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• W2953833247 abstract "What kinds of functions are learnable from their satisfying assignments? Motivated by this simple question, we extend the framework of De, Diakonikolas, and Servedio [DDS15], which studied the learnability of probability distributions over ${0,1}^n$ defined by the set of satisfying assignments to Boolean functions, to Boolean-valued functions defined over continuous domains. In our learning scenario there is a known $mathcal{D}$ over $mathbb{R}^n$ (such as a known normal distribution or a known log-concave distribution) and the learner is given i.i.d. samples drawn from a target distribution $mathcal{D}_f$, where $mathcal{D}_f$ is $mathcal{D}$ restricted to the satisfying assignments of an unknown low-complexity Boolean-valued function $f$. The problem is to learn an approximation $mathcal{D}'$ of the target distribution $mathcal{D}_f$ which has small error as measured in total variation distance. We give a range of efficient algorithms and hardness results for this problem, focusing on the case when $f$ is a low-degree polynomial threshold function (PTF). When the background distribution $mathcal{D}$ is log-concave, we show that this learning problem is efficiently solvable for degree-1 PTFs (i.e.,~linear threshold functions) but not for degree-2 PTFs. In contrast, when $mathcal{D}$ is a normal distribution, we show that this learning problem is efficiently solvable for degree-2 PTFs but not for degree-4 PTFs. Our hardness results rely on standard assumptions about secure signature schemes." @default.
• W2953833247 created "2019-07-12" @default.
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• W2953833247 date "2019-07-02" @default.
• W2953833247 modified "2023-09-27" @default.
• W2953833247 title "Learning from satisfying assignments under continuous distributions" @default.
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