论文标题
在二进制模型中推断的尖锐渐近和最佳性能
Sharp Asymptotics and Optimal Performance for Inference in Binary Models
论文作者
论文摘要
我们研究了二进制模型中高维推断的凸经验风险最小化。我们的第一个结果彻底预测了在各向同性高斯特征下线性渐近状态下此类估计量的统计性能。重要的是，对广泛的凸损失功能的预测构成，我们利用这些函数，以证明它们之间的最佳可实现性能。值得注意的是，我们表明，通过构造适当的损失函数来实现它的流行二进制模型（例如签名，逻辑或概率），所提出的界限是紧密的。更有趣的是，对于逻辑和概率模型下的二进制线性分类，我们证明最小二乘的性能不得差于最佳的0.997和0.98倍。数值模拟证实了我们的理论发现，即使对于相对较小的问题维度，它们也是准确的。
We study convex empirical risk minimization for high-dimensional inference in binary models. Our first result sharply predicts the statistical performance of such estimators in the linear asymptotic regime under isotropic Gaussian features. Importantly, the predictions hold for a wide class of convex loss functions, which we exploit in order to prove a bound on the best achievable performance among them. Notably, we show that the proposed bound is tight for popular binary models (such as Signed, Logistic or Probit), by constructing appropriate loss functions that achieve it. More interestingly, for binary linear classification under the Logistic and Probit models, we prove that the performance of least-squares is no worse than 0.997 and 0.98 times the optimal one. Numerical simulations corroborate our theoretical findings and suggest they are accurate even for relatively small problem dimensions.
