Abstract
Consider a $k$-SAT formula $\Phi$ where every variable appears at most $d$ times. Let $\sigma$ be a satisfying assignment, sampled proportionally to $e^{\beta m(\sigma)}$ where $m(\sigma)$ is the number of true variables and $\beta$ is a real parameter. Given $\Phi$ and $\sigma$, can we efficiently learn $\beta$? 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 $\beta$ is sufficiently large, and bootstrap this to small values of $\beta$ 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 $\beta$. For the uniform case $\beta\rightarrow 0$, 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 $d\gtrsim 2^{k/2}$.