The Quasi-Polynomial Low-Degree Conjecture is False

Abstract: There is a growing body of work on proving hardness results for average-case estimation problems by bounding the low-degree advantage (LDA) - a quantitative estimate of the closeness of low-degree moments - between a null distribution and a related planted distribution. Such hardness results are now ubiquitous not only for foundational average-case problems but also central questions in statistics and cryptography. This line of work is supported by the low-degree conjecture of Hopkins, which postulates that a vanishing degree-$D$ LDA implies the absence of any noise-tolerant distinguishing algorithm with runtime $n^{\widetilde{O}(D)}$ whenever 1) the null distribution is product on ${0,1}^{\binom{n}{k}}$, and 2) the planted distribution is permutation invariant, that is, invariant under any relabeling $[n] \rightarrow [n]$. In this paper, we disprove this conjecture. Specifically, we show that for any fixed $\varepsilon>0$ and $k\geq 2$, there is a permutation-invariant planted distribution on ${0,1}^{\binom{n}{k}}$ that has a vanishing degree-$n^{1-O(\varepsilon)}$ LDA with respect to the uniform distribution on ${0,1}^{\binom{n}{k}}$, yet the corresponding $\varepsilon$-noisy distinguishing problem can be solved in $n^{O(\log^{1/(k-1)}(n))}$ time. Our construction relies on algorithms for list-decoding for noisy polynomial interpolation in the high-error regime. We also give another construction of a pair of planted and (non-product) null distributions on $\mathbb{R}^{n \times n}$ with a vanishing $n^{\Omega(1)}$-degree LDA while the largest eigenvalue serves as an efficient noise-tolerant distinguisher. Our results suggest that while a vanishing LDA may still be interpreted as evidence of hardness, developing a theory of average-case complexity based on such heuristics requires a more careful approach.