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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.07135 |
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| _version_ | 1866909929466494976 |
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| author | Galanis, Andreas Goldberg, Leslie Ann Zhang, Xusheng |
| author_facet | Galanis, Andreas Goldberg, Leslie Ann Zhang, Xusheng |
| contents | 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 $d\gtrsim 2^{k/2}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_07135 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | One-Shot Learning for k-SAT Galanis, Andreas Goldberg, Leslie Ann Zhang, Xusheng Data Structures and Algorithms Machine Learning Statistics Theory 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 $d\gtrsim 2^{k/2}$. |
| title | One-Shot Learning for k-SAT |
| topic | Data Structures and Algorithms Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2502.07135 |