# Balancing Depth for Robustness: A Study on Reincarnating Reinforcement Learning Models
## 2 Materials and Methods
Output and Feature Representation Layers: Finally, a feature representation layer outputs a 128-dimensional feature vector, which effectively connects to the output layer to generate the final Q-values, guiding the agent's decisions in the reinforcement learning environment. This study constructs models with varying depths by incrementally increasing network depth d while maintaining other parameters unchanged, thereby isolating depth as the primary factor affecting performance. The network output \(Q_{d}(s,a)\) is expressed as:  
\[Q_{d}(s,a)=f_{d}(s,a;\theta) \tag{3}\]  
where \(f_{d}\) is a neural network function with depth \(d\), and \(\theta\) represents model parameters. The network depth directly affects the expressiveness and robustness of the model. Appropriately increasing the depth can enhance the feature extraction ability of the model, but a network that is too deep may lead to overfitting and unstable training. Therefore, the impact of network depth on model robustness is a key factor. According to the Universal Approximation Theorem, any continuous function can be approximated by a sufficiently deep neural network. However, the approximation ability alone cannot fully explain the effect of deep networks in actual tasks, especially when the complexity of the task and the uncertainty of the environment increase; the performance of deep networks may decrease due to overfitting. In order to better understand the impact of network depth on robustness, we consider the following model of depth and performance:  
\[f_{L}(x)=\sum\
olimits_{i=1}^{N_{L}}\alpha_{i}\sigma(W_{L}x+b_{L}) \tag{4}\]  
Among them, \(f_{L}(x)\) represents the output of a neural network with a depth of \(L\), \(N_{L}\) is the number of neurons in the Lth layer of the network, and \(W_{L}\) and \(b_{L}\) are the weight and  
Figure 2: Network Architecture. bias of the layer, respectively. A deeper network can represent a more complex mapping, but the higher the complexity, the easier it is to capture noise during training. This can be reflected in the following mathematical model:  
\[E_{gen}(L)=E_{bias}(L)+E_{variance}(L)+E_{complexity}(L) \tag{5}\]  
Expression error \(E_{bias}(L)\): As the depth \(L\) increases, the representation ability of the network improves, and the expression error decreases. Variance error \(E_{variance}(L)\): When the depth is too large, the network may begin to fit the noise in the training data, the variance increases, and overfitting occurs. Complexity error \(E_{complexity}(L)\): When the depth is too large, the network complexity is too high, which may also lead to a decrease in the generalization ability on the test data. In order to quantify the impact of network depth on generalization ability, this study introduced the Rademacher Complexity model [39, 40]. Rademacher complexity measures the sensitivity of the network structure to random noise and reflects the complexity and robustness of the model. Let the network model be \(f_{\theta}(x)\) where \(x\) is the input, \(\theta\) is the network parameter, \(\mathrm{S}=\{x1,x2,\ldots,xn\}\) is the training data set, and the Rademacher complexity of the network is:  
\[\mathcal{R}_{n}(f)=\mathbb{E}\left[\underset{\theta}{sup}\ \frac{1}{n}\sum_{i=1}^{n} \sigma(f_{\theta}(x_{i}))\right] \tag{6}\]  
where \(\sigma\) is the activation function and \(\mathrm{E}\) represents the expectation for all input data. According to this formula, as the network depth increases, the Rademacher complexity increases, which means that the network may be more sensitive to noise in the training data, leading to overfitting. However, within a reasonable range of network depth, the Rademacher complexity increases with depth until it reaches a balance point. This also explains why a network depth of seven layers may be at the optimal balance point between generalization ability and noise sensitivity. Feature Extraction Ability and Depth: The network depth \(d\) is closely related to the model's feature extraction capability.