一个最优的“直到结束才结束”定理
An Optimal "It Ain't Over Till It's Over" Theorem
摘要：
研究了最大影响值小的布尔函数在随机约束下成为常数的概率。令$f$为布尔函数，使得$f$的方差为$Ω(1)$，并且其所有个体影响均以$τ$为界。我们表明，当限制所有，但$ρ=\tildeΩ((\log(1/τ))^{-1})$分数的坐标，限制功能保持不恒定的压倒性的概率。这个界限本质上是最优的，正如部落函数$\mathrm{TRIBES}=\mathrm{AND}_{n/C\log n}\circ\mathrm{OR}_{C\log n}$所证明的那样。我们将其扩展到一个反浓度的结果，表明限制函数具有非平凡的方差概率$1-o(1)$。这给出了Mossel、O'Donnell和Oleszkiewicz的“直到结束才结束”定理的尖锐版本。我们的证明是离散的，避免了使用不变性原理。我们还证明了上述结果的两个推论：（i）作为推论，我们证明了对于均匀随机输入$x$，$f$在$x$处的块灵敏度是$\tildeΩ(\log(1/τ))$，概率为$1-o(1)$。这应该与Kahn、Kalai和Linial的结果的含义进行比较，后者暗示$f$的平均块敏感性为$Ω(\log(1/τ))$。(ii)将我们的证明与O'Donnell、Saks、Schramm和Servedio的一个著名结果相结合，我们还可以得出结论：限制单调函数$f$的坐标的$ρ=\tildeΩ(1/\sqrt{\log (1/τ) })$部分之外的所有部分，则受限函数具有决策树复杂度$Ω(τ^{-Θ(ρ)})$，概率为$Ω(1)$。
英文摘要：
We study the probability of Boolean functions with small max influence to become constant under random restrictions. Let $f$ be a Boolean function such that the variance of $f$ is $Ω(1)$ and all its individual influences are bounded by $τ$. We show that when restricting all but a $ρ=\tildeΩ((\log(1/τ))^{-1})$ fraction of the coordinates, the restricted function remains nonconstant with overwhelming probability. This bound is essentially optimal, as witnessed by the tribes function $\mathrm{TRIBES}=\mathrm{AND}_{n/C\log n}\circ\mathrm{OR}_{C\log n}$. We extend it to an anti-concentration result, showing that the restricted function has nontrivial variance with probability $1-o(1)$. This gives a sharp version of the "it ain't over till it's over" theorem due to Mossel, O'Donnell, and Oleszkiewicz. Our proof is discrete, and avoids the use of the invariance principle. We also show two consequences of our above result: (i) As a corollary, we prove that for a uniformly random input $x$, the block sensitivity of $f$ at $x$ is $\tildeΩ(\log(1/τ))$ with probability $1-o(1)$. This should be compared with the implication of Kahn, Kalai, and Linial's result, which implies that the average block sensitivity of $f$ is $Ω(\log(1/τ))$. (ii) Combining our proof with a well-known result due to O'Donnell, Saks, Schramm, and Servedio, one can also conclude that: Restricting all but a $ρ=\tildeΩ(1/\sqrt{\log (1/τ) })$ fraction of the coordinates of a monotone function $f$, then the restricted function has decision tree complexity $Ω(τ^{-Θ(ρ)})$ with probability $Ω(1)$.
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