Counterexamples to the Low-Degree Conjecture
This work clarifies the applicability of the low-degree framework for computational lower bounds in statistical problems, addressing foundational issues in theoretical machine learning.
The authors refute Hopkins' low-degree conjecture, which posits that polynomial-time algorithms cannot outperform low-degree polynomials in certain high-dimensional hypothesis testing problems, by constructing counterexamples that exploit specific noise operators and symmetry assumptions.
A conjecture of Hopkins (2018) posits that for certain high-dimensional hypothesis testing problems, no polynomial-time algorithm can outperform so-called "simple statistics", which are low-degree polynomials in the data. This conjecture formalizes the beliefs surrounding a line of recent work that seeks to understand statistical-versus-computational tradeoffs via the low-degree likelihood ratio. In this work, we refute the conjecture of Hopkins. However, our counterexample crucially exploits the specifics of the noise operator used in the conjecture, and we point out a simple way to modify the conjecture to rule out our counterexample. We also give an example illustrating that (even after the above modification), the symmetry assumption in the conjecture is necessary. These results do not undermine the low-degree framework for computational lower bounds, but rather aim to better understand what class of problems it is applicable to.