Active Subsampling for Measurement-Constrained M-Estimation of Individualized Thresholds with High-Dimensional Data
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| Format: | Preprint |
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2024
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| author | Duan, Jingyi Ning, Yang |
| author_facet | Duan, Jingyi Ning, Yang |
| contents | In the measurement-constrained problems, despite the availability of large datasets, we may be only affordable to observe the labels on a small portion of the large dataset. This poses a critical question that which data points are most beneficial to label given a budget constraint. In this paper, we focus on the estimation of the optimal individualized threshold in a measurement-constrained M-estimation framework. Our goal is to estimate a high-dimensional parameter $θ$ in a linear threshold $θ^T Z$ for a continuous variable $X$ such that the discrepancy between whether $X$ exceeds the threshold $θ^T Z$ and a binary outcome $Y$ is minimized. We propose a novel $K$-step active subsampling algorithm to estimate $θ$, which iteratively samples the most informative observations and solves a regularized M-estimator. The theoretical properties of our estimator demonstrate a phase transition phenomenon with respect to $β\geq 1$, the smoothness of the conditional density of $X$ given $Y$ and $Z$. For $β>(1+\sqrt{3})/2$, we show that the two-step algorithm yields an estimator with the parametric convergence rate $O_p((s \log d /N)^{1/2})$ in $l_2$ norm. The rate of our estimator is strictly faster than the minimax optimal rate with $N$ i.i.d. samples drawn from the population. For the other two scenarios $1<β\leq (1+\sqrt{3})/2$ and $β=1$, the estimator from the two-step algorithm is sub-optimal. The former requires to run $K>2$ steps to attain the same parametric rate, whereas in the latter case only a near parametric rate can be obtained. Furthermore, we formulate a minimax framework for the measurement-constrained M-estimation problem and prove that our estimator is minimax rate optimal up to a logarithmic factor. Finally, we demonstrate the performance of our method in simulation studies and apply the method to analyze a large diabetes dataset. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_13763 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Active Subsampling for Measurement-Constrained M-Estimation of Individualized Thresholds with High-Dimensional Data Duan, Jingyi Ning, Yang Statistics Theory Methodology Machine Learning In the measurement-constrained problems, despite the availability of large datasets, we may be only affordable to observe the labels on a small portion of the large dataset. This poses a critical question that which data points are most beneficial to label given a budget constraint. In this paper, we focus on the estimation of the optimal individualized threshold in a measurement-constrained M-estimation framework. Our goal is to estimate a high-dimensional parameter $θ$ in a linear threshold $θ^T Z$ for a continuous variable $X$ such that the discrepancy between whether $X$ exceeds the threshold $θ^T Z$ and a binary outcome $Y$ is minimized. We propose a novel $K$-step active subsampling algorithm to estimate $θ$, which iteratively samples the most informative observations and solves a regularized M-estimator. The theoretical properties of our estimator demonstrate a phase transition phenomenon with respect to $β\geq 1$, the smoothness of the conditional density of $X$ given $Y$ and $Z$. For $β>(1+\sqrt{3})/2$, we show that the two-step algorithm yields an estimator with the parametric convergence rate $O_p((s \log d /N)^{1/2})$ in $l_2$ norm. The rate of our estimator is strictly faster than the minimax optimal rate with $N$ i.i.d. samples drawn from the population. For the other two scenarios $1<β\leq (1+\sqrt{3})/2$ and $β=1$, the estimator from the two-step algorithm is sub-optimal. The former requires to run $K>2$ steps to attain the same parametric rate, whereas in the latter case only a near parametric rate can be obtained. Furthermore, we formulate a minimax framework for the measurement-constrained M-estimation problem and prove that our estimator is minimax rate optimal up to a logarithmic factor. Finally, we demonstrate the performance of our method in simulation studies and apply the method to analyze a large diabetes dataset. |
| title | Active Subsampling for Measurement-Constrained M-Estimation of Individualized Thresholds with High-Dimensional Data |
| topic | Statistics Theory Methodology Machine Learning |
| url | https://arxiv.org/abs/2411.13763 |