学习单调电路及其规模近似的ETH难度
ETH-Hardness of Learning Monotone Circuits and Approximating Their Size
AI总结：
在随机指数时间假设下，研究用特定规模单调电路对单调公式PAC学习及对单调函数最小电路规模近似的难度，通过提升论证在证明和通信复杂度中的新颖应用得出结论，给出学习和近似所需时间下限。
AI中文摘要：
我们展示了关于单调学习和单调电路规模近似的如下难度结果：首先，在随机指数时间假设（rETH）下，对于每个\(\epsilon > 0\)，用规模为\(n^{(\log n)^{1 - \epsilon}}\)的单调电路对具有\(n\)个输入位且规模\(s(n) = n\)的单调公式进行PAC学习需要\(n^{\Omega(\log n)}\)时间。其次，在rETH下，对于任意\(\delta > 0\)，存在多项式有界函数\(m\)，使得对与\(n\)位输入上\(m(n)\)个带标记示例\(\{(x_i, b_i)\}\)一致的单调函数的最小单调电路规模进行\(m^{1 - \delta}\)乘法近似需要\(m^{\Omega(\log(m))}\)时间。我们的结果通过在证明和通信复杂度中提升论证在单调学习难度上的新颖应用得出，基于Atserias和Müller（J. ACM，2020）关于自动化归结证明难度的开创性结果。
英文摘要：
We show the following hardness results for monotone learning and approximation of monotone circuit size: 1. Under the Randomised Exponential-Time Hypothesis (rETH), it requires time $n^{Ω(\log n)}$ to PAC-learn monotone formulas with $n$ input bits and size $s(n) = n$ by monotone circuits of size $n^{(\log n)^{1-ε}}$, for every $ε> 0$. 2. Under the Randomised Exponential-Time Hypothesis (rETH), for any $δ> 0$, there is a polynomially bounded function $m$ such that $m^{1-δ}$-multiplicatively approximating the minimum monotone circuit size of a monotone function consistent with a sequence of $m(n)$ labelled examples $\{(x_i, b_i)\}$ over $n$-bit inputs requires time $m^{Ω(\log(m))}$. Our results are shown by a novel application of lifting arguments in proof and communication complexity to hardness of monotone learning, by building on the seminal result of Atserias and Müller (J. ACM, 2020) on hardness of automating Resolution proofs.
