Continuous LWE

TL;DR

This paper introduces the Continuous Learning with Errors (CLWE) problem, establishing its computational hardness via quantum reductions, and applies it to learning mixtures of Gaussians and robust machine learning, bridging gaps in theoretical understanding.

Contribution

It defines CLWE as a continuous analogue of LWE, proves its hardness through quantum reductions, and addresses open problems in learning Gaussian mixtures and robust ML.

Findings

CLWE has similar hardness to LWE under quantum reductions

Resolves open problem on learning Gaussian mixtures without separability

Addresses computational hardness in robust machine learning models

Abstract

We introduce a continuous analogue of the Learning with Errors (LWE) problem, which we name CLWE. We give a polynomial-time quantum reduction from worst-case lattice problems to CLWE, showing that CLWE enjoys similar hardness guarantees to those of LWE. Alternatively, our result can also be seen as opening new avenues of (quantum) attacks on lattice problems. Our work resolves an open problem regarding the computational complexity of learning mixtures of Gaussians without separability assumptions (Diakonikolas 2016, Moitra 2018). As an additional motivation, (a slight variant of) CLWE was considered in the context of robust machine learning (Diakonikolas et al.~FOCS 2017), where hardness in the statistical query (SQ) model was shown; our work addresses the open question regarding its computational hardness (Bubeck et al.~ICML 2019).