Optimal Rates for Regularization Operators in Learning Theory

By Andrea Caponnetto. 2006

40 citations recorded by the source corpus.

Abstract : We develop some new error bounds for learning algorithms induced by regularization methods in the regression setting. The hardness of the problem is characterized in terms of the parameters r and s, the first related to the complexity of the target function, the second connected to the effective dimension of the marginal probability measure over the input space. We show, extending previous results, that by a suitable choice of the regularization parameter as a function of the number of the available examples, it is possible attain the optimal minimax rates of convergence for the expected squared loss of the estimators, over the family of priors fulfilling the constraint r + s > 1/2. The setting considers both labelled and unlabelled examples, the latter being crucial for the optimality results on the priors in the range r < 1/2 .

Subfields: Computational Mechanics, Mathematical Physics, Statistics and Probability

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Citations in the corpus

Learning Sets with Separating Kernels (2012, methodology): “The a priori conditions we consider in the present paper are given by the following two assumptions, which involve both the reproducing kernel K and the probability measure ρ (compare with [12, 11]).”
Multi-output learning via spectral filtering (2012, methodology): “In fact, if the kernel is universal (Caponnetto et al. 2008) such an assumption can be dropped and (universal) Bayes consistency can be proven with similar techniques (Caponnetto 2006).”
Spectral Algorithms for Supervised Learning (2008, methodology): “…the theoretical analysis proposed in Bauer, Pereverzev, and Rosasco (2007), De Vito, Rosasco, and Verri (2005), Caponnetto and De Vito (2007), and Caponnetto (2006), showing that many regularization methods originally proposed in the context of inverse problems give rise to consistent kernel…”
Early stopping for non-parametric regression: An optimal data-dependent stopping rule (2011, methodology): “A line of past work [42, 7, 13, 11, 12, 37]) has analyzed stopping rules based on this type of holdout rule.”