arXiv subjects

Zhifan Li

Publications and source records attributed to Zhifan Li.

10 recordsLinked to original sources

arXiv2026

Sharp convergence rates for Spectral methods via the feature space decomposition method

In this paper, we apply the Feature Space Decomposition (FSD) method developed in [LS24, GLS25, LSSW26, ALSS26] to obtain, under fairly general conditions, matching upper and lower bounds for the population excess risk of spectral methods in linear regression under the squared loss, for every covariance and every signal. This result enables us, for a given linear regression problem, to define a pre-order on the set of spectral methods according to their convergence rates, thereby characterizing which spectral algorithm is superior for that specific problem. Furthermore, this allows us to generalize the saturation effect proposed in inverse problems and to provide necessary and sufficient conditions for its occurrence. Our method also shows that, under broad conditions, any spectral algorithm cannot overcome the barrier of the information exponent in problems such as single-index learning.

math.ST↗

arXiv2026

Alignment-Sensitive Minimax Rates for Spectral Algorithms with Learned Kernels

We study spectral algorithms in the setting where kernels are learned from data. We introduce the effective span dimension (ESD), an alignment-sensitive complexity measure that depends jointly on the signal, spectrum, and noise level $σ^2$. The ESD is well-defined for arbitrary kernels and signals without requiring eigen-decay conditions or source conditions. We prove that for sequence models whose ESD is at most $K$, the minimax excess risk scales as $σ^2 K$. Furthermore, we analyze over-parameterized gradient flow and prove that it can reduce the ESD. This finding establishes a connection between adaptive feature learning and provable improvements in generalization of spectral algorithms. We demonstrate the generality of the ESD framework by extending it to linear models and RKHS regression, and we support the theory with numerical experiments. This framework provides a novel perspective on generalization beyond traditional fixed-kernel theories.

cs.LG↗

arXiv2026

High-dimensional online learning via asynchronous decomposition: Non-divergent results, dynamic regularization, and beyond

Existing high-dimensional online learning methods often face the challenge that their error bounds, or per-batch sample sizes, diverge as the number of data batches increases. To address this issue, we propose an asynchronous decomposition framework that leverages summary statistics to construct a surrogate score function for current-batch learning. This framework is implemented via a dynamic-regularized iterative hard thresholding algorithm, providing a computationally and memory-efficient solution for sparse online optimization. We provide a unified theoretical analysis that accounts for both the streaming computational error and statistical accuracy, establishing that our estimator maintains non-divergent error bounds and $\ell_0$ sparsity across all batches. Furthermore, the proposed estimator adaptively achieves additional gains as batches accumulate, attaining the oracle accuracy as if the entire historical dataset were accessible and the true support were known. These theoretical properties are further illustrated through an example of the generalized linear model.

stat.ML↗

arXiv2025

Learning Joint Graphical Model with Computational Efficiency, Dynamic Regularization, and Adaptation

Multi-sourced datasets are common in studies of variable interactions, for example, individual-level fMRI integration, cross-domain recommendation, etc, where each source induces a related but distinct dependency structure. Joint learning of multiple graphical models (i.e., multiple precision matrices) has emerged as an important tool in analyzing such data. Unlike separate learning, joint learning can leverage shared structural patterns across graphs to yield more accurate results. In this paper, we present an efficient and adaptive method named MIGHT (\textbf{M}ulti-task \textbf{I}terative \textbf{G}raphical \textbf{H}ard \textbf{T}hresholding) to estimate multiple graphs jointly. We reformulate the joint model into a series of multi-task learning problems through a column-by-column manner, and solve these problems using a dynamic regularized algorithm based on iterative hard thresholding. This framework is inherently parallelizable and therefore efficient in computation. Theoretically, we derive the non-asymptotic error bound for the resulting estimator. Furthermore, the proposed algorithm is adaptive to heterogeneous column-wise signal strengths: for nodes with strong signals, our estimator achieves improved error bounds and selection consistency adaptively, and also exhibits asymptotic normality -- properties rarely explored in existing joint learning methods. The performance of our method is illustrated through numerical simulations and real data analysis on a cancer gene-expression RNA-seq dataset.

stat.ME↗

arXiv2025

Exact recovery in the double sparse model: sufficient and necessary signal conditions

The double sparse linear model, which has both group-wise and element-wise sparsity in regression coefficients, has attracted lots of attention recently. This paper establishes the sufficient and necessary relationship between the exact support recovery and the optimal minimum signal conditions in the double sparse model. Specifically, sharply under the proposed signal conditions, a two-stage double sparse iterative hard thresholding procedure achieves exact support recovery with a suitably chosen threshold parameter. Also, this procedure maintains asymptotic normality aligning with an OLS estimator given true support, hence holding the oracle properties. Conversely, we prove that no method can achieve exact support recovery if these signal conditions are violated. This fills a critical gap in the minimax optimality theory on support recovery of the double sparse model. Finally, numerical experiments are provided to support our theoretical findings.

math.ST↗

arXiv2025

Rethinking Hard Thresholding Pursuit: Full Adaptation and Sharp Estimation

Hard Thresholding Pursuit (HTP) has aroused increasing attention for its robust theoretical guarantees and impressive numerical performance in non-convex optimization. In this paper, we introduce a novel tuning-free procedure, named Full-Adaptive HTP (FAHTP), that simultaneously adapts to both the unknown sparsity and signal strength of the underlying model. We provide an in-depth analysis of the iterative thresholding dynamics of FAHTP, offering refined theoretical insights. In specific, under the beta-min condition $\min_{i \in S^*}|{\boldsymbolβ}^*_i| \ge Cσ(\log p/n)^{1/2}$, we show that the FAHTP achieves oracle estimation rate $σ(s^*/n)^{1/2}$, highlighting its theoretical superiority over convex competitors such as LASSO and SLOPE, and recovers the true support set exactly. More importantly, even without the beta-min condition, our method achieves a tighter error bound than the classical minimax rate with high probability. The comprehensive numerical experiments substantiate our theoretical findings, underscoring the effectiveness and robustness of the proposed FAHTP.

math.ST↗

arXiv2024

A minimax optimal approach to high-dimensional double sparse linear regression

In this paper, we focus our attention on the high-dimensional double sparse linear regression, that is, a combination of element-wise and group-wise sparsity. To address this problem, we propose an IHT-style (iterative hard thresholding) procedure that dynamically updates the threshold at each step. We establish the matching upper and lower bounds for parameter estimation, showing the optimality of our proposal in the minimax sense. More importantly, we introduce a fully adaptive optimal procedure designed to address unknown sparsity and noise levels. Our adaptive procedure demonstrates optimal statistical accuracy with fast convergence. Additionally, we elucidate the significance of the element-wise sparsity level $s_0$ as the trade-off between IHT and group IHT, underscoring the superior performance of our method over both. Leveraging the beta-min condition, we establish that our IHT-style procedure can attain the oracle estimation rate and achieve almost full recovery of the true support set at both the element level and group level. Finally, we demonstrate the superiority of our method by comparing it with several state-of-the-art algorithms on both synthetic and real-world datasets.

math.ST↗

arXiv2024

The Optimality of Kernel Classifiers in Sobolev Space

Kernel methods are widely used in machine learning, especially for classification problems. However, the theoretical analysis of kernel classification is still limited. This paper investigates the statistical performances of kernel classifiers. With some mild assumptions on the conditional probability $η(x)=\mathbb{P}(Y=1\mid X=x)$, we derive an upper bound on the classification excess risk of a kernel classifier using recent advances in the theory of kernel regression. We also obtain a minimax lower bound for Sobolev spaces, which shows the optimality of the proposed classifier. Our theoretical results can be extended to the generalization error of overparameterized neural network classifiers. To make our theoretical results more applicable in realistic settings, we also propose a simple method to estimate the interpolation smoothness of $2η(x)-1$ and apply the method to real datasets.

math.ST↗

arXiv2023

Sharp minimax optimality of LASSO and SLOPE under double sparsity assumption

This paper introduces a rigorous approach to establish the sharp minimax optimalities of both LASSO and SLOPE within the framework of double sparse structures, notably without relying on RIP-type conditions. Crucially, our findings illuminate that the achievement of these optimalities is fundamentally anchored in a sparse group normalization condition, complemented by several novel sparse group restricted eigenvalue (RE)-type conditions introduced in this study. We further provide a comprehensive comparative analysis of these eigenvalue conditions. Furthermore, we demonstrate that these conditions hold with high probability across a wide range of random matrices. Our exploration extends to encompass the random design, where we prove the random design properties and optimal sample complexity under both weak moment distribution and sub-Gaussian distribution.

math.ST↗

arXiv2023

Minimax Rates for High-dimensional Double Sparse Structure over $\ell_u(\ell_q)$-balls

In this paper, we focus on the high-dimensional double sparse structure, where the parameter of interest simultaneously encourages group-wise sparsity and element-wise sparsity in each group. By combining the Gilbert-Varshamov bound and its variants, we develop a novel lower bound technique for the metric entropy of the parameter space, specifically tailored for the double sparse structure over $\ell_u(\ell_q)$-balls with $u,q \in [0,1]$. We prove lower bounds on the estimation error using an information-theoretic approach, leveraging our proposed lower bound technique and Fano's inequality. To complement the lower bounds, we establish matching upper bounds through a direct analysis of constrained least-squares estimators and utilize results from empirical processes. A significant finding of our study is the discovery of a phase transition phenomenon in the minimax rates for $u,q \in (0, 1]$. Furthermore, we extend the theoretical results to the double sparse regression model and determine its minimax rate for estimation error. To tackle double sparse linear regression, we develop the DSIHT (Double Sparse Iterative Hard Thresholding) algorithm, demonstrating its optimality in the minimax sense. Finally, we demonstrate the superiority of our method through numerical experiments.

math.ST↗