2026-07-23T00:26:23Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23461 2025-10-02T12:55:49Z ddc:004 open_access

One-Shot Learning for k-SAT Galanis, Andreas Goldberg, Leslie Ann Zhang, Xusheng Computational Learning Theory k-SAT Maximum likelihood estimation Consider a k-SAT formula Φ where every variable appears at most d times, and let σ be a satisfying assignment of Φ sampled proportionally to e^{β m(σ)} where m(σ) is the number of variables set to true and β is a real parameter. Given Φ and σ, can we learn the value of β efficiently? This problem falls into a recent line of works about single-sample ("one-shot") learning of Markov random fields. The k-SAT setting we consider here was recently studied by Galanis, Kandiros, and Kalavasis (SODA'24) where they showed that single-sample learning is possible when roughly d ≤ 2^{k/6.45} and impossible when d ≥ (k+1) 2^{k-1}. Crucially, for their impossibility results they used the existence of unsatisfiable instances which, aside from the gap in d, left open the question of whether the feasibility threshold for one-shot learning is dictated by the satisfiability threshold of k-SAT formulas of bounded degree. Our main contribution is to answer this question negatively. We show that one-shot learning for k-SAT is infeasible well below the satisfiability threshold; in fact, we obtain impossibility results for degrees d as low as k² when β is sufficiently large, and bootstrap this to small values of β when d scales exponentially with k, via a probabilistic construction. On the positive side, we simplify the analysis of the learning algorithm and obtain significantly stronger bounds on d in terms of β. In particular, for the uniform case β → 0 that has been studied extensively in the sampling literature, our analysis shows that learning is possible under the condition d≲ 2^{k/2}. This is nearly optimal (up to constant factors) in the sense that it is known that sampling a uniformly-distributed satisfying assignment is NP-hard for d≳ 2^{k/2}. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Andreas Galanis and Leslie Ann Goldberg and Xusheng Zhang 2025 Is Part Of LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2025.84 urn:nbn:de:0030-drops-234610 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.84 eng https://creativecommons.org/licenses/by/4.0/legalcode