On the Complexity of Learning Sparse Functions with Statistical and Gradient Queries

Nirmit Joshi, Theodor Misiakiewicz, Nathan Srebro

Keywords: Statistical Query Complexity, Differtiable Learning, Sparse Functions, Leap Exponent

TL;DR: We study the power of differentiable learning compared to the statistical query (SQ) and correlation statistical query (CSQ) methods for the problem of learning sparse support of size P out of d coordinates with d>>P.

Abstract: The goal of this paper is to investigate the complexity of gradient algorithms when learning sparse functions (juntas). We introduce a type of Statistical Queries ($\mathsf{SQ}$), which we call Differentiable Learning Queries ($\mathsf{DLQ}$), to model gradient queries on a specified loss with respect to an arbitrary model. We provide a tight characterization of the query complexity of $\mathsf{DLQ}$ for learning the support of a sparse function over generic product distributions. This complexity crucially depends on the loss function. For the squared loss, $\mathsf{DLQ}$ matches the complexity of Correlation Statistical Queries $(\mathsf{CSQ})$—potentially much worse than $\mathsf{SQ}$. But for other simple loss functions, including the $\ell_1$ loss, $\mathsf{DLQ}$ always achieves the same complexity as $\mathsf{SQ}$. We also provide evidence that $\mathsf{DLQ}$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer neural network in the mean field regime and linear scaling.

Primary Area: Learning theory

Submission Number: 11669