发表机构
Georgia Institute of Technology; The University of Texas at Austin（佐治亚理工学院; 德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了非对称二元感知机在线问题的精确阈值$\alpha_{\mathrm{on}}(\kappa)$，给出零间隔下的数值范围，证明其在线算法密度远优于此前结果，还分析了阈值随$\kappa$的渐近行为。
AI 中文摘要
设$G\in\mathbb{R}^{M\times N}$具有独立标准高斯项。对于固定间隔$\kappa\in\mathbb{R}$，非对称二元感知机问题要求找到$\sigma\in\{\pm1\}^N$，使得$G\sigma/\sqrt{N}\ge\kappa\mathbf{1}_M$。我们研究该问题的在线版本，其中$G$的列会依次到达，且在后续列揭示前必须不可撤销地选择每个符号。我们确定了每个固定$\kappa$对应的精确阈值$\alpha_{\mathrm{on}}(\kappa)$：当$M/N\to\alpha$且$\alpha<\alpha_{\mathrm{on}}(\kappa)$时，存在一种确定性在线算法，使用$O(MN)$次算术运算和多项式比特复杂度，能以高概率成功；而当$\alpha>\alpha_{\mathrm{on}}(\kappa)$时，没有在线算法能以高概率成功。该阈值由布朗运动的一维随机控制问题刻画。主要难点在于将单坐标布朗极限提升至所有$M=\Theta(N)$约束同时满足的可行性，我们通过半直线单调性和一个短的最终校正块解决了这一问题。在零间隔下，我们给出计算机辅助证明，得出$0.32747<\alpha_{\mathrm{on}}(0)<0.36664$。特别地，低于$0.32747$的密度可通过此类在线算法实现，这是之前任何多项式时间算法（无论在线或离线）可实现的最佳密度的三倍多，之前的界限为$\alpha\le0.1$，由Li、Schramm和Zhou给出。当$\kappa\to+\infty$时，在线阈值与离线存储容量的一阶近似一致；当$\kappa\to-\infty$时，其渐近尺度与已知最佳离线多项式时间保证相同，而存储容量大了约$\kappa^2$倍。
英文摘要
Let $G\in\mathbb{R}^{M\times N}$ have independent standard Gaussian entries. For a fixed margin $κ\in\mathbb{R}$, the asymmetric binary perceptron asks for $σ\in\{\pm1\}^N$ such that $Gσ/\sqrt{N}\geκ\mathbf{1}_M$. We study the online version of this problem, in which the columns of $G$ arrive sequentially and each sign must be chosen irrevocably before future columns are revealed. We determine the exact threshold $α_{\mathrm{on}}(κ)$ for every fixed $κ$: for $M/N\toα$ with $α<α_{\mathrm{on}}(κ)$, there is a deterministic online algorithm, using $O(MN)$ arithmetic operations and polynomial bit complexity, that succeeds with high probability, while for $α>α_{\mathrm{on}}(κ)$, no online algorithm succeeds with high probability. The threshold is characterized by a one-dimensional stochastic control problem for Brownian motion. The main difficulty is to upgrade a single-coordinate Brownian limit to simultaneous feasibility of all $M=Θ(N)$ constraints, which we do with half-line monotonicity and a short final correction block. At zero margin, we give a computer-assisted proof that $0.32747<α_{\mathrm{on}}(0)<0.36664$. In particular, every density below $0.32747$ is achievable online by such an algorithm, more than tripling the best density previously proved attainable by any polynomial-time algorithm, online or offline (the previous bound was $α\le0.1$, due to Li, Schramm, and Zhou). As $κ\to+\infty$, the online threshold agrees to first order with the offline storage capacity. As $κ\to-\infty$, it has the same asymptotic scale as the best known offline polynomial-time guarantee, while the storage capacity is larger by a factor of order $κ^2$.
Comments56 pages
