Identifying the relevant variables for a classification model with correct confidence levels is a central but difficult task in high-dimension. Despite the core role of sparse logistic regression in statistics and machine learning, it still lacks a good solution for accurate inference in the regime where the number of features $p$ is as large as or larger than the number of samples $n$. Here we tackle this problem by improving the Conditional Randomization Test (CRT). The original CRT algorithm shows promise as a way to output p-values while making few assumptions on the distribution of the test statistics. As it comes with a prohibitive computational cost even in mildly high-dimensional problems, faster solutions based on distillation have been proposed. Yet, they rely on unrealistic hypotheses and result in low-power solutions. To improve this, we propose \emph{CRT-logit}, an algorithm that combines a variable-distillation step and a decorrelation step that takes into account the geometry of $\ell_1$-penalized logistic regression problem. We provide a theoretical analysis of this procedure, and demonstrate its effectiveness on simulations, along with experiments on large-scale brain-imaging and genomics datasets.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The authors propose a new screening rule Slores for pre-filtering variables for logistic regression. This statement though sounds too simple and doesn't give the paper justice at all. The paper provides a rigorous and theoretically well founded derivation of a novel pre-screening rule which could in principle be extended to other settings as well. The method is also efficient compared to other safe rules that guarantee to discard only non-zero entries.

The l1-regularized logistic regression (or sparse logistic regression) is a widely used method for simultaneous classification and feature selection. Although many recent efforts have been devoted to its efficient implementation, its application to high dimensional data still poses significant challenges. In this paper, we present a fast and effective sparse logistic regression screening rule (Slores) to identify the zero components in the solution vector, which may lead to a substantial reduction in the number of features to be entered to the optimization. An appealing feature of Slores is that the data set needs to be scanned only once to run the screening and its computational cost is negligible compared to that of solving the sparse logistic regression problem. Moreover, Slores is independent of solvers for sparse logistic regression, thus Slores can be integrated with any existing solver to improve the efficiency. We have evaluated Slores using high-dimensional data sets from different applications. Extensive experimental results demonstrate that Slores outperforms the existing state-of-the-art screening rules and the efficiency of solving sparse logistic regression is improved by one magnitude in general.

We propose an enhanced zeroth-order stochastic Frank-Wolfe framework to address constrained finite-sum optimization problems, a structure prevalent in large-scale machine-learning applications. Our method introduces a novel double variance reduction framework that effectively reduces the gradient approximation variance induced by zeroth-order oracles and the stochastic sampling variance from finite-sum objectives. By leveraging this framework, our algorithm achieves significant improvements in query efficiency, making it particularly well-suited for high-dimensional optimization tasks. Specifically, for convex objectives, the algorithm achieves a query complexity of O(d \sqrt{n}/\epsilon ) to find an epsilon-suboptimal solution, where d is the dimensionality and n is the number of functions in the finite-sum objective. For non-convex objectives, it achieves a query complexity of O(d^{3/2}\sqrt{n}/\epsilon^2 ) without requiring the computation ofd partial derivatives at each iteration. These complexities are the best known among zeroth-order stochastic Frank-Wolfe algorithms that avoid explicit gradient calculations. Empirical experiments on convex and non-convex machine learning tasks, including sparse logistic regression, robust classification, and adversarial attacks on deep networks, validate the computational efficiency and scalability of our approach. Our algorithm demonstrates superior performance in both convergence rate and query complexity compared to existing methods.

Identifying the relevant variables for a classification model with correct confidence levels is a central but difficult task in high-dimension. Despite the core role of sparse logistic regression in statistics and machine learning, it still lacks a good solution for accurate inference in the regime where the number of features p is as large as or larger than the number of samples n . Here we tackle this problem by improving the Conditional Randomization Test (CRT). The original CRT algorithm shows promise as a way to output p-values while making few assumptions on the distribution of the test statistics. As it comes with a prohibitive computational cost even in mildly high-dimensional problems, faster solutions based on distillation have been proposed.

Sparse logistic regression is for classification and feature selection simultaneously. Although many studies have been done to solve $\ell_1$-regularized logistic regression, there is no equivalently abundant work on solving sparse logistic regression with nonconvex regularization term. In this paper, we propose a unified framework to solve $\ell_1$-regularized logistic regression, which can be naturally extended to nonconvex regularization term, as long as certain requirement is satisfied. In addition, we also utilize a different line search criteria to guarantee monotone convergence for various regularization terms. Empirical experiments on binary classification tasks with real-world datasets demonstrate our proposed algorithms are capable of performing classification and feature selection effectively at a lower computational cost.

Sparsity regularized loss minimization problems play an important role in various fields including machine learning, data mining, and modern statistics. Proximal gradient descent method and coordinate descent method are the most popular approaches to solving the minimization problem. Although existing methods can achieve implicit model identification, aka support set identification, in a finite number of iterations, these methods still suffer from huge computational costs and memory burdens in high-dimensional scenarios. The reason is that the support set identification in these methods is implicit and thus cannot explicitly identify the low-complexity structure in practice, namely, they cannot discard useless coefficients of the associated features to achieve algorithmic acceleration via dimension reduction. To address this challenge, we propose a novel accelerated doubly stochastic gradient descent (ADSGD) method for sparsity regularized loss minimization problems, which can reduce the number of block iterations by eliminating inactive coefficients during the optimization process and eventually achieve faster explicit model identification and improve the algorithm efficiency. Theoretically, we first prove that ADSGD can achieve a linear convergence rate and lower overall computational complexity. More importantly, we prove that ADSGD can achieve a linear rate of explicit model identification. Numerically, experimental results on benchmark datasets confirm the efficiency of our proposed method.

Identifying the relevant variables for a classification model with correct confidence levels is a central but difficult task in high-dimension. Despite the core role of sparse logistic regression in statistics and machine learning, it still lacks a good solution for accurate inference in the regime where the number of features $p$ is as large as or larger than the number of samples $n$. Here, we tackle this problem by improving the Conditional Randomization Test (CRT). The original CRT algorithm shows promise as a way to output p-values while making few assumptions on the distribution of the test statistics. As it comes with a prohibitive computational cost even in mildly high-dimensional problems, faster solutions based on distillation have been proposed. Yet, they rely on unrealistic hypotheses and result in low-power solutions. To improve this, we propose \emph{CRT-logit}, an algorithm that combines a variable-distillation step and a decorrelation step that takes into account the geometry of $\ell_1$-penalized logistic regression problem. We provide a theoretical analysis of this procedure, and demonstrate its effectiveness on simulations, along with experiments on large-scale brain-imaging and genomics datasets.