Active Subsampling for Measurement-Constrained M-Estimation of Individualized Thresholds with High-Dimensional Data

• URL: http://arxiv.org/abs/2411.13763v1
• Date: Thu, 21 Nov 2024 00:21:17 GMT
• Title: Active Subsampling for Measurement-Constrained M-Estimation of Individualized Thresholds with High-Dimensional Data
• Authors: Jingyi Duan, Yang Ning,
• Abstract summary: 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.
• Score: 3.1138411427556445
• License:
• Abstract: 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 $\theta$ in a linear threshold $\theta^T Z$ for a continuous variable $X$ such that the discrepancy between whether $X$ exceeds the threshold $\theta^T Z$ and a binary outcome $Y$ is minimized. We propose a novel $K$-step active subsampling algorithm to estimate $\theta$, 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 $\beta\geq 1$, the smoothness of the conditional density of $X$ given $Y$ and $Z$. For $\beta>(1+\sqrt{3})/2$, 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 $1<\beta\leq (1+\sqrt{3})/2$ and $\beta=1$, 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.

Related papers

• Multiple-policy Evaluation via Density Estimation [30.914344538340412]
  We propose an algorithm named $mathrmCAESAR$ for this problem. Up to low order and logarithmic terms $mathrmCAESAR$ achieves a sample complexity $tildeOleft(fracH4epsilon2sum_h=1Hmax_kin[K]sum_s,afrac(d_hpik(s,a))2mu*_h(s,a)right)$, where $dpi
  arXiv  Detail & Related papers  (2024-03-29T23:55:25Z)
• Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path [80.60592344361073]
  We study the Shortest Path (SSP) problem with a linear mixture transition kernel. An agent repeatedly interacts with a environment and seeks to reach certain goal state while minimizing the cumulative cost. Existing works often assume a strictly positive lower bound of the iteration cost function or an upper bound of the expected length for the optimal policy.
  arXiv  Detail & Related papers  (2024-02-14T07:52:00Z)
• Memory Efficient And Minimax Distribution Estimation Under Wasserstein Distance Using Bayesian Histograms [6.21295508577576]
  We show that when $d 2v$, histograms possess a special textitmemory efficiency property, in reference to the sample size $n, order $nd/2v$ bins are needed to obtain minimax rate optimality. The attained memory footprint overcomes existing minimax optimal procedures by a factor in $n$; for example an $n1 - d/2v$ factor reduction in the footprint when compared to the empirical measure, a minimax estimator in the Borel probability measure class.
  arXiv  Detail & Related papers  (2023-07-19T16:13:20Z)
• Data Structures for Density Estimation [66.36971978162461]
  Given a sublinear (in $n$) number of samples from $p$, our main result is the first data structure that identifies $v_i$ in time sublinear in $k$. We also give an improved version of the algorithm of Acharya et al. that reports $v_i$ in time linear in $k$.
  arXiv  Detail & Related papers  (2023-06-20T06:13:56Z)
• Robust Nonparametric Regression under Poisoning Attack [13.470899588917716]
  An adversarial attacker can modify the values of up to $q$ samples from a training dataset of size $N$. Our initial solution is an M-estimator based on Huber loss minimization. The final estimate is nearly minimax optimal for arbitrary $q$, up to a $ln N$ factor.
  arXiv  Detail & Related papers  (2023-05-26T09:33:17Z)
• Robust Sparse Mean Estimation via Incremental Learning [15.536082641659423]
  In this paper, we study the problem of robust mean estimation, where the goal is to estimate a $k$-sparse mean from a collection of partially corrupted samples. We present a simple mean estimator that overcomes both challenges under moderate conditions. Our method does not need any prior knowledge of the sparsity level $k$.
  arXiv  Detail & Related papers  (2023-05-24T16:02:28Z)
• Estimating the minimizer and the minimum value of a regression function under passive design [72.85024381807466]
  We propose a new method for estimating the minimizer $boldsymbolx*$ and the minimum value $f*$ of a smooth and strongly convex regression function $f$. We derive non-asymptotic upper bounds for the quadratic risk and optimization error of $boldsymbolz_n$, and for the risk of estimating $f*$.
  arXiv  Detail & Related papers  (2022-11-29T18:38:40Z)
• Best Policy Identification in Linear MDPs [70.57916977441262]
  We investigate the problem of best identification in discounted linear Markov+Delta Decision in the fixed confidence setting under a generative model. The lower bound as the solution of an intricate non- optimization program can be used as the starting point to devise such algorithms.
  arXiv  Detail & Related papers  (2022-08-11T04:12:50Z)
• Localization in 1D non-parametric latent space models from pairwise affinities [6.982738885923206]
  We consider the problem of estimating latent positions in a one-dimensional torus from pairwise affinities. We introduce an estimation procedure that provably localizes all the latent positions with a maximum error of the order of $sqrtlog(n)/n$, with high-probability.
  arXiv  Detail & Related papers  (2021-08-06T13:05:30Z)
• Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret [144.6358229217845]
  We study the problem of learning in the shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus. We prove that EB-SSP achieves the minimax regret rate $widetildeO(B_star sqrtS A K)$, where $K$ is the number of episodes, $S$ is the number of states, $A$
  arXiv  Detail & Related papers  (2021-04-22T17:20:48Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.