Conditional quantiles with varying Gaussians

Convex risk minimization is a commonly used setting in learning theory. In this paper, we firstly give a perturbation analysis for such algorithms, and then we apply this result to differential private learning algorithms. Our analysis needs the objective functions to be strongly convex. This leads to an extension of our previous analysis to the non-differentiable loss functions, when constructing differential private algorithms. Finally, an error analysis is then provided to show the selection for the parameters.

show abstract

“…So we use the natural assumption on , which is for some and . This assumption is trivial in concrete algorithms; see [25–27], etc.…”

Section: Error Analysismentioning

confidence: 99%

Perturbation of convex risk minimization and its application in differential private learning algorithms

Nie

Wang

2017

J Inequal Appl

View full text Add to dashboard Cite

show abstract

“…It is unknown whether the above learning rate can be derived by existing approaches in the literature (e.g. [28,29,41,42,7]) even after projection. Note that the kernel in the above example is independent of the sample size.…”

Section: Comparison Of Learning Ratesmentioning

confidence: 99%

Learning rates for the risk of kernel-based quantile regression estimators in additive models

Christmann

Zhou

2016

Anal. Appl.

View full text Add to dashboard Cite

Additive models play an important role in semiparametric statistics. This paper gives learning rates for regularized kernel based methods for additive models. These learning rates compare favourably in particular in high dimensions to recent results on optimal learning rates for purely nonparametric regularized kernel based quantile regression using the Gaussian radial basis function kernel, provided the assumption of an additive model is valid. Additionally, a concrete example is presented to show that a Gaussian function depending only on one variable lies in a reproducing kernel Hilbert space generated by an additive Gaussian kernel, but does not belong to the reproducing kernel Hilbert space generated by the multivariate Gaussian kernel of the same variance.

show abstract

“…A family of kernel based learning algorithms for quantile regression has been widely studied in a large literature [1][2][3][4] and references therein. The form of the algorithms is a regularized scheme in a reproducing kernel Hilbert space H (RKHS, see [5] for details) associated with a Mercer kernel .…”

Section: The Pinball Lossmentioning

confidence: 99%

“…In [1,3,4], error analysis for general H has been done. Learning with varying Gaussian kernel was studied in [2]. ERM scheme (3) is very different from kernel based regularized scheme (4).…”

Section: The Pinball Lossmentioning

confidence: 99%