Towards Lower Bounds on the Depth of ReLU Neural Networks
This work addresses foundational theoretical questions in deep learning for researchers, providing a mathematical counterbalance to universal approximation theorems, though it is incremental in building on existing theories.
The paper tackles the problem of understanding the representational capacity of ReLU neural networks by investigating whether adding more layers strictly increases the class of exactly representable functions, and it settles an old conjecture about piecewise linear functions in the affirmative while providing upper bounds on network sizes for logarithmic depth.
We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical geometry, we provide a mathematical counterbalance to the universal approximation theorems which suggest that a single hidden layer is sufficient for learning any function. In particular, we investigate whether the class of exactly representable functions strictly increases by adding more layers (with no restrictions on size). As a by-product of our investigations, we settle an old conjecture about piecewise linear functions by Wang and Sun (2005) in the affirmative. We also present upper bounds on the sizes of neural networks required to represent functions with logarithmic depth.