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Nonasymptotic analysis of robust regression with modified Huber's loss
Affiliation:1. Department of Mathematical and Statistical Sciences, University of Alberta Edmonton, Alberta T6G 2G1, Canada;2. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada;3. Department of Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK;4. University of South Carolina, 1523 Greene St., Columbia SC, 29208, USA;5. Moscow Center for Fundamental and Applied Mathematics, Russian Federation;6. Steklov Institute of Mathematics, Russian Federation;7. Lomonosov Moscow State University, Russian Federation;8. Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra (Barcelona), Spain;9. ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain;10. Universitat Autònoma de Barcelona, Spain;1. KAIST, School of Computing, Daejeon, Republic of Korea;2. LIX, CNRS, École Polytechnique, Institute Polytechnique de Paris, France;1. School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan;2. Graduate School of Advanced Science and Engineering, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima City, Hiroshima 739-8526, Japan
Abstract:To achieve robustness against the outliers or heavy-tailed sampling distribution, we consider an Ivanov regularized empirical risk minimization scheme associated with a modified Huber's loss for nonparametric regression in reproducing kernel Hilbert space. By tuning the scaling and regularization parameters in accordance with the sample size, we develop nonasymptotic concentration results for such an adaptive estimator. Specifically, we establish the best convergence rates for prediction error when the conditional distribution satisfies a weak moment condition.
Keywords:Nonparametric regression  Nonasymptotic analysis  Reproducing kernel Hilbert space  Concentration inequality  Empirical risk minimization
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