Statistical Query Algorithms and Low-Degree Tests Are Almost Equivalent

TL;DR

This paper demonstrates that statistical query algorithms and low-degree polynomial tests are nearly equivalent in power for high-dimensional hypothesis testing, providing new lower bounds for several complex problems.

Contribution

It establishes the near-equivalence of two major computational models in high-dimensional testing, leading to new lower bounds for key problems.

Findings

Statistical query and low-degree tests are essentially equivalent in power.

New lower bounds for sparse PCA, tensor PCA, and planted clique variants.

Unified framework for understanding computational limits in high-dimensional inference.

Abstract

Researchers currently use a number of approaches to predict and substantiate information-computation gaps in high-dimensional statistical estimation problems. A prominent approach is to characterize the limits of restricted models of computation, which on the one hand yields strong computational lower bounds for powerful classes of algorithms and on the other hand helps guide the development of efficient algorithms. In this paper, we study two of the most popular restricted computational models, the statistical query framework and low-degree polynomials, in the context of high-dimensional hypothesis testing. Our main result is that under mild conditions on the testing problem, the two classes of algorithms are essentially equivalent in power. As corollaries, we obtain new statistical query lower bounds for sparse PCA, tensor PCA and several variants of the planted clique problem.