# Balancing Depth for Robustness: A Study on Reincarnating Reinforcement Learning Models
## 2 Materials and Methods
For an input \(X\), the output of the depth \(\mathrm{d}\) layer can be represented as:  
\[Z_{d}=W_{d}\cdot f_{d-1}(X)+b_{d} \tag{7}\]  
Here, \(W_{d}\) and \(b_{d}\) denote the weight matrix and bias vector of the depth \(\mathrm{d}\) layer, respectively, and \(f_{d-1}(X)\) represents the output of the depth \(\mathrm{d}\) layer. To evaluate the robustness of different network depths against perturbations, we define the robustness loss \(L_{robust}\) as:  
Figure 1.0  
This loss function measures the extent of changes in the network's output before and after input perturbations. Higher robustness should correspond to a smaller \(L_{robust}(d)\). This loss function measures the change in network output before and after input perturbation, with lower values indicating higher robustness. The total evaluation combines task loss \(L_{Qdagger}(d)\) and robustness loss: where \(\lambda\) is a weighting parameter balancing task performance and robustness. Figure 1.1  
This architecture allows automatic depth optimization tailored to specific task environments, achieving a balance between feature extraction and training efficiency. Many studies have explored the impact of network depth on model performance. Generally speaking, deeper networks can represent more complex functional relationships. However, how to strike a balance between different depths to optimize the robustness of the model remains an open question. Especially in the context of regenerative reinforcement learning, different network structures may have different effects on the performance of the model, so it is of great research value to deeply explore the relationship between network structure and model robustness. In addition, regenerative reinforcement learning also faces the challenge of how to effectively utilize historical models. According to the research of [PERSON] et al., RRL can not only accelerate the learning of new tasks by reusing calculation results but also cope with the uncertainty in task changes by adjusting the model architecture. Therefore, the design of the network structure is particularly important in RRL. It not only affects the learning speed and performance of the model but also plays a key role in the robustness of the model when facing complex environments. To investigate the impact of network depth, this study designed several sets of regenerative neural networks (Q-Dagger) with different depths, which are adaptively selected based on task complexity. This approach allows us to explore how different network depths tailored to each task affect model performance. The Algorithm 1 outlines the methods used in our study. ```
1. Initialization: Use Xavier uniform initializer for network parameters. 2. Preprocess Input: If inputs_preprocessed is False, preprocess input x. 3. Convolutional Layers: Adaptively choose conv_layers (1-5) based on task complexity (state space size, reward sparsity, action space size). 4. For each layer, randomly select features from [32; 64; 128; 256], and use (3, 3) kernel, (1, 1) strides. 5. Apply convolution, ReLU, and BatchNorm. 6. Flatten: Reshape the output tensor to prepare for fully connected layers. 7. Fully Connected Layers: Adaptively choose fc layers (1-3) based on task complexity and training feedback. 8. For each layer (except last), randomly select units from [64; 128; 256], apply dense and ReLU. 9. Representation Layer: Add a dense layer with 128 units, apply ReLU. 10. Output Layer: Add a dense layer to produce num_actions Q-values. 11. Return: Return Q-values and representation. ```  
**Algorithm 1**Adaptive Deep Network Architecture for Reinforcement Learning. To evaluate the performance of networks of different depths in regenerative reinforcement learning (RRL), this study developed a systematic experimental framework covering several classical reinforcement learning environments. The experiments adopted a controlled variable approach to ensure that the observed performance differences mainly come from the adjustment of network depth.