Organizations: Department of Statistics and Data Science, National University of Singapore. · Department of Statistics and Data Science, Tsinghua University.
Existing large-dimensional theory for spectral algorithms resolves either the optimally tuned point or the interpolation limit, but leaves the under-regularized regime unexplored. We study the learning curve and benign overfitting of spectral algorithms in the large-dimensional setting where the sample size and dimension are of comparable order, i.e., n≍dγ for some γ>0. We first consider inner-product kernels on the sphere Sd−1 and establish a sharp asymptotic characterization of the excess risk across the full regularization path under various source conditions s≥0, where s measures the relative smoothness of the regression function. Our results reveal that the learning curve is not simply U-shaped but instead consists of three distinct regimes: over-regularized, under-regularized, and interpolation regimes. This characterization allows us to fully capture the benign overfitting phenomenon, demonstrating that benign overfitting arises consistently across both the under-regularized and interpolation regimes whenever s is positive but no larger than a critical threshold. We further show that, in the sufficiently regularized regime, the kernel learning curve is recovered by an associated sequence model. Finally, we extend the learning-curve analysis to large-dimensional KRR for a class of kernels on general domains in Rd whose low-degree eigenspaces satisfy spectral-scaling and hyper-contractivity conditions.
The principal objective of this work is twofold within nonparametric regression settings: (1) to establish the minimax optimal convergence rates for fixed-bandwidth Gaussian kernel spectral algorithms when the true regression function resides in a Sobolev space, and (2) to apply Gaussian spectral algorithms for achieving robust and adaptive transfer learning under concept shift. While minimax optimality of misspecified spectral algorithms has been established, existing guarantees are typically restricted to the non-saturation regime. We demonstrate that the infinite smoothness of fixed-bandwidth Gaussian kernels provides universal robustness to model misspecification by showing that this kernel choice enables any spectral algorithm to attain minimax optimal rates, provided the regularization parameter decays exponentially. This result effectively decouples optimality from the algorithm's inherent qualification. Building on this, we then advocate Gaussian spectral algorithms as powerful components in a learning framework for robust and adaptive transfer. Specifically, we derive the adaptive convergence rate of the excess risk for this framework and show that the rates are optimal up to logarithmic factors. Our results also reveal the impact of the magnitude of the concept shift and the sample size on the generalization error.
Recent studies have reported saturation effects and multiple descent behavior in large dimensional kernel ridge regression (KRR). However, these findings are predominantly derived under restrictive settings, such as inner product kernels on sphere or strong eigenfunction assumptions like hypercontractivity. Whether such behaviors hold for other kernels remains an open question. In this paper, we establish a broad, new family of large dimensional kernels and derive the corresponding convergence rates of the generalization error. As a result, we recover key phenomena previously associated with inner product kernels on sphere, including: i) the minimax optimality when the source condition s≤1; ii) the saturation effect when s>1; iii) a periodic plateau phenomenon in the convergence rate and a multiple-descent behavior with respect to the sample size n.
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.