New algorithm improves regression error bounds and accelerates performance for low noise.
problem Nonparametric least square regression in RKHS with optimal error bounds.
method Kernel Truncated Randomized Ridge Regression (KTRRR) with optimal generalization error bounds.
result Faster finite-time and asymptotic rates on low noise problems.
TKRR improves KRR performance by aligning target functions with kernels.
problem Improving kernel ridge regression performance through target alignment.
method Focuses on truncated kernel ridge regression (TKRR) with an additional spectral truncation parameter.
result TKRR can achieve faster rates than full KRR, reaching parametric rates.
Spectrally-truncated KRR outperforms full KRR for large data.
problem Computational intensity of KRR for large datasets.
method Spectrally truncating the kernel matrix to its largest r eigenvalues. result Spectrally-truncated KRR can outperform full KRR for all finite samples above a threshold.
Improved kernel ridge regression for large datasets using weighted random binning.
problem Efficiently approximating kernel matrices for large-scale datasets.
method Introduced weighted random binning features for locality sensitive hashing.
result Weighted random binning features generate Gaussian processes of any desired smoothness.
Bayes-optimal learning of deep random networks with Gaussian weights is studied.
problem Learning a target function corresponding to a deep, extensive-width, non-linear neural network with random Gaussian weights.
method Closed-form expressions for Bayes-optimal test error, ridge regression, kernel and random features regression are computed.
result Optimally regularized ridge regression and kernel regression achieve Bayes-optimal performances, while logistic loss yields a near-optimal test error for classification.
ParK efficiently solves kernel ridge regression for large datasets.
problem Large-scale kernel ridge regression efficiency and accuracy.
method Partitioning feature space with random projections and iterative optimization.
result Provably maintains statistical accuracy with reduced space and time complexity.
Random Fourier features is one of the most popular techniques for scaling up kernel methods, such as kernel ridge regression. However, despite impressive empirical results, the statistical properties of random Fourier features are still not well understood. In this paper we take steps toward filling this gap. Specifica…
Kernel ridge regression for causal inference with missing data.
problem Estimating treatment effects with missing data in selected samples.
method Kernel ridge regression estimators for nonparametric dose response curves and semiparametric treatment effects.
result Uniform consistency and finite sample rates for continuous treatment, root-n consistency for discrete treatment.
This paper improves computational efficiency in kernel ridge regression under covariate shift.
problem Covariate shift in nonparametric regression.
method Random projections in RKHS to reduce computational demands.
result Significant computational savings can be achieved without compromising learning performance under covariate shift.
We analyzed optimism in linear and kernel regression models.
problem Understanding predictive complexity in regression models.
method Derived closed-form asymptotic optimism for linear and kernel regression models.
result Scaled optimism is a useful measure for model complexity.
In this paper, we propose a random projection approach to estimate variance in kernel ridge regression. Our approach leads to a consistent estimator of the true variance, while being computationally more efficient. Our variance estimator is optimal for a large family of kernels, including cubic splines and Gaussian ker…
RF models implicitly regularize kernel methods as feature count increases.
problem Understanding implicit regularization in RF models.
method Random matrix theory applied to Gaussian RF models and KRR.
result The average RF predictor is close to a KRR predictor with an effective ridge.
Two methods solve kernel ridge regression problems efficiently.
problem Solving kernel ridge regression problems with large datasets.
method RPCholesky and KRILL preconditioning techniques.
result Efficient solutions to KRR problems with strong guarantees.
Random features and KRR generalize similarly when N is large enough.
problem Understanding the generalization error of random features and KRR methods.
method Analyzing spectral conditions and hypercontractivity on kernel eigenfunctions.
result The test error of random features is larger than KRR when N is small, but they achieve the same error when N is large.
Study shows minimizing the norm of the ERM solution stabilizes kernel ridge-less regression.
problem Stability of kernel ridge-less regression.
method Minimizing the norm of the ERM solution to minimize CV stability.
result Interpolating solution with minimum norm minimizes CV stability.
This paper carries out a large dimensional analysis of a variation of kernel ridge regression that we call \emph{centered kernel ridge regression} (CKRR), also known in the literature as kernel ridge regression with offset. This modified technique is obtained by accounting for the bias in the regression problem resulti…
Analyzes neural networks using linear models to understand their behavior.
problem Understanding multi-layer neural networks through linear models.
method Recalls and reviews four models: linear regression with concentrated features, kernel ridge regression, random feature model, and neural tangent model.
result Highlights limitations of linear theory and discusses approaches to overcome them.
Paper proposes robust estimators for heavy-tailed data with infinite variance.
problem Developing robust estimators for heavy-tailed data with infinite variance.
method Proposes two robust estimators: ridge log-truncated M-estimator and elastic net log-truncated M-estimator.
result Demonstrates robustness of log-truncated estimations over standard estimations through simulations and real data analysis.
Kernel ridge regression imputation with consistent variance estimation for handling missing data.
problem Handling missing data in statistical analysis.
method Kernel ridge regression imputation combined with entropy method for variance estimation.
result Root-n consistency of the imputation estimator in a Sobolev space setting.
Novel algorithm identifies nonlinear Granger causal relationships using kernel ridge regression.
problem Identification of nonlinear Granger causal relationships.
method Flexible plug-in architecture with kernel ridge regression using radial basis function.
result Kernel ridge regression in mlcausality achieves competitive AUC scores and more finely calibrated p-values.
pGMM kernel outperforms ordinary ridge regression and RBF kernel ridge regression without tuning.
problem Comparing pGMM kernel regression with other ridge regression methods.
method Implemented and compared pGMM kernel regression with ordinary ridge regression and RBF kernel ridge regression.
result pGMM kernel performs well without tuning and can match boosted trees with parameter tuning.
Generalizes leverage score sampling for neural networks, accelerating kernel methods and deep learning.
problem Accelerating kernel methods and deep learning training.
method Generalizes leverage score sampling to neural networks and proves equivalence to neural tangent kernel ridge regression.
result Equivalence between regularized neural network and neural tangent kernel ridge regression under leverage score sampling initialization.
New decentralized KRR algorithm adapts to node-specific data.
problem Consistent node-specific data in decentralized KRR.
method Data-dependent random features for adaptive RF generation.
result Average regression accuracy improved by 25.5% across six datasets.
Paper proves convergence rates for Gaussian kernel ridge regression.
problem Understanding convergence rates for Gaussian kernel ridge regression.
method Establishes polynomial convergence rates for KRR with fixed hyperparameters.
result First polynomial convergence rates for Gaussian kernel ridge regression.
New method shows supervised learning can mimic unsupervised learning effectively.
problem The fundamental difference between supervised and unsupervised learning.
method A two-stage procedure where unsupervised model selection is followed by adding outputs without changing parameters.
result Asymptotic out-of-sample risk bounds for various models trained without access to labels.
Improves kernel ridge regression by optimizing scale and feature parameters.
problem Kernel ridge regression with fixed kernel.
method Introduces a matrix parameter U to optimize scale and feature parameters.
result Solves a nonlinear variational problem to optimize U.
A new sampling strategy for random Fourier features reduces computation time and improves prediction performance.
problem Efficient generation of random Fourier features for kernel approximation.
method Surrogate leverage weighted sampling guided by kernel alignment, avoiding matrix inversion.
result Time complexity reduced from O(ns^2+s^3) to O(ns^2), comparable or slightly better prediction performance.
Study shows RFRR's effectiveness with nearly orthogonal data in overparameterized settings.
problem Understanding the effectiveness of random feature regression with nearly orthogonal data.
method Investigates RFRR with nearly orthogonal deterministic unit-length input data vectors in the overparameterized regime.
result Shows high-probability non-asymptotic concentration results for RFRR's training, cross-validation, and generalization errors.
Proposes landmark selection for kernel methods.
problem Selecting important landmarks from large training sets.
method Deterministic and randomized adaptive algorithm for landmark selection.
result Landmarks are related to the minima of kernelized Christoffel functions.
New method approximates complex kernel norms with random features, making learning tractable.
problem Complexity of learning with kernel methods in high dimensions.
method Random features approximations to Fp norms, focusing on p>1. result For p>1, the number of random features required is polynomial in the sample size, making learning tractable. Kernel methods are an extremely popular set of techniques used for many important machine learning and data analysis applications. In addition to having good practical performances, these methods are supported by a well-developed theory. Kernel methods use an implicit mapping of the input data into a high dimensional f…
New findings on kernel regression in the quadratic regime, improving understanding of machine learning models.
problem Understanding kernel ridge regression in the quadratic asymptotic regime.
method Extended study of kernel regression to the quadratic regime, establishing approximation bounds and spectral distributions.
result Broad class of inner-product kernels exhibit behavior similar to a quadratic kernel, with precise asymptotic training and test errors characterized.
Kernel ridge regression (KRR) is a standard method for performing non-parametric regression over reproducing kernel Hilbert spaces. Given n samples, the time and space complexity of computing the KRR estimate scale as O(n3) and O(n2) respectively, and so is prohibitive in many cases. We prop…
Kernel ridge regression inference for nonstandard data.
problem Inferential theory for kernel ridge regression with nonstandard data.
method Constructs valid and sharp confidence sets using anti-symmetric multipliers.
result Develops a test for match effects in school matching mechanisms.
Paper analyzes distributed learning with non-i.i.d. samples.
problem Learning rate analysis for distributed kernel ridge regression with dependent samples.
method Integral operator approach and covariance inequality for strong mixing sequences.
result Derives optimal learning rates for distributed kernel ridge regression.
Ridge regression performs optimally in noisy environments with heavy-tailed distributions.
problem Performance of ridge regression in noisy environments with heavy-tailed noise.
method Established excess risk bounds using integral operator framework and Fuk-Nagaev inequality.
result Ridge regression achieves optimal convergence rates under heavy-tailed noise, demonstrating robustness.
Estimates KRR risk from training data for various kernels and hyperparameters.
problem Predicting the generalization error of Kernel Ridge Regression.
method Introduces SCT and KARE to approximate KRR risk from training data.
result KARE provides an excellent approximation of KRR risk and helps select good kernels.
Kernel balancing weights are generalized as KRRR, providing better confidence intervals for treatment effects.
problem Lack of generalization error, correct feature specification, and limited to average effects.
method Interpreting kernel balancing weights as KRRR, relaxing feature specification, and extending Gaussian approximation.
result KRRR provides strong generalization properties and justifies confidence sets for causal functions.
Sharp bounds derived for test error of finite-rank kernel ridge regression.
problem Loose bounds on test error for finite-rank kernels in machine learning.
method Sharp non-asymptotic upper and lower bounds for KRR test error.
result Tighter bounds on finite-rank KRR test error, valid for any regularization parameters.
New bounds for RFM-KRR with weak assumptions and easy verification.
problem Establishing accurate out-of-sample bounds for RFM-KRR.
method Elementary linear algebra and weak assumptions.
result Novel out-of-sample error upper and lower bounds with weak assumptions.
Short proof shows how ridge regression works with random data.
problem Understanding prediction error in ridge regression with random design.
method Combination of exchangeability arguments, matrix perturbation, and operator convexity.
result Elementary proof of prediction error without complex inequalities.
A new method for faster bandwidth selection in Gaussian kernel ridge regression.
problem Efficiently selecting the bandwidth in Gaussian kernel ridge regression.
method Formulated an approximate Jacobian expression for bandwidth selection, proposing a closed-form heuristic.
result Our method is as accurate as cross-validation and marginal likelihood maximization but up to six orders of magnitude faster.
The paper extends kernel ridge regression to product kernels and reveals new convergence behaviors.
problem Understanding kernel ridge regression in large dimensions with various kernels.
method Established a broad family of large dimensional kernels and derived convergence rates.
result Revealed new phenomena including minimax optimality, saturation effect, and multiple descent behavior.
Many machine learning problems can be formulated as predicting labels for a pair of objects. Problems of that kind are often referred to as pairwise learning, dyadic prediction or network inference problems. During the last decade kernel methods have played a dominant role in pairwise learning. They still obtain a stat…
Improved kernel ridge regression using conjugate gradients.
problem Efficiently solving large-scale kernel ridge regression problems.
method Structured Gaussian regression model with low-rank approximation and conjugate gradients.
result Enhanced approximation of kernel ridge regressor/Gaussian process posterior mean.
The paper analyzes learning curves for kernel ridge regression with dot-product kernels.
problem Understanding the learning curves for different scaling regimes of data and model.
method Precise formulas for mean test error, bias, and variance in the mo∞ with m/dr constant regime. result A peak in the learning curve at m≈dr/r! for any integer r. We improve kernel ridge regression for skewed responses using oversampling and adaptive partitioning.
problem Kernel ridge regression struggles with skewed response variables, leading to poor estimates.
method Combines adaptive partitioning with oversampling to address skewed responses in kernel ridge regression.
result The proposed method yields estimates with smaller risk compared to classical methods under mild conditions.
The paper analyzes and improves the learning rates of distributed kernel ridge regression.
problem Generalization performance and learning rates of distributed kernel ridge regression.
method The paper derives optimal learning rates for DKRR in expectation and probability, proposes a communication strategy to improve learning performance, and evaluates these through theory and experiments.
result The communication strategy significantly improves the learning performance of DKRR, as demonstrated by both theoretical assessments and numerical experiments.