Is Deeper Better only when Shallow is Good?
This addresses the theoretical challenge of understanding when deep networks can be trained effectively, with implications for deep learning theory, though it is incremental as it builds on existing depth separation arguments.
The paper investigates whether the benefits of depth in neural networks are realized during gradient-based optimization, showing that distributions with fractal structure can be efficiently expressed by deep but not shallow networks, and proving that gradient-based algorithms fail when distributions are concentrated on fine details.
Understanding the power of depth in feed-forward neural networks is an ongoing challenge in the field of deep learning theory. While current works account for the importance of depth for the expressive power of neural-networks, it remains an open question whether these benefits are exploited during a gradient-based optimization process. In this work we explore the relation between expressivity properties of deep networks and the ability to train them efficiently using gradient-based algorithms. We give a depth separation argument for distributions with fractal structure, showing that they can be expressed efficiently by deep networks, but not with shallow ones. These distributions have a natural coarse-to-fine structure, and we show that the balance between the coarse and fine details has a crucial effect on whether the optimization process is likely to succeed. We prove that when the distribution is concentrated on the fine details, gradient-based algorithms are likely to fail. Using this result we prove that, at least in some distributions, the success of learning deep networks depends on whether the distribution can be well approximated by shallower networks, and we conjecture that this property holds in general.