This paper studies high-dimensional M-estimation in the proportional asymptotic regime (p/n -> gamma > 0) when the noise distribution has infinite variance. For noise with regularly-varying tails of index alpha in (1,2), we establish that the asymptotic behavior of a regularized M-estimator is governed by a single geometric property of the loss function: the boundedness of the domain of its Fenchel conjugate. When this conjugate domain is bounded -- as is the case for the Huber, absolute-value, and quantile loss functions -- the dual variable in the min-max formulation of the estimator is confined, the effective noise reduces to the finite first absolute moment of the noise distribution, and the estimator achieves bounded risk without recourse to external information. When the conjugate domain is unbounded -- as for the squared loss -- the dual variable scales with the noise, the effective noise involves the diverging second moment, and bounded risk can be achieved only through transfer regularization toward an external prior. For the squared-loss class specifically, we derive the exact asymptotic risk via the Convex Gaussian Minimax Theorem under a noise-adapted regularization scaling. The resulting risk converges to a universal floor that is independent of the regularizer, yielding a loss-risk trichotomy: squared-loss estimators without transfer diverge; Huber-loss estimators achieve bounded but non-vanishing risk; transfer-regularized estimators attain the floor.

翻译：本文研究比例渐近机制（p/n → γ > 0）下噪声分布具有无限方差时的高维M估计。对于尾部指数α∈(1,2)的规则变化噪声，我们证明正则化M估计量的渐近行为由损失函数的单一几何性质决定：其Fenchel共轭域的有界性。当共轭域有界时——如Huber损失、绝对值损失和分位数损失函数——估计量极小极大对偶形式中的对偶变量被约束，有效噪声简化为噪声分布的一阶绝对矩，无需借助外部信息即可实现有界风险。当共轭域无界时——如平方损失——对偶变量随噪声尺度变化，有效噪声涉及发散的二阶矩，唯有通过向外部先验的迁移正则化才能实现有界风险。针对平方损失类，我们通过凸高斯极小极大定理推导了噪声自适应正则化尺度下的精确渐近风险。所得风险收敛至与正则化器无关的通用下界，从而形成损失-风险三分法：无迁移的平方损失估计量发散；Huber损失估计量实现有界但非零风险；迁移正则化估计量达到风险下界。