Hardness of Noise-Free Learning for Two-Hidden-Layer Neural Networks

TL;DR

This paper establishes superpolynomial statistical query lower bounds for noise-free learning of two-hidden-layer ReLU neural networks with Gaussian inputs, revealing fundamental computational hardness in this setting.

Contribution

It provides the first general SQ lower bounds for noise-free learning of ReLU networks of any depth, extending previous results and introducing new reduction techniques.

Findings

Superpolynomial SQ lower bounds for two-hidden-layer ReLU networks

New cryptographic hardness results for PAC learning ReLU networks

Lower bounds for learning constant-depth ReLU networks from label queries

Abstract

We give superpolynomial statistical query (SQ) lower bounds for learning two-hidden-layer ReLU networks with respect to Gaussian inputs in the standard (noise-free) model. No general SQ lower bounds were known for learning ReLU networks of any depth in this setting: previous SQ lower bounds held only for adversarial noise models (agnostic learning) or restricted models such as correlational SQ. Prior work hinted at the impossibility of our result: Vempala and Wilmes showed that general SQ lower bounds cannot apply to any real-valued family of functions that satisfies a simple non-degeneracy condition. To circumvent their result, we refine a lifting procedure due to Daniely and Vardi that reduces Boolean PAC learning problems to Gaussian ones. We show how to extend their technique to other learning models and, in many well-studied cases, obtain a more efficient reduction. As such, we also prove new cryptographic hardness results for PAC learning two-hidden-layer ReLU networks, as well as new lower bounds for learning constant-depth ReLU networks from label queries.