A mathematical theory of semantic development in deep neural networks

By Andrew M. Saxe, James L. McClelland, Surya Ganguli. 2018

357 citations recorded by the source corpus.

Significance Over the course of development, humans learn myriad facts about items in the world, and naturally group these items into useful categories and structures. This semantic knowledge is essential for diverse behaviors and inferences in adulthood. How is this richly structured semantic knowledge acquired, organized, deployed, and represented by neuronal networks in the brain? We address this question by studying how the nonlinear learning dynamics of deep linear networks acquires information about complex environmental structures. Our results show that this deep learning dynamics can self-organize emergent hidden representations in a manner that recapitulates many empirical phenomena in human semantic development. Such deep networks thus provide a mathematically tractable window into the development of internal neural representations through experience. An extensive body of empirical research has revealed remarkable regularities in the acquisition, organization, deployment, and neural representation of human semantic knowledge, thereby raising a fundamental conceptual question: What are the theoretical principles governing the ability of neural networks to acquire, organize, and deploy abstract knowledge by integrating across many individual experiences? We address this question by mathematically analyzing the nonlinear dynamics of learning in deep linear networks. We find exact solutions to this learning dynamics that yield a conceptual explanation for the prevalence of many disparate phenomena in semantic cognition, including the hierarchical differentiation of concepts through rapid developmental transitions, the ubiquity of semantic illusions between such transitions, the emergence of item typicality and category coherence as factors controlling the speed of semantic processing, changing patterns of inductive projection over development, and the conservation of semantic similarity in neural representations across species. Thus, surprisingly, our simple neural model qualitatively recapitulates many diverse regularities underlying semantic development, while providing analytic insight into how the statistical structure of an environment can interact with nonlinear deep-learning dynamics to give rise to these regularities.

Subfields: Artificial Intelligence, Cognitive Neuroscience, Statistical and Nonlinear Physics, Scaling & Learning Theory

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Cognitive Neuroscience

In Statistical and Nonlinear Physics

In Scaling & Learning Theory

Citations in the corpus

Implicit Regularization of Discrete Gradient Dynamics in Deep Linear Neural Networks (2019, methodology): “Prior work [Saxe et al., 2013, 2014, Advani and Saxe, 2017, Saxe et al., 2018, Lampinen and Ganguli, 2019] studied the gradient dynamics of two-layer linear networks and proved a result similar to our Thm. 2.”
Orthogonal representations for robust context-dependent task performance in brains and neural networks (2022, positive): “In the rich regime, which occurs under low initial variance, the hidden units instead learn highly structured representations that are tailored to the task demands (Saxe et al., 2019; Geiger et al., 2020; Woodworth et al., 2020; Paccolat et al., 2021).”
Anatomy of Catastrophic Forgetting: Hidden Representations and Task Semantics (2020, background): “A fundamental property of biological and artiﬁcial neural networks is that their predictive output is critically aﬀected by semantic properties of their training data [39, 40].”
Learning Transferable Knowledge Through Embedding Spaces (2019, background): “Work on semantic cognition within the parallel distributed processing framework hypothesizes that abstract semantic concepts are formed in higher-level layers of the nervous system [8; 221].”
When MAML Can Adapt Fast and How to Assist When It Cannot (2019, background): “While linear layers are often cited for implicit regularization to improve generalization of models, (Gidel et al., 2019; Saxe et al., 2019) in our settings, there is no overfitting.”
What shapes feature representations? Exploring datasets, architectures, and training (2020, background): “or the most variance in the data before learning features that account for less variance [35]. Given hierarchical data, models learn high-level semantic features before proceeding to lower-level ones [36]. This observation that models progressively differentiate task structure has been leveraged to ”
If deep learning is the answer, what is the question? (2020, methodology): “These network idealizations have generated mathematical insight into perplexing questions about network behaviour, including why deep networks are often slower to train 138 (Fig.”
A Sample Complexity Separation between Non-Convex and Convex Meta-Learning (2020, result): “Our results draw upon work motivated by understanding deep learning that analyzes trajectories and implicit regularization in deep linear neural networks (Saxe et al., 2014; Gunasekar et al., 2018; Saxe et al., 2019; Gidel et al., 2019).”
Few-Shot Adaptation of Generative Adversarial Networks (2020, methodology): “SVD is also used for a wide number of other applications, including regularization (Sedghi et al., 2018), and quantiﬁcation of entanglement (Martyn et al., 2020), and has also been used to build theoretical background for semantic development in neural networks (Saxe et al., 2019).”
Gradient Starvation: A Learning Proclivity in Neural Networks (2020, background): “Several works including [90, 91, 1, 60] investigate the dynamics of deep linear networks trained with squared-error loss.”
Understanding self-supervised Learning Dynamics without Contrastive Pairs (2021, background): “…tractable theoretical model of nonconvex loss landscapes (Kawaguchi, 2016; Du & Hu, 2019; Laurent & Brecht, 2018) and nonlinear learning dynamics (Saxe et al., 2013; 2019; Lampinen & Ganguli, 2018; Arora et al., 2018) in these landscapes, yielding insights like dynamical isome-try (Saxe et al.,…”