Generalization Performance Of Kernelized Elastic Net Regularization Algorithm

Kernelized elastic net regularization(KENReg)is a kernelization of the well-known elastic net regularization.KENReg has some good properties such as gener-alization,sparseness,stability.We are more concerned about the generalization per-formance,The appearance of KENReg not only can solve the problems that can not be solved under some classical system,but also can improve the performance of the learning algorithm greatly.KENReg combines the Lasso and ridge regression,which not only can overcomes the limitations of the Lasso and Ridge,but also can make use of their advantages.Therefore,it makes sense to study KENReg.So far,a lot of work about KENReg was based on the assumption that the ob-served data was independent and identically.However,the assumption of independent is a restrictive concept,there is two reasons:On the one hand,it is difficult to justify independent samples in real-world applications;On the other hand,in many statisti-cal learning applications,for example,system diagnosis,speech recognition and web search,the samples are dependent.Therefore,it is necessary to generalize the indepen-dent and identically process to dependent.In this paper,we study KENReg based on dependent observations:First,we derive the error analysis of KENReg based on expo-nentially strongly mixing observations,and we get generalization bounds and learning rate.In reality,it is difficult to get exponentially strongly mixing observations,by contrast,we can obtain uniformly ergodic Markov chain easier.Therefore,We can generalize samples from exponentially strongly mixing observations to uniformly er-godic Markov chain,and we get generalization bounds and learning rate of KENReg based on uniformly ergodic Markov chain.Exponentially strongly mixing and uniformly ergodic Markov chain are weaker than independent and identically observations.In this paper,the learning rate of KEN-Reg for exponentially strongly mixing observations and uniformly ergodic Markov chain is close to m-1,which is the same as the result based on independent and i-dentically observations.Compared to previous research work,we can get better results of KENReg based on dependent.