## 3 Methodological Framework for Meta-Learning
### Theoretical Models of Meta-Learning  
Building on top of the conventional model given in (1), meta-learning seeks to optimize meta-parameters across a distribution of tasks \(k\) sampled from a task distribution \(P_{k}\) where \((k=1,2,\dots\ K\in P_{k})\). s are typically a function of the model parameters \(\theta\) and inductive bias \(\omega\). Each task \(k\) has its own dataset \(D_{k}\) that consists of the training set (\(D_{k}^{tr}\)) and testing set (\(D_{k}^{te}\)). The training set is sampled with \(k\) data points from each of the classes present in the dataset and is also called a support set. Similarly, the testing set is sampled with different \(k\) data points from each of the classes and is also referred to as query sets.  
Figure 3: Meta-learning pipeline illustrating pilot symbols over an OFDM symbol drawn from a 4-PAM constellation with \(N^{tr}=4\), \(N^{te}=4\), and P = 2 for the meta-test device.  
Figure 2: A meta-learning illustration where a learner trains a model to solve a visual object recognition task. The base learner minimizes a loss function that measures the difference between the true label of an input and the label predicted by a neural network model. The meta-learner minimizes a meta-loss function, such as the generalization error of the trained neural network on novel data.  
For each task \(k\), the model with parameters \(\theta\) is trained on \(D_{k}^{tr}\). The training updates the parameters to \(\theta_{k}^{*}\).  
\[\theta_{k}^{*}=\underset{\theta}{argmin}\operatorname*{\mathbb{E}}_{k\sim P_{k} }L(D_{k}^{tr},\theta) \tag{2}\]  
Here, \(\theta_{k}^{*}\) represents the optimal parameters for the task \(k\). The meta-parameters \(\omega\) are optimized based on the model's performance across multiple tasks. This performance is evaluated on the query sets of each task, using the task-specific parameters \(\theta_{k}^{*}\). The meta-training can then be mathematically defined as  
\[\omega^{*}=\underset{\omega}{argmin}\operatorname*{\mathbb{E}}_{k\sim P_{k} }L(D_{k}^{test},\theta_{k}^{*}(\omega)) \tag{3}\]  
The expectation \(\operatorname*{\mathbb{E}}\) is over the task distribution, and \(L(D_{k}^{test},\theta_{k}^{*}(\omega))\) measures the loss on the query set of task \(k\) after learning with parameters \(\theta_{k}^{*}\) that were adapted in (2).  
After the completion of meta-training, the performance of the meta-parameters is assessed by drawing a new meta-test task (\(k^{Q}\)), which is drawn from the same distribution as that of the tasks used in the meta-training phase. The inductive bias is applied to the corresponding dataset (\(D_{kQ}^{tr}\)) and tested on (\(D_{kQ}^{te}\)). During the meta-test step, the new inductive bias \(\omega^{*}\) is used to train the base model (inner loop) on each previously unseen task (\(q\sim Q\)):  
\[\theta^{*}=\underset{\theta}{argmax}\ logp(\theta|\omega^{*},D_{kQ}^{tr}) \tag{4}\]  
The previous discussion sums up the flow of meta-learning in multiple-task scenarios. Meta-learning algorithms are differentiated based on how inductive bias is estimated, e.g., [PERSON]. [42] proposed a Bayesian fast-adaption method for meta-learning and [PERSON]. [43] formulated the problem as a fully Bayesian inference problem. Primarily, the goal is to simplify the implementation of meta-learning which has been a major research topic over the past few years; see, e.g., [44, 45, 46, 41]. Based on these works, meta-learning can be categorized into three distinct types: model-based, metric-based, and optimization-based meta-learning, each with its unique strategies for inducing rapid learning adaptation.