How Neural Architectures and Regularization Shape the Learning Landscape of Machine Models

What Do Neural Networks Really Learn? Every machine learning model, whether deep or shallow, learns by exploring a "hypothesis space" — the set of functions it can theoretically represent. However, this space is far from neutral. It is shaped and influenced by two key factors: architecture and regularization. As models become more sophisticated and specialized, understanding these factors is no longer just an academic exercise but a fundamental aspect of intelligent model design. The Interaction Between Architecture and Regularization Our goal here is to delve into how different neural architectures influence the geometry and topology of hypothesis spaces, and how regularization can be seen as a method for prioritizing certain regions over others within these spaces. By framing the problem geometrically, we aim to develop a deeper intuition for what models prefer to learn and why. A Tale of Two Learners Consider two neural networks trained on the same dataset: a shallow Multi-Layer Perceptron (MLP) and a Convolutional Neural Network (CNN). Both achieve low training errors, yet their generalization performance varies significantly. Why is this the case? Although both are universal approximators, capable of representing a wide range of functions, their hypothesis spaces have different structures. The MLP lacks any built-in notion of locality or translation invariance, which means it must learn these properties from scratch. On the other hand, the CNN starts with a geometry where spatial locality is inherent. This difference affects not only the types of functions that can be represented but also the ease with which the optimizer can find and prefer certain solutions. The architecture not only sets the boundaries of the space but also determines its gradient-weighted landscape. From Functions to Manifolds To understand this precisely, consider the hypothesis space as a manifold embedded within a larger function space. The architecture of a model defines a specific submanifold of functions it can express. This submanifold is not flat or uniform; instead, it has a complex geometry with peaks, valleys, and other topological features. Through a geometric deep learning perspective, architectural priors shape the metric and topology of the hypothesis space. For instance, CNNs favor translationally equivariant functions, Graph Neural Networks (GNNs) favor permutation invariance, and Transformers emphasize attention-weighted global interactions. The optimizer navigates along this curved and structured manifold, rather than exploring the entire function space. Regularization as a Measure Over Hypothesis Space Regularization is often seen as a mechanism for penalizing complexity, but this view is incomplete. More fundamentally, regularization defines a measure over the hypothesis space, influencing which functions are considered more probable or desirable. For example: Dropout encourages more distributed representations by reducing reliance on specific units. Spectral norm regularization constrains Lipschitz continuity, promoting smoother functions. Bayesian neural networks explicitly define a prior over the weights, which in turn induces a prior over the functions. From this geometric perspective, regularization is not a mere constraint on learning but a powerful shaping force. It modifies the energy landscape of the hypothesis space, altering the likelihood of the optimizer settling into certain valleys. This interaction between architecture and regularization is particularly intriguing, as different regularizers can have varying effects depending on the curvature and composition of the underlying hypothesis space. For instance, a regularizer that enhances generalization in one architecture might hinder it in another. This nonlinearity underscores the importance of tailoring both architecture and regularization techniques to the specific learning task and dataset. Understanding how these elements interact can lead to more effective and efficient model designs, ultimately improving performance in real-world applications.