Algorithmic Thresholds for Refuting Random Polynomial Systems

Abstract

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Algorithmic Thresholds for Refuting Random Polynomial SystemsJun-Ting Hsieh and Pravesh K. KothariJun-Ting Hsieh and Pravesh K. Kotharipp.1154 - 1203Chapter DOI:https://doi.org/10.1137/1.9781611977073.49PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Consider a system of m polynomial equations {pi(x) = bi}i≤m of degree D ≥ 2 in n-dimensional variable x ∊ ℝn such that each coefficient of every pi and bis are chosen at random and independently from some continuous distribution. We study the basic question of determining the smallest m–the algorithmic threshold–for which efficient algorithms can find refutations (i.e. certificates of unsatisfiability) for such systems. This setting generalizes problems such as refuting random SAT instances, low-rank matrix sensing and certifying pseudo-randomness of Goldreich's candidate generators and generalizations. We show that for every d ∊ ℕ, the (n + m)O(d)-time canonical sum-of-squares (SoS) relaxation refutes such a system with high probability whenever . We prove a lower bound in the restricted low-degree polynomial model of computation which suggests that this trade-off between SoS degree and the number of equations is nearly tight for all d. We also confirm the predictions of this lower bound in a limited setting by showing a lower bound on the canonical degree-4 sum-of-squares relaxation for refuting random quadratic polynomials. Together, our results provide evidence for an algorithmic threshold for the problem at -time algorithms for all δ. Our upper-bound relies on establishing a sharp bound on the smallest integer d such that degree d–D polynomial combinations of the input pis generate all degree-d polynomials in the ideal generated by the pis. Our lower bound actually holds for the easier problem of distinguishing random polynomial systems as above from a distribution on polynomial systems with a “planted” solution. Our choice of planted distribution is slightly (and necessarily) subtle: it turns out that the natural and well-studied planted distribution for quadratic systems (studied as the matrix sensing problem in machine learning) is easily distinguishable whenever m ≥ Õ(n)–a factor n smaller than the threshold in our upper bound above. Thus, our setting provides an example where refutation is harder than search in the natural planted model. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771