## 3 RMSprop SGD-Based LFT Model
### RLFT Model  
In the SGD-based LFT model, if the objective function's surface exhibits complex topography (such as elongated valleys), the gradient direction may deviate from the optimal path, leading to slow convergence or even oscillations. The RMSprop algorithm addresses this by employing exponentially weighted moving averages of squared gradients to dynamically adapt the learning rate. This approach mitigates gradient fluctuations and accommodates sparse gradients, thereby optimizing the path toward more stable and efficient convergence. As illustrated in Figure 1, RMSprop demonstrates a more direct trajectory toward the optimal solution, achieving faster convergence with reduced oscillations. Therefore, adopting the RMSprop algorithm is essential for enhancing model training efficiency in complex optimization problems.  
Figure 1: Training process by LFT with/without RMSprop.  
When integrating the adaptive learning rate method (i.e., the RMSprop algorithm) into the SGD-based LFT model, it is also necessary to update the decision parameters for each training instance during every iteration. According to Formula (6), the update rules for \(t_{ir},l_{jp}\), and \(w_{br}\) can be derived by following Equations (10)-(12) [23]:  
The update of \(t_{ir}\) is as follows:  
\[\begin{split}& A^{0}_{t_{ir}}=0;A^{s}_{t_{ir}}=\gamma A^{s-1}_{t_{ ir}}+(1-\gamma)(\frac{\partial\epsilon^{s-1}_{ijk}}{\partial t^{s-1}_{ir}})\\ & t^{s}_{ir}=t^{s-1}_{ir}-\frac{\eta_{ir}}{\sqrt{A^{s}_{t_{ir}}+ \delta}}\cdot\frac{\partial\epsilon^{s-1}_{ijk}}{\partial t^{s-1}_{ir}}\end{split} \tag{10}\]  
The update of \(l_{jp}\) is as follows:  
\[\begin{split}& A^{0}_{t_{ir}}=0;A^{s}_{t_{ir}}=\gamma A^{s-1}_{t_ {ir}}+(1-\gamma)(\frac{\partial\epsilon^{s-1}_{ijk}}{\partial t^{s-1}_{ir}})\\ & l^{s}_{jp}=l^{s-1}_{jp}-\frac{\eta_{ir}}{\sqrt{A^{s}_{t_{ir}}+ \delta}}\cdot\frac{\partial\epsilon^{s-1}_{ijk}}{\partial t^{s-1}_{jp}}\end{split} \tag{11}\]  
The update of \(w_{kr}\) is as follows:  
\[\begin{split}& A^{0}_{w_{br}}=0;A^{s}_{w_{br}}=\gamma A^{s-1}_{ w_{br}}+(1-\gamma)(\frac{\partial\epsilon^{s-1}_{ijk}}{\partial t^{s-1}_{kr}}) ^{2}\\ & w^{s}_{kr}=w^{s-1}_{kr}-\frac{\eta_{ir}}{\sqrt{A^{s}_{w_{br}}+ \delta}}\cdot\frac{\partial\epsilon^{s-1}_{ijk}}{\partial w^{s-1}_{kr}}\end{split} \tag{12}\]  
By combining Formulas (10)-(12) with Formula (8), the objective loss function is minimized, as shown in Equation (13) [23]:  
\[\begin{split}&\arg\min_{T,L,W}\left(T,L,W\right)\overset{ RLFT}{\Rightarrow}\forall i\in I,j\in J,k\in K,r\in\{1,2,\ldots,R\}:\\ &\left\{\begin{array}{l}A^{0}_{t_{ir}}=0;A^{s}_{t_{ir}}=\gamma A ^{s-1}_{t_{ir}}+(1-\gamma)(\lambda_{t}t^{s-1}_{ir}-l^{s-1}_{ir}w^{s-1}_{kr}(y _{ijk}-\delta^{s-1}_{ijk}))^{2}\\ t^{s}_{ir}\leftarrow t^{s-1}_{ir}-\frac{\eta_{ir}}{\sqrt{A^{s}_{t_{ir}}+ \delta}}\cdot(\lambda_{t}t^{s-1}_{ir}-l^{s-1}_{ir}w^{s-1}_{jk}(y_{ijk}-\delta^ {s-1}_{ijk}))\\ A^{0}_{t_{ir}}=0;A^{s}_{t_{ir}}=\gamma A^{s-1}_{t_{ir}}+(1-\gamma)(\lambda_{ t}l^{s-1}_{ir}-t^{s-1}_{ir}w^{s-1}_{kr}(y_{ijk}-\delta^{s-1}_{ijk}))^{2}\\ l^{s}_{jp}\leftarrow l^{s-1}_{jp}-\frac{\eta_{ir}}{\sqrt{A^{s}_{t_{ir}}+ \delta}}\cdot(\lambda_{t}l^{s-1}_{jp}-t^{s-1}_{ir}w^{s-1}_{kr}(y_{jk}-\delta^ {s-1}_{ijk}))\\ A^{0}_{w_{br}}=0;A^{s}_{w_{br}}=\gamma A^{s-1}_{w_{br}}+(1-\gamma)(\lambda_{ w}w^{s-1}_{kr}-t^{s-1}_{ir}l^{s-1}_{jp}(y_{ijk}-\delta^{s-1}_{ijk}))^{2}\\ w^{s}_{kr}\leftarrow w^{s-1}_{kr}-\frac{\eta_{ir}}{\sqrt{A^{s}_{w_{br}}+ \delta}}\cdot(\lambda_{w}w^{s-1}_{kr}-t^{s-1}_{ir}l^{s-1}_{jp}(y_{ijk}-\delta^ {s-1}_{ijk}))\end{array}\right.\end{split} \tag{13}\]