## 2 Meta-Learning
Optimal parameters should maximize the likelihood of identifying true labels in multiple training batches \(B\subset D\):  
\[\theta^{*}=argmax_{\theta}\mathbb{E}_{(x,y)\in D}[P_{\theta}(y|x)] \tag{5}\]  
\[\theta^{*}=argmax_{\theta}\mathbb{E}_{B\subset D}\left[\sum_{(x,y)\in B}P_{ \theta}(y|x)\right] \tag{6}\]  
The aim is to reduce the prediction error in data samples with unknown labels, in which there is a small set of \"fast learning\" support which can be used for \"fine-tuning\". Fast learning is a trick which creates a \"fake\" dataset containing a small subset of labels (to avoid exposing all labels to the model). During the optimization process, various modifications take place, aiming to achieve rapid learning. A brief step-by-step description of the whole process is presented below [11]:  
1. Development of a subset of labels \(L_{s}\subset L\)
2. Development of a training subset \(S^{L}\subset D\) and a forecast subset \(B^{L}\subset D\). Both of them include data points with labels belonging to the subset \(L_{s}\), \(y\in L_{s}\), \(\forall\) (\(x,y\)) \(\in S^{L}\), \(B^{L}\). 3. The optimization procedure uses \(B^{L}\) to calculate the error and to update the model parameters via back propagation. This is done in the same way as it is used in a simple supervised learning model. In this way each sample pair (\(S^{L},B^{L}\)) can be considered to be a data point. The model is trained so that it can generalize to new unknown datasets. The following function (Equation (7)) is a modification of the supervised learning model. The symbols of the meta-learning process have been added:  
\[\theta^{*}=argmax_{0}\mathbb{E}_{L_{s}\subset L}\left[\mathbb{E}_{S^{L}\subset D,B^{L}\subset D}\left[\sum_{(x,y)\in B^{L}}P_{0}\Big{(}x,y,S^{L}\Big{)}\right]\right] \tag{7}\]  
There are three _meta-learning_ modeling approaches, as presented below [11] and the Table 1:  
1. Model-based: These are techniques based on the use of circular networks with external or internal memory. They update their parameters quickly with minimal training steps. This can be achieved through their internal architecture or by using other control models. _Memory-augmented neural networks_ and _meta networks_ are characteristic cases of _model-based meta-learning_ techniques. 2. Metrics-based: These are techniques based on learning effective distance measurements which can offer generalization. The core concept of their operation is similar to that of the nearest neighbors algorithms, where they aim to learn a measurement or a distance from objects. The concept of a good metric depends on the problem, as it should _represent_ the relationship between inputs to the site, facilitating problem solving. _Convolutional Siamese neural networks_, _matching networks, relation networks and prototypical networks_ are characteristic metrics-based meta-learning techniques. 3. Optimization-based: These are techniques based on optimizing the parameters of the model for quick learning. _LSTM Meta-Learners, temporal discreteness and the reptile plus Model-Agnostic Meta-Learning_ (MAML) algorithms are typical cases of optimization-based meta-Learning. The _Recurrent Neural Networks_ (RNNs) which use only internal memory, and also the _Long-Short-Term Memory_ approaches (LSTM), are not considered meta-learning techniques [11]. Meta-learning can be achieved through a variety of learning examples. In this case, the _supervised gradient-based_ learning can be considered as the most effective method [11]. More specifically, the _gradient-based_, _end-to-end differentiable meta-learning_, provides a wide framework for the application of effective _meta-learning_ techniques. This research proposes an _optimization-based_, _gradient-based_, _end-to-end_ differentiable meta-learning architecture, based on an innovative evolution of the MAML algorithm [10]. MAML is one of the most