Abstract
Deep learning has been immensely successful at a variety of tasks, ranging
from classification to AI. Learning corresponds to fitting training data, which
is implemented by descending a very high-dimensional loss function.
Understanding under which conditions neural networks do not get stuck in poor
minima of the loss, and how the landscape of that loss evolves as depth is
increased remains a challenge. Here we predict, and test empirically, an
analogy between this landscape and the energy landscape of repulsive ellipses.
We argue that in FC networks a phase transition delimits the over- and
under-parametrized regimes where fitting can or cannot be achieved. In the
vicinity of this transition, properties of the curvature of the minima of the
loss are critical. This transition shares direct similarities with the jamming
transition by which particles form a disordered solid as the density is
increased, which also occurs in certain classes of computational optimization
and learning problems such as the perceptron. Our analysis gives a simple
explanation as to why poor minima of the loss cannot be encountered in the
overparametrized regime, and puts forward the surprising result that the
ability of fully connected networks to fit random data is independent of their
depth. Our observations suggests that this independence also holds for real
data. We also study a quantity $\Delta$ which characterizes how well
($\Delta<0$) or badly ($\Delta>0$) a datum is learned. At the critical point it
is power-law distributed, $P_+(\Delta)\sim\Delta^þeta$ for $\Delta>0$ and
$P_-(\Delta)\sim(-\Delta)^-\gamma$ for $\Delta<0$, with $þeta\approx0.3$
and $\gamma\approx0.2$. This observation suggests that near the transition the
loss landscape has a hierarchical structure and that the learning dynamics is
prone to avalanche-like dynamics, with abrupt changes in the set of patterns
that are learned.
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