Optimal Rates for the Regularized Learning Algorithms under General Source Condition

Rastogi, Abhishake; Sampath, Sivananthan

doi:10.3389/fams.2017.00003

Cited by 21 publications

(25 citation statements)

References 29 publications

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“…Proof of estimate (4.13). We use arguments similar to those appearing in [5,24,16,7]. The treatment of generalized source conditions has been considered in [24] and we use arguments very close in spirit to that reference, deriving here inequalities that are tailored for our needs.…”

Section: Bounds In the Supervised Learning Settingmentioning

confidence: 99%

Lepskii Principle in Supervised Learning

Blanchard,

Mathé,

Mücke

2019

Preprint

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In the setting of supervised learning using reproducing kernel methods, we propose a data-dependent regularization parameter selection rule that is adaptive to the unknown regularity of the target function and is optimal both for the least-square (prediction) error and for the reproducing kernel Hilbert space (reconstruction) norm error. It is based on a modified Lepskiȋ balancing principle using a varying family of norms.

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Section: Bounds In the Supervised Learning Settingmentioning

confidence: 99%

Lepskii Principle in Supervised Learning

Blanchard,

Mathé,

Mücke

2019

Preprint

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“…, Theorem 2.1 share the upper convergence rates with the lower convergence rates of Theorem 3.11, 3.12 [44]. Therefore the choice of parameters is optimal.…”

Section: Convergence Analysismentioning

confidence: 99%

Error Estimates for Multi-Penalty Regularization Under General Source Condition

Rastogi

2017

Computer Science &Amp; Information Technology (CS &Amp; IT)

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In learning theory, the convergence issues of the regression problem are investigated with the least square Tikhonov regularization schemes in both the RKHS-norm and the L 2 -norm.We consider the multi-penalized least square regularization scheme under the general source condition with the polynomial decay of the eigenvalues of the integral operator. One of the motivation for this work is to discuss the convergence issues for widely considered manifold regularization scheme. The optimal convergence rates of multi-penalty regularizer is achieved in the interpolation norm using the concept of effective dimension. Further we also propose the penalty balancing principle based on augmented Tikhonov regularization for the choice of regularization parameters. The superiority of multi-penalty regularization over single-penalty regularization is shown using the academic example and moon data set.

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“…Effective dimension. Now we introduce the concept of the effective dimension which is an important ingredient to derive the rates of convergence rates [7,10,14,20,22,30]. The effective dimension is defined as…”

Section: 3mentioning

confidence: 99%

Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

Rastogi¹

2020

Communications on Pure &Amp; Applied Analysis

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In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.

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Optimal Rates for the Regularized Learning Algorithms under General Source Condition

Cited by 21 publications

References 29 publications

Lepskii Principle in Supervised Learning

Lepskii Principle in Supervised Learning

Error Estimates for Multi-Penalty Regularization Under General Source Condition

Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

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