# An Identity For Kernel Ridge Regression

This paper describes an approach to prove theoretical guarantees on online kernelized Ridge Regression in the form of equality. It approaches the results described in Zhdanov and Vovk, 2009, from the probabilistic perspective.

The method of kernel Ridge Regression with a kernel and a real regularisation parameter suggests the function , where is the column vector of outcomes,

is the kernel matrix and

**Theorem.** Take a kernel on a domain and a parameter . Let
be the RKHS for the kernel . For a sample
let
be the predictions output by ridge
regression with the kernel and the parameter in the
on-line mode. Then

where .

The idea of the proof is very elegant: the density of the joint distribution of the variables at the point is calculated in three different ways: by decomposing the density into a chain of conditional densities, marginalisation, and, finally, direct calculation. Each method will give us a different expression corresponding to a term in the identity. Since all the three terms express the same density, they must be equal.

The following corollary is also proven on the cumulative square loss of the clipped kernel Ridge Regression.

**Corollary.**
Take a kernel on a domain and a parameter . Let
be the RKHS for the kernel . For a sample
such that for
all , let
be the
predictions output by clipped ridge regression with the kernel
and the parameter in the on-line mode. Then

where is as above.

- Fedor Zhdanov and Vladimir Vovk. Competing with Gaussian linear experts. Technical report, arXiv:0910.4683 [cs.LG], arXiv.org e-Print archive, 2009.
- Fedor Zhdanov and Yuri Kalnishkan. An identity for kernel ridge regression. In Proceedings of the 21st International Conference on Algorithmic Learning Theory, 2010.