A Simple Approximation to the Distribution of the Ridge Regression Estimator

We present a simple Gaussian approximation to the finite-sample distribution of the classical ridge regression estimator. Our approximation captures the fact that, in finite samples, the ridge regression estimator trades off bias and variance to reduce estimation and prediction error. Our approximation is based on nonstandard asymptotics where $i)$ we let the estimator's regularization parameter grow proportionally to the sample size; and $ii)$ we treat the population regression coefficients as \emph{local} to the reference vector that defines the estimator's direction of shrinkage. In contrast to other asymptotic approximations in the literature, we allow for general forms of heteroskedasticity and autocorrelation in the data generating process (at the cost of considering a low-dimensional model where the number of covariates is not allowed to grow with the sample size). We use our simple Gaussian approximation to propose two new strategies to select the regularization parameter for the ridge regression estimator. The suggested strategies select the regularization parameter to minimize either average or worst-case excess prediction risk, where risk is computed using our suggested Gaussian approximation.

Prediction-Only Distillation in Linear and Logistic Regression

Self-distillation (SD) is typically studied when the student is retrained on the teacher's original training inputs. In many practical deployments, however, the labeled training data are no longer available, and one has access only to the trained predictor and fresh unlabeled covariates. We study SD in this prediction-only regime through a fresh-X prediction-mixed scheme: a pure-distilled student is trained on fresh covariates pseudo-labeled by the teacher, and the final predictor is an affine combination of the teacher and student predictions. For ridge regression under proportional asymptotics, we derive deterministic equivalents for the optimally mixed prediction risk under general anisotropic covariance and deterministic signal. We show that this risk is strictly smaller than the teacher risk for almost every pair of teacher and student regularization levels, including when the fresh covariates are out-of-distribution and even when their covariance is isotropic. We further show that the optimal mixing weight cannot be identified from unlabeled data alone, but can be consistently estimated in a single post-training step using a small independent labeled calibration set, without additional model fitting. Finally, for binary logistic regression, we show that prediction mixing can outperform both the teacher and the pure-distilled classifier.

Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization

Bernstein--Schur kernels are products of a finite-feature kernel and a completely monotone shift-invariant kernel: nonstationary kernels falling between the shift-invariant and dot-product templates random features exploit, so neither Bochner sampling nor polynomial sketching applies to the full kernel directly. We give one random-feature construction for the whole class that randomizes both factors: it sketches the finite modulation and samples the radial factor's one-dimensional Bernstein--Widder scale before applying Gaussian random Fourier features, giving feature dimension $Dm$, free of the $O(d^2)$ size of the exact modulation feature. With the modulation kept exact (the $m\to\infty$ limit), we prove unbiasedness, an exact variance, and a matrix-Bernstein operator-norm bound controlled by the top kernel and modulation eigenvalues and an intrinsic dimension rather than the crude $N\max_{ij}$ route. Whitening this argument at the ridge makes the effective dimension $d_{\mathrm{eff}}(λ)$ the \emph{exact} intrinsic dimension of the matrix variance, so $O((1+\|P\|_{\mathrm{op}}/λ)\log(d_{\mathrm{eff}}/δ))$ radial draws preserve the kernel-ridge solution; tilting the draw by a closed-form whitened leverage improves this to the effective-dimension count $O((1+d_{\mathrm{eff}})\log(d_{\mathrm{eff}}/δ))$. Conditioning on the sketch carries every guarantee to the deployed doubly-randomized estimator up to one additive sketch term, and all hold for the whole class with the modulation Gram in place of the polynomial one. The flagship instance is the biased $yat$-kernel $k_{yat,b}(w,x)=(w^\top x+b)^2/(\|w-x\|^2+\varepsilon)$, whose family span contains the inverse-multiquadric kernel by finite differences in $b$.