Statistical Query Algorithms and Low-Degree Tests Are Almost Equivalent
arXiv:2009.06107
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.
Version 3 fixes typos and adds note on presentation at COLT 2021
References in corpus (7)
- Scaling Laws for Neural Language Models
- Exact Recovery of Sparsely-Used Dictionaries
- Computational Barriers to Estimation from Low-Degree Polynomials
- Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs
- Counterexamples to the Low-Degree Conjecture
- How Hard Is Robust Mean Estimation?
- Statistical Query Lower Bounds for Tensor PCA
Cited by in corpus (6)
- Computational Barriers to Estimation from Low-Degree Polynomials
- Statistical Query Lower Bounds for Tensor PCA
- The Average-Case Time Complexity of Certifying the Restricted Isometry Property
- Statistical Query Lower Bounds for List-Decodable Linear Regression
- Machinery for Proving Sum-of-Squares Lower Bounds on Certification Problems
- Near-Optimal Statistical Query Hardness of Learning Halfspaces with Massart Noise