发表机构
Stanford University; Harvard University; Massachusetts Institute of Technology（斯坦福大学; 哈佛大学; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对高维投影追踪问题，在具有维度无关Lipschitz依赖的算法框架下，通过随机控制理论与分支OGP，得到了感知机模型优化的精确算法阈值。
AI 中文摘要
我们研究高维投影追踪的零模型：给定M个独立同分布采样自N维标准高斯分布的点，其中M、N均趋于无穷大，且M/N趋近于α∈(0,∞)。我们的目标是刻画这些点沿依赖于数据的方向x投影的可能经验分布，x的取值范围要么是球面S_N=√N S^{N-1}，要么是立方体Σ_N={-1,+1}^N。我们在算法语境下考虑该问题，其中x必须是对输入具有与维度无关的Lipschitz依赖的算法的输出；这类算法包括基于梯度的通用方法，如朗之万动力学和近似消息传递（AMP）。我们的主要结果通过一维随机控制问题精确刻画了该算法类可实现的经验分布集合。作为主要结果的推论，我们得到了具有一般有界连续激活函数的球形或伊辛感知机模型的哈密顿量优化的精确算法阈值。对于球形问题，Montanari和Zhou（2024）的独立工作也通过随机控制刻画了相关两阶段AMP算法可实现的经验分布。我们的硬度证明基于前两位作者在早期工作中引入的分支重叠间隙性质（branching overlap gap property）。我们的主要创新是在分支OGP框架内发展随机控制理论，显著扩展了其能定位精确算法阈值的场景。值得注意的是，尽管刻画所有可行投影的非算法问题仍是一个重大未解决挑战，我们的方法依然适用。为得到匹配的算法结果，我们构造了一种新的增量AMP算法，该算法作用于高斯无序的布朗桥揭示（Brownian-bridge revelation）并模拟同一类受控随机微分方程（SDE）。
英文摘要
We study a null model of high-dimensional projection pursuit: we are given $M$ points sampled i.i.d. from a standard gaussian in $N$ dimensions, where $M,N\to\infty$ with $M/N\toα\in(0,\infty)$. Our goal is to characterize the possible empirical distributions of these points' projections along a data-dependent direction $x$, which ranges over either the sphere $S_N=\sqrt{N}\mathbb{S}^{N-1}$ or cube $Σ_N=\{-1,+1\}^N$. We consider this problem in an algorithmic setting, where $x$ must be the output of an algorithm with dimension-free Lipschitz dependence on the input; this class of algorithms includes general gradient-based methods such as Langevin dynamics and approximate message passing (AMP). Our main result exactly characterizes the set of empirical distributions attainable by this class in terms of a one-dimensional stochastic control problem. As a consequence of our main result, we obtain exact algorithmic thresholds for optimizing the Hamiltonian of a spherical or Ising perceptron model with general bounded continuous activation. For the spherical problem, independent work of Montanari and Zhou (2024) characterized the empirical distributions attainable by a related two-stage AMP algorithm, also in terms of stochastic control. Our proof of hardness builds on the branching overlap gap property introduced in earlier work by the first two authors. Our main innovation is to develop stochastic control theory within the branching OGP framework, significantly expanding the settings in which it locates an exact algorithmic threshold. Notably, our methods apply even though the non-algorithmic problem of characterizing all feasible projections remains a major outstanding challenge. For the matching algorithmic result, we construct a new incremental AMP algorithm that acts on a Brownian-bridge revelation of the gaussian disorder and simulates the same family of controlled SDEs.
Comments295 pages. All ideas in this paper are human-generated, and all the writing was done by the human authors. AI was used in the writing of this paper only for light proofreading and copy-editing
