Neural Networks and Prior Probabilities: How Architecture Shapes Learning Before Training

Neural networks (NNs) are powerful tools in the field of machine learning, capable of solving a wide range of problems. However, it is often surprising to discover that these networks come with biases and assumptions even before they begin to learn from data. These biases are primarily incurred through the architecture, initialization, and regularization techniques employed before the training process begins. In this article, we explore how neural networks contain a form of prior probability before training, and discuss the mechanisms that contribute to this phenomenon.

Architecture as a Source of Prior Probabilities

The choice of architecture in a neural network dictates many of its properties and biases. For instance, a convolutional neural network (CNN) assumes a spatial hierarchy in image data, reflecting a prior belief about the organization of features in images. Similarly, other architectures like recurrent neural networks (RNNs) and transformers can be seen as embodying prior beliefs about the nature of data they will process.

Weight Sharing in CNNs

Weight sharing is a key feature of CNNs, where filters are applied across the entire input space. This technique reflects the assumption that there are spatially coherent features in images and other structured data. For example, edges and textures are likely to appear in similar locations across different parts of an image.

Residual and Skip Connections

Residual and skip connections are designed to alleviate the vanishing gradient problem in deep networks. They encourage the flow of gradients through the network by adding short cuts, which reflect a prior belief that deep layers should capture more complex features but also have access to information from earlier layers.

Initialization as a Weak Prior

Before training, the weights of a neural network are often initialized randomly using techniques like Xavier or He initialization. While these initial weights can be interpreted as embodying a prior belief about the model's parameters, the strong assumptions are typically discarded during the training process. This is because the optimization landscape of neural networks is such that there are often large valleys of good local minima, and stochastic gradient descent (SGD) effectively explores these regions.

Impact of Initialization at Test Time

At test time, the initialization values play almost no role in determining the inference outputs. This is due to the strength and complexity of the training process, which can override the initial conditions. The stochastic nature of SGD further ensures that any initial biases are diluted or eliminated.

Regularization Techniques and Priors

Regularization techniques such as L2 regularization (also known as weight decay) can be seen as imposing a prior distribution on the weights. These techniques encourage weights to remain small, which can be interpreted as a Bayesian prior. This helps in preventing overfitting by penalizing large weights, thereby fostering a belief that simpler models are preferable.

Learning Rate and Batch Size

The choices made regarding hyperparameters like learning rate and batch size can also influence the model's behavior before training begins. These choices set up a prior belief about the learning process, guiding how the model interacts with the training data.

Conclusion

In summary, while neural networks do not explicitly encode a prior probability in the same way Bayesian models do, their architecture, initialization, and regularization techniques collectively provide a framework of assumptions that guide their learning process. Understanding these priors can help in designing more effective and efficient neural networks.