## 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”:

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*.

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

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.

*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, and the -pinball loss, 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:

*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

*f*such that is close to the minimal risk . Our next goal is to formalize this idea. We begin with the following definition:

_{D}Let *X* be a set and be a closed subset. A learning method on maps every set , , to a function .

*f*produced by a specific learning method satisfy If this convergence takes place for all

_{D}*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.

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

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

*L*, , and .

## 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 *L _{p}*-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

*B*for its closed unit ball.

_{E}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:

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).

*E*of bounded measurable functions , we define the -norm by and denote the space of all bounded -functions by .

**-Mixing Process**. Let be a stationary stochastic process. For , the -mixing coefficients are defined by and, similarly, the time-reversed -mixing coefficients are defined by 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 for some constants , , and , then is called geometrically (time-reversed) -mixing. If is of the form 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).

### 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.

*C*is a constant independent of

*n*, and ,

*c*are positive constants.

_{B}Note that in assumption ^{18}, the generalized Bernstein-type inequality, equation 3.3, is assumed with respect to *n _{eff}* 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

*n*in section 3.4.

_{eff}### 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

#### 3.3.2 Geometrically -Mixing Processes

*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 we have . Hence, the right-hand side of equation 3.3 takes the form It is easily seen that this bound is of the generalized form of equation 3.3 with

*n*

_{0}given in equation 3.4: , , , and .

#### 3.3.3 Restricted Geometrically -Mixing Processes

^{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, for all and , where

*c*is some constant depending only on

_{b}*b*,

*c*is some constant depending only on

_{c}*c*, and

*v*

^{2}is defined by

#### 3.3.4 Geometrically -Mixing Markov Chains

*v*

^{2}is defined in equation 3.6 and is given in equation 3.7. Consequently, the following inequality, holds for with 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

#### 3.3.6 Geometrically -Mixing Processes

^{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: 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 the right-hand side of equation 3.3 takes the form It is easy to see that equation 3.9 is also of the generalized form of equation 3.3 with

*n*

_{0}given in equation 3.8,

*C*= 2, , , and .

#### 3.3.7 Polynomially -Mixing Processes

*h*as in the geometrically -mixing case, it states that when with the right-hand side of equation 3.3 takes the form An easy computation shows that it is also of the generalized form of equation 3.3 with

*n*

_{0}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 *n _{eff}*. As verified above, the assumed generalized Bernstein-type inequality indeed holds for many mixing processes, whereas

*n*may take different values in different circumstances. Supposing that we have

_{eff}*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.

Examples . | Effective 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 |

Examples . | Effective 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 *n _{eff}* turns out to be the choice. An intuitive illustration of this procedure is shown in Figure 3.

## 4 A Generalized Sharp Oracle Inequality

^{6}. For and , we write Since , , and , we have , Furthermore, we assume that there exists a function that satisfies 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:

^{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 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 with , every learning method defined by equation 2.4 satisfies with probability not less than :

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 *B*_{0} is necessary since the assumed boundedness of *L* guarantees only , while *B*_{0} bounds the function for an unclipped . We do not assume that all satisfy ; therefore, in general *B*_{0} is necessary. We refer to examples ^{13}, ^{14}, and ^{15} for situations, where *B*_{0} 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 *n _{eff}*, 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:

**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 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.

*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 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, and denote as the population version of , which is given by The next example discusses learning rates for LS-SVMs using a given generic kernel.

**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

*B*of

_{H}*H*satisfies for some constants and . In addition, assume that the approximation error function satisfies for some , , and all .

*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 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

**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 where and is some constant. Let us additionally assume that the marginal distribution

*P*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 for the same

_{X}*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

*m*, that is, a sufficiently smooth kernel

*k*. Moreover, in this case, for geometrically -mixing processes, the rate, equation 5.6, becomes 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 .

^{11}as Moreover, Eberts and Steinwart (2013) show that there exists a constant such that for all and all , there is an with and

**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 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 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 learns with the rate

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).

**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 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}.

^{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 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).

^{11}, we need to reformulate equation 3.3. Setting , with some simple transformations, we obtain for all and .

*Estimating the first stochastic term.*We first bound the term . To this end, we further split this difference into Now, implies , and hence we obtain Inequality A.1 applied to thus shows that holds with probability not less than . Moreover, using , we find and consequently we have with probability not less than that 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 In addition, if , the first inequality in lemma 7.1 of Steinwart and Christmann (2008) implies for , , , and that Since , this inequality also holds for , and hence equation A.1 shows that we have 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 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 with probability not less than . It thus remains to consider the case .

*Introduction of the quotients.*To establish a nontrivial bound on the term in equation A.2, we define functions For