Symmetric Single Index Learning
This work addresses the challenge of extending provable learning guarantees to more complex neural architectures, though it is incremental as it builds on prior single-index model analyses.
The paper tackles the problem of provable learning for symmetric neural networks in the single-index model, proving that gradient flow recovers the hidden direction under analytic and degree assumptions, with sample complexity governed by an adapted information exponent.
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.