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math.ST updates on arXiv.org

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 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Non-asymptotic Bayesian Minimax Adaptation
Keisuke Yano, Fumiyasu Komaki · 2016-09-04 · via math.ST updates on arXiv.org

This paper studies a Bayesian approach to non-asymptotic minimax adaptation in nonparametric estimation. Estimating an input function on the basis of output functions in a Gaussian white-noise model is discussed. The input function is assumed to be in a Sobolev ellipsoid with an unknown smoothness and an unknown radius. Our purpose in this paper is to present a Bayesian approach attaining minimaxity up to a universal constant without any knowledge regarding the smoothness and the radius. Our Bayesian approach provides not only a rate-exact minimax adaptive estimator in large sample asymptotics but also a risk bound for the Bayes estimator quantifying the effects of both the smoothness and the ratio of the squared radius to the noise variance, where the smoothness and the ratio are the key parameters to describe the minimax risk in this model. Application to non-parametric regression models is also discussed.

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