超越负岭端点：通过负移位梯度下降的混合符号谱正则化

中文摘要 AI 辅助

在过参数化线性回归中，许多弱谱方向对承载信号的谱起到岭罚作用；负岭是自然校正，将滤波器推到大于一的值。然而，稳定的负岭端点在结构上受限：其极点必须低于最小非零经验特征值，且它对较小特征值的反收缩作用大于较大特征值。早期停止的负移位梯度下降摆脱了此约束。其滤波器在潜在极点处平滑且具有混合符号能力。在高斯尖峰加平坦模型中发现了马尔琴科 - 帕斯特尔障碍，停止路径在明确条件下按风险的多项式因子改进每个可允许端点。主要定理允许一般的高效秩尾部，处理非收缩移位动力学是核心技术挑战，通过局部杜哈梅尔积分控制它们，有限网格留出不等式将分离转移到验证选择的算法。

英文摘要

In overparameterized linear regression, many weak spectral directions act like a ridge penalty on the signal-bearing spectrum; negative ridge is the natural correction, pushing filters above one. The stable negative-ridge endpoint, however, is structurally limited: its pole must stay below the smallest nonzero empirical eigenvalue, and it anti-shrinks smaller eigenvalues more than larger ones. Early-stopped negative-shifted gradient descent escapes this constraint. Its filter is smooth at the would-be pole and mixed-sign-capable: above-ridgeless directions form a leading prefix, with lower directions shrunk or exposure-controlled while stopping sets the crossover. In a Gaussian spike-plus-flat model we discover a Marchenko-Pastur barrier: the shift that cancels the implicit penalty lies a bulk width above the smallest empirical eigenvalue, and the stopped path improves on every admissible endpoint by a polynomial factor in risk under explicit conditions. Our main theorem permits a general high-effective-rank tail: its trace sets the implicit floor, its squared spectrum controls exposure, and the floor-critical path recovers all head scales at once, beyond positive shrinkage and, once scales separate, every uniform rescaling of ridgeless. Handling the noncontractive shifted dynamics is the central technical challenge; localized Duhamel integrals control them. A finite-grid hold-out inequality transfers the separations to the validation-selected algorithm.

发表机构

• University of Delaware（特拉华大学）

机构由 AI 辅助整理，请以论文原文为准。