# Fixed-Gaussian Spectral Algorithms: Minimax Optimal Rates for Misspecified Learning and Transfer

arXiv: https://arxiv.org/abs/2501.10870 · HTML: https://mlsanity.com/paper/2501.10870/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 85 references that are in the map.
- First posted on arXiv 18 Jan 2025.

## Builds on (most cited first, 30 of 45)

- [How transferable are features in deep neural networks?](https://mlsanity.com/paper/1411.1792/) (2014)
- [A Comprehensive Survey on Transfer Learning](https://mlsanity.com/paper/1911.02685/) (2019)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Transfer Learning for High-dimensional Linear Regression: Prediction, Estimation, and Minimax Optimality](https://mlsanity.com/paper/2006.10593/) (2020)
- [Transfer Learning under High-dimensional Generalized Linear Models](https://mlsanity.com/paper/2105.14328/) (2021)
- [Statistical Optimality of Stochastic Gradient Descent on Hard Learning Problems through Multiple Passes](https://mlsanity.com/paper/1805.10074/) (2018)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates](https://mlsanity.com/paper/1305.5029/) (2013)
- [Marginal Singularity, and the Benefits of Labels in Covariate-Shift](https://mlsanity.com/paper/1803.01833/) (2018)
- [Transfer Learning for Nonparametric Classification: Minimax Rate and Adaptive Classifier](https://mlsanity.com/paper/1906.02903/) (2019)
- [Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces](https://mlsanity.com/paper/1801.06720/) (2018)
- [Optimally tackling covariate shift in RKHS-based nonparametric regression](https://mlsanity.com/paper/2205.02986/) (2022)
- [Distributed learning with regularized least squares](https://mlsanity.com/paper/1608.03339/) (2016)
- [Near-Optimal Linear Regression under Distribution Shift](https://mlsanity.com/paper/2106.12108/) (2021)
- [Transfer Learning in Large-scale Gaussian Graphical Models with False Discovery Rate Control](https://mlsanity.com/paper/2010.11037/) (2020)
- [On the Optimality of Misspecified Spectral Algorithms](https://mlsanity.com/paper/2303.14942/) (2023)
- [Hypothesis Transfer Learning via Transformation Functions](https://mlsanity.com/paper/1612.01020/) (2016)
- [A Generalized Neyman-Pearson Criterion for Optimal Domain Adaptation](https://mlsanity.com/paper/1810.01545/) (2018)
- [Optimal Convergence for Distributed Learning with Stochastic Gradient Methods and Spectral Algorithms](https://mlsanity.com/paper/1801.07226/) (2018)
- [Transfer Learning for Nonparametric Regression: Non-asymptotic Minimax Analysis and Adaptive Procedure](https://mlsanity.com/paper/2401.12272/) (2024)
- [Beating SGD Saturation with Tail-Averaging and Minibatching](https://mlsanity.com/paper/1902.08668/) (2019)
- [Minimax optimal approaches to the label shift problem in non-parametric settings](https://mlsanity.com/paper/2003.10443/) (2020)
- [Spectral Regularized Kernel Two-Sample Tests](https://mlsanity.com/paper/2212.09201/) (2022)
- [On the Optimality of Misspecified Kernel Ridge Regression](https://mlsanity.com/paper/2305.07241/) (2023)
- [A linear adjustment based approach to posterior drift in transfer learning](https://mlsanity.com/paper/2111.10841/) (2021)
- [Pseudo-Labeling for Kernel Ridge Regression under Covariate Shift](https://mlsanity.com/paper/2302.10160/) (2023)
- [Fast Rates by Transferring from Auxiliary Hypotheses](https://mlsanity.com/paper/1412.1619/) (2014)
- [Doubly Robust Augmented Model Accuracy Transfer Inference with High Dimensional Features](https://mlsanity.com/paper/2208.05134/) (2022)
- [On the Saturation Effect of Kernel Ridge Regression](https://mlsanity.com/paper/2405.09362/) (2024)

## Built on by (most cited first, 2 of 2)

- [Transfer Learning Through Conditional Quantile Matching](https://mlsanity.com/paper/2602.02358/) (2026)
- [Black-Box Assisted Regression: Phase Transitions and Minimax Optimality](https://mlsanity.com/paper/2606.25743/) (2026)

## Abstract (arXiv)

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.