The jamming transition as a paradigm to understand the loss landscape of deep neural networks
This provides a foundational insight into the loss landscape of deep learning, potentially benefiting researchers in machine learning theory by explaining why poor minima are avoided in overparametrized regimes, though it is incremental as it builds on prior analogies like the perceptron.
The paper tackles the problem of understanding when deep neural networks avoid poor minima in their loss landscape by drawing an analogy to the jamming transition in repulsive ellipses, predicting that a phase transition separates over- and under-parametrized regimes where fitting is possible or not. It finds that the ability of fully connected networks to fit random data is independent of depth, with critical exponents θ≈0.3 and γ≈0.2 characterizing the distribution of learning quality near the transition.
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 $Δ$ which characterizes how well ($Δ<0$) or badly ($Δ>0$) a datum is learned. At the critical point it is power-law distributed, $P_+(Δ)\simΔ^θ$ for $Δ>0$ and $P_-(Δ)\sim(-Δ)^{-γ}$ for $Δ<0$, with $θ\approx0.3$ and $γ\approx0.2$. 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.