Fractal Structure and Generalization Properties of Stochastic Optimization Algorithms

Understanding generalization in deep learning has been one of the major challenges in statistical learning theory over the last decade. While recent work has illustrated that the dataset and the training algorithm must be taken into account in order to obtain meaningful generalization bounds, it is still theoretically not clear which properties of the data and the algorithm determine the generalization performance. In this study, we approach this problem from a dynamical systems theory perspective and represent stochastic optimization algorithms as \emph{random iterated function systems} (IFS). Well studied in the dynamical systems literature, under mild assumptions, such IFSs can be shown to be ergodic with an invariant measure that is often supported on sets with a \emph{fractal structure}. As our main contribution, we prove that the generalization error of a stochastic optimization algorithm can be bounded based on the complexity' of the fractal structure that underlies its invariant measure.