sentences
stringlengths 1
243k
|
---|
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 . |
*
key words and phrases . |
* additive model , kernel , quantile regression , semiparametric , rate of convergence , support vector machine . |
additive models @xcite provide an important family of models for semiparametric regression or classification . |
some reasons for the success of additive models are their increased flexibility when compared to linear or generalized linear models and their increased interpretability when compared to fully nonparametric models . |
it is well - known that good estimators in additive models are in general less prone to the curse of high dimensionality than good estimators in fully nonparametric models . |
many examples of such estimators belong to the large class of regularized kernel based methods over a reproducing kernel hilbert space @xmath0 , see e.g. |
@xcite . |
in the last years
many interesting results on learning rates of regularized kernel based models for additive models have been published when the focus is on sparsity and when the classical least squares loss function is used , see e.g. |
@xcite , @xcite , @xcite , @xcite , @xcite , @xcite and the references therein . |
of course , the least squares loss function is differentiable and has many nice mathematical properties , but it is only locally lipschitz continuous and therefore regularized kernel based methods based on this loss function typically suffer on bad statistical robustness properties , even if the kernel is bounded . |
this is in sharp contrast to kernel methods based on a lipschitz continuous loss function and on a bounded loss function , where results on upper bounds for the maxbias bias and on a bounded influence function are known , see e.g. |
@xcite for the general case and @xcite for additive models . |
therefore , we will here consider the case of regularized kernel based methods based on a general convex and lipschitz continuous loss function , on a general kernel , and on the classical regularizing term @xmath1 for some @xmath2 which is a smoothness penalty but not a sparsity penalty , see e.g. |
@xcite . |
such regularized kernel based methods are now often called support vector machines ( svms ) , although the notation was historically used for such methods based on the special hinge loss function and for special kernels only , we refer to @xcite . |
in this paper we address the open question , whether an svm with an additive kernel can provide a substantially better learning rate in high dimensions than an svm with a general kernel , say a classical gaussian rbf kernel , if the assumption of an additive model is satisfied . |
our leading example covers learning rates for quantile regression based on the lipschitz continuous but non - differentiable pinball loss function , which is also called check function in the literature , see e.g. |
@xcite and @xcite for parametric quantile regression and @xcite , @xcite , and @xcite for kernel based quantile regression . |
we will not address the question how to check whether the assumption of an additive model is satisfied because this would be a topic of a paper of its own . |
of course , a practical approach might be to fit both models and compare their risks evaluated for test data . |
for the same reason we will also not cover sparsity . |
consistency of support vector machines generated by additive kernels for additive models was considered in @xcite . |
in this paper
we establish learning rates for these algorithms . |
let us recall the framework with a complete separable metric space @xmath3 as the input space and a closed subset @xmath4 of @xmath5 as the output space . |
a borel probability measure @xmath6 on @xmath7 is used to model the learning problem and an independent and identically distributed sample @xmath8 is drawn according to @xmath6 for learning . |
a loss function @xmath9 is used to measure the quality of a prediction function @xmath10 by the local error @xmath11 . |
_ throughout the paper we assume that @xmath12 is measurable , @xmath13 , convex with respect to the third variable , and uniformly lipschitz continuous satisfying @xmath14 with a finite constant @xmath15 . |
_ support vector machines ( svms ) considered here are kernel - based regularization schemes in a reproducing kernel hilbert space ( rkhs ) @xmath0 generated by a mercer kernel @xmath16 . |
with a shifted loss function @xmath17 introduced for dealing
even with heavy - tailed distributions as @xmath18 , they take the form @xmath19 where for a general borel measure @xmath20 on @xmath21 , the function @xmath22 is defined by @xmath23 where @xmath24 is a regularization parameter . |
the idea to shift a loss function has a long history , see e.g. |
@xcite in the context of m - estimators . |
it was shown in @xcite that @xmath22 is also a minimizer of the following optimization problem involving the original loss function @xmath12 if a minimizer exists : @xmath25 the additive model we consider consists of the _ input space decomposition _
@xmath26 with each @xmath27 a complete separable metric space and a _ hypothesis space _
@xmath28 where @xmath29 is a set of functions @xmath30 each of which is also identified as a map @xmath31 from @xmath3 to @xmath5 . |
hence the functions from @xmath32 take the additive form @xmath33 . |
we mention , that there is strictly speaking a notational problem here , because in the previous formula each quantity @xmath34 is an element of the set @xmath35 which is a subset of the full input space @xmath36 , @xmath37 , whereas in the definition of sample @xmath8 each quantity @xmath38 is an element of the full input space @xmath36 , where @xmath39 . |
because these notations will only be used in different places and because we do not expect any misunderstandings , we think this notation is easier and more intuitive than specifying these quantities with different symbols . |
the additive kernel @xmath40 is defined in terms of mercer kernels @xmath41 on @xmath27 as @xmath42 it generates an rkhs @xmath0 which can be written in terms of the rkhs @xmath43 generated by @xmath41 on @xmath27 corresponding to the form ( [ additive ] ) as @xmath44 with norm given by @xmath45 the norm of @xmath46 satisfies @xmath47 to illustrate advantages of additive models , we provide two examples of comparing additive with product kernels . |
the first example deals with gaussian rbf kernels . |
all proofs will be given in section [ proofsection ] . |
[ gaussadd ] let @xmath48 , @xmath49 $ ] and @xmath50 ^ 2.$ ] let @xmath51 and @xmath52.\ ] ] the additive kernel @xmath53 is given by @xmath54 furthermore , the product kernel @xmath55 is the standard gaussian kernel given by @xmath56 define a gaussian function @xmath57 on @xmath58 ^ 2 $ ] depending only on one variable by @xmath59 then @xmath60 but @xmath61 where @xmath62 denotes the rkhs generated by the standard gaussian rbf kernel @xmath63 . |
the second example is about sobolev kernels . |
[ sobolvadd ] let @xmath64 , @xmath65 $ ] and @xmath58^s.$ ] let @xmath66 : = \bigl\{u\in l_2([0,1 ] ) ; d^\alpha u \in l_2([0,1 ] ) \mbox{~for~all~}|\alpha|\le 1\bigr\}\ ] ] be the sobolev space consisting of all square integrable univariate functions whose derivative is also square integrable . |
it is an rkhs with a mercer kernel @xmath67 defined on @xmath68 ^ 2 $ ] . |
if we take all the mercer kernels @xmath69 to be @xmath67 , then @xmath70 $ ] for each @xmath71 . |
the additive kernel @xmath72 is also a mercer kernel and defines an rkhs @xmath73\right\}.\ ] ] however , the multivariate sobolev space @xmath74^s)$ ] , consisting of all square integrable functions whose partial derivatives are all square integrable , contains discontinuous functions and is not an rkhs . |
denote the marginal distribution of @xmath6 on @xmath27 as @xmath75 . |
under the assumption that @xmath76 for each @xmath71 and that @xmath43 is dense in @xmath29 in the @xmath77-metric , it was proved in @xcite that @xmath78 in probability as long as @xmath79 satisfies @xmath80 and @xmath81 . |
the rest of the paper has the following structure . |
section [ ratessection ] contains our main results on learning rates for svms based on additive kernels . |
learning rates for quantile regression
are treated as important special cases . |
section [ comparisonsection ] contains a comparison of our results with other learning rates published recently . |
section [ proofsection ] contains all the proofs and some results which can be interesting in their own . |
in this paper we provide some learning rates for the support vector machines generated by additive kernels for additive models which helps improve the quantitative understanding presented in @xcite . |
the rates are about asymptotic behaviors of the excess risk @xmath82 and take the form @xmath83 with @xmath84 . |
they will be stated under three kinds of conditions involving the hypothesis space @xmath0 , the measure @xmath6 , the loss @xmath12 , and the choice of the regularization parameter @xmath85 . |
the first condition is about the approximation ability of the hypothesis space @xmath0 . |
since the output function @xmath19 is from the hypothesis space , the learning rates of the learning algorithm depend on the approximation ability of the hypothesis space @xmath0 with respect to the optimal risk @xmath86 measured by the following approximation error . |
[ defapprox ] the approximation error of the triple @xmath87 is defined as @xmath88 to estimate the approximation error , we make an assumption about the minimizer of the risk @xmath89 for each @xmath90 , define the integral operator @xmath91 associated with the kernel @xmath41 by @xmath92 we mention that @xmath93 is a compact and positive operator on @xmath94 . |
hence we can find its normalized eigenpairs @xmath95 such that @xmath96 is an orthonormal basis of @xmath94 and @xmath97 as @xmath98 . |
fix @xmath99 . |
then we can define the @xmath100-th power @xmath101 of @xmath93 by @xmath102 this is a positive and bounded operator and its range is well - defined . |
the assumption @xmath103 means @xmath104 lies in this range . |
[ assumption1 ] we assume @xmath105 and @xmath106 where for some @xmath107 and each @xmath108 , @xmath109 is a function of the form @xmath110 with some @xmath111 . |
the case @xmath112 of assumption [ assumption1 ] means each @xmath113 lies in the rkhs @xmath43 . |
a standard condition in the literature ( e.g. |
, @xcite ) for achieving decays of the form @xmath114 for the approximation error ( [ approxerrordef ] ) is @xmath115 with some @xmath116 . |
here
the operator @xmath117 is defined by @xmath118 in general , this can not be written in an additive form . |
however , the hypothesis space ( [ additive ] ) takes an additive form @xmath119 . |
so it is natural for us to impose an additive expression @xmath120 for the target function @xmath121 with the component functions @xmath113 satisfying the power condition @xmath110 . |
the above natural assumption leads to a technical difficulty in estimating the approximation error : the function @xmath113 has no direct connection to the marginal distribution @xmath122 projected onto @xmath27 , hence existing methods in the literature ( e.g. |
, @xcite ) can not be applied directly . |
note that on the product space @xmath123 , there is no natural probability measure projected from @xmath6 , and the risk on @xmath124 is not defined . |
our idea to overcome the difficulty is to introduce an intermediate function @xmath125 . |
it may not minimize a risk ( which is not even defined ) . |
however , it approximates the component function @xmath113 well . |
when we add up such functions @xmath126 , we get a good approximation of the target function @xmath121 , and thereby a good estimate of the approximation error . |
this is the first novelty of the paper . |
[ approxerrorthm ] under assumption [ assumption1 ] , we have @xmath127 where @xmath128 is the constant given by @xmath129 the second condition for our learning rates is about the capacity of the hypothesis space measured by @xmath130-empirical covering numbers . |
let @xmath131 be a set of functions on @xmath21 and @xmath132 for every @xmath133 the * covering number of @xmath131 * with respect to the empirical metric @xmath134 , given by @xmath135 is defined as @xmath136 and the * @xmath130-empirical covering number * of @xmath137 is defined as @xmath138 [ assumption2 ] we assume @xmath139 and that for some @xmath140 , @xmath141 and every @xmath142 , the @xmath130-empirical covering number of the unit ball of @xmath43 satisfies @xmath143 the second novelty of this paper is to observe that the additive nature of the hypothesis space yields the following nice bound with a dimension - independent power exponent for the covering numbers of the balls of the hypothesis space @xmath0 , to be proved in section [ samplesection ] . |
[ capacitythm ] under assumption [ assumption2 ] , for any @xmath144 and @xmath145 , we have @xmath146 the bound for the covering numbers stated in theorem [ capacitythm ] is special : the power @xmath147 is independent of the number @xmath148 of the components in the additive model . |
it is well - known @xcite in the literature of function spaces that the covering numbers of balls of the sobolev space @xmath149 on the cube @xmath150^s$ ] of the euclidean space @xmath151 with regularity index @xmath152 has the following asymptotic behavior with @xmath153 : @xmath154 here the power @xmath155 depends linearly on the dimension @xmath148 . |
similar dimension - dependent bounds for the covering numbers of the rkhss associated with gaussian rbf - kernels can be found in @xcite . |
the special bound in theorem [ capacitythm ] demonstrates an advantage of the additive model in terms of capacity of the additive hypothesis space . |
the third condition for our learning rates is about the noise level in the measure @xmath6 with respect to the hypothesis space . |
before stating the general condition
, we consider a special case for quantile regression , to illustrate our general results . |
let @xmath156 be a quantile parameter . |
the quantile regression function @xmath157 is defined by its value @xmath158 to be a @xmath159-quantile of @xmath160 , i.e. |
, a value @xmath161 satisfying @xmath162 the regularization scheme for quantile regression considered here takes the form ( [ algor ] ) with the loss function @xmath12 given by the pinball loss as @xmath163 a noise condition on @xmath6 for quantile regression is defined in @xcite as follows . |
to this end , let @xmath164 be a probability measure on @xmath165 and @xmath166 . |
then a real number @xmath167 is called @xmath159-quantile of @xmath164 , if and only if @xmath167 belongs to the set @xmath168\bigr ) \ge
\tau \mbox{~~and~~ } q\bigl([t , \infty)\bigr ) \ge 1-\tau\bigr\}\,.\ ] ] it is well - known that @xmath169 is a compact interval . |
[ noisecond ] let @xmath166 . |
1 . |
a probability measure @xmath164 on @xmath165 is said to have a * @xmath159-quantile of type @xmath170 * , if there exist a @xmath159-quantile @xmath171 and a constant @xmath172 such that , for all @xmath173 $ ] , we have @xmath174 2 . |
let @xmath175 $ ] . |
we say that a probability measure @xmath20 on @xmath176 has a * @xmath159-quantile of @xmath177-average type @xmath170 * if the conditional probability measure @xmath178 has @xmath179-almost surely a @xmath159-quantile of type @xmath170 and the function @xmath180 where @xmath181 is the constant defined in part ( 1 ) , satisfies @xmath182 . |
one can show that a distribution @xmath164 having a @xmath159-quantile of type @xmath170 has a unique @xmath159-quantile @xmath183 . |
moreover , if @xmath164 has a lebesgue density @xmath184 then @xmath164 has a @xmath159-quantile of type @xmath170 if @xmath184 is bounded away from zero on @xmath185 $ ] since we can use @xmath186\}$ ] in ( [ tauquantileoftype2formula ] ) . |
End of preview. Expand
in Dataset Viewer.
- Downloads last month
- 5