Optimal SQ Lower Bounds for Robustly Learning Discrete Product Distributions and Ising Models

TL;DR

This paper proves optimal statistical query lower bounds for robustly learning certain high-dimensional discrete distributions, showing that existing algorithms are essentially optimal in terms of error guarantees.

Contribution

The paper introduces tight SQ lower bounds for robust learning of discrete product distributions and Ising models, matching known algorithm guarantees.

Findings

No efficient SQ algorithm can learn the mean within o(ε√log(1/ε)) for corrupted product distributions.

No efficient SQ algorithm can learn the Ising model within o(ε log(1/ε)) in total variation distance.

Develops a generic SQ lower bound framework from low-dimensional moment matching constructions.

Abstract

We establish optimal Statistical Query (SQ) lower bounds for robustly learning certain families of discrete high-dimensional distributions. In particular, we show that no efficient SQ algorithm with access to an $\epsilon$-corrupted binary product distribution can learn its mean within $\ell_2$-error $o(\epsilon \sqrt{\log(1/\epsilon)})$. Similarly, we show that no efficient SQ algorithm with access to an $\epsilon$-corrupted ferromagnetic high-temperature Ising model can learn the model to total variation distance $o(\epsilon \log(1/\epsilon))$. Our SQ lower bounds match the error guarantees of known algorithms for these problems, providing evidence that current upper bounds for these tasks are best possible. At the technical level, we develop a generic SQ lower bound for discrete high-dimensional distributions starting from low dimensional moment matching constructions that we believe will find other applications. Additionally, we introduce new ideas to analyze these moment-matching constructions for discrete univariate distributions.