Symmetric Single Index Learning
Aaron Zweig Joan Bruna
Few neural architectures lend themselves to provable learning with gradient
based methods. One popular model is the single-index model, in which labels are
produced by composing an unknown linear projection with a possibly unknown
scalar link function. Learning this model with SGD is relatively
well-understood, whereby the so-called information exponent of the link
function governs a polynomial sample complexity rate. However, extending this
analysis to deeper or more complicated architectures remains challenging.
  In this work, we consider single index learning in the setting of symmetric
neural networks. Under analytic assumptions on the activation and maximum
degree assumptions on the link function, we prove that gradient flow recovers
the hidden planted direction, represented as a finitely supported vector in the
feature space of power sum polynomials. We characterize a notion of information
exponent adapted to our setting that controls the efficiency of learning.