# Local Back-Propagation for Forward-Forward Networks: Independent Unsupervised Layer-Wise Training
## 3 Local Back-Propagation Algorithm
### Architecture  
The overall structure of our model is shown in Figure 1. The original data are fed into the first cell, and each cell performs unsupervised learning, similar to approaches used in Auto-Encoders or GANs, by attempting to reconstruct its input. This design is flexible and can be applied to any system capable of calculating a local loss independently in each cell. Although we focus on reconstruction-based models here, the approach can be extended to any method that produces latent vectors of a consistent size at each cell.  
Figure 3 illustrates the basic form of a single cell using these models. In each cell, the encoder receives input data or latent vectors from the previous cell, compressing it into a lower-dimensional latent representation. The decoder attempts to reconstruct the original input from the latent representation, generating reconstruction data. The reconstruction loss between input data and reconstruction data is minimized independently in each cell, providing local supervision.  
The unsupervised learning models we employ in this study include AE, DAE [25], CAE, and GAN. Figure 3a shows a cell using an AE, while Figure 3b depicts a DAE-based cell, where random noise is introduced into the input. Figure 3c presents a CAE cell, which uses a convolutional encoder and a transposed convolutional decoder. Figure 3d illustrates a cell incorporating a GAN structure, in which the generator is built like an AE, and the discriminator compresses both the real and decoder-produced (fake) data into one-dimensional vectors to judge authenticity.  
Each unsupervised learning model is implemented with as few layers as possible for simplicity. We fix the latent vector size at half the size of the input for computational  
Figure 3: Structure of each Local Back-Propagation cell, clearly illustrating the input and output data. Each cell receives the latent output from the previous cell as input, except for the first cell, which takes the original image as input. (**a**) Auto-Encoder and (**c**) Convolutional Auto-Encoder cell both reconstruct the input. (**b**) Denoising Auto-Encoder cell receives a noisy version of the input. (**d**) GAN cell takes random noise and latent input to generate data.  
convenience. While there are no strict limits on the number of cells, if additional cells are needed and the latent vector size cannot be further reduced, we maintain the same latent vector size in subsequent cells.  
The final classifier layer provides the model's overall output. Unlike the cells, this layer is a standard fully connected layer (or potentially another type, depending on the task) and does not rely on unsupervised learning. It takes as input the concatenation of the latent vectors from all preceding cells.  
To better illustrate the positioning of our proposed LBP method within the broader context of deep learning training strategies, we provide a comparative summary of its key properties relative to traditional BP and the FF algorithm.  
Table 2 highlights the fundamental differences across core aspects such as training mechanics, input requirements, memory usage, and compatibility with conventional pipelines. This comparison clarifies how LBP inherits the forward-only advantage of FF while addressing its limitations through general-purpose loss functions and improved modularity.