奇偶决策树的Fourier增长
Fourier Growth of Parity Decision Trees
摘要：
我们证明了对于$n$变量上的深度$d$的每一个奇偶决策树，$\ell$层上的傅立叶系数的绝对值之和至多为$d^{\ell/2} \cdot O(\ell \cdot \log(n))^\ell$。我们的结果对于$\ell$的小值几乎是紧的，并且扩展了Sherstov，Storozhenko和Wu（STOC，2021）之前标准决策树的傅立叶界。作为我们的傅立叶边界的应用，使用Bansal和Sinha（STOC，2021）的结果，我们证明了$k$ -fold Forrelation问题具有（随机化）奇偶校验决策树复杂度$\tildeΩ\left(n^{1-1/k}\right)$，同时具有量子查询复杂度$\lceil k/2\rceil$。我们的证明遵循随机游走的方法，分析决策树中的随机路径的贡献的水平- $\ell$傅立叶表达式。为了进行论证，我们对奇偶校验决策树应用了仔细的清理过程，确保随机游走的值以高概率有界。我们观察到，级别$\ell$行走的步长可以通过级别$\le \ell-1$行走的中间值来计算，这需要一个归纳参数。我们的方法不同于Tal（FOCS，2020）和Sherstov，Storozhenko和Wu（STOC，2021）之前的证明，这些证明依赖于树的分解。特别是，对于标准决策树的特殊情况，我们认为我们的证明稍微简单和更直观。此外，我们证明了一个类似的界限噪声决策树的成本最多$d$ -一个模型，最近介绍了本大卫和布莱斯（FOCS，2020）。
英文摘要：
We prove that for every parity decision tree of depth $d$ on $n$ variables, the sum of absolute values of Fourier coefficients at level $\ell$ is at most $d^{\ell/2} \cdot O(\ell \cdot \log(n))^\ell$. Our result is nearly tight for small values of $\ell$ and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021). As an application of our Fourier bounds, using the results of Bansal and Sinha (STOC, 2021), we show that the $k$-fold Forrelation problem has (randomized) parity decision tree complexity $\tildeΩ\left(n^{1-1/k}\right)$, while having quantum query complexity $\lceil k/2\rceil$. Our proof follows a random-walk approach, analyzing the contribution of a random path in the decision tree to the level-$\ell$ Fourier expression. To carry the argument, we apply a careful cleanup procedure to the parity decision tree, ensuring that the value of the random walk is bounded with high probability. We observe that step sizes for the level-$\ell$ walks can be computed by the intermediate values of level $\le \ell-1$ walks, which calls for an inductive argument. Our approach differs from previous proofs of Tal (FOCS, 2020) and Sherstov, Storozhenko, and Wu (STOC, 2021) that relied on decompositions of the tree. In particular, for the special case of standard decision trees we view our proof as slightly simpler and more intuitive. In addition, we prove a similar bound for noisy decision trees of cost at most $d$ -- a model that was recently introduced by Ben-David and Blais (FOCS, 2020).
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