Counterexample to majority optimality in NICD with erasures
Paata Ivanisvili, Xinyuan Xie
TL;DR
The paper investigates whether the majority function maximizes the NICD with erasures objective $\Phi_p(f)=\mathbb{E}_z[|f(z)|]$ over unbiased Boolean functions for small erasure rates $p<1/2$. It provides a finite counterexample at $n=5$, $p=0.40$ with a linear-threshold function $f(x)=\mathrm{sgn}(x_1-3x_2+x_3-x_4+3x_5)$, showing $\Phi_{0.40}(f)=\frac{2689}{6250}$ exceeds $\Phi_{0.40}(\mathrm{Maj}_5)=\frac{5363}{12500}$, thus majority is not optimal in this NICD setting. The authors also discuss the related “Majority is Least Stable” (L2) conjecture and contrast it with the L1 nature of $\Phi_p$, noting the subtlety in comparing these two loss functions. Finally, they prove that for each fixed odd $n$, majority is optimal in a neighborhood of $p=0$ by deriving a quadratic error bound $|\Phi_p(f)-p\sum_i|\hat f(i)|| \le C_n p^2$, establishing a positive gap $\delta_n$ and an explicit $p_0(n)$ guaranteeing the optimality of Maj_n for small $p$. The work highlights the complementary roles of AI-assisted discovery and analytical bounds in addressing open questions in theoretical computer science.
Abstract
We asked GPT-5 Pro to look for counterexamples among a public list of open problems (the Simons ``Real Analysis in Computer Science'' collection). After several numerical experiments, it suggested a counterexample for the Non-Interactive Correlation Distillation (NICD) with erasures question: namely, a Boolean function on 5 bits that achieves a strictly larger value of $\mathbb{E}|f(z)|$ than the 5-bit majority function when the erasure parameter is $p=0.40.$ In this very short note we record the finding, state the problem precisely, give the explicit function, and verify the computation step by step by hand so that it can be checked without a computer. In addition, we show that for each fixed odd $n$ the majority is optimal (among unbiased Boolean functions) in a neighborhood of $p=0$. We view this as a little spark of an AI contribution in Theoretical Computer Science: while modern Large Language Models (LLMs) often assist with literature and numerics, here a concrete finite counterexample emerged.