This paper investigates the asymptotic behavior of the out-of-sample prediction risk of the high-dimensional ridgeless least-squares estimator when the feature dimension p and the sample size n grow proportionally. We consider a generalized spiked population covariance model with multiple latent factors, where the number of spiked eigenvalues may remain finite or increase with n, and the spiked eigenvalues may be bounded or diverge at arbitrary rates. Beyond characterizing the impact of covariance spectra, we reveal a new mechanism underlying benign overfitting: the prediction behavior of ridgeless interpolation is fundamentally governed by the alignment between the regression coefficient β and the spiked eigenspaces of the population covariance matrix. In particular, we show that the signal energy distributed along latent spike directions determines whether interpolation leads to benign, tempered, or catastrophic overfitting. Our theoretical framework establishes sharp prediction risk limits under minimal moment conditions, requiring only finite fourth moments rather than Gaussianity. We characterize how the number, strength, and geometric structure of the spikes jointly influence the double-descent phenomenon. These results provide a unified understanding of when latent covariance structures facilitate or hinder generalization in overparameterized regression.
Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage--Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.
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.
We consider L2-regularized linear (ridge) regression over a finite data sample X with bounded covariance and linear prediction targets y with additive isotropic noise of finite variance. We present an iterative procedure to compute the optimal regularization strength numerically from the generative parameters in the fixed-X setting and prove its convergence at limited noise levels. Our experimental evaluation over synthetic data shows that the proposed procedure combined with sample-based parameter estimates attains near-optimal random-X generalization across a wide range of sample sizes, aspect ratios, and noise levels, at an added computational cost equivalent to one preliminary ridge regression in the underparameterized regime and two in the overparameterized case.