Nonregular and Minimax Estimation of Individualized Thresholds in High Dimension with Binary Responses
This work addresses a nonregular estimation problem in high-dimensional statistics with binary responses, which is incremental as it builds on M-estimation frameworks by smoothing and regularization to handle computational and theoretical issues.
The paper tackles the estimation of a high-dimensional individualized linear threshold parameter for binary responses, addressing computational intractability and nonregular statistical challenges by proposing a regularized smoothed loss function. It achieves a finite sample error bound of (s log d/n)^{β/(2β+1)} and proves minimax rate optimality up to a logarithmic factor, with an efficient path-following algorithm showing geometric convergence.
Given a large number of covariates $Z$, we consider the estimation of a high-dimensional parameter $θ$ in an individualized linear threshold $θ^T Z$ for a continuous variable $X$, which minimizes the disagreement between $\text{sign}(X-θ^TZ)$ and a binary response $Y$. While the problem can be formulated into the M-estimation framework, minimizing the corresponding empirical risk function is computationally intractable due to discontinuity of the sign function. Moreover, estimating $θ$ even in the fixed-dimensional setting is known as a nonregular problem leading to nonstandard asymptotic theory. To tackle the computational and theoretical challenges in the estimation of the high-dimensional parameter $θ$, we propose an empirical risk minimization approach based on a regularized smoothed loss function. The statistical and computational trade-off of the algorithm is investigated. Statistically, we show that the finite sample error bound for estimating $θ$ in $\ell_2$ norm is $(s\log d/n)^{β/(2β+1)}$, where $d$ is the dimension of $θ$, $s$ is the sparsity level, $n$ is the sample size and $β$ is the smoothness of the conditional density of $X$ given the response $Y$ and the covariates $Z$. The convergence rate is nonstandard and slower than that in the classical Lasso problems. Furthermore, we prove that the resulting estimator is minimax rate optimal up to a logarithmic factor. The Lepski's method is developed to achieve the adaption to the unknown sparsity $s$ and smoothness $β$. Computationally, an efficient path-following algorithm is proposed to compute the solution path. We show that this algorithm achieves geometric rate of convergence for computing the whole path. Finally, we evaluate the finite sample performance of the proposed estimator in simulation studies and a real data analysis.