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Understanding the power of depth in feed-forward neural networks is anongoing challenge in the field of deep learning theory. While current worksaccount for the importance of depth for the expressive power ofneural-networks, it remains an open question whether these benefits areexploited during a gradient-based optimization process. In this work we explorethe relation between expressivity properties of deep networks and the abilityto train them efficiently using gradient-based algorithms. We give a depthseparation argument for distributions with fractal structure, showing that theycan be expressed efficiently by deep networks, but not with shallow ones. Thesedistributions have a natural coarse-to-fine structure, and we show that thebalance between the coarse and fine details has a crucial effect on whether theoptimization process is likely to succeed. We prove that when the distributionis concentrated on the fine details, gradient-based algorithms are likely tofail. Using this result we prove that, at least in some distributions, thesuccess of learning deep networks depends on whether the distribution can bewell approximated by shallower networks, and we conjecture that this propertyholds in general.

Is Deeper Better only when Shallow is Good?