Abstract
PACBayesian learning bounds are of the utmost interest to the learning community. Their role is to connect the generalization ability of an aggregation distribution \(\rho \) to its empirical risk and to its KullbackLeibler divergence with respect to some prior distribution \(\pi \). Unfortunately, most of the available bounds typically rely on heavy assumptions such as boundedness and independence of the observations. This paper aims at relaxing these constraints and provides PACBayesian learning bounds that hold for dependent, heavytailed observations (hereafter referred to as hostile data). In these bounds the KullackLeibler divergence is replaced with a general version of Csiszár’s fdivergence. We prove a general PACBayesian bound, and show how to use it in various hostile settings.
Keywords
PACBayesian theory Dependent and unbounded data Oracle inequalities fdivergence1 Introduction
Learning theory can be traced back to the late 60s and has attracted a great attention since. We refer to the monographs Devroye et al. (1996) and Vapnik (2000) for a survey. Most of the literature addresses the simplified case of i.i.d observations coupled with bounded loss functions. Many bounds on the excess risk holding with large probability were provided  these bounds are refered to as PAC learning bounds since Valiant (1984).^{1}
In the late 90s, the PACBayesian approach was pioneered by ShaweTaylor and Williamson (1997) and McAllester (1998, 1999). It consists of producing PAC bounds for a specific class of Bayesianflavored estimators. Similar to classical PAC results, most PACBayesian bounds have been obtained with bounded loss functions (see Catoni (2007), for some of the most accurate results). Note that Catoni (2004) provides bounds for unbouded loss, but still under very strong exponential moment assumptions. Different types of PACBayesian bounds were proved in very various models (Seeger 2002; Langford and ShaweTaylor 2002; Seldin and Tishby 2010; Seldin et al. 2012, 2011; Guedj and Alquier 2013; Bégin et al. 2016; Alquier et al. 2016; Oneto et al. 2016) but the boundedness or exponential moment assumptions were essentially not improved in these papers.
The relaxation of the exponential moment assumption is however a theoretical challenge, with huge practical implications: in many applications of regression, there is no reason to believe that the noise is bounded or subexponential. Actually, the belief that the noise is subexponential leads to an overconfidence in the prediction that is actually very harmful in practice, see for example the discussion in Taleb (2007) on finance. Still, thanks to the aforementionned works, the road to obtain PAC bounds for bounded observations has now become so nice and comfortable that it might refrain inclination to explore different settings.

Using the socalled smallball property, Mendelson and several coauthors developed in a striking series of papers tools to study the Empirical Risk Minimizer (ERM) and penalized variants without an exponential moment assumption: we refer to their most recent works (Mendelson 2015; Lecuè and Mendelson 2016). Under a quite similar assumption, Grünwald and Mehta (2016) derived PACBayesian learning bounds (“empirical witness of badness” assumption). Other assumptions were introduced in order to derive fast rates for unbounded losses, like the multiscale Bernstein assumption (Dinh et al. 2016).

Another idea consists in using robust loss functions. This leads to better confidence bounds than the previous approach, but at the price of replacing the ERM by a more complex estimator, usually building on PACBayesian approaches (Audibert and Catoni 2011; Catoni 2012; Oliveira 2013; Giulini 2015; Catoni 2016).

Finally, Devroye et al. (2015), using medianofmeans, provide bounds in probability for the estimation of the mean without exponential moment assumption. It is possible to extend this technique to more general learning problems (Minsker 2015; Hsu and Sabato 2016; Lugosi and Mendelson 2016; Guillaume and Matthieu 2017; Lugosi and Mendelson 2017).
Regarding dependent observations, like time series or random fields, PAC and/or PACBayesian bounds were provided in various settings (Modha and Masry 1998; Steinwart and Christmann 2009; Mohri and Rostamizadeh 2010; Ralaivola et al. 2010; Seldin et al. 2012; Alquier and Wintenberger 2012; Alquier and Li 2012; Agarwal and Duchi 2013; Alquier et al. 2013; Kuznetsov and Mohri 2014; Giraud et al. 2015; Zimin and Lampert 2015; London et al. 2016). However these works massively relied on concentration inequalities for or limit theorems for time series (Yu 1994; Doukhan 1994; Rio 2000; Kontorovich and Ramanan 2008), for which boundedness or exponential moments are crucial.
This paper shows that a proof scheme of PACBayesian bounds proposed by Bégin et al. (2016) can be extended to a very general setting, without independence nor exponential moments assumptions. We would like to stress that this approach is not comparable to the aforementionned work, and in particular it is technically far less sophisticated. However, while it leads to suboptimal rates in many cases, it allows to derive PACBayesian bounds in settings where no PAC learning bounds were available before: for example heavytailed time series.
Let us now introduce the two following key quantities.
Definition 1
Definition 2
Csiszár introduced fdivergences in the 60s, see his recent monograph Csiszàr and Shields (2004, Chapter 4) for a survey.
We use the following notation for recurring functions: \(\phi _p(x) = x^p\). Consequently \(\mathcal {M}_{\phi _p,n} = \int \mathbb {E}\left( r_n(\theta )R(\theta )^{p} \right) \pi (\mathrm{d}\theta )\). Thus \(\mathcal {M}_{\phi _p,n}\) is a moment of order p. As for divergences, we denote the KullbackLeibler divergence by \(\mathcal {K}(\rho ,\pi )=D_{f}(\rho ,\pi )\) when \(f(x)=x\log (x)\), and the chisquare divergence \(\chi ^2(\rho ,\pi )=D_{\phi _21}(\rho ,\pi )\).
Theorem 1
Proof of Theorem 1
In Sect. 2 we discuss the divergence term \(D_{\phi _p1}(\rho ,\pi )\). In particular, we derive an explicit bound on this term when \(\rho \) is chosen in order to concentrate around the ERM (empirical risk minimizer) \(\hat{\theta }_{\mathrm{ERM}}=\arg \min _{\theta \in \Theta }\ r_n(\theta )\). This is meant to provide the reader some intuition on the order of magnitude of the divergence term. In Sect. 3 we discuss how to control the moment \(\mathcal {M}_{\phi _{q},n}\). We derive explicit bounds in various examples: bounded and unbounded losses, i.i.d and dependent observations. The most important result of the section is a risk bound for autoregression with heavytailed time series, something new up to our knowledge. In Sect. 4 we come back to the general case. We show that it is possible to explicitely minimize the righthand side in (2). We then show that Theorem 1 leads to powerful oracle inequalities in the various statistical settings discussed above, exhibiting explicit rates of convergence.
2 Calculation of the divergence term
The aim of this section is to provide some hints on the order of magnitude of the divergence term \(D_{\phi _p1}(\rho ,\pi )\). We start with the example of a finite parameter space \(\Theta \). The following proposition results from straightforward calculations.
Proposition 1
Proposition 2
Remark that \(D_{\phi _p1}(\rho ,\pi )\) seems to be related to the complexity K of the parameter space \(\Theta \). This intuition can be extended to an infinite parameter space, for example using the empirical complexity parameter introduced in Catoni (2007).
Assumption 1
Proposition 3
So the bound is in \(\mathcal {O}((\mathcal {M}_{\phi _{q},n}/\delta )^{1/(1+d/q)})\). In order to understand the order of magnitude of the bound, it is now crucial to understand the moment term \(\mathcal {M}_{\phi _{q},n}\). This is the object of the next section.
3 Bounding the moments
In this section, we show how to control \(\mathcal {M}_{\phi _{q},n}\) depending on the assumptions on the data.
3.1 The i.i.d setting
First, let us assume that the observations \((X_i, Y_i)\) are independent and identically distributed. In general, when the observations are possibly heavytailed, we recommend to use Theorem 1 with \(q\le 2\) (which implies \(p\ge 2)\).
Proposition 4
Proof of Proposition 4
Corollary 1
Note that a similar upper bound was proved in Honorio and Jaakkola (2014), yet only in the case of the 0–1 loss (which is bounded). Also, note that the assumption on the moments of order 4 is comparable to the one in Audibert and Catoni (2011) and allow heavytailed distributions. Still, in our result, the dependence in \(\delta \) is less good than in Audibert and Catoni (2011). So, we end this subsection with a study of the subGaussian case (wich also includes the bounded case). In this case, we can use any \(q\ge 2\) in Theorem 1. The larger q, the better will be the dependence with respect to \(\delta \).
Definition 3
Proposition 5
A straighforward consequence is the following result.
Proposition 6
3.2 Dependent observations
Here we propose to analyze the harder and more realistic case where the observations \((X_i,Y_i)\) are possibly dependent. It includes the autoregressive case where \(X_i=Y_{i1}\) or \(X_i=(Y_{i1},\ldots ,Y_{ip})\). Note that in this setting, different notions of risks were used in the literature. The risk \(R(\theta )\) considered in this paper is the same as the one used in many references given in the introduction, Modha and Masry (1998), Steinwart and Christmann (2009), Alquier and Wintenberger (2012) and Alquier and Li (2012) among others. Alternative notions of risk were proposed, for example by Zimin and Lampert (2015).
We remind the following definition.
Definition 4
We refer the reader to Doukhan (1994) and Rio (2000) (among others) for more details. We still provide a basic interpretation of this definition. First, when \(\mathcal {F}\) and \(\mathcal {G}\) are independent, then for all \(A\in \mathcal {F}\) and \(B\in \mathcal {G}\), \(\mathbb {P}(A \cap B) = \mathbb {P}(A) \mathbb {P}(B)\) by definition of independence, and so \(\alpha (\mathcal {F},\mathcal {G})=0\). On the other hand, when \(\mathcal {F}=\mathcal {G}\), as soon as these \(\sigma \)algebras contain an event A with \(\mathbb {P}(A)=1/2\) then \(\alpha (\mathcal {F},\mathcal {G})= \mathbb {P}(A \cap A)  \mathbb {P}(A) \mathbb {P}(A) = 1/21/4=1/4\). More generally, \(\alpha (\mathcal {F},\mathcal {G})\) is a measure of the dependence of the information provided by \(\mathcal {F}\) and \(\mathcal {G}\), ranging from 0 (independance) to 1 / 4 (maximal dependence). We provide another interpretation in terms of covariances.
Proposition 7
Let us first consider the bounded case.
Proposition 8
Proof of Proposition 8
Remark 1
Other assumptions than \(\alpha \)mixing can be used. Actually, we see from the proof that the only requirement to get a bound on \(\mathcal {M}_{\phi _{2},n}\) is to control the covariance \(\mathrm{Cov}[\ell _{i}(\theta ),\ell _{j}(\theta )]\); \(\alpha \)mixing is very stringent as it imposes that we can control this for any function \(\ell _i(\theta )\). In the case of a Lipschitz loss, we could actually consider more general conditions like the weak dependence conditions in Dedecker et al. (2007) and Alquier and Wintenberger (2012).
We now turn to the unbounded case.
Proposition 9
Proof of Proposition 9
The proof relies on the following property. \(\square \)
Proposition 10
Corollary 2
This is, up to our knowledge, the first PAC(Bayesian) bound in the case of a time series without any boundedess nor exponential moment assumption.
4 Optimal aggregation distribution and oracle inequalities
We have now gone through the way to control the different terms in our PACBayesian inequality (Theorem 1). We now come back to this result to derive which predictor minimizes the bound, and which statistical guarantees can be achieved by this predictor.
Definition 5
The following proposition states that \(\hat{\rho }_n\) is actually the minimizer of the righthand side in inequality (5).
Proposition 11
Proof of Proposition 11
A direct consequence of (5) and (6) is the following result, which provides theoretical guarantees for \(\hat{\rho }_n\).
Proposition 12
Proof of Proposition 12
Example 1
The last example shows that it is possible in some cases to deduce from Proposition 12 an oracle inequality, that is, a comparison to the performance of the optimal parameter. The end of this section is devoted to a systematic derivation of such oracle inequalities, using the complexity parameter introduced in Sect. 3, first in its empirical version, and then in its theoretical form.
Theorem 2
Proof of Theorem 2
We can also perform an explicit minimization of the oracletype bound (8), which leads to a variant of Theorem 2 under a nonempirical complexity assumption.
Definition 6
Assumption 2
Theorem 3
The proof is a direct adaptation of the proofs of Propostion 11 and Theorem 2.
5 Discussion and perspectives
We proposed a new type of PACBayesian bounds, which makes use of Csiszár’s fdivergence to generalize the KullbackLeibler divergence. This is an extension of the results in Bégin et al. (2016). In favourable contexts, there exists sophisticated approaches to get better bounds, as discussed in the introduction. However, the major contribution of our work is that our bounds hold in hostile situations where no PAC bounds at all were available, such as heavytailed time series. We plan to study the connections between our PACBayesian bounds and aforementionned approaches by Mendelson (2015) and Grünwald and Mehta (2016) in future works.
Footnotes
 1.
PAC stands for Probably Approximately Correct.
Notes
Acknowledgements
We would like to thank Pascal Germain for fruitful discussions, along with two anonymous Referees and the Editor for insightful comments.This author gratefully acknowledges financial support from the research programme New Challenges for New Data from LCL and GENES, hosted by the Fondation du Risque, from Labex ECODEC (ANR11LABEX0047) and from Labex CEMPI (ANR11LABX000701).
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