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What is Learnable in Valiant's Theory of the Learnable? Learning Perturbations to Extrapolate Your LLM Byzantine-Robust Distributed Sparse Learning Revisited The Sample Complexity of Multiple Change Point Identification under Bandit Feedback A proximal gradient algorithm for composite log-concave sampling Model-based Bootstrap of Controlled Markov Chains Approximation of Maximally Monotone Operators : A Graph Convergence Perspective Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces MIST: Reliable Streaming Decision Trees for Online Class-Incremental Learning via McDiarmid Bound A Spectral Framework for Closed-Form Relative Density Estimation Fast Rates for Offline Contextual Bandits with Forward-KL Regularization under Single-Policy Concentrability Higher-Order Equilibrium Tracking for EM-Compressible Online Estimation Scaling Limits of Long-Context Transformers A Note on Non-Negative $L_1$-Approximating Polynomials Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning Linear Response Estimators for Singular Statistical Models Statistical inference with belief functions: A survey Robust stochastic first order methods in heavy-tailed noise via medoid mini-batch gradient sampling Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexity Adaptive auditing of AI systems with anytime-valid guarantees Locally Near Optimal Piecewise Linear Regression in High Dimensions via Difference of Max-Affine Functions Risk-Controlled Post-Processing of Decision Policies Covariate Balancing and Riesz Regression Should Be Guided by the Neyman Orthogonal Score in Debiased Machine Learning A Unified Pair-GRPO Family: From Implicit to Explicit Preference Constraints for Stable and General RL Alignment Time-Inhomogeneous Preconditioned Langevin Dynamics A Fine-Grained Understanding of Uniform Convergence for Halfspaces CITE: Anytime-Valid Statistical Inference in LLM Self-Consistency Ratio-based Loss Functions Optimal Confidence Band for Kernel Gradient Flow Estimator A renormalization-group inspired lattice-based framework for piecewise generalized linear models
A Geometrical Analysis of Kernel Ridge Regression and its...
Georgios Gavrilopoulos, Guillaume Lecué, Zong Shang · 2024-04-11 · via math.ST updates on arXiv.org

We obtain upper bounds for the estimation error of Kernel Ridge Regression (KRR) for all non-negative regularization parameters, offering a geometric perspective on various phenomena in KRR. As applications: 1. We address the multiple descent problem, unifying the proofs of arxiv:1908.10292 and arxiv:1904.12191 for polynomial kernels and we establish multiple descent for the upper bound of estimation error of KRR under sub-Gaussian design and non-asymptotic regimes. 2. For a sub-Gaussian design vector and for non-asymptotic scenario, we prove a one-sided isomorphic version of the Gaussian Equivalent Conjecture. 3. We offer a novel perspective on the linearization of kernel matrices of non-linear kernel, extending it to the power regime for polynomial kernels. 4. Our theory is applicable to data-dependent kernels, providing a convenient and accurate tool for the feature learning regime in deep learning theory. 5. Our theory extends the results in arxiv:2009.14286 under weak moment assumption. Our proof is based on three mathematical tools developed in this paper that can be of independent interest: 1. Dvoretzky-Milman theorem for ellipsoids under (very) weak moment assumptions. 2. Restricted Isomorphic Property in Reproducing Kernel Hilbert Spaces with embedding index conditions. 3. A concentration inequality for finite-degree polynomial kernel functions.