One-Shot Learning for k-SAT

TL;DR

This work analyzes one-shot learning of the β-parameter in weighted k-SAT under bounded-degree constraints. It provides a near-linear-time MPLE-based estimator that achieves $O(n^{-1/2})$ accuracy under a precise degree bound that improves prior results and aligns with sampling-threshold intuition for small β. Complementing the positive result, it proves strong impossibility results showing non-identifiability of β well below the satisfiability threshold, including regimes with large and small β, and near the unique-satisfiability threshold via gadget constructions. Overall, the paper strengthens the link between one-shot learnability and sampling thresholds, narrowing the gap to the conjectured true boundary and clarifying the role of hard constraints in parameter recovery.

Abstract

Consider a $k$-SAT formula $Φ$ where every variable appears at most $d$ times. Let $σ$ be a satisfying assignment, sampled proportionally to $e^{βm(σ)}$ where $m(σ)$ is the number of true variables and $β$ is a real parameter. Given $Φ$ and $σ$, can we efficiently learn $β$? This problem falls into a recent line of work about single-sample (``one-shot'') learning of Markov random fields. Our $k$-SAT setting was recently studied by Galanis, Kalavasis, Kandiros (SODA24). They showed that single-sample learning is possible when roughly $d\leq 2^{k/6.45}$ and impossible when $d\geq (k+1) 2^{k-1}$. In addition to the gap in~$d$, their impossibility result left open the question of whether the feasibility threshold for one-shot learning is dictated by the satisfiability threshold for bounded-degree $k$-SAT formulas. 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 $k^2$ 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, obtaining significantly stronger bounds on $d$ in terms of $β$. For the uniform case $β\rightarrow 0$, we show that learning is possible under the condition $d\lesssim 2^{k/2}$. This is (up to constant factors) all the way to the sampling threshold -- it is known that sampling a uniformly-distributed satisfying assignment is NP-hard for $d\gtrsim 2^{k/2}$.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3: Lemma 26, GJL19
• proof : Proof of Lemma \ref{['lem: flippable']}
• Lemma 3.1
• proof : Proof of Theorem \ref{['thm:impossible2']}