Explicit Bounds on Hoffman Constants for Sparse Polyhedral Regularizers
稀疏多面体正则化中Hoffman常数的显式界AI 解读 ↓
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稀疏多面体正则化中Hoffman常数的显式界
- 研究问题
- 摘要称，Hoffman常数刻画多面体系统到解集的距离，但因其计算复杂而少有显式取值；本文研究稀疏正则化中一类结构化多面体系统的Hoffman常数能否给出显式界及其尺度行为。
- 方法与思路
- 作者提出将Hoffman常数分解为谱、几何与交互三个分量，用以解释最坏情形维度依赖界与结构化问题中经验良好条件数之间的差距；并以ℓ1正则最小二乘为主要应用，再推广到弹性网与自适应ℓ1正则化。
- 材料陈述的贡献
- 摘要称，该框架给出结构化回归中Hoffman常数的显式可解释形式，说明有效常数取决于低维活跃子空间而非环境维度；并指出受限特征值条件只约束活跃集的原曲率、未约束对偶交互几何，标准ℓ∞相干条件也不足以给出维度无关的欧氏界。
- 阅读关注点
- 应核对分解中各分量的定义与推导假设、ℓ1最小二乘有效常数依赖低维活跃子空间的具体条件、弹性网与自适应ℓ1下几何如何被改变，以及受限特征值条件与ℓ∞相干条件不足的论证范围。
- 阅读边界
- 仅依据arXiv预印本摘要，未读全文；这是预印本，不能据此声称已同行评审或期刊接收。摘要未说明具体常数数值、证明细节、实验设置与所提后验特征剪枝和超参数选择工具的实际表现。
原文摘要
Hoffman constants quantify the distance to the solution set of a polyhedral system in terms of its residual; they control error bounds, geometric conditioning, the convergence rate of first-order iterative algorithms, and the stability of regression estimators to target perturbations. Despite their importance, explicit values of Hoffman constants are rarely available due to their computational complexity. In this paper, we derive explicit bounds on the Hoffman constant for a class of structured polyhedral systems arising in sparse regularization. We show that the constants admit a decomposition into spectral, geometric, and interaction components, which together determine their scaling behavior. This framework explains the gap between worst-case dimension-dependent bounds and the favorable conditioning observed empirically in structured problems. As a primary application, we analyze $\ell_1$-regularized least squares, demonstrating that the effective constants depend on low-dimensional active subspaces rather than the ambient dimension. We then extend to the elastic net and adaptive $\ell_1$-regularized formulations, demonstrating how the underlying optimization geometry is altered by the $\ell_2$ penalty and reweighting scheme. Furthermore, we relate our results to classical statistical assumptions and show that the restricted eigenvalue condition bounds only the primal curvature of the active set, leaving the dual interaction geometry unconstrained and is thus insufficient alone for fast theoretical convergence rates. We further show standard $\ell_\infty$ coherence conditions do not yield dimension-independent Euclidean bounds to close this gap. Our theoretical results show that Hoffman constants admit explicit, interpretable forms in structured regression settings and, in practice, provide actionable tools for post-hoc feature pruning and a priori optimal hyperparameter selection.