ESTIMATING THE APPROXIMATION ERROR IN LEARNING THEORY

By Steve Smale, Ding-Xuan Zhou. 2003

285 citations recorded by the source corpus.

Subfields: Mathematical Physics, Numerical Analysis, Statistics and Probability

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Citations in the corpus

On the mathematical foundations of learning (2001, methodology): “The definition of H s (X) can be extended to s ∈ R, s ≥ 0, by using a Fourier transform argument (see also [38]).”
Quasi-oracle estimation of heterogeneous treatment effects (2017, background): “Their argument for choosing the optimal value of c n to use in (23) leverages the approximation error bound established by Smale and Zhou (2003) given Assumption 3.”
Ranking and empirical minimization of U-statistics (2006, methodology): “For the approximation properties of such kernel classes we refer the reader to Cucker and Smale [14], Scovel and Steinwart [36], Smale and Zhou [38], Steinwart [39] etc.”
Optimal regression rates for SVMs using Gaussian kernels (2013, background): “Note that one of the reasons for these learning rates is the fact that the approximation error decays polynomially fast if (and only if) the regression function f L,P is contained in some interpolation space [L2(X),W m 2 (X)]β,∞, m ∈ Z+, β ∈ (0, 1), see [26] for more details.”
Regularized Policy Iteration with Nonparametric Function Spaces (2016, background): “The behavior of the function approximation error for certain classes of " small " RKHS has been discussed by Smale and Zhou (2003); Steinwart and Christmann (2008).”
Regularization in reinforcement learning (2011, background): “The behavior of the function approximation error for certain classes of “small” RKHS has been discussed by Smale and Zhou [2003]; Steinwart and Christmann [2008].”
Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs (2012, background): “What happens with the images of other fractional powers T β k of Tk? It turns out, see [34, 14], that (4) can be used to show that these images can be closely, though not exactly, described by the real interpolation spaces [L2(ν), [H]∼]p,∞, where [H]∼ denotes the RKHS H when considered as a subset o”
Learning the Kernel Function via Regularization (2005, background): “…of effective error bounds and stability analysis relative to perturbations ofthe data (see, for example, Bousquet and Elisseeff, 2002; Cucker and Smale, 2002; Mukherjee et al., in press; Scovel and Steinwart, 2004; Smale and Zhou, 2003; Vapnik, 1998; Ying and Zhou, 2004; Zhang, 2004; Zhou, 2002).”
Shannon sampling and function reconstruction from point values (2004, background): “One may study the error even for f∗ to be outside HK , as done for the approximation error in [22], [26].”
Capacity of reproducing kernel spaces in learning theory (2003, methodology): “a Mercer kernel on , the RKHS can be embedded into the Sobolev space for any , using the results on approximation errors given by Smale and Zhou in [18].”
A note on application of integral operator in learning theory (2009, methodology): “By the discussion in [12], it is in fact equivalent to the assumption L−β K fρ ∈ LρX (X) which has been used in [13,16] and can be characterized by interpolation spaces.”
Shannon sampling II: Connections to learning theory (2005, result): “We follow the same line as we did in [11].”
Fast cross-validation via sequential testing (2012, background): “Approximation errors have been studied for example in the papers by Smale and Zhou (2003) and Steinwart and Scovel (2007), but the papers only prove upper bounds, and the rates are also worst-case rates which are likely not close enough to the true errors.”
Generalization Properties of Learning with Random Features (2016, background): “Condition (13) allows to control the bias of the estimator and is common in approximation theory [28].”