Star-Shaped Space of Solutions of the Spherical Negative Perceptron.

Empirical studies on the landscape of neural networks have shown that low-energy configurations are often found in complex connected structures, where zero-energy paths between pairs of distant solutions can be constructed. Here, we consider the spherical negative perceptron, a prototypical nonconvex neural network model framed as a continuous constraint satisfaction problem. We introduce a general analytical method for computing energy barriers in the simplex with vertex configurations sampled from the equilibrium. We find that in the overparametrized regime the solution manifold displays simple connectivity properties. There exists a large geodesically convex component that is attractive for a wide range of optimization dynamics. Inside this region we identify a subset of atypical high-margin solutions that are geodesically connected with most other solutions, giving rise to a star-shaped geometry. We analytically characterize the organization of the connected space of solutions and show numerical evidence of a transition, at larger constraint densities, where the aforementioned simple geodesic connectivity breaks down.

Shaping the learning landscape in neural networks around wide flat minima.

Learning in deep neural networks takes place by minimizing a nonconvex high-dimensional loss function, typically by a stochastic gradient descent (SGD) strategy. The learning process is observed to be able to find good minimizers without getting stuck in local critical points and such minimizers are often satisfactory at avoiding overfitting. How these 2 features can be kept under control in nonlinear devices composed of millions of tunable connections is a profound and far-reaching open question. In this paper we study basic nonconvex 1- and 2-layer neural network models that learn random patterns and derive a number of basic geometrical and algorithmic features which suggest some answers. We first show that the error loss function presents few extremely wide flat minima (WFM) which coexist with narrower minima and critical points. We then show that the minimizers of the cross-entropy loss function overlap with the WFM of the error loss. We also show examples of learning devices for which WFM do not exist. From the algorithmic perspective we derive entropy-driven greedy and message-passing algorithms that focus their search on wide flat regions of minimizers. In the case of SGD and cross-entropy loss, we show that a slow reduction of the norm of the weights along the learning process also leads to WFM. We corroborate the results by a numerical study of the correlations between the volumes of the minimizers, their Hessian, and their generalization performance on real data.

Chaos and Correlated Avalanches in Excitatory Neural Networks with Synaptic Plasticity.

A collective chaotic phase with power law scaling of activity events is observed in a disordered mean field network of purely excitatory leaky integrate-and-fire neurons with short-term synaptic plasticity. The dynamical phase diagram exhibits two transitions from quasisynchronous and asynchronous regimes to the nontrivial, collective, bursty regime with avalanches. In the homogeneous case without disorder, the system synchronizes and the bursty behavior is reflected into a period doubling transition to chaos for a two dimensional discrete map. Numerical simulations show that the bursty chaotic phase with avalanches exhibits a spontaneous emergence of persistent time correlations and enhanced Kolmogorov complexity. Our analysis reveals a mechanism for the generation of irregular avalanches that emerges from the combination of disorder and deterministic underlying chaotic dynamics.