Rethinking Hard Thresholding Pursuit: Full Adaptation and Sharp Estimation

TL;DR

This paper introduces a fully adaptive, tuning-free version of Hard Thresholding Pursuit (FAHTP) that effectively estimates sparse signals, adapts to unknown parameters, and outperforms traditional methods like LASSO in theory and practice.

Contribution

The paper proposes a novel, tuning-free FAHTP algorithm that adaptively estimates sparsity and signal strength, with refined theoretical analysis and superior performance over convex competitors.

Findings

FAHTP achieves oracle estimation rate under beta-min condition.

FAHTP recovers the true support set exactly.

Numerical experiments confirm robustness and effectiveness.

Abstract

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 $\min_{i \in S^*}|{\boldsymbol{\beta}}^*_i| \ge C\sigma (\log p/n)^{1/2}$, we show that the FAHTP achieves oracle estimation rate $\sigma (s^*/n)^{1/2}$, 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.