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

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.