Beyond the Lazy versus Rich Dichotomy: Geometry Insights in Feature Learning from Task-Relevant Manifold Untangling
Keywords: Computational neuroscience, storage capacity, neural manifolds, representational geometry, rich and lazy learning, training dynamics, feature learning
TL;DR: Quantifying feature learning beyond lazy versus rich through manifold capacity and representational geometry.
Abstract: The ability to integrate task-relevant information into neural representations is a fundamental aspect of both human and machine intelligence. Recent studies have explored the transition of neural networks from the *lazy* training regime (where the trained network is equivalent to a linear model of initial random features) to the *rich* feature learning regime (where the network learns task-relevant features). However, most approaches focus on weight matrices or neural tangent kernels, limiting their relevance for neuroscience due to the lack of representation-based methods to study feature learning. Furthermore, the simple lazy-versus-rich dichotomy overlooks the potential for richer subtypes of feature learning driven by variations in learning algorithms, network architectures, and data properties.
In this work, we present a framework based on representational geometry to study feature learning. The key idea is to use the untangling of task-relevant neural manifolds as a signature of rich learning. We employ manifold capacity—a representation-based measure—to quantify this untangling, along with geometric metrics to uncover structural differences in feature learning. Our contributions are threefold: First, we show both theoretically and empirically that task-relevant manifolds untangle during rich learning, and that manifold capacity quantifies the degree of richness. Second, we use manifold geometric measures to reveal distinct learning stages and strategies driven by network and data properties, demonstrating that feature learning is richer than the lazy-versus-rich dichotomy. Finally, we apply our method to problems in neuroscience and machine learning, providing geometric insights into structural inductive biases and out-of-distribution generalization. Our work introduces a novel perspective for understanding and quantifying feature learning through the lens of representational geometry.
Primary Area: applications to neuroscience & cognitive science
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Submission Number: 7330
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