Problem
Research questions and friction points this paper is trying to address.
logistic regression
Gaussian design
parameter estimation
maximum likelihood estimator (MLE)
finite-sample error rate
Minimax Optimal Estimator and Improved Error Rate for the MLE in Logistic Regression with Gaussian Design
🤖 AI Summary
本文解决了逻辑回归中参数估计问题，通过提出最小最大最优估计器改进了MLE的误差率。
📝 Abstract
We study finite-sample parameter estimation in logistic regression with Gaussian design, where the goal is to estimate $\mathbfθ^*\in \mathbb{R}^d$ with $R=\|\mathbfθ^*\|_2\ge 1$ from i.i.d. samples $\{(\mathbf{x}_i,y_i)\}_{i=1}^n,$ $\mathbf{x}_i \sim N(0,\mathbf{I}_d)$, $y_i\mid \mathbf{x}_i \sim \mathrm{Bernoulli}((1+\exp(-\mathbf{x}_i^\top \mathbfθ^*))^{-1})$. In this paper, we provide the first minimax optimal estimator, and improve on the best known finite-sample error rate for the maximum likelihood estimator (MLE). These two accomplishments are due to a minimax optimal estimator for the parameter norm $R$. First, we establish the minimax lower bound $Ω(\sqrt{R^3/n})$ for norm estimation. We then improve the best known norm estimation error rate of the MLE, i.e., $O(\sqrt{R^3d/n})$ from Chardon, Lerasle and Mourtada (2024), to $\tilde{O}(\sqrt{R^3/n}+R^2d/n)$. The additional term, $R^2d/n$, appears to be the intrinsic bias of the MLE, as evidenced by the high-dimensional asymptotic theory of Zhao, Sur and Candes (2022) and numerical examples. We show that, however, this additional term is not information-theoretically necessary. To this end, we construct an efficient debiased norm estimator that achieves the error rate $O(\sqrt{R^3/n})$ and is therefore minimax optimal. Combining this with the optimal direction estimator given by the MLE, we establish the minimax optimal rate $Θ(\sqrt{Rd/n}+\sqrt{R^3/n})$ for estimating $\mathbfθ^*$, as well as the improved finite-sample error rate $\tilde{O}(\sqrt{Rd/n}+\sqrt{R^3/n}+R^2d/n)$ for the MLE. Numerical experiments demonstrate that the proposed minimax optimal estimators outperform the MLE.
Innovation
Methods, ideas, or system contributions that make the work stand out.
minimax optimal estimator
improved error rate
MLE
Gaussian design
logistic regression
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