Position: A Theory of Deep Learning Must Include Compositional Sparsity
This is a foundational position paper proposing a key principle for understanding deep learning, which could impact the entire field of ML/AI if validated.
The authors argue that the success of deep neural networks stems from their ability to exploit compositional sparsity in target functions, where functions are composed from a small set of low-dimensional components, and they claim this property is present in all efficiently computable functions.
Overparametrized Deep Neural Networks (DNNs) have demonstrated remarkable success in a wide variety of domains too high-dimensional for classical shallow networks subject to the curse of dimensionality. However, open questions about fundamental principles, that govern the learning dynamics of DNNs, remain. In this position paper we argue that it is the ability of DNNs to exploit the compositionally sparse structure of the target function driving their success. As such, DNNs can leverage the property that most practically relevant functions can be composed from a small set of constituent functions, each of which relies only on a low-dimensional subset of all inputs. We show that this property is shared by all efficiently Turing-computable functions and is therefore highly likely present in all current learning problems. While some promising theoretical insights on questions concerned with approximation and generalization exist in the setting of compositionally sparse functions, several important questions on the learnability and optimization of DNNs remain. Completing the picture of the role of compositional sparsity in deep learning is essential to a comprehensive theory of artificial, and even general, intelligence.