partial\mathcal{C}(y_{t},S(x_{l};\hat{w}_{s }^{\prime}))}{\partial w_{s}}\cdot\mathbb{E}_{\hat{y}_{u}\sim T(x_{s};w_{t})} \bigg{[}g_{s}(\hat{y}_{u})\cdot\frac{\partial\mathcal{C}(\hat{y}_{u},T(x_{u};w _{t}))}{\partial w_{t}}\bigg{]}\end{split}  \tag{17}\]  
For simplicity but without loss of generality, we can calculate the terms in (17) by utilizing Monte Carlo approximation with samples \(\hat{y}_{u}\). Furthermore, we can approximate \(\hat{w}_{s}^{\prime}\) by updating the student parameter on unlabeled samples through (12). With those approximations, final gradient \(\
abla_{w_{t}}L_{l}\) can be calculated by  
\[