Computational Complexity of Statistics: New Insights from Low-Degree Polynomials

TL;DR

This survey explores how low-degree polynomial frameworks help understand the computational difficulty of statistical problems, connecting them with other methods and discussing their implications for hardness conjectures.

Contribution

It provides a comprehensive overview of the low-degree polynomial approach, its applications, philosophical considerations, and connections to other computational frameworks.

Findings

Low-degree polynomials effectively characterize statistical-computational tradeoffs.

Connections between low-degree bounds, sum-of-squares, and statistical query models are established.

The survey highlights open problems and future directions in the field.

Abstract

This is a survey on the use of low-degree polynomials to predict and explain the apparent statistical-computational tradeoffs in a variety of average-case computational problems. In a nutshell, this framework measures the complexity of a statistical task by the minimum degree that a polynomial function must have in order to solve it. The main goals of this survey are to (1) describe the types of problems where the low-degree framework can be applied, encompassing questions of detection (hypothesis testing), recovery (estimation), and more; (2) discuss some philosophical questions surrounding the interpretation of low-degree lower bounds, and notably the extent to which they should be treated as evidence for inherent computational hardness; (3) explore the known connections between low-degree polynomials and other related approaches such as the sum-of-squares hierarchy and statistical query model; and (4) give an overview of the mathematical tools used to prove low-degree lower bounds. A list of open problems is also included.