ETH-Hardness of Learning Monotone Circuits and Approximating Their Size

2026-07-14

AI Summary

This study investigates the computational complexity lower bounds for learning monotone circuits and approximating their size under the randomized Exponential Time Hypothesis (rETH). By introducing, for the first time, lifting techniques from proof and communication complexity into monotone learning theory, the work establishes a novel connection between this learning problem and the hardness of automating resolution proofs. Leveraging this connection, it is shown that, assuming rETH, learning a monotone formula of size $n$ requires time $n^{\Omega(\log n)}$, and approximating the minimum size of a monotone circuit also demands time $m^{\Omega(\log m)}$. This result extends the hardness of automating proofs established by Atserias and Müller to the realm of learning theory, yielding the first super-polynomial time lower bound for monotone PAC learning.

Abstract

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.

Problem

• monotone circuits
• PAC learning
• circuit size approximation
• ETH-hardness
• monotone functions

Innovation

• monotone circuits
• PAC learning
• rETH
• lifting arguments
• circuit size approximation