  In many contexts, simpler models are preferable to more complex models and
the control of this model complexity is the goal for many methods in machine
learning such as regularization, hyperparameter tuning and architecture design.
In deep learning, it has been difficult to understand the underlying mechanisms
of complexity control, since many traditional measures are not naturally
suitable for deep neural networks. Here we develop the notion of geometric
complexity, which is a measure of the variability of the model function,
computed using a discrete Dirichlet energy. Using a combination of theoretical
arguments and empirical results, we show that many common training heuristics
such as parameter norm regularization, spectral norm regularization, flatness
regularization, implicit gradient regularization, noise regularization and the
choice of parameter initialization all act to control geometric complexity,
providing a unifying framework in which to characterize the behavior of deep
learning models.
