发表机构
Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出ℓ-局部决策树模型，证明其强下界可蕴含突破性电路下界，并利用和集结构新技术为oblivious变体证明最优下界，同时开创次数为2的决策树研究。
AI 中文摘要
我们引入并研究了一种新的决策树模型，该模型处于可证明的电路下界研究的前沿。我们研究了ℓ-局部决策树，其中每个内部节点查询输入的一个ℓ-局部函数。我们证明，对于ℓ-局部决策树足够强的下界将蕴含若干突破性的电路下界，包括对数深度电路的超线性规模下界以及无限制深度电路的改进下界。此前，已知这些结论可由深度为3的电路的强下界推导得出，而这正是先前试图证明这些结果的主要方法。由于局部决策树严格弱于深度为3的电路，这为获得相同的电路下界结论提供了一条形式上更简单的途径。我们为一种较弱的变体（称为 oblivious ℓ-局部决策树，其中同一深度的所有节点查询相同的函数）证明了本质上最优的下界。我们的下界源于一种揭示局部映射中和集结构的新技术，我们的困难函数是和集分散器、和集压缩器和方向仿射分散器。在此过程中，我们给出了一个具有小熵损失的和集压缩器的显式构造。作为额外贡献，我们开创了次数为2的决策树的研究，其中每个查询是输入的二次多项式。我们证明，在此模型中证明下界是构建次数为2的品种源分散器的自然垫脚石，而后者蕴含无限制深度电路的改进电路下界。我们为该模型的 oblivious 变体证明了近乎最大的下界。
英文摘要
We introduce and study a new model of decision trees that lies at the frontier of provable circuit lower bounds. We study $\ell$-local decision trees, in which each internal node queries an $\ell$-local function of the input. We show that sufficiently strong lower bounds for $\ell$-local decision trees would imply several breakthrough circuit lower bounds, including super-linear size lower bounds for log-depth circuits and improved bounds for unrestricted-depth circuits. Previously, such consequences were known to follow from strong lower bounds for depth-$3$ circuits, which has been the main prior approach in attempting to prove these results. Since local decision trees are strictly weaker than depth-$3$ circuits, this provides a formally easier route to the same circuit lower-bound consequences. We prove essentially optimal lower bounds for a weaker variant that we call oblivious $\ell$-local decision trees, where all nodes at the same depth query the same function. Our lower bounds follow from a new technique that uncovers sumset structure in local maps, and our hard functions are sumset dispersers, sumset condensers, and directional affine dispersers. Along the way, we give an explicit construction of a sumset condenser with small entropy loss. As an additional contribution, we initiate the study of degree-$2$ decision trees, in which each query is a quadratic polynomial of the input. We show that proving lower bounds in this model is a natural stepping stone toward constructing dispersers for degree-$2$ variety sources, which imply improved circuit lower bounds for unrestricted depth circuits. We prove nearly maximal lower bounds for the oblivious variant of the model.
