## 3 Single-Antenna Frequency-Flat Channels
### Meta-Learning  
Unlike transfer learning, which utilizes all the available datasets \(\{\mathcal{Z}_{f}\}_{f=1}^{F}\) from the previous frames at once as in (15), meta-learning allows for the separate adaptation of the predictor in each frame. To this end, for each frame \(f\), we split the \(L\) data points into \(L^{\text{tr}}\) training pairs \(\{(x_{i,f},y_{i,f})\}_{i=1}^{L^{\text{tr}}}\equiv\{(x_{i,f}^{\text{tr}},y_{i, f}^{\text{tr}})\}_{i=1}^{L^{\text{tr}}}=\mathcal{Z}_{f}^{\text{tr}}\) and \(L^{\text{te}}=L-L^{\text{tr}}\) test pairs \(\{(x_{i,f},y_{i,f})\}_{i=L^{\text{tr}}+1}^{L}\equiv\{(x_{i,f}^{\text{te}},y_{ i,f}^{\text{te}})\}_{i=1}^{L^{\text{tr}}}=\mathcal{Z}_{f}^{\text{te}}\), resulting in two separate datasets, \(\mathcal{Z}_{f}^{\text{tr}}\) and \(\mathcal{Z}_{f}^{\text{te}}\). We correspondingly define the \(L^{\text{tr}}\times N\) input matrix \(X_{f}^{\text{tr}}\) and the \(L^{\text{tr}}\times 1\) target vector \(y_{f}^{\text{tr}}\), as well as the \(L^{\text{te}}\times N\) input matrix \(X_{f}^{\text{te}}\) and the \(L^{\text{te}}\times 1\) target vector \(y_{f}^{\text{te}}\).  
The hyperparameter vector \(\tilde{v}\) is then optimized by minimizing the sum loss of the predictors \(v^{*}(\mathcal{Z}_{f}^{\text{tr}}|\tilde{v})\) in (13) that are adapted separately for each frame \(f=1,\ldots,F\) given the bias vector \(\tilde{v}\). Accordingly, estimating the loss in each frame \(f\) via the test set \(\mathcal{Z}_{f}^{\text{te}}\) yields the meta-learning problem  
\[\tilde{v}^{\text{meta}}=\operatorname*{arg\,min}_{\sigma\in\mathcal{C}^{N \times 1}}\Bigg{\{}\sum_{f=1}^{F}\left|v^{*}(\mathcal{Z}_{f}^{\text{tr}}| \tilde{v})^{\dagger}x_{i,f}^{\text{te}}-y_{i,f}^{\text{te}}\right|^{2}\Bigg{\}}. \tag{17}\]  
As studied in [52], the minimization in (17) is a least squares problem that can be solved in closed form as  
\[\tilde{v}^{\text{meta}} =\operatorname*{arg\,min}_{\sigma\in\mathcal{C}^{N\times 1}}\sum_{f=1 }^{F}\left\|\tilde{X}_{f}^{\text{te}}\tilde{v}-\tilde{y}_{f}^{\text{te}}\right\| ^{2}\] \[=(\widehat{X}^{\dagger}\widehat{X})^{-1}\widehat{X}^{\dagger} \tilde{y}, \tag{18}\]  
where \(L^{\text{te}}\times N\) matrix \(\widehat{X}_{f}^{\text{te}}\) contains by row the Hermitian transpose of the \(N\times 1\) preconditioned input vectors \(\{\lambda(A_{f}^{\text{tr}})^{-1}x_{i,f}^{\text{te}}\}_{i=1}^{L^{\text{te}}}\), with \(A_{f}^{\text{tr}}=(X_{f}^{\text{tr}})^{\dagger}X_{f}^{\text{tr}}+\lambda I_{i}\), \(\tilde{y}_{f}^{\text{te}}\) is \(L^{\text{te}}\times 1\) vector containing vertically the complex conjugate of the transformed outputs \(\{(y_{i,f}^{\text{te}}-(y_{f}^{\text{tr}})^{\dagger}X_{f}^{\text{tr}}(A_{f}^{ \text{tr}})^{-1}x_{i,f}^{\text{te}}\}_{i=1}^{L^{\text{te}}}\), the \(FL^{\text{te}}\times N\) matrix \(\widehat{X}=[\widehat{X}_{1}^{\text{te}},\ldots,\widehat{X}_{F}^{\text{te}}] ^{\top}\) stacks vertically the \(L^{\text{te}}\times N\)\(N\) matrices \(\{\tilde{\mathcal{X}}_{f}^{\text{te}}\}_{f=1}^{F}\), and the \(FL^{\text{te}}\times 1\) vector \(\tilde{y}=[y_{1}^{\text{te}},\ldots,y_{F}^{\text{te}}]^{\top}\) stacks vertically the \(L^{\text{te}}\times 1\) vectors \(\{\tilde{y}_{f}^{\text{te}}\}_{f=1}^{F}\). Unlike standard meta-learning algorithms used by most papers on communications [25; 29; 30; 32; 33; 34], the proposed meta-learning procedure adopts linear models, significantly reducing the computational complexity of meta-learning [52].  
After meta-learning, similar to transfer learning, based on the meta-learned hyperparameter \(\tilde{\sigma}_{\lambda}^{\text{meta}}\), we train a channel predictor via ridge regression (13), obtaining  
\[v_{f^{\text{new}}}^{*}=v^{*}(\mathcal{Z}_{f^{\text{new}}}|\sigma^{\text{meta} }). \tag{19}\]