One-Shot Learning for k-SAT

TL;DR

This paper investigates the limits of one-shot learning of the parameter  in biased distributions over solutions to bounded-degree k-SAT formulas, showing it is infeasible below the satisfiability threshold and providing stronger bounds for feasible cases.

Contribution

It demonstrates that one-shot learning is impossible at degrees much lower than the satisfiability threshold and improves analysis for when learning is feasible, especially in the uniform case.

Findings

Impossibility results for degrees as low as k^2 when  is large.

Bootstrap impossibility to small  with exponential degree scaling.

Learning is feasible for d \u2264 2^{k/2} in the uniform case.

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 $d\gtrsim 2^{k/2}$.