Abstract

This letter investigates the supervised learning problem with observations drawn from certain general stationary stochastic processes. Here by general, we mean that many stationary stochastic processes can be included. We show that when the stochastic processes satisfy a generalized Bernstein-type inequality, a unified treatment on analyzing the learning schemes with various mixing processes can be conducted and a sharp oracle inequality for generic regularized empirical risk minimization schemes can be established. The obtained oracle inequality is then applied to derive convergence rates for several learning schemes such as empirical risk minimization (ERM), least squares support vector machines (LS-SVMs) using given generic kernels, and SVMs using gaussian kernels for both least squares and quantile regression. It turns out that for independent and identically distributed (i.i.d.) processes, our learning rates for ERM recover the optimal rates. For non-i.i.d. processes, including geometrically -mixing Markov processes, geometrically -mixing processes with restricted decay, -mixing processes, and (time-reversed) geometrically -mixing processes, our learning rates for SVMs with gaussian kernels match, up to some arbitrarily small extra term in the exponent, the optimal rates. For the remaining cases, our rates are at least close to the optimal rates. As a by-product, the assumed generalized Bernstein-type inequality also provides an interpretation of the so-called effective number of observations for various mixing processes.

1  Introduction

In this letter, we study the supervised learning problem, which aims at inferring a functional relation between explanatory variables and response variables (Vapnik, 1998). In the literature of statistical learning theory, one of the main research topics is the generalization ability of different learning schemes, which indicates their learnabilities on future observations. It is now well understood that Bernstein-type inequalities play an important role in deriving fast learning rates. For example, the analysis of various algorithms from nonparametric statistics and machine learning crucially depends on these inequalities (see Devroye, Györfi, & Lugosi, 1996; Devroye & Lugosi, 2001; Györfi, Kohler, Krzyżak, & Walk, 2002; Steinwart & Christmann, 2008). Here, stronger results can typically be achieved since Bernstein-type inequality allows for localization due to its specific dependence on the variance. In particular, most derivations of minimax optimal learning rates are based on it.

The classical Bernstein inequality assumes that the data are generated by an i.i.d. process. Unfortunately, this assumption is often violated in many real-world applications, including financial prediction, signal processing, system identification and diagnosis, text and speech recognition, and time series forecasting, among others. For this and other reasons, there has been some effort to establish Bernstein-type inequalities for non-i.i.d. processes. For instance, generalizations of Bernstein-type inequalities to the cases of -mixing (Rosenblatt, 1956) and -mixing (Hang & Steinwart, in press) processes have been found in Bosq (1993), Modha and Masry (1996), Merlevède, Peligrad, and Rio (2009), Samson (2000), Hang and Steinwart (in press), and Hang, Feng, and Suykens (2016). These Bernstein-type inequalities have been applied to derive various convergence rates. For example, the Bernstein-type inequality established in Bosq (1993) was employed in Zhang (2004) to derive convergence rates for sieve estimates from strictly stationary -mixing processes in the special case of neural networks. Hang and Steinwart (2014) applied the Bernstein-type inequality in Modha and Masry (1996) to derive an oracle inequality (Steinwart & Christmann, 2008, for the meaning of the oracle inequality) for generic regularized empirical risk minimization algorithms with stationary -mixing processes. By applying the Bernstein-type inequality in Merlevède et al. (2009), Belomestny (2011) derived almost certain uniform convergence rates for the estimated Lévy density in both mixed-frequency and low-frequency setups and proved their optimality in the minimax sense. Particularly, concerning the least squares loss, Alquier and Wintenberger (2012) obtained optimal learning rates for -mixing processes by applying the Bernstein-type inequality established in Samson (2000). By developing a Bernstein-type inequality for -mixing processes that include -mixing processes and many discrete-time dynamical systems, Hang and Steinwart (in press) established an oracle inequality as well as fast learning rates for generic regularized empirical risk minimization algorithms with observations from -mixing processes.

The inequalities noted are termed Bernstein type since they rely on the variance of the random variables. However, these inequalities are usually presented in similar but rather complicated forms, which are not easy to apply directly in analyzing the performance of statistical learning schemes and maybe also lack interpretability. On the other hand, existing studies on learning from mixing processes may diverge from one to another since they may be conducted under different assumptions and notations, which leads to barriers in comparing the learnability of these learning algorithms.

In this work, we first introduce a generalized Bernstein-type inequality and show that it can be instantiated to various stationary mixing processes. Based on the generalized Bernstein-type inequality, we establish an oracle inequality for a class of learning algorithms including ERM (Steinwart & Christmann, 2008) and SVMs. On the technical side, the oracle inequality is derived by refining and extending the analysis of Steinwart and Christmann (2009). To be more precise, the analysis in Steinwart and Christmann (2009) partially ignored localization with respect to the regularization term, which in our study is addressed by a carefully arranged peeling approach inspired by Steinwart and Christmann (2008). This leads to a sharper stochastic error bound and, consequently, a sharper bound for the oracle inequality compared with that of Steinwart and Christmann (2009). Besides, based on the assumed generalized Bernstein-type inequality, we also provide an interpretation and comparison of the effective numbers of observations when learning from various mixing processes.

Our second main contribution in this study lies in that we present a unified treatment on analyzing learning schemes with various mixing processes. For example, we establish fast learning rates for -mixing and (time-reversed) -mixing processes by tailoring the generalized oracle inequality. For ERM, our results match those in the i.i.d. case, if one replaces the number of observations with the effective number of observations. For LS-SVMs, as far as we know, the best learning rates for the case of geometrically -mixing process are those derived in Xu and Chen (2008) and Sun and Wu (2010), and Feng (2012). When applied to LS-SVMs, it turns out that our oracle inequality leads to faster learning rates that those reported in Xu and Chen (2008) and Feng (2012). For sufficiently smooth kernels, our rates are also faster than those in Sun & Wu (2010). For other mixing processes including geometrically -mixing Markov chains, geometrically -mixing processes, and geometrically -mixing processes, our rates for LS-SVMs with gaussian kernels essentially match the optimal learning rates, while for LS-SVMs with a given generic kernel, we obtain rates that are close to the optimal rates.

The rest of this letter is organized as follows. In section 2, we introduce some basics of statistical learning theory. Section 3 presents the key assumption of a generalized Bernstein-type inequality for stationary mixing processes and some concrete examples that satisfy this assumption. Based on the generalized Bernstein-type inequality, a sharp oracle inequality is developed in section 4; its proof is deferred to the appendix. Section 5 provides some applications of the newly developed oracle inequality. The letter concludes in section 6.

2  A Primer in Learning Theory

Let be a measurable space and be a closed subset. The goal of (supervised) statistical learning is to find a function such that for the value is a good prediction of y at x. The following definition will help us define what we mean by “good”:

Definition 1.

Let be a measurable space and be a closed subset. Then a function is called a loss function, or simply a loss, if it is measurable.

In this study, we are interested in loss functions that in some sense can be restricted to domains of the form as defined below, which is typical in learning theory (Steinwart & Christmann, 2008) and is in fact motivated by the boundedness of Y.

Definition 2.
We say that a loss can be clipped at , if, for all , we have
formula
where denotes the clipped value of t at , that is,
formula

Throughout this work, we make the following assumptions on the loss function L:

Assumption 1.
The loss function can be clipped at some . Moreover, it is both bounded in the sense of and locally Lipschitz continuous, that is,
formula
2.1
Here both inequalites are supposed to hold for all and .

Note that assumption 17 with Lipschitz constant equal to one can typically be enforced by scaling. To illustrate the generality of the above assumptions on L, we first consider the case of binary classification, that is, . For this learning problem, one often uses a convex surrogate for the original discontinuous classification loss , since the latter may lead to computationally infeasible approaches. Typical surrogates L belong to the class of margin-based losses, that is, L is of the form , where is a suitable, convex function. Then L can be clipped if and only if has a global minimum (see lemma 2.23 in Steinwart & Christmann, 2008). In particular, the hinge loss, the least squares loss for classification, and the squared hinge loss can be clipped, but the logistic loss for classification and the AdaBoost loss cannot be clipped. Steinwart (2009) established a simple technique, which is similar to inserting a small amount of noise into the labeling process, to construct a clippable modification of an arbitrary convex, margin-based loss. Finally, both the Lipschitz continuity and the boundedness of L can easily be verified for these losses, where for the latter, it may be necessary to suitably scale the loss.

Bounded regression is another class of learning problems where the assumptions made on L are often satisfied. Indeed, if and L is a convex, distance-based loss represented by some , that is, , then L can be clipped whenever (see again lemma 2.23 in Steinwart & Christmann, 2008). In particular, the least squares loss,
formula
2.2
and the -pinball loss,
formula
2.3
used for quantile regression can be clipped. Again, for both losses, the Lipschitz continuity and the boundedness can be easily enforced by a suitable scaling of the loss.

Given a loss function L and an , we often use the notation for the function . Our major goal is to have a small average loss for future unseen observations . This leads to the following definition:

Definition 3.
Let be a loss function and P be a probability measure on . Then, for a measurable function , the L-risk is defined by
formula
Moreover, the minimal L-risk,
formula
is called the Bayes risk with respect to P and L. In addition, a measurable function satisfying is called a Bayes decision function.
Letting be a probability space and be an -valued stochastic process on , we write
formula
for a training set of length n that is distributed according to the first n components of . Informally, the goal of learning from a training set D is to find a decision function fD such that is close to the minimal risk . Our next goal is to formalize this idea. We begin with the following definition:
Definition 4.

Let X be a set and be a closed subset. A learning method on maps every set , , to a function .

Now a natural question is whether the functions fD produced by a specific learning method satisfy
formula
If this convergence takes place for all P, then the learning method is called universally consistent. In the i.i.d. case, many learning methods are known to be universally consistent—for example, see Devroye et al. (1996) for classification methods, Györfi et al. (2002) for regression methods, and Steinwart and Christmann (2008) for generic SVMs. For consistent methods, it is natural to ask how fast the convergence rate is. Unfortunately, in most situations, uniform convergence rates are impossible (see theorem 7.2 in Devroye et al., 1996), and hence establishing learning rates requires some assumptions on the underlying distribution P. Again, results in this direction can be found in the above-mentioned books. In the non-i.i.d. case, Nobel (1999) showed that no uniform consistency is possible if one only assumes that the data-generating process is stationary and ergodic. But if some further assumptions of the dependence structure of are made, then consistency is possible (see, e.g., Steinwart, Hush, & Scovel, 2009a).

We now describe the learning algorithms of particular interest to us. To this end, we assume that we have a hypothesis set consisting of bounded measurable functions , which is precompact with respect to the supremum norm . Since the cardinality of can be infinite, we need to recall the following concept, which will enable us to approximate by using finite subsets.

Definition 5.
Let be a metric space and . We call an -net of T if for all there exists an with . Moreover, the -covering number of T is defined by
formula
where and denotes the closed ball with center and radius .

Note that our hypothesis set is assumed to be precompact, and hence for all , the covering number is finite.

Denote , where denotes the (random) Dirac measure at . In other words, Dn is the empirical measure associated with the data set D. Then the risk of a function with respect to this measure,
formula
is called the empirical L-risk.

With these preparations, we now introduce the class of learning methods of interest:

Definition 6.
Let be a loss that can be clipped at some ; a hypothesis set, that is, a set of measurable functions , with ; and a regularizer on , that is, with . Then, for , a learning method whose decision functions satisfy
formula
2.4
for all and is called -approximate clipped regularized empirical risk minimization (-CR-ERM) with respect to L, , and .
In the case , we simply speak of clipped regularized empirical risk minimization (CR-ERM). In this case, in fact can be also defined as follows:
formula
Note that on the right-hand side of equation 2.4, the unclipped loss is considered, and hence CR-ERMs do not necessarily minimize the regularized clipped empirical risk . Moreover, in general, CR-ERMs do not minimize the regularized risk either, because on the left-hand side of equation 2.4, the clipped function is considered. However, if we have a minimizer of the unclipped regularized risk, then it automatically satisfies equation 2.4. In particular, ERM decision functions satisfy equation 2.4 for the regularizer and , and SVM decision functions satisfy equation 2.4 for the regularizer and . In other words, ERM and SVMs are CR-ERMs.

3  Mixing Processes and a Generalized Bernstein-Type Inequality

In this section, we introduce a generalized Bernstein-type inequality. Here the inequality is said to be generalized in that it depends on the effective number of observations instead of the number of observations, which, as we see later, makes it applicable to various stationary stochastic processes. To this end, we first introduce several mixing processes.

3.1  Several Stationary Mixing Processes

We begin with introducing some notations. Recall that is a measurable space and is closed. We further denote as a probability space, as an -valued stochastic process on , and and as the -algebras generated by and , respectively. Throughout, we assume that is stationary, that is, the -valued random variables and have the same distribution for all n, i, . Let be a measurable map. is denoted as the -image measure of , which is defined as , measurable. We denote as the space of (equivalence classes of) measurable functions with finite Lp-norm . Then , together with , forms a Banach space. Moreover, if is a sub--algebra, then denotes the space of all -measurable functions . We further denote as the space of d-dimensional sequences with finite Euclidean norm. Finally, for a Banach space E, we write BE for its closed unit ball.

In order to characterize the mixing property of a stationary stochastic process, various notions have been introduced in the literature (Bradley, 2005). Several frequently considered examples are -mixing, -mixing, and -mixing, which are, respectively, defined as follows:

Definition 7: -Mixing Process.
A stochastic process is called -mixing if there holds
formula
where is the -mixing coefficient defined by
formula
Moreover, a stochastic process is called geometrically -mixing, if
formula
3.1
for some constants , , and .
Definition 8: -Mixing Process.
A stochastic process is called -mixing if there holds
formula
where is the -mixing coefficient defined by
formula
Definition 9: -Mixing Process.
A stochastic process is called -mixing if there holds
formula
where is the -mixing coefficient defined by
formula

The -mixing concept was introduced by Rosenblatt (1956), while the -mixing coefficient was introduced by Volkonskii and Rozanov (1959, 1961) and was attributed there to Kolmogorov. Moreover, Ibragimov (1962) introduced the -coefficient (see also Ibragimov & Rozanov, 1978). An extensive and thorough account on mixing concepts including - and -mixing is also provided by Bradley (2007). It is well known that (see e.g., section 2 in Hang & Steinwart, 2014), the - and -mixing sequences are also -mixing (see Figure 1). From the above definition, it is obvious that i.i.d. processes are also geometrically -mixing processes since equation 3.1 is satisfied for c = 0 and all . Moreover, several time series models such as ARMA and GARCH, which are often used to describe, for example, financial data, satisfy equation 3.1 under natural conditions (Fan & Yao, 2003), and the same is true for many Markov chains including some dynamical systems perturbed by dynamic noise (see, e.g., Vidyasagar, 2003).

Figure 1:

Relations among -, -, and -mixing processes.

Figure 1:

Relations among -, -, and -mixing processes.

Another important class of mixing processes called (time-reversed) -mixing processes was introduced in Maume-Deschamps (2006) and recently investigated in Hang and Steinwart (in press). As shown below, it is defined in association with a function class that takes into account the smoothness of functions and therefore could be more general in the dynamical system context. As illustrated in Maume-Deschamps (2006) and Hang and Steinwart (in press), the -mixing process encounters a large family of dynamical systems. Given a seminorm on a vector space E of bounded measurable functions , we define the -norm by
formula
3.2
and denote the space of all bounded -functions by .
Definition 10:
-Mixing Process. Let be a stationary stochastic process. For , the -mixing coefficients are defined by
formula
and, similarly, the time-reversed -mixing coefficients are defined by
formula
Let be a strictly positive sequence converging to 0. Then we say that is (time-reversed) -mixing with rate , if we have for all . Moreover, if is of the form
formula
for some constants , , and , then is called geometrically (time-reversed) -mixing. If is of the form
formula
for some constants , and , then is called polynomial (time-reversed) -mixing.

Figure 2 illustrates the relations among -mixing processes, -mixing processes, and -mixing processes. Clearly, -mixing processes are -mixing (Hang & Steinwart, in press). Furthermore, various discrete-time dynamical systems, including Lasota-Yorke maps, unimodal maps, and piecewise expanding maps in higher dimension, are -mixing (see Maume-Deschamps, 2006). Moreover, smooth expanding maps on manifolds, piecewise expanding maps, uniformly hyperbolic attractors, and nonuniformly hyperbolic unimodal maps are time-reversed geometrically -mixing (see propositions 2.7 and 3.8, corollary 4.11, and theorem 5.15 in Viana, 1997, respectively).

Figure 2:

Relations among -, -, and -mixing processes.

Figure 2:

Relations among -, -, and -mixing processes.

3.2  A Generalized Bernstein-Type Inequality

As discussed in section 1, the Bernstein-type inequality plays an important role in many areas of probability and statistics. In the statistical learning theory literature, it is also crucial in conducting concentrated estimation for learning schemes. As mentioned previously, these inequalities are usually presented in rather complicated forms under different assumptions, which therefore limit their portability to other contexts. However, what is common behind these inequalities is their reliance on the boundedness assumption of the variance. Given the above discussions, we introduce in this section the following generalized Bernstein-type inequality, with the hope of making it an off-the-shelf tool for various mixing processes.

Assumption 2.
Let be an -valued, stationary stochastic process on and . Furthermore, let be a bounded measurable function for which there exist constants and such that , , and . Assume that for all , there exist constants independent of and such that for all , we have
formula
3.3
where is the effective number of observations, C is a constant independent of n, and , cB are positive constants.

Note that in assumption 18, the generalized Bernstein-type inequality, equation 3.3, is assumed with respect to neff instead of n, which is a function of n and is termed the effective number of observations. The term effective number of observations “provides a heuristic understanding of the fact that the statistical properties of autocorrelated data are similar to a suitably defined number of independent observations” (Zieba, 2010). We continue our discussion on the effective number of observations neff in section 3.4.

3.3  Instantiation to Various Mixing Processes

We now show that the generalized Bernstein-type inequality in assumption 18 can be instantiated to various mixing processes: i.i.d processes, geometrically -mixing processes, restricted geometrically -mixing processes, geometrically -mixing Markov chains, -mixing processes, geometrically -mixing processes, and polynomially -mixing processes, among others.

3.3.1  I.I.D Processes

Clearly the classical Bernstein inequality (Bernstein, 1946) satisfies equation 3.3 with , C = 1, , , and .

3.3.2  Geometrically -Mixing Processes

For stationary geometrically -mixing processes , theorem 4.3 in Modha and Masry (1996) bounds the left-hand side of equation 3.3 by
formula
for any , and , where
formula
where is the largest integer less than or equal to t and is the smallest integer greater than or equal to t for . Observe that for all and for all . From this, it is easy to conclude that for all with
formula
3.4
we have . Hence, the right-hand side of equation 3.3 takes the form
formula
It is easily seen that this bound is of the generalized form of equation 3.3 with n0 given in equation 3.4: , , , and .

3.3.3  Restricted Geometrically -Mixing Processes

A restricted geometrically -mixing process is referred to as a geometrically -mixing process (see definition 10) with . For this kind of -mixing processes, theorem 2 in Merlevède et al. (2009) established a bound for the right-hand side of equation 3.3 that takes the following form,
formula
3.5
for all and , where cb is some constant depending only on b, cc is some constant depending only on c, and v2 is defined by
formula
3.6
In fact, for any , by using Davydov’s covariance inequality (Davydov, 1968, corollary to lemma 2.1) with and , we obtain for ,
formula
Consequently, we have
formula
Setting
formula
3.7
then the probability bound, equation 3.5, can be reformulated as
formula
When with
formula
it can be further upper bounded by
formula
Therefore, the Bernstein-type inequality for the restricted -mixing process is also of the generalized form, equation 3.3, where , , , and .

3.3.4  Geometrically -Mixing Markov Chains

For the stationary geometrically -mixing Markov chain with centered and bounded random variables, Adamczak (2008) bounds the left-hand side of equation 3.3 by
formula
where .
Following similar arguments as in the restricted geometrically -mixing case, we know that for an arbitrary , there holds
formula
where v2 is defined in equation 3.6 and is given in equation 3.7. Consequently, the following inequality,
formula
holds for with
formula
That is, when , the Bernstein-type inequality for the geometrically -mixing Markov chain can be also formulated as the generalized form, equation 3.3, with C = 1, , , and .

3.3.5  -Mixing Processes

For a -mixing process , Samson (2000) provides the following bound for the left-hand side of equation 3.3,
formula
where . Obviously it is of the general form of equation 3.3 with , C = 1, , , and .

3.3.6  Geometrically -Mixing Processes

For the geometrically -mixing process in definition 10, Hang and Steinwart (in press) recently developed a Bernstein-type inequality. To state the inequality, the following assumption on the seminorm in equation 3.2 is needed:
formula
Under the above restriction on the seminorm in equation 3.2 and the assumptions that , , and , Hang and Steinwart (in press) state that when with
formula
3.8
the right-hand side of equation 3.3 takes the form
formula
3.9
It is easy to see that equation 3.9 is also of the generalized form of equation 3.3 with n0 given in equation 3.8, C = 2, , , and .

3.3.7  Polynomially -Mixing Processes

For the polynomially -mixing processes, a Bernstein-type inequality was established recently in Hang et al. (2016). Under the same restriction on the seminorm and assumption on h as in the geometrically -mixing case, it states that when with
formula
3.10
the right-hand side of equation 3.3 takes the form
formula
An easy computation shows that it is also of the generalized form of equation 3.3 with n0 given in equation 3.10, C = 2, , , and with .

3.4  From Observations to Effective Observations

The generalized Bernstein-type inequality in assumption 18 is assumed with respect to the effective number of observations neff. As verified above, the assumed generalized Bernstein-type inequality indeed holds for many mixing processes, whereas neff may take different values in different circumstances. Supposing that we have n observations drawn from a certain mixing process discussed above, Table 1 reports its effective number of observations. As mentioned above, it can be roughly treated as the number of independent observations when inferring the statistical properties of correlated data. In this section, we make some effort in presenting an intuitive understanding toward the meaning of the effective number of observations.

Table 1:
Effective Number of Observations for Different Mixing Processes.
ExamplesEffective Number of Observations
i.i.d processes n 
Geometrically -mixing processes  
Restricted geometrically -mixing processes  
Geometrically -mixing Markov chains  
-mixing processes n 
Geometrically -mixing processes  
Polynomially -mixing processes  with  
ExamplesEffective Number of Observations
i.i.d processes n 
Geometrically -mixing processes  
Restricted geometrically -mixing processes  
Geometrically -mixing Markov chains  
-mixing processes n 
Geometrically -mixing processes  
Polynomially -mixing processes  with  

The term effective observations, or effective number of observations, depending on the context, probably appeared first in Bayley and Hammersley (1946) when studying autocorrelated time series data. In fact, many similar concepts can be found in the literature of statistical learning from mixing processes (see, e.g., Lubman, 1969; Yaglom, 1987; Modha & Masry, 1996; Şen, 1998; Zieba, 2010). For stochastic processes, mixing indicates asymptotic independence. In some sense, the effective observations can be taken as the independent observations that can contribute when learning from a certain mixing process.

In fact, when inferring statistical properties with data drawn from mixing processes, a frequently employed technique is to split the data of size n into k blocks, each of size (see Yu, 1994; Modha & Masry, 1996; Bosq, 2012; Mohri & Rostamizadeh, 2010; Hang & Steinwart, in press). Each block may be constructed by choosing consecutive points in the original observation set or by a jump selection (see, Modha & Masry, 1996; Hang & Steinwart, in press). With the constructed blocks, one can then introduce a new sequence of blocks that are independent between the blocks by using the coupling technique. Due to the mixing assumption, the difference between the two sequences of blocks can be measured with respect to a certain metric. Therefore, one can deal with the independent blocks instead of dependent blocks now. For observations in each originally constructed block, one can again apply the coupling technique (Bosq, 2012; Duchi, Agarwal, Johansson, & Jordan, 2012) to tackle, for example, introducing new i.i.d. observations and bounding the difference between the newly introduced observations and the original observations with respect to a certain metric. During this process, one tries to ensure that the number of blocks k is as large as possible, for which neff turns out to be the choice. An intuitive illustration of this procedure is shown in Figure 3.

Figure 3:

An illustration of the effective number of observations when inferring the statistical properties of the data drawn from mixing processes. The data of size n are split into neff blocks, each of size .

Figure 3:

An illustration of the effective number of observations when inferring the statistical properties of the data drawn from mixing processes. The data of size n are split into neff blocks, each of size .

4  A Generalized Sharp Oracle Inequality

In this section, we present one of our main results: an oracle inequality for learning from mixing processes satisfying the generalized Bernstein-type inequality, equation 3.3. We first introduce a few more notations. Let be a hypothesis set in the sense of definition 6. For
formula
4.1
and , we write
formula
4.2
Since , , and , we have , Furthermore, we assume that there exists a function that satisfies
formula
4.3
for all , and a suitable constant . Note that there are actually many hypothesis sets satisfying assumption 4.3 (see section 5 for some examples).

Now, we present the oracle inequality as follows:

Theorem 1.
Let be a stochastic process satisfying assumption 18 with constants , , , and . Furthermore, let L be a loss satisfying assumption 17. Moreover, assume that there exists a Bayes decision function and constants and such that
formula
4.4
where is a hypothesis set with . We define and by equations 4.1 and 4.2, respectively, and assume that equation 4.3 is satisfied. Finally, let be a regularizer with , be a fixed function, and be a constant such that . Then, for all fixed , , , , and satisfying
formula
4.5
with , every learning method defined by equation 2.4 satisfies with probability not less than :
formula
4.6

The proof of theorem 11 is provided in the appendix. Before we illustrate this oracle inequality in the next section with various examples, we briefly discuss the variance bound, equation 4.4. For example, if and L is the least squares loss, then it is well known that equation 4.4 is satisfied for and (see, e.g., example 7.3 in Steinwart & Christmann, 2008). Moreover, under some assumptions on the distribution P, Steinwart and Christmann (2011) established a variance bound of the form equation 4.4 for the so-called pinball loss used for quantile regression. In addition, for the hinge loss, equation 4.4 is satisfied for , if Tsybakov’s noise assumption (Tsybakov, 2004, proposition 1) holds for q (see theorem 8.24 in Steinwart & Christmann, 2008). Finally, based on Blanchard, Lugosi, and Vayatis (2003), Steinwart (2009) established a variance bound with for the earlier mentioned clippable modifications of strictly convex, twice continuously differentiable margin-based loss functions.

We remark that in theorem 11 the constant B0 is necessary since the assumed boundedness of L guarantees only , while B0 bounds the function for an unclipped . We do not assume that all satisfy ; therefore, in general B0 is necessary. We refer to examples 13, 14, and 15 for situations, where B0 is significantly larger than 1.

5  Applications to Statistical Learning

To illustrate the oracle inequality developed in section 4, we now apply it to establish learning rates for some algorithms including ERM over finite sets and SVMs using either a given generic kernel or a gaussian kernel with varying widths. In the ERM case, our results match those in the i.i.d. case if one replaces the number of observations n with the effective number of observations neff, while for LS-SVMs with given generic kernels, our rates are slightly worse than the recently obtained optimal rates (Steinwart et al., 2009b) for i.i.d. observations. The latter difference is not surprising when considering the fact that Steinwart et al. (2009b) used heavy machinery from empirical process theory such as Talagrand’s inequality and localized Rademacher averages, while our results use only a lightweight argument based on the generalized Bernstein-type inequality and the peeling method. However, when using gaussian kernels, we indeed recover the optimal rates for LS-SVMs and SVMs for quantile regression with i.i.d. observations.

We now present the first example: the empirical risk minimization scheme over a finite hypothesis set:

Example 1:
ERM. Let be a stochastic process satisfying assumption 18 and the hypothesis set be finite with and for all . Moreover, assume that for all . Then, for accuracy, , the learning method described by equation 2.4 is ERM, and theorem 11 shows by some simple estimates that
formula
hold with probability not less than .

Recalling that for the i.i.d. case we have , therefore, in example 12, the oracle inequality, equation 4.6, is thus an exact analogue to standard oracle inequality for ERM learning from i.i.d. processes (see, e.g., theorem 7.2 in Steinwart & Christmann, 2008), albeit with different constants.

For further examples let us begin by briefly recalling SVMs (see Steinwart & Christmann, 2008). To this end, let X be a measurable space, , and k be a measurable (reproducing) kernel on X with reproducing kernel Hilbert space (RKHS) H. Given a regularization parameter and a convex loss L, SVMs find the unique solution
formula
In particular, SVMs using the least squares loss, equation 2.2, are called least squares SVMs (LS-SVMs) (Suykens, Van Gestel, De Brabanter, De Moor, & Vandewalle, 2002) where a primal-dual characterization is given, and also termed as kernel ridge regression in the case of zero bias term as studied in Steinwart and Christmann (2008). SVMs using the -pinball loss, equation 2.3, are called SVMs for quantile regression. To describe the approximation properties of H, we need the approximation error function,
formula
5.1
and denote as the population version of , which is given by
formula
5.2
The next example discusses learning rates for LS-SVMs using a given generic kernel.
Example 2:
Generic kernels. Let be a measurable space, , and be a stochastic process satisfying assumption 18. Furthermore, let L be the least squares loss and H be an RKHS over X such that the closed unit ball BH of H satisfies
formula
for some constants and . In addition, assume that the approximation error function satisfies for some , , and all .
Recall that for SVMs we always have (see equation 5.4 in Steinwart & Christmann, 2008). Consequently we only need to consider the hypothesis set . Then equation 4.2 implies that and, consequently, we find
formula
5.3
Thus, we can define the function in equation 4.3 as . For the least squares loss, the variance bound, equation 4.4, is valid with ; hence, condition 4.5 is satisfied if
formula
5.4
Therefore, let r be the sum of the terms on the right-hand side. Since for large n, the first and next-to-last term in equation 5.4 dominate, the oracle inequality, equation 4.6, becomes
formula
where is defined in equation 5.2 and C is a constant independent of n, , , or . Now optimizing over , we then see from lemma A.1.7 in Steinwart and Christmann (2008) that the LS-SVM using learns with the rate , where
formula
5.5
In particular, for geometrically -mixing processes, we obtain the learning rate , where and as in equation 5.5. Let us compare this rate with the ones previously established for LS-SVMs in the literature. For example, Steinwart and Christmann (2009) proved a rate of the form
formula
under exactly the same assumptions. Since and , our rate is always better than that of Steinwart and Christmann (2009). In addition, Feng (2012) generalized the rates of Steinwart and Christmann (2009) to regularization terms of the form with . The resulting rates are again always slower than the ones established in this work. For the standard regularization term, that is, , Xu and Chen (2008) established the rate , which is always slower than ours too. Finally, in the case , Sun and Wu (2009) established the rate , which was subsequently improved to in Sun and Wu (2010). The latter rate is worse than ours if and only if . In particular, for we always get better rates. Furthermore, the analyses of Sun and Wu (2009, 2010) are restricted to LS-SVMs, while our results hold for rather generic learning algorithms.
Example 3:
Smooth kernels. Let be a compact subset, , and be a stochastic process satisfying assumption 18. Furthermore, let L be the least squares loss and be a Sobolev space with smoothness . Then it is well known (see, e.g., Steinwart, Hush, & Scovel, 2009b, or theorem 6.26 in Steinwart & Christmann, 2008) that
formula
where and is some constant. Let us additionally assume that the marginal distribution PX is absolutely continuous with respect to the uniform distribution, where the corresponding density is bounded away from 0 and . Then there exists a constant such that
formula
for the same p (see Mendelson & Neeman, 2010, and corollary 3 in Steinwart et al., 2009b). Consequently, we can bound as in Steinwart et al. (2009b). Moreover, the assumption on the approximation error function is satisfied for , whenever and . Therefore, the resulting learning rate is
formula
5.6
Note that in the i.i.d. case, where , this rate is worse than the optimal rate , where the discrepancy is the term in the denominator. However, this difference can be made arbitrarily small by picking a sufficiently large m, that is, a sufficiently smooth kernel k. Moreover, in this case, for geometrically -mixing processes, the rate, equation 5.6, becomes
formula
where . Comparing this rate with the one from Sun and Wu (2010), it turns out that their rate is worse than ours if . Note that by the constraint , the latter is always satisfied for .
In the following, we are mainly interested in the commonly used gaussian kernels defined by
formula
where is a nonempty subset and is a free parameter called the width. We write for the corresponding RKHSs, which are described in some detail in Steinwart, Hush, and Scovel (2006). The entropy numbers for gaussian kernels (Steinwart & Christmann, 2008, theorem 6.27) and the equivalence of covering and entropy numbers (Steinwart & Christmann, 2008, lemma 6.21) yield that
formula
5.7
for some constants and . Then equation 4.2 implies and, consequently,
formula
Therefore, we can define the function in theorem 11 as
formula
5.8
Moreover, Eberts and Steinwart (2013) show that there exists a constant such that for all and all , there is an with and
formula
Example 4:
Gaussian kernels. Let for some , , be a stochastic process satisfying assumption 18, and P be a distribution on Z whose marginal distribution on is concentrated on and absolutely continuous with regard to the Lebesgue measure on . We denote the corresponding density and assume and for some . Moreover, assume that the Bayes decision function satisfies as well as for some and with . Here, denotes the Besov space with the smoothness parameter t (see also section 2 in Eberts & Steinwart, 2013). Recall that for the least squares loss, the variance bound, equation 4.4, is valid with . Consequently, condition 4.5 is satisfied if
formula
5.9
Note that on the right-hand side of equation 5.9, the first term dominates when n goes large. In this context, the oracle inequality, equation 4.6, becomes
formula
Here C is a constant independent of n, , , , or . Again, optimizing over together with some standard techniques (see lemma A.1.7 in Steinwart & Christmann, 2008), we then see that for all , the LS-SVM using gaussian RKHS and
formula
learns with the rate
formula
5.10
In the i.i.d. case, we have , and hence the learning rate, equation 5.10, becomes
formula
5.11
Recall that modulo the arbitrarily small , these learning rates are essentially optimal (see, e.g., theorem 13 in Steinwart et al., 2009b or theorem 3.2 in Györfi et al., 2002). Moreover, for geometrically -mixing processes, the rate, equation 5.10, becomes
formula
where . This rate is optimal up to the factor and the additional in the exponent. Particularly for restricted geometrically -mixing processes, geometrically -mixing Markov chains, and -mixing processes, we obtain the essentially optimal learning rates (see equation 5.11). Moreover, the same essentially optimal learning rates can be achieved for (time-reversed) geometrically -mixing processes if we additionally assume (see also example 4.7 in Hang & Steinwart, in press).

In the final example, we briefly discuss learning rates for SVMs for quantile regression. For more information on such SVMs we refer to section 4 in Eberts and Steinwart (2013).

Example 5:
Quantile regression with gaussian kernels. Let , , be a stochastic process satisfying assumption 18, P be a distribution on Z, and Q be the marginal distribution of P on . Assume that and that for Q-almost all , the conditional probability is absolutely continuous with regard to the Lebesgue measure on Y and the conditional densities of are uniformly bounded away from 0 and (see also example 4.5 in Eberts & Steinwart, 2013). Moreover, assume that Q is absolutely continuous with regard to the Lebesgue measure on X with associated density for some . For , let be a conditional -quantile function that satisfies . In addition, we assume that for some and such that . Then theorem 2.8 in Steinwart and Christmann (2011) yields a variance bound of the form
formula
5.12
for all , where V is a suitable constant and is the -pinball loss. Then, following similar arguments as those in example 15, with the same choices of and , the same rates can be obtained as in example 15.
Here, we give two remarks on example 16. First, it is noted that the Bernstein condition, equation 5.12, holds when the distribution P is of a -quantile of p-average type q in the sense of definition in Steinwart and Christmann (2011). Two distributions of this type can be found in examples and in Steinwart and Christmann (2007). On the other hand, the rates obtained in example 16 are in fact for the excess -risk. However, since theorem 2.7 in Steinwart and Christmann (2011) shows
formula
for some constant and all , we also obtain the same rates for . Last but not least, optimality for various mixing processes can be discussed along the lines of LS-SVMs.

6  Conclusion

In this letter, we proposed a unified learning theory approach to studying learning schemes, sampling from various commonly investigated stationary mixing processes that include geometrically -mixing processes, geometrically -mixing Markov chains, -mixing processes, and geometrically -mixing processes. The proposed approach is considered to be unified in the following sense. First, in our study, the empirical processes of these mixing processes were assumed to satisfy a generalized Bernstein-type inequality, which includes many commonly considered cases. Second, by instantiating the generalized Bernstein-type inequality to different scenarios, we illustrated an effective number of observations for different mixing processes. Third, based on the generalized Bernstein-type concentration assumption, a generalized sharp oracle inequality was established within the statistical learning theory framework. Finally, faster, or at least comparable, learning rates can be obtained by applying the established oracle inequality to various learning schemes with different mixing processes.

Appendix:  Proof of Theorem 11 in Section 4

Since the proof of theorem 11 is rather complicated, we describe its main steps briefly. First, we decompose the regularized excess risk into an approximation error term and two stochastic error terms. The approximation error and the first stochastic error term can be estimated by standard techniques. Similarly, the first step in the estimation of the second error term is a rather standard quotient approach (see, e.g., theorem 7.20 in Steinwart & Christmann, 2008), which allows for localization with respect to both the variance and the regularization. Due to the absence of tools from empirical process theory, however, the remaining estimation steps become more involved. To be more precise, we split the “unit ball” of the hypothesis space into disjoint “spheres.” For each sphere, we then use localized covering numbers and the generalized Bernstein-type inequality from assumption 18, and the resulting estimates are then combined using the peeling method. This yields a quasi-geometric series with rate smaller than 1 if the radius of the innermost ball is sufficiently large. As a result, the estimated error probability on the whole unit ball nearly equals the estimated error probability of the innermost ball, which unsurprisingly leads to a significant improvement compared to Steinwart and Christmann (2009).

Before we prove theorem 11, we need to reformulate equation 3.3. Setting , with some simple transformations, we obtain
formula
A.1
for all and .
Proof of Theorem 1.
Main Decomposition: For we define . By the definition of , we then have
formula
and consequently we obtain
formula
A.2
Estimating the first stochastic term. We first bound the term . To this end, we further split this difference into
formula
A.3
Now, implies , and hence we obtain
formula
Inequality A.1 applied to thus shows that
formula
holds with probability not less than . Moreover, using , we find
formula
and consequently we have with probability not less than that
formula
A.4
In order to bound the remaining term in equation A.3, that is, , we first observe that equation 2.1 implies , and hence we have . Moreover, equation 4.4 yields
formula
In addition, if , the first inequality in lemma 7.1 of Steinwart and Christmann (2008) implies for , , , and that
formula
Since , this inequality also holds for , and hence equation A.1 shows that we have
formula
with probability not less than . By combining this estimate with equations A.4 and A.3, we now obtain that with probability not less than , we have
formula
A.5
since ; that is, we have established a bound on the second term in equation A.2.
Estimating the second stochastic term. For the third term in equation A.2, we first consider the case . Combining equation A.5 with A.2 and using , , , , and , we find that
formula
with probability not less than . It thus remains to consider the case .
Introduction of the quotients. To establish a nontrivial bound on the term