Analytic Study of Families of Spurious Minima in Two-Layer ReLU Neural Networks

We study the optimization problem associated with fitting two-layer ReLU neural networks with respect to the squared loss, where labels are generated by a target network. We make use of the rich symmetry structure to develop a novel set of tools for studying families of spurious minima. In contrast to existing approaches which operate in limiting regimes, our technique directly addresses the nonconvex loss landscape for a finite number of inputs d and neurons k, and provides analytic, rather than heuristic, information. In particular, we derive analytic estimates for the loss at different minima, and prove that modulo O(d−1/2)-terms the Hessian spectrum concentrates near small positive constants, with the exception of Θ(d) eigenvalues which grow linearly with d. We further show that the Hessian spectrum at global and spurious minima coincide to O(d−1/2)-order, thus challenging our ability to argue about statistical generalization through local curvature. Lastly, our technique provides the exact fractional dimensionality at which families of critical points turn from saddles into spurious minima. This makes possible the study of the creation and the annihilation of spurious minima using powerful tools from equivariant bifurcation theory. One of the outstanding conundrums of deep learning concerns the ability of simple gradient-based methods to successfully train neural networks despite the nonconvexity of the associated optimization problems. Indeed, generic nonconvex optimization landscapes can exhibit wide and flat basins of attraction around poor local minima which may lead to a complete failure of such methods. The nature by which nonconvex problems associated with neural networks deviate from generic ones is currently not well-understood. In particular, much of the dynamics of gradient-based methods follows from the curvature of the loss landscape around local minima. It is therefore vital to study the local geometry of spurious (i.e., non-global local) and global minima in order to understand the mysterious mechanism which drives gradient-based methods towards minima of high quality. However, already establishing the very existence of spurious minima seems to be beyond reach of existing analytic tools; let alone rigorously arguing about their height, curvature and structure—the aim of this work. In this paper, we focus on two-layer ReLU neural networks of the form