高斯老虎机中的自适应随机矩阵：谱普适性与选择诱导的离群值
Adaptive Random Matrices in Gaussian Bandits: Spectral Universality and Selection-Induced Outliers
- Abstract Math Institute（抽象数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究高斯老虎机中自适应臂选择对观测谱的影响，证明在臂数对数次线性增长时谱收敛于Marchenko-Pastur定律，并揭示选择诱导的离群值与方向效应。
AI中文摘要：
自适应臂选择会改变老虎机算法所收集观测的分布，但未必会改变其极限经验谱。我们研究维度和观测数量按比例增长的高斯老虎机设计。一个定量耦合定理将任何因果选择规则生成的设计与独立高斯设计进行比较。如果可用臂数量的对数在维度上是次线性的，则经验谱分布收敛到Marchenko-Pastur定律，且对选择规则一致成立。因此，高斯贝叶斯老虎机在后验均方不确定性、平方后验协方差和信息获取方面具有与策略无关的一阶极限。对于线性分数选择，我们获得了精确的条件臂分布，并表明两臂选择在每一维度下都生成精确的Wishart Gram矩阵，尽管其条件均值非零。对于固定选择方向，我们识别出一个由高斯最大值的二阶矩控制的显式特征值和特征向量转变。一个反例表明，方向与参考信号的重叠并不决定这一转变。最后，指数大的臂池允许不同的体极限，确立了臂增长条件的阶精确性。这些结果区分了自适应老虎机数据中的全局谱稳定性与方向效应。
英文摘要：
Adaptive arm selection changes the distribution of the observations collected by a bandit algorithm, but it need not change their limiting empirical spectrum. We study Gaussian bandit designs in which the dimension and the number of observations grow proportionally. A quantitative coupling theorem compares the design generated by any causal selection rule with an independent Gaussian design. If the logarithm of the number of available arms is sublinear in the dimension, the empirical spectral distribution converges to the Marchenko-Pastur law, uniformly over the selection rule. Consequently, Gaussian Bayesian bandits have policy-independent first-order limits for posterior mean-square uncertainty, squared posterior covariance, and information acquisition. For linear-score selection, we obtain the exact conditional arm distribution and show that two-arm selection produces an exactly Wishart Gram matrix in every dimension, despite its nonzero conditional mean. For a fixed selection direction, we identify an explicit eigenvalue and eigenvector transition governed by the second moment of a Gaussian maximum. A counterexample shows that a direction's overlap with a reference signal does not determine this transition. Finally, an exponentially large arm pool permits a different bulk limit, establishing the order-sharpness of the arm-growth condition. These results distinguish global spectral stability from directional effects in adaptive bandit data.
