Why does deep and cheap learning work so well?
This work addresses the fundamental question of why deep learning is successful for researchers in machine learning and physics, offering a theoretical explanation rather than an incremental improvement.
The paper argues that the effectiveness of deep learning stems from physical principles like symmetry and locality, which allow neural networks to approximate practical functions with exponentially fewer parameters than generic ones, and proves that deep networks can be more efficient than shallow ones for hierarchical data, with examples such as requiring at least 2^n neurons in a shallow network to multiply n variables.
We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well, the class of functions of practical interest can frequently be approximated through "cheap learning" with exponentially fewer parameters than generic ones. We explore how properties frequently encountered in physics such as symmetry, locality, compositionality, and polynomial log-probability translate into exceptionally simple neural networks. We further argue that when the statistical process generating the data is of a certain hierarchical form prevalent in physics and machine-learning, a deep neural network can be more efficient than a shallow one. We formalize these claims using information theory and discuss the relation to the renormalization group. We prove various "no-flattening theorems" showing when efficient linear deep networks cannot be accurately approximated by shallow ones without efficiency loss, for example, we show that $n$ variables cannot be multiplied using fewer than 2^n neurons in a single hidden layer.