Pascal Germain

Amaury Habrard

François Laviolette

Emilie Morvant

A New PAC-Bayesian Perspective on Domain Adaptation

We study the issue of PAC-Bayesian domain adaptation: We want to learn, from a source domain, a majority vote model dedicated to a target one. Our theoretical contribution brings a new perspective by deriving an upper-bound on the target risk where the distributions' divergence-expressed as a ratio-controls the trade-off between a source error measure and the target voters' disagreement. Our bound suggests that one has to focus on regions where the source data is informative. From this result, we derive a PAC-Bayesian generalization bound, and specialize it to linear classifiers. Then, we infer a learning algorithm and perform experiments on real data.

Introduction

Machine learning practitioners are commonly exposed to the issue of domain adaptation1 [START_REF] Jiang | A literature survey on domain adaptation of statistical classifiers[END_REF][START_REF] Margolis | A literature review of domain adaptation with unlabeled data[END_REF]: One usually learns a model from a corpus, i.e., a fixed yet unknown source distribution, then wants to apply it on a new corpus, i.e., a related but slightly different target distribution. Therefore, domain adaptation is widely studied in a lot of application fields like computer vision [START_REF] Patel | Visual domain adaptation: A survey of recent advances[END_REF][START_REF] Ganin | Unsupervised domain adaptation by backpropagation[END_REF], bioinformatics [START_REF] Liu | a hidden variable model for integrating gene evidence for eukaryotic gene prediction[END_REF], natural language processing [START_REF] Blitzer | Domain adaptation of natural language processing systems[END_REF][START_REF] Daumé | Frustratingly easy domain adaptation[END_REF], etc. A simple example is the common spam filtering problem, where a model needs to be adapted from one user mailbox to another receiving significantly different emails.

Several approaches exist to address domain adaptation, often with the same idea: If we are able to apply a transformation in order to "move closer" the distributions, then we can learn a model with the available labels. This process is generally performed by reweighting the importance of labeled data [START_REF] Huang | Correcting sample selection bias by unlabeled data[END_REF][START_REF] Sugiyama | Direct importance estimation with model selection and its application to covariate shift adaptation[END_REF][START_REF] Cortes | Learning bounds for importance weighting[END_REF]2015), and/or by learning a new common representation for the source and target distributions [START_REF] Chen | Marginalized denoising autoencoders for domain adaptation[END_REF][START_REF] Ganin | Unsupervised domain adaptation by backpropagation[END_REF], and/or by minimizing a measure of divergence between the distributions [START_REF] Cortes | Domain adaptation and sample bias correction theory and algorithm for regression[END_REF][START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF]. The divergence-based approach has especially been explored to derive generalization bounds for domain adaptation (e.g., [START_REF] Ben-David | Analysis of representations for domain adaptation[END_REF]2010;[START_REF] Mansour | Domain adaptation: Learning bounds and algorithms[END_REF][START_REF] Li | A bayesian divergence prior for classiffier adaptation[END_REF][START_REF] Zhang | Generalization bounds for domain adaptation[END_REF]. Recently, this issue has been studied through the PAC-Bayesian framework [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF], which focuses on learning weighted majority votes 2 , without any target label. Even if their result clearly opens the door to tackle domain adaptation in a PAC-Bayesian fashion, it shares the same philosophy as the seminal works of [START_REF] Ben-David | Analysis of representations for domain adaptation[END_REF]2010) and [START_REF] Mansour | Domain adaptation: Learning bounds and algorithms[END_REF]: The risk of the target model is upper-bounded jointly by the risk of the model on the source distribution, the divergence between the marginal distributions, and a non-estimable term 3 related to the ability to adapt in the current space. Note that [START_REF] Li | A bayesian divergence prior for classiffier adaptation[END_REF] previously proposed a PAC-Bayesian generalization bound for domain adaptation but they considered target labels.

In this paper, we derive a novel domain adaptation bound for the weighted majority vote framework. Concretely, the risk of the target model is still upper-bounded by three terms, but these terms differ in the information they capture. The first term is estimable from unlabeled data and relies on a notion of expected voters' disagreement on the target domain. The second term depends on the expected accuracy of the voters on the source domain. Interestingly, this latter is weighted by a divergence between the source and the target domains that enable controlling the relationship between domains. The third term estimates the "volume" of the target domain living apart from the source one4 , which has to be small for ensuring adaptation. From our bound, we deduce that a good adaptation strategy consists in finding a weighted majority vote leading to a suitable trade-off-controlled by the domains' divergence-between the first two terms: Minimizing the first one corresponds to look for voters that disagree on the target domain, and minimizing the second one to seek accurate voters on the source. Thereafter, we provide PAC-Bayesian generalization guarantees to justify the empirical minimization of our new domain adaptation bound, and specialize it to linear classifiers (following a methodology known to give rise to tight bound values). This allows to design dalc, a learning algorithm that improves the performances of the previous PAC-Bayesian domain adaptation algorithm.

The rest of the paper is organized as follows. Section 2 presents the PAC-Bayesian domain adaptation setting. Section 3 reviews previous theoretical results on domain adaptation. Section 4 states our new analysis of domain adaptation for majority votes, that we relate to other works in Section 5. Then, Section 6 provides generalization bounds, specialized to linear classifiers in Section 7 to motivate the dalc learning algorithm, evaluated in Section 8.

Unsupervised Domain Adaptation Setting

We tackle domain adaptation in a binary classification setting, from a d-dimensional input space X ⊆ R d to an output space Y = {-1, 1}. Our goal is to perform domain adaptation from a distribution S-the source domain-to another (related) distribution T -the target domain-on X×Y ; S X and T X being the associated marginal distributions on X. Given a distribution D, we denote (D) m the distribution of a msample constituted by m elements drawn i.i.d. from D. We consider the unsupervised domain adaptation setting in which the algorithm is provided with a labeled source m s -sample S = {(x i , y i )} ms i=1 ∼ (S) ms , and with an unlabeled target m t -sample

T = {x i } mt i=1 ∼ (T X ) mt .
PAC-Bayesian domain adaptation. Our work is inspired by the PAC-Bayesian theory (first introduced by [START_REF] Mcallester | Some PAC-Bayesian theorems[END_REF]. More precisely, we adopt the PAC-Bayesian domain adaptation setting studied by [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF]. Given H, a set of voters h : X → Y , the elements of this approach are a prior distribution π on H, a pair of source-target learning samples (S, T ) and a posterior distribution ρ on H. The prior distribution π models an a priori belief-before observing (S, T )-of the voters' accuracy.

Then, given the information provided by (S, T ), we aim at learning a posterior distribution ρ leading to a ρ-weighted majority vote over H,

B ρ (•) = sign E h∼ρ h(•) ,
with nice generalization guarantees on the target domain T . In other words, we want to find the posterior distribution ρ minimizing the true target risk of B ρ :

R T (B ρ ) = E (x,y)∼T I B ρ (x) = y ,
where I a = 1 if a is true, and 0 otherwise. However, in most PAC-Bayesian analyses one does not directly focus on this majority vote risk, but studies the expectation of the risks over H according to ρ, designed as the Gibbs risk : [START_REF] Herbrich | A PAC-Bayesian margin bound for linear classifiers: Why svms work[END_REF]. Unfortunately, this worst case bound often leads to poor generalization guarantees on the majority vote risk. To address this issue, [START_REF] Lacasse | PAC-Bayes bounds for the risk of the majority vote and the variance of the Gibbs classifier[END_REF] have exhibited that one can obtain a tighter bound on R D (B ρ ) by studying the expected disagreement d D (ρ) of pairs of voters, defined as

R D (G ρ ) = E (x,y)∼D E h∼ρ I h(x) = y . (1) It is well-known in the PAC-Bayesian literature that R D (B ρ ) ≤ 2 R D (G ρ ) (e.g.,
d D (ρ) = E x∼D X E h∼ρ E h ∼ρ I h(x) = h (x) , (2) as R D (B ρ ) ≤ 1- (1-2 R D (Gρ)) 2
1-2 d D (ρ) . Note that, although relying on d D (ρ), our present work does not reuse the latter result. 5 Instead, we adopt another well-known strategy to obtain tight majority vote bounds, by specializing our PAC-Bayesian bound to linear classifiers. We describe this approach, and refer to related works, in Section 7.

Some Previous Domain Adaptation Bounds

Many approaches tackling domain adaptation share the same underlying "philosophy", pulling its origins in the work of [START_REF] Ben-David | Analysis of representations for domain adaptation[END_REF]2010) which proposed a domain adaptation bound (Theorem 1, below). To summarize, the domain adaptation bounds reviewed in this section (see [START_REF] Zhang | Generalization bounds for domain adaptation[END_REF][START_REF] Cortes | Learning bounds for importance weighting[END_REF]2015, for other bounds) express a similar trade-off between three terms: (i) the source risk, (ii) the distance between source and target marginal distributions over X, (iii) a non-estimable term (without target label) quantifying the difficulty of the task.

In [START_REF] Ben-David | Analysis of representations for domain adaptation[END_REF], the domains are first assumed to be related, in the sense that there exists a (unknown) model that performs well on both domains. Formally, their domain adaptation bound depends on the error µ h * = R S (h * )+R T (h * ) of the best hypothesis overall h * = argmin h∈H R S (h) + R T (h) . In practice, when no target label is available, µ h * is non-estimable and is assumed to be low when domain adaptation is achievable (or at least that there exists a representation space in which this assumption can be verified). In such a scenario, the domain adaptation strategy is then to look for a set H of possible models that behave "similarly" on both the source and target data, and to learn a model in H with a good accuracy on the source data. This similarity notion, called the H∆H-distance d H∆H (S X , T X ), is defined as 2 sup

(h,h )∈H 2 E x∼S X I h(x) = h (x) -E x∼T X I h(x) = h (x) ,
and gives rise to the following domain adaptation bound.

Theorem 1 [START_REF] Ben-David | Analysis of representations for domain adaptation[END_REF]2010). Let H be a (symmetric6 ) hypothesis class. We have,

∀h ∈ H, R T (h) ≤ R S (h) + 1 2 d H∆H (S X , T X ) + µ h * . (3)
Pursuing in the same line of research, [START_REF] Mansour | Domain adaptation: Learning bounds and algorithms[END_REF] generalizes the H∆H-distance to realvalued loss functions L : [-1, 1] 2 → R + , to express a similar theorem for regression. Their discrepancy disc L (S X , T X ) is defined as

sup (h,h )∈H 2 E x∼S X L h(x), h (x) -E x∼T X L h(x), h (x) .
The accuracy of the [START_REF] Mansour | Domain adaptation: Learning bounds and algorithms[END_REF]'s bound also relies on a non-estimable term assumed to be low when adaptation is achievable. Roughly, this term depends on the risk of the best target hypothesis and its agreement with the best source hypothesis on the source domain.

Building on previous domain adaptation analyses, [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF] derived a PAC-Bayesian domain adaptation bound. This bound is based on a divergence suitable for PAC-Bayes, i.e., for the risk of a ρ-weighted majority vote of the voters of H (instead of a single classifier h ∈ H). This domain disagreement dis ρ (S X , T X ) is defined as

dis ρ (S X , T X ) = d S (ρ) -d T (ρ) . (4)
In favorable adaptation situations, they make the strong assumption that the learned posterior agrees with the best target one ρ T * = argmin ρ R T (G ρ ). Indeed, their bound (Theorem 2, below) relies on the following non-estimable term:

λ(ρ) = RT (Gρ T * ) +E h∼ρ E h ∼ρ T * Ex∼S X I h(x) =h (x) +E h∼ρ E h ∼ρ T * Ex∼T X I h(x) =h (x) .
Theorem 2 [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF]. Let H be a set of voters. For any domains S and T over X×Y , we have,

∀ρ on H, R T (G ρ ) ≤ R S (G ρ ) + dis ρ (S X , T X ) + λ(ρ).
A compelling aspect of this PAC-Bayesian analysis is the suggested trade-off, which is function of ρ. Indeed, given a fixed instance space X and a fixed class H, apart from using importance weighting methods, the only way to minimize the bound of Theorem 1 is to find h ∈ H that minimizes R S (h). Conversely, in [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF], the bound of Theorem 2 inspired an algorithm-named pbda-selecting ρ over H that achieves a trade-off between R S (G ρ ) and dis ρ (S X , T X ). However, the term λ(ρ) does not appear in the optimization process of pbda, even if it relies on the learned weight distribution ρ. [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF] argued that the value of λ(ρ) should be negligible (uniformly for all ρ) when adaptation is achievable. Nevertheless, this strong assumption cannot be verified because the best target posterior distribution ρ T * is unknown. We claim this is a major weakness of the work of [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF] that the approach of this paper overcomes.

A New Domain Adaptation Perspective

In this section, we introduce an original approach to upper-bound the non-estimable risk of a ρ-weighted majority vote on a target distribution T thanks to a term depending on its marginal distribution T X , another one on a related source domain S, and a term capturing the "volume" of the source distribution uninformative for the target task. We base our bound on the expected disagreement d D (ρ) of Equation ( 2) and the expected joint error e D (ρ), defined as

e D (ρ) = E (x,y)∼D E h∼ρ E h ∼ρ I h(x) = y I h (x) = y .
(5) Indeed, [START_REF] Lacasse | PAC-Bayes bounds for the risk of the majority vote and the variance of the Gibbs classifier[END_REF] observed that, given a domain D on X×Y and a distribution ρ on H, we can decompose the Gibbs risk as

R D (G ρ ) = 1 2 E (x,y)∼D E h∼ρ E h ∼ρ I h(x) =y +I h (x) =y = E (x,y)∼D E h∼ρ E h ∼ρ I h(x) =h (x) +2 I h(x) =y∧h (x) =y 2 = 1 2 d D (ρ) + e D (ρ) . (6)
A key observation is that the voters' disagreement does not rely on labels; we can compute d D (ρ) using the marginal distribution D X . Thus, in the present domain adaptation context, we have access to d T (ρ) even if the target labels are unknown. However, the expected joint error can only be computed on the labeled source domain.

Domains' divergence.

In order to link the target joint error e T (ρ) with the source one e S (ρ), we weight the latter thanks to a divergence measure between the domains β q (T S) parametrized by a real value q > 0 :

β q (T S) = E (x,y)∼S T (x, y) S(x, y) q 1 q . ( 7
)
It is worth noting that considering some q values allow us to recover well-known divergences. For instance, choosing q = 2 relates our result to the χ 2 -distance, as β 2 (T S) = χ 2 (T S) + 1 . Moreover, we can link β q (T S) to the Rényi divergence7 , which has led to generalization bounds in the context of importance weighting [START_REF] Cortes | Learning bounds for importance weighting[END_REF]. We denote the limit case q → ∞ by

β ∞ (T S) = sup (x,y)∈supp(S) T (x, y) S(x, y) , (8)
where supp(S) is the support of domain S. This divergence β q (T S) handles the input space areas where the source distribution support supp(S) is included in the target one supp(T ). It seems reasonable to assume that, when adaptation is achievable, such areas are fairly large. However, it is also likely that supp(T ) is not entirely included in supp(S).

We denote T \S the distribution of (x, y)∼T conditional to (x, y)∈supp(T )\supp(S). Since it is hardly conceivable to estimate the joint error e T \S (ρ) without making extra assumptions, we consider the worst possible risk for this unknown area. To do so, we define

η T \S = Pr (x,y)∼T (x, y) / ∈ supp(S) sup h∈H R T \S (h) . (9)
Even if we cannot evaluate sup H R T \S (h), the value of η T \S is necessarily lower than Pr T ((x, y) / ∈supp(S)).

The domain adaptation bound. Let us state the result underlying the domain adaptation perspective of this paper.

Theorem 3. Let H be a hypothesis space, let S and T respectively be the source and the target domains on X×Y . Let q > 0 be a constant. We have, for all ρ on H,

R T (G ρ ) ≤ 1 2 d T (ρ) + β q (T S)× e S (ρ) 1-1 q + η T \S ,
where d T (ρ), e S (ρ), β q (T S) and η T \S are respectively defined by Equations (2), ( 5), ( 7) and (9).

Proof. Let us define t = E (x,y)∼T I (x, y) / ∈supp(S) , then

η ρ = E (x,y)∼T I (x, y) / ∈supp(S) E h∼ρ E h ∼ρ I h(x) =y I h (x) =y = t E (x,y)∼T \S E h∼ρ E h ∼ρ I h(x) =y I h (x) =y = t e T \S (ρ) = t R T \S (G ρ )-1 2 d T \S (ρ) ≤ t sup h∈H R T \S (h) = η T \S .
Then, with β q = β q (T S) and p such that

1 p =1-1 q , e T (ρ) = E (x,y)∼T E h∼ρ E h ∼ρ I h(x) =y I h (x) =y = E (x,y)∼S T (x,y) S(x,y) E h∼ρ E h ∼ρ I h(x) =y I h (x) =y +η ρ (10) ≤ β q E h∼ρ E h ∼ρ E (x,y)∼S I h(x) =y I h (x) =y p 1 p +η ρ .
Last line is due to Hölder inequality. Finally, we remove the exponent from expression (I h(x) = y I h (x) = y ) p without affecting its value, which is either 1 or 0, and the final result follows from Equation ( 6).

Note that the bound of Theorem 3 is reached whenever the domains are equal (S = T ). Thus, when adaptation is not necessary, our analysis is still sound and non-degenerated:

R S (G ρ ) = R T (G ρ ) ≤ 1 2 d T (ρ) + 1 × [e S (ρ)] 1 + 0 = 1 2 d S (ρ) + e S (ρ) = R S (G ρ ) .
Meaningful quantities. Similarly to the previous results recalled in Section 3, our domain adaptation theorem bounds the target risk by a sum of three terms. However, our approach breaks the problem into atypical quantities: (i) The expected disagreement d T (ρ) captures second degree information about the target domain. (ii) The domains' divergence β q (T S) weights the influence of the expected joint error e S (ρ) of the source domain; the parameter q allows us to consider different relationships between β q (T S) and e S (ρ). (iii) The term η T \S quantifies the worst feasible target error on the regions where the source domain is uninformative for the target one. In the current work, we assume that this area is small.

Comparison With Related Works

In this section, we discuss how our domain adaptation bound can be related to some previous works.

On the previous PAC-Bayesian bound.

It is instructive to compare our Theorem 3 with the previous PAC-Bayesian domain adaptation bound of Theorem 2. In our bound, the non-estimable terms are the domain divergence β q (T S) and the term η T \S . Contrary to the non-controllable term λ(ρ) of Theorem 2, these terms do not depend on the learned posterior distribution ρ: For every ρ on H, β q (T S) and η T \S are constant values measuring the relation between the domains. Moreover, the fact that the domain divergence β q (T S) is not an additive term but a multiplicative one (as opposed to disρ(S X , T X )+λ(ρ) in Theorem 2) is a contribution of our analysis. Consequently, β q (T S) can be viewed as a hyperparameter allowing us to tune the trade-off between the target voters' disagreement and the source joint error. Experiments of Section 8 confirm that this hyperparameter can be successfully selected.

On some domain adaptation assumptions

In order to characterize which domain adaptation task may be learnable, Ben-David et al. ( 2012) presented three assumptions that can help domain adaptation. Our Theorem 3 does not rely on these assumptions, but they can be interpreted in our framework as discussed below.

On the covariate shift. A domain adaptation task fulfills the covariate shift assumption [START_REF] Shimodaira | Improving predictive inference under covariate shift by weighting the log-likelihood function[END_REF] if the source and target domains only differ in their marginals according to the input space, i.e., T Y |x (y) = S Y |x (y). In this scenario, one may estimate β q (T X S X ), and even η T \S , by using unsupervised density estimation methods. Interestingly, by also assuming that the domains share the same support, we have η T \S = 0. Then from Line (10) we obtain

RT (Gρ) = 1 2 dT (ρ)+ E x∼S X T X (x) S X (x) E h∼ρ E h ∼ρ I h(x) =y I h (x) =y ,
which suggests a way to correct the shift between the domains by reweighting the labeled source distribution, while considering the information from the target disagreement.

On the weight ratio. The weight ratio (Ben-David et al., 2012) of source and target domains, with respect to a collection of input space subsets B ⊆ 2 X , is given by

C B (S, T ) = inf b∈B T X (b) =0 S X (b) T X (b)
.

When C B (S, T ) is bounded away from 0, adaptation should be achievable under covariate shift. In this context, and when supp(S) = supp(T ), the limit case of β ∞ (T S) is equal to the inverse of the point-wise weight ratio obtained by letting B = {{x} : x ∈ X} in C B (S, T ). Indeed, both β q and C B compare the densities of source and target domains, but provide distinct strategies to relax the point-wise weight ratio; the former by lowering the value of q and the latter by considering larger subspaces B.

On the cluster assumption. A target domain fulfills the cluster assumption when examples of the same labels belong to a common "area" of the input space, and when the differently labeled "areas" are well separated by low-density regions (this concept has been mathematically formalized by the probabilistic Lipschitzness notion in [START_REF] Urner | Access to unlabeled data can speed up prediction time[END_REF]. As we shall see in Section 7, once specialized to linear classifiers, d T (ρ) behaves nicely in this context.

On representation learning

The main assumption underlying our domain adaptation algorithm exhibited in Section 7 is that the support of the target domain is mostly included in the support of the source domain, i.e., the value of the term η T \S is small. When T \S is sufficiently large to prevent proper adaptation, one could try to reduce its volume, while taking care to preserve a good compromise between d T (ρ) and e S (ρ), using a representation learning approach, i.e., by projecting source and target examples into a new common input space, as done for example by [START_REF] Chen | Marginalized denoising autoencoders for domain adaptation[END_REF]; [START_REF] Ganin | Unsupervised domain adaptation by backpropagation[END_REF].

PAC-Bayesian Generalization Guarantees

To compute our domain adaptation bound, one needs to know the distributions S and T X , which is never the case in real life tasks. The PAC-Bayesian theory provides tools to convert the bound of Theorem 3 into a generalization bound on the target risk computable from a pair of source-target samples (S, T )∼(S) ms ×(T X ) mt . To achieve this, we first provide generalization guarantees for d T (ρ) and e S (ρ). These results are presented as corollaries of Theorem 4 below, that generalizes a PAC-Bayesian theorem of [START_REF] Catoni | PAC-Bayesian supervised classification: the thermodynamics of statistical learning[END_REF] to arbitrary loss functions. Indeed, Theorem 4, with (h, x, y) = I h(x) =y and Equation (1), gives the usual bound on the Gibbs risk.

Theorem 4. For any domain D over X×Y , any set of voters H, any prior π over H, any loss : H×X×Y →[0, 1], any real number c>0, with a probability at least 1-δ over the choice of {(x i , y i )} m i=1 ∼(D) m , we have for all ρ on H:

E (x,y)∼D E h∼ρ (h, x, y) ≤ c 1-e -c 1 m m i=1 E h∼ρ (h, x i , y i ) + KL(ρ π) + ln 1 δ m × c .
Proof. Reported in Appendix A.

Note that, similarly to [START_REF] Mcallester | Generalization bounds and consistency for latent structural probit and ramp loss[END_REF], we could choose to restrict c ∈ (0, 2) to obtain a slightly looser but simpler bound. Using e -c ≤ 1 -c -1 2 c 2 , an upper bound on the right hand side of above equation is given by

1 1-1 2 c 1 m m i=1 E h∼ρ (h, x i , y i ) + KL(ρ π)+ln 1 δ m×c
. We now exploit Theorem 4 to obtain generalization guarantees on the expected disagreement and the expected joint error. PAC-Bayesian bounds on these quantities appeared in [START_REF] Lacasse | PAC-Bayes bounds for the risk of the majority vote and the variance of the Gibbs classifier[END_REF], but under different forms. In Corollary 5 below, we are especially interested in the possibility of controlling the trade-off-between the empirical estimate computed on the samples and the complexity term KL(ρ π)with the help of parameters b and c.

Corollary 5. For any domains S and T over X×Y , any set of voters H, any prior π over H, any δ∈(0, 1], any real numbers b > 0 and c > 0, we have: -with a probability at least 1-δ over T ∼ (T X ) mt ,

∀ρ on H, d T (ρ) ≤ c 1-e -c d T (ρ)+ 2KL(ρ π)+ln 1 δ m t × c ,
-with a probability at least 1-δ over S ∼ (S) ms ,

∀ρ on H, e S (ρ) ≤ b 1-e -b e S (ρ)+ 2KL(ρ π)+ln 1 δ m s × b ,
where d T (ρ) and e S (ρ) are the empirical estimations of the target voters' disagreement and the source joint error.

Proof. Given π and ρ over H, we consider a new prior π 2 and a new posterior ρ 2 , both over H 2 , such that: [START_REF] Lacasse | PAC-Bayes bounds for the risk of the majority vote and the variance of the Gibbs classifier[END_REF]. Let us define two new loss functions for a "paired voter"

∀ h ij = (h i , h j ) ∈ H 2 , π 2 (h ij ) = π(h i )π(h j ), and ρ 2 (h ij ) = ρ(h i )ρ(h j ). Thus, KL(ρ 2 π 2 ) = 2KL(ρ 2 π 2 ) (see
h ij ∈ H 2 : d (h ij , x, y) = I h i (x) = h j (x) , and e (h ij , x, y) = I h i (x) = y ×I h j (x) = y .
Then, the bound on d T (ρ) is obtained from Theorem 4 with := d , and Equation (2). The bound on e S (ρ) is similarly obtained with := e and using Equation (5).

For algorithmic simplicity, we deal with Theorem 3 when q→∞. Thanks to Corollary 5, we obtain the following generalization bound defined with respect to the empirical estimates of the target disagreement and the source joint error. Theorem 6. For any domains S and T over X×Y , any set of voters H, any prior π over H, any δ∈(0, 1], any b>0 and c>0, with a probability at least 1-δ over the choices of S∼(S) ms and T ∼(T X ) mt , we have

∀ρ on H, R T (G ρ ) ≤ c 1 2 d T (ρ) + b e S (ρ) + η T \S + c mt×c + b ms×b 2 KL(ρ π) + ln 2 δ ,
where d T (ρ) and e S (ρ) are the empirical estimations of the target voters' disagreement and the source joint error, and b = b 1-e -b β ∞ (T S), and c = c 1-e -c . Proof. We bound separately d T (ρ) and e S (ρ) using Corollary 5 (with probability 1-δ 2 each), and then combine the two upper bounds according to Theorem 3.

From an optimization perspective, the problem suggested by the bound of Theorem 6 is much more convenient to minimize than the PAC-Bayesian bound that [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF] derived from Theorem 2. The former is smoother than the latter: the absolute value related to the domain disagreement dis ρ (S X , T X ) of Equation ( 4) disappears in benefit of the domain divergence β ∞ (T S), which is constant and can be considered as an hyperparameter of the algorithm. Additionally, Theorem 2 requires equal source and target sample sizes while Theorem 6 allows m s =m t . Moreover, recall that [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF] choose to ignore the ρ-dependent non-constant term λ(ρ). In our case such compromise is not mandatory in order to apply the theoretical result to real problems, since our non-estimable term η T \S is constant and does not depend on the learned ρ. Hence, we can neglect η T \S without any impact on the optimization problem described in the next section. Moreover, it is realistic to consider this is a small quantity in situations where the source and target supports are similar. negative, source points are red, resp. green. The blue dashed line shows the decision boundaries of algorithm pbda [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF].

Specialization to Linear Classifiers

In order to derive an algorithm, we now specialize the bounds of Theorems 3 and 6 to the risk of a linear classifier h w , defined by a weight vector w ∈ R d :

∀x ∈ X, h w (x) = sign (w • x) .
The taken approach is the one privileged in numerous PAC-Bayesian works (e.g., [START_REF] Langford | PAC-Bayes & margins[END_REF][START_REF] Ambroladze | Tighter PAC-Bayes bounds[END_REF][START_REF] Mcallester | Generalization bounds and consistency for latent structural probit and ramp loss[END_REF][START_REF] Parrado-Hernández | PAC-Bayes bounds with data dependent priors[END_REF]Germain et al., 2009a;[START_REF] Mcallester | A PAC-Bayesian tutorial with a dropout bound[END_REF], as it makes the risk of the linear classifier h w and the risk of a (properly parametrized) majority vote coincide, while in the same time promoting large margin classifiers. To this end, let H be the set of all linear classifiers over the input space, H = h w | w ∈ R d , and let ρ w over H be a posterior distribution, respectively a prior distribution π 0 , that is constrained to be a spherical Gaussian with identity covariance matrix centered on vector w, respectively 0,

∀h w ∈ H, ρ w (h w ) = 1 √ 2π d e -1 2 w -w 2 , and π 0 (h w ) = 1 √ 2π d e -1 2 w 2 .
The KL-divergence between ρ w and π 0 simply is

KL(ρ w π 0 ) = 1 2 w 2 . (11)
Thanks to this parameterization, the majority vote classifier B ρw corresponds to the one of the linear classifier h w (see above cited PAC-Bayesian works). That is,

∀ x∈X, w∈H, h w (x) = sign E h w ∼ρw h w (x) = B ρw (x) . Then, R D (h w ) = R D (B ρw ) for any data distribution D.
Moreover, [START_REF] Langford | PAC-Bayes & margins[END_REF] showed that the closely related Gibbs risk (Equation ( 1)) is related to the linear classifier margin y w•x

x , as follows:

R D (G ρw ) = E (x,y)∼D Φ y w • x x , (12)
where Φ

(x)= 1 2 -1 2 Erf x √ 2 , and Erf(x) = 2 √ π
x 0 e -t 2 dt is the Gauss error function. Here, Φ(x) can be seen as a smooth surrogate-sometimes called the probit loss (e.g., [START_REF] Mcallester | Generalization bounds and consistency for latent structural probit and ramp loss[END_REF])-of the zero-one loss function I x ≤ 0 relying on y w•x

x . Note that w plays an important role on the value of R D (G ρw ), but not on R D (h w ). Indeed, R D (G ρw ) tends to R D (h w ) as w grows, which can provide very tight bounds (see the empirical analyses of [START_REF] Ambroladze | Tighter PAC-Bayes bounds[END_REF]Germain et al., 2009b). In the PAC-Bayesian context, w turns out to be a measure of complexity of the learned classifier, as Equation ( 11) shows.

Germain, Habrard, Laviolette, Morvant A New PAC-Bayesian Perspective on Domain Adaptation

We now seek to express the expected disagreement d D (ρ w ) and the expected joint error e D (ρ w ) of Equations ( 2) and ( 5) related to the parameterized distribution ρ w . As shown by [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF] the former is given by

d D (ρ w ) = E x∼D X Φ dis w • x x ,
where Φ dis (x) = 2×Φ(x)×Φ(-x). Following a similar approach, we obtain, for all w ∈ R,

e D (ρ w ) = E (x,y)∼D E h∼ρw E h ∼ρw I h(x) =y I h (x) =y = E (x,y)∼D E h∼ρw I h(x) =y E h ∼ρw I h (x) =y = E (x,y)∼D Φ err y w • x x , with Φ err (x) = Φ(x) 2 .
As function Φ in Equation ( 12), functions Φ err and Φ dis defined above can be interpreted as loss functions for linear classifiers (illustrated by Figure 1).

Domain adaptation bound.

Theorem 3 specialized to linear classifiers gives the following corollary. Note that, as mentioned above, R

T (h w ) = R T (B ρw ) ≤ 2 R T (G ρw ).
Corollary 7. Let S and T respectively be the source and the target domains on X×Y . For all w ∈ R, we have :

R T (h w ) ≤ d T (ρ w ) + 2 β ∞ (T S) × e S (ρ w ) + 2 η T \S ,
Figure 1 leads to an insightful geometric interpretation of the domain adaptation trade-off promoted by Corollary 7. For fixed values of β ∞ (T S) and η T \S , the target risk R T (h w ) is upper-bounded by a (β ∞weighted) sum of two losses. The expected Φ err -loss (i.e., the joint error) is computed on the (labeled) source domain; it aims to label the source examples correctly, but is more permissive on the required margin than the Φ-loss (i.e., the Gibbs risk). The expected Φ dis -loss (i.e., the disagreement) is computed on the target (unlabeled) domain; it promotes large unsigned target margins. Thus, if a target domain fulfills the cluster assumption (described in Section 5.2), d T (ρ w ) will be low when the decision boundary crosses a low-density region between the homogeneous labeled clusters. Hence, Corollary 7 reflects that some source errors may be allowed if, doing so, the separation of the target domain is improved.

Generalization bound and learning algorithm. Theorem 6 specialized to linear classifiers gives the following.

Corollary 8. For any domains S and T over X×Y , any δ∈(0, 1], any a>0 and b>0, with a probability at least 1-δ over the choices of S∼(S) ms and T ∼(T X ) mt , we have

∀w ∈ R : R T (h w ) ≤ c d T (ρ w ) + 2 b e S (ρ w ) + 2 η T \S + 2 c mt×c + b ms×b w 2 + ln 2 δ .
For a source S={(x i , y i )} ms i=1 and a target T ={(x i )} mt i=1 samples of potentially different size, and some hyperparameters C>0, B>0, minimizing the next objective function w.r.t w∈R is equivalent to minimize the above bound.

C d T (ρ w ) + B e S (ρ w ) + w 2 (13) = C mt i=1 Φ dis w•x i x i + B ms i=1 Φ err y i w•xi xi + w 2 .
We call the optimization of Equation ( 13) by gradient descent the dalc algorithm, for Domain Adaptation of Linear Classifiers. The kernel trick applies to dalc. That is, given a kernel k : R d ×R d →R, one can express a linear classifier in a RKHS8 by a dual weight vector α α α ∈ R ms+mt : Even though the objective function is highly non-convex, we achieved good empirical results by minimizing the "kernelized" version of Equation ( 13) by gradient descent, with a uniform weight vector as a starting point. More details are given in Appendix B.

h w (•) = sign ms i=1 α i k(x i , •) + mt i=1 α i+ms k(x i , •) .

Experimental Results

Firstly, Figure 2 illustrates the behavior of the decision boundary of our algorithm dalc on an intertwining moons toy problem9 , where each moon corresponds to a label. The target domain, for which we have no label, is a rotation of the source domain. The figure shows clearly that dalc succeeds to adapt to the target domain, even for a rotation angle of 50 • . We see that dalc does not rely on the restrictive covariate shift assumption, as some source examples are misclassified. This behavior illustrates the dalc trade-off in action, that concedes some errors on the source sample to lower the disagreement on the target sample.

Secondly, we evaluate dalc on the classical Amazon.com Reviews benchmark [START_REF] Blitzer | Domain adaptation with structural correspondence learning[END_REF] according to the setting used by [START_REF] Chen | Co-training for domain adaptation[END_REF][START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF]. This dataset contains reviews of four types of products (books, DVDs, electronics, and kitchen appliances) described with about 100, 000 attributes. Originally, the reviews were labeled with a rating from 1 to 5. [START_REF] Chen | Co-training for domain adaptation[END_REF] proposed a simplified binary setting by regrouping ratings into two classes (products rated lower than 3 and products rated higher than 4). Moreover, they reduced the dimensionality to about 40, 000 by only keeping the features appearing at least ten times for a given domain adaptation task. Finally, the data are preprocessed with a tf-idf re-weighting. A domain corresponds to a kind of product. Therefore, we perform twelve domain adaptation tasks. For instance, "books→DVD's" is the task for which the source domain is "books" and the target one is "DVDs". We compare dalc with the classical non-adaptive algorithm svm (trained only on the source sample), the adaptive algorithm dasvm [START_REF] Bruzzone | Domain adaptation problems: A DASVM classification technique and a circular validation strategy[END_REF], the adaptive co-training coda [START_REF] Chen | Co-training for domain adaptation[END_REF], and the PAC-Bayesian domain adaptation algorithm pbda [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF] based on Theorem 2. Note that, in [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF], dasvm has shown better accuracy than svm, coda and pbda. Each parameter is selected with a grid search thanks to a usual cross-validation (CV) on the source sample for svm, and thanks to a reverse validation procedure10 (RCV) for coda, dasvm, pbda, and dalc. The algorithms use a linear kernel and consider 2, 000 labeled source examples and 2, 000 unlabeled target examples. Table 1 reports the error rates of all the methods evaluated on the same separate target test sets proposed by [START_REF] Chen | Co-training for domain adaptation[END_REF].

Above all, we observe that the adaptive approaches show the best result, implying that tackling this problem with a domain adaptation method is reasonable. Then, our new method dalc is the best algorithm overall on this task. Except for the two adaptive tasks between "electronics" and "DVDs", dalc is either the best one (six times), or the second one (four times). Moreover, dalc clearly increases the performance over the previous PAC-Bayesian algorithm (pbda), which confirms that our new bound improves the analysis done by [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF].

Conclusion

We propose a new domain adaptation analysis for majority vote learning. It relies on an upper bound on the target risk, expressed as a trade-off between the voters' disagreement on the target domain, the voters' joint errors on the source one, and a term reflecting the worst case error in regions where the source domain is non-informative. To the best of our knowledge, a crucial novelty of our contribution is that the trade-off is controlled by the divergence β q (T S) (Equation 7) between the domains: The divergence is not an additive term (as in many domain adaptation bounds) but is a factor weighting the importance of the source information. Our analysis, combined with a PAC-Bayesian generalization bound, leads to a new domain adaptation algorithm for linear classifiers. The empirical experiments show that our new algorithm outperforms the previous PAC-Bayesian approach of [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF].

As future work, we first aim at investigating the case where the domains' divergence β q (T S) can be estimated, i.e., when the covariate shift assumption holds or when some target labels are available. In these scenarios, β q (T S) might not be considered as a hyperparameter to tune. Last but not least, the term η T \S of our bound-suggesting that the two domains should live in the same regions-can be dealt with a representation learning approach. As mentioned in Section 5.3, this could be an incentive to combine our learning algorithm with existing representation learning techniques. In another vein, considering an active learning setup (as in [START_REF] Berlind | Active nearest neighbors in changing environments[END_REF], one could query the labels of target examples to estimate the value bounded by η T \S . We see this as a great source of inspiration for new algorithms for this learning paradigm.

Appendix A Proof of Theorem 4

Proof. We use the following shorthand notation: The final result is obtained by using the inequality 1-exp(-z) ≤ z.

L D (h) = E

B Using dalc with a kernel function

Let S = {(x i , y i )} ms i=1 , T = {x i } mt i=1 and m = m s + m t . We will denote

x # = x i if # ≤ m s (source examples) x #-ms otherwise.
(target examples)

The kernel trick allows us to work with dual weight vector α α α ∈ R m that is a linear classifier in an augmented space. Given a kernel k : R d × R d → R, we have

h w (•) = sign m i=1 α i k(x i , •) .
Let us denote K the kernel matrix of size m × m such as K i,j = k(x i , x j ) . In that case, the objective function-Equation ( 13) of the main paper-can be rewritten in term of the vector For our experiments, we minimize this objective function using a Broyden-Fletcher-Goldfarb-Shanno method (BFGS) implemented in the scipy python library [START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF].

We initialize the optimization procedure at α i = 1 m for all i ∈ {1, . . . , m}.

C Experimental Protocol

For obtaining the dalc RCV results of Table 1, the reverse validation procedure searches on a 20 × 20 parameter grid for a C between 0.01 and 10 6 and a parameter B between 1.0 and 10 8 , both on a logarithm scale. The results of the other algorithms are reported from [START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF].

Figure 1 :

 1 Figure 1: Graphical representation of the loss functions given by the specialization to linear classifiers.

Figure 2 :

 2 Figure 2: Decision boundaries of dalc on the intertwining moons toy problem, for fixed parameters B=C=1, and a RBF kernel k(x, x ) = exp(-xx 2 ). The target points are black. The positive, resp. negative, source points are red, resp. green. The blue dashed line shows the decision boundaries of algorithm pbda (Germain et al., 2013).

  Consider any convex function ∆ : [0, 1]×[0, 1] → R. Applying consecutively Jensen's Inequality and the change of measure inequality (seeSeldin & Tishby (2010, Lemma 4) andMcAllester (2013, Equation (20))), we obtain∀ρ on H : m×∆ E h∼ρ L S (h), E h∼ρ L D (h) ≤ E h∼ρ m×∆ (L S (h), L D (h)) ≤ KL(ρ π) + ln X π (S) , with X π (S) = E h∼π e m×∆(L S (h), L D (h)) .Then, Markov's Inequality givesPr S∼D m X π (S) ≤ 1 δ E S ∼D m X π (S ) ≥ 1-δ , and E S ∼D m X π (S ) = E S ∼D m E h∼π e m×∆(L S (h), L D (h)) = E h∼π E S ∼D m e m×∆(L S (h), L D (h)) (h)) k (1-L D (h)) m-k e m×∆( k m , L D (h)) ,(14)where the last inequality is due to Maurer (2004, Lemma 3) (we have an equality when the output of is in {0, 1}). As shown inGermain et al. (2009a, Corollary 2.2), by fixing∆(q, p) = -c×q -ln[1-p (1-e -c )] ,Line 14 becomes equal to 1, and then E S ∼D m X π (S ) ≤ 1. Hence,Pr S∼D m ∀ρ on H : -c E h∼ρ L S (h) -ln[1-E h∼ρ L D (h) (1-e -c )] ≤ KL(ρ π) + ln 1 δ m ≥ 1-δ .By reorganizing the terms, we have, with probability 1-δ over the choice of S ∈ D m , ∀ρ on H : E h∼ρ L D (h) ≤ 1 1-e -c 1 -exp -c E h∼ρ L S (h) -KL(ρ π) + ln 1 δ m .

  j K i,j .

Table 1 :

 1 Error rates on Amazon dataset. Best risks appear in bold and seconds are in italic.

		svm	dasvm	coda	pbda	dalc
		(CV)	(RCV)	(RCV)	(RCV)	(RCV)
	books→DVDs	0 .179	0.193	0.181	0.183	0.178
	books→electro	0.290	0 .226	0.232	0.263	0.212
	books→kitchen	0.251	0.179 0.215	0.229	0 .194
	DVDs→books	0.203	0.202	0.217 0 .197	0.186
	DVDs→electro	0.269	0.186 0 .214	0.241	0.245
	DVDs→kitchen	0.232	0.183 0 .181	0.186	0.175
	electro→books	0.287	0.305	0.275 0.232	0 .240
	electro→DVDs	0.267	0.214 0.239 0 .221	0.256
	electro→kitchen	0 .129	0.149	0.134	0.141	0.123
	kitchen→books	0.267	0.259 0 .247 0 .247	0.236
	kitchen→DVDs	0.253	0.198 0.238	0.233	0 .225
	kitchen→electro	0.149	0.157	0.153 0.129	0 .131
	Average	0.231	0 .204	0.210	0.208	0.200

Domain adaptation is associated with transfer learning(Pan & Yang,

2010;[START_REF] Quionero-Candela | Dataset shift in machine learning[END_REF]. 2 This setting is not too restrictive since many machine learning algorithms can be seen as a majority vote learning. For instance. ensemble learning and kernel methods output classifiers which can be interpreted as majority votes.3 More precisely, this term can only be estimated in the presence of labeled data from both the source and the target domains.

Note that in this paper, we do not focus on learning a new representation to help the adaptation: We directly aim at adapting in the current representation space.

The quantity d D (ρ) is also used in the domain adaptation bound of[START_REF] Germain | A PAC-Bayesian approach for domain adaptation with specialization to linear classifiers[END_REF] to measure divergence between distributions. See forthcoming Theorem 2.

In a symmetric H, for all h ∈ H, its inverse -h is also in H.

For q ≥ 0, we can show βq(T S) = 2 q-1 q Dq (T S) , where Dq(T S) is the Rényi divergence between T and S.

Technical Report V 3

It is non-trivial to show that the kernel trick holds when π 0 and ρw are Gaussian over infinite-dimensional feature space. As mentioned by[START_REF] Mcallester | Generalization bounds and consistency for latent structural probit and ramp loss[END_REF], it is, however, the case provided we consider Gaussian processes as measure of distributions π 0 and ρw over (infinite) H.

We generate each pair of moons with the make moons function provided in scikit-learn[START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF].

For details on the reverse validation procedure, see[START_REF] Bruzzone | Domain adaptation problems: A DASVM classification technique and a circular validation strategy[END_REF];[START_REF] Zhong | Cross validation framework to choose amongst models and datasets for transfer learning[END_REF]. Other details on our experimental protocol are given in Appendix C.Technical Report V 3


Jonatan Barrera-Chimal

Sonia Prince

Fouad Fadel

Soumaya El Moghrabi

David G Warnock

Peter Kolkhof

Dr Frédéric Jaisser
email: frederic.jaisser@inserm.fr

Sulfenic acid modification of ET-B receptor is responsible for the benefit of a non-steroidal MR antagonist in renal ischemia

Acute kidney injury is associated with high mortality rates and chronic kidney disease development. Ischemia/reperfusion (IR) is an important cause of acute kidney injury.

Unfortunately, there is no available pharmacological approach to prevent/limit renal IR injury in common clinical practice. Renal IR is characterized by diminished nitric oxide bioavailability and a reduction in the renal blood flow; however, the mechanisms leading to these alterations are poorly understood. In a rat model of renal IR, we show that the administration of the novel non-steroidal MR antagonist BR-4628 can prevent or treat the renal dysfunction and tubular injury induced by IR. Renal injury induced by ischemia was associated with increased oxidant damage, which leads to a cysteine sulfenic acid modification in ET B receptor and consequently decreased eNOS activation. These modifications were efficiently prevented by non-steroidal MR antagonism. Furthermore, we demonstrate that the protective effect of BR-4628 against IR was lost when a selective ET B receptor antagonist was co-administered. These data describes a new mechanism for reduced eNOS activation during renal IR that can be blocked by MR antagonism with BR-4628.

Acute kidney injury (AKI) is one of the most common complications in hospitalized and intensive care unit patients and is associated with adverse outcomes such as increased mortality rates and chronic kidney disease development. [START_REF] Lewington | Raising awareness of acute kidney injury: a global perspective of a silent killer[END_REF] An important cause of AKI is renal ischemia/reperfusion (IR). [START_REF] Mehta | Program to Improve Care in Acute Renal Disease: Spectrum of acute renal failure in the intensive care unit: the PICARD experience[END_REF] Currently, there are not any effective pharmacological approaches to treating or preventing AKI despite recent advances in the understanding of the underlying mechanisms involved. [START_REF] Kinsey | Pathogenesis of acute kidney injury: foundation for clinical practice[END_REF][START_REF] Jo | Pharmacologic treatment of acute kidney injury: why drugs haven't worked and what is on the horizon[END_REF] Ischemic AKI is a complex entity in which multiple processes are involved including: tubular cell injury, endothelial alterations, increased oxidant stress and inflammation. [START_REF] Bonventre | Cellular pathophysiology of ischemic acute kidney injury[END_REF][START_REF] Sharfuddin | Pathophysiology of ischemic acute kidney injury[END_REF] One of the hallmarks of ischemic AKI is the reduction in the renal blood flow. [START_REF] Sutton | Microvascular endothelial injury and dysfunction during ischemic acute renal failure[END_REF] This alteration is mediated by numerous factors such as an increase in the production of vasoconstrictor mediators and/or a reduction in the levels of vasodilator agents. [START_REF] Bonventre | Cellular pathophysiology of ischemic acute kidney injury[END_REF] A reduction in the NO bioavailability is proposed to play an important contribution to the sustained vasoconstriction observed in renal IR, which perpetuates the injury as a consequence of worsened renal ischemia. [START_REF] Phillips | Nitric oxide mechanism of protection in ischemia and reperfusion injury[END_REF] Of particular interest is the role of mineralocorticoid receptor (MR) activation in modulating the blood flow during renal ischemia: previous studies have shown that MR antagonism with spironolactone protects against renal IR injury by preventing the reduction in the renal blood flow. [START_REF] Mejía-Vilet | Renal ischemia-reperfusion injury is prevented by the mineralocorticoid receptor blocker spironolactone[END_REF][START_REF] Sánchez-Pozos | Recovery from ischemic acute kidney injury by spironolactone administration[END_REF] The mechanisms by which MR antagonism leads to kidney protection remain unclear. Although the blockade of MR may be a promising therapeutic approach, the use of spironolactone in renal ischemia is limited the increased risk of hyperkalemia especially in patients with compromised renal function. [START_REF] Weir | Potassium homeostasis and renin-angiotensin-aldosterone system inhibitors[END_REF][START_REF] Palmer | Managing hyperkalemia caused by inhibitors of the renin-angiotensinaldosterone system[END_REF] Recently novel non-steroidal MR antagonists (MRA) such as BR-4628 have been described. BR-4628 is derived from the dihydropyridine and displays a greater selectivity and potency for MR inhibition. [START_REF] Fagart | A new mode of mineralocorticoid receptor antagonism by a potent and selective nonsteroidal molecule[END_REF] Furthermore, the risk of hyperkalemia may be less with this class of agents, at least in the chronic setting. [START_REF] Pitt | Safety and tolerability of the novel non-steroidal mineralocorticoid receptor antagonist BAY 94-8862 in patients with chronic heart failure and mild or moderate chronic kidney disease: a randomized, double-blind trial[END_REF] In this study we tested the efficiency of the non-steroidal MRA BR-4628 to protect against renal injury induced by IR and provide insights into the mechanism of kidney protection conferred by MRA. First, we evaluated whether BR-4628 can prevent or treat renal injury induced by IR. AKI induced by bilateral renal IR was assessed by the increase in plasma levels of creatinine (fig. 1A) and urea (fig. 1B). These alterations were prevented by BR-4628 when administered before or up to 3 hours after the ischemic insult. We determined the effects on sensitive biomarkers of tubular injury such as Hsp72, Kim-1 and NGAL. The increased urinary Hsp72 induced by IR was significantly reduced in rats treated with BR-4648 (fig. 1C). A similar pattern was observed for renal Kim-1 (fig. 1D) and NGAL (fig. 1E) mRNA levels. The tubular lesions induced by IR were documented by histological analysis.

Kidneys from rats subjected to IR displayed tubular casts, tubular dilation and cell detachment (fig. 1H). These structural alterations were also prevented by BR-4628 administration before (fig. 1I) or after (fig. 1J) the induction of the renal ischemia. This effect was corroborated by the quantification of injured tubules (fig. 1F). As an indicator of apoptosis in the kidneys with bilateral renal IR, we analyzed the protein levels of cleaved caspase-3. Renal IR was characterized by an increase in cleaved caspase-3 levels, which was prevented by MR antagonism (Supplementary fig. 1). To evaluate if the treatment with BR-4628 reduced inflammation induced by renal IR, we performed immunohistochemistry for CD68 positive cells to evaluate macrophage infiltration in renal sections. As shown in supplementary figure 2 renal IR was associated with a marked increase in macrophage infiltration, an effect that was efficiently prevented by the administration of BR-4628 either before or after the induction of the renal injury. The levels of potassium remained unchanged either by IR or by MR antagonism with BR-4628 (Supplementary fig. 3). Therefore, the antagonism of MR by BR-4628 prevented or treated the renal dysfunction and tubular lesions induced by IR. These functional and histological benefits were associated to the prevention of renal oxidant stress. Indeed, the production of reactive oxygen species is a hallmark of IR and may induce damage to biomolecules such as proteins and alter their function. [START_REF] Himmelfarb | Oxidative stress is increased in critically ill patients with acute renal failure[END_REF] In addition to the ROS released upon renal reperfusion, it has been shown that MR activation contributes to increased ROS production by activation of the NADPH oxidase in both endothelial and smooth muscle cells. [START_REF] Caprio | Functional mineralocorticoid receptors in human vascular endothelial cells regulate intercellular adhesion molecule-1 expression and promote leukocyte adhesion[END_REF][START_REF] Callera | Aldosterone activates vascular p38MAP kinase and NADPH oxidase via c-Src[END_REF] Moreover in endothelial cells aldosterone reduced the expression of glucose-6phosphate dehydrogenase, a key regulator of the redox balance, therefore increasing oxidative stress. [START_REF] Leopold | Aldosterone impairs vascular reactivity by decreasing glucose-6phosphate dehydrogenase activity[END_REF] Renal IR increased the amount of oxidized proteins in the kidney, an effect that is prevented by MR antagonism before or after renal IR (Supplementary fig. 4A). BR-4628 administration also prevented the IR-induced increase in the urinary levels of hydrogen peroxide, a marker of oxidant stress (Supplementary fig. 4B).

The underlying mechanisms of the benefit of MRA in renal IR are unclear; we hypothesized that the improvement in renal blood flow may be associated with the effects of ET-1 on renal hemodynamics. ET-1 is one of the vasoactive substances that are altered during renal IR and that may contribute to the enhanced vasoconstriction. [START_REF] Wilhelm | Endothelin upregulation and localization following renal ischemia and reperfusion[END_REF] ET-1 induces constriction when it binds on the ET A receptor in the smooth muscle cells. On the contrary, when it binds the ET B receptor in endothelial cells, it exerts vasodilator actions through increased NO production by eNOS. [START_REF] Liu | Endothelin-1 activates endothelial cell nitric-oxide synthase via heterotrimeric G-protein betagamma subunit signaling to protein kinase B/Akt[END_REF][START_REF] Mazzuca | Vascular endothelin receptor type B: structure, function and dysregulation in vascular disease[END_REF] As expected, renal IR induced an increase in mRNA levels of prepro-endothelin-1 (Fig 2A). This was not prevented by MR antagonism suggesting that the effect of MRA is downstream of ET-1 production (Fig. 2A). The mRNA levels of ET A receptor were not altered (data not shown). However, ET B receptor expression was reduced in the setting of IR (Fig. 2B). This effect was prevented by BR-4628 administration before or after IR. To analyze whether the mechanism of kidney protection from IR by MR antagonism is dependent on the ET B receptor, we included another groups of rats in which a selective ET B receptor antagonist (BQ-788) [START_REF] Ishikawa | Biochemical and pharmacological profile of a potent and selective endothelin B-receptor antagonist, BQ-788[END_REF] was co-administered with BR-4628. Renal IR was accompanied by increased plasma creatinine (Fig. 3A) and renal vascular resistance (Fig. 3B), significantly reduced renal blood flow (Supplementary fig. 5A) and tubular injury (Fig. 3C).

No changes were observed for the mean arterial pressure among the groups (Supplementary fig. 5B). As previously reported, [START_REF] Kuro | Selective antagonism of the ETA receptor, but not the ETB receptor, is protective against ischemic acute renal failure in rats[END_REF] ET B antagonism had no effect on renal IR severity (similar levels of creatinine, renal blood flow and renal injury in the IR+ BQ-788 group when compared to untreated IR rats) (Figs. 3 A-C). Again, MR antagonism with BR-4628 was associated protection against IR (Figs. 3 A-C). Importantly, ET B receptor antagonism fully blunted the protective effect of BR-4628: creatinine levels remains elevated, renal vascular resistance was augmented and histology analysis indicated similar degree of renal injury than in the non-treated IR group, showing that the protective effect of MRA is lost when an ET B antagonist is co-administered.

To understand the underlying mechanisms linking the ET B receptor and the benefit of MR antagonism in IR-induced renal injury, we evaluated the presence of a cysteine sulfenic acid modification, a post-translational modification of ET B . It has been previously reported that Cysteine (Cys405) of ET B can be modified, preventing its activation and therefore preventing increased eNOS activity mediated by the ET B receptor when stimulated by ET-1. [START_REF] Okamoto | Palmitoylation of human endothelinB. Its critical role in G protein coupling and a differential requirement for the cytoplasmic tail by G protein subtypes[END_REF] In cultured pulmonary artery endothelial cells, aldosterone induces inactivation of the ET B receptor through a mechanism involving MR activation, increased oxidant stress and sulfenic acid modification in Cys405, thus leading to reduced NO production. [START_REF] Maron | Aldosterone inactivates the endothelin-B receptor via a cysteinyl thiol redox switch to decrease pulmonary endothelial nitric oxide levels and modulate pulmonary arterial hypertension[END_REF] We hypothesized that this mechanism could be translated in vivo and underlies the benefit of the BR-4628 MRA in IR. ET B sulfenic acid modification was analyzed in kidneys following IR, treated or not with the BR-4628 MRA. Renal IR induced ET B receptor sulfonilation in vivo (Fig. 4A). These data suggest that NO bioavailability may be reduced during renal IR due to ET B inactivation by a cysteine sulfenic acid modification. To support this idea we analyzed the phosphorylation of eNOS at Serine residue 1177 (S1177) that is known to activate the production of NO by eNOS, [START_REF] Rafikov | eNOS activation and NO function: structural motifs responsible for the posttranslational control of endothelial nitric oxide synthase activity[END_REF] as well as urinary nitrites and nitrates excretion as an indirect measure of NO production. [START_REF] Barrera-Chimal | Intra-renal transfection of heat shock protein 90 alpha or beta (Hsp90α or Hsp90β) protects against ischemia/reperfusion injury[END_REF][START_REF] Kwon | Diminished NO generation by injured endothelium and loss of macula densa nNOS may contribute to sustained acute kidney injury after ischemia-reperfusion[END_REF] During renal IR, phosphorylation of the S1177 residue is decreased (Fig. 4B), and this effect was associated with a reduction in urinary nitrites and nitrates excretion (Fig. 4C). MR antagonism with BR-4628 prevented the sulfenic acid modification of the ET B receptor (Fig. 4A), and prevented the decrease of S1177 eNOS phosphorylation (Fig. 4B). Moreover, the reduction of urinary nitrites and nitrates was prevented by MRA given before or after the induction of renal IR (Fig. 4C).

Therefore, during renal IR there is an increase in the ROS, which in turn induces a sulfenic acid modification in ET B receptor, preventing its ability to stimulate eNOS and NO production. This results in renal vasoconstriction and decreased blood flow (Fig. 4E). On the other hand, when MR is antagonized with BR-4628 the oxidant injury and the sulfenic acid modification on ET B receptor are prevented, thus maintaining normal eNOS activation and NO balance which appears to blunt the deleterious effects of IR (Fig. 4D).

In summary, we report that BR-4628, a novel non-steroidal MR antagonist is a novel pharmacological approach for prevention or treatment of AKI in a rat model of renal IR injury. Moreover, we provide evidence that during renal IR, the ET B receptor undergoes sulfenic acid modification of a critical cysteine residue that decreases the activation of eNOS and hence, NO production. Finally we demonstrated that the benefit of the non-steroidal MR antagonist BR-4628 against renal IR is dependent on ET B receptor activation and increased NO bioavailability. Altogether, these data supports the use of non-steroidal MR antagonists in the clinical setting for the prevention and/or treatment of ischemic AKI.

CONCISE METHODS

Experimental protocols

All the experiments involving animal manipulation were performed according to the ethical guidelines of INSERM for the care and use of laboratory animals. Male Wistar rats (Janvier Labs, France) weighting 270-300 g were included in the study. The BR-4628 was administered at a dose of 10 mg/kg by oral gavage in vehicle (40% kolliphor, 10% ethanol and 50% water); the dose was selected according to previous studies performed in rats. [START_REF] Schupp | Mineralocorticoid receptor-mediated DNA damage in kidneys of DOCA-salt hypertensive rats[END_REF] In the first set of experiments 20 rats were divided in 4 groups: sham-operated rats, rats with 25 min of bilateral ischemia (IR), rats treated with BR-4628 at day -2 and 1h before the induction of IR (BR-4628 pre) and rats receiving BR-4628 (10 mg/kg) 3 hours after the IR (BR-4628 post). For the second set of experiments 30 rats were divided in 5 groups: sham, rats underwent 25 min of bilateral ischemia (IR), rats receiving the ET B antagonist BQ-788 for 2 days before IR (ET B A), rats receiving BR-4628 for 2 days before the induction of IR (MRA) and rats receiving BR-4628 and BQ-788 2 days before renal ischemia (ET B A + MRA). Rats were sacrificed 24h after IR.

Kidney ischemia/reperfusion injury model

The rats were anesthetized by an intra-peritoneal injection of sodium pentobarbital (30 mg/kg) and placed on a heating pad with a rectal probe to maintain a constant body temperature around 37 °C. An abdominal incision was performed and both renal pedicles were exposed and dissected. Renal ischemia was induced by placing non-traumatic vascular clamps over the pedicles and was visually verified by change in kidney color. After 25 min the clamps were released and the reperfusion was corroborated by the return of oxygenated blood to the kidney. The abdominal incision was closed in two layers with 3-0 sutures and the reperfusion was allowed for 24 h.

Evaluation of functional parameters

After 24 h of reperfusion the rats were anesthetized with an intra-peritoneal injection of sodium pentobarbital (30 mg/kg) and placed on a heating pad. The right femoral artery was catheterized with polyethylene tubing (PE-50) in order to monitor the mean arterial pressure.

The left renal artery was dissected and the renal blood flow was recorded by placing an ultrasound flow probe (Transonic) filled with ultrasonic coupling gel around the artery. A blood sample was taken and the plasma creatinine and urea concentrations were determined by an automatic analyzer (Konelab 20i, Thermoscientific). The levels of potassium were analyzed by flame photometer. At the end of the experiment the right kidney was removed and fixed in Bouin Fixative solution and the left kidney was quickly frozen for molecular studies.

Histological analysis

After fixation of the tissue, the kidney slices were dehydrated and embedded in paraffin. Sections of 4 m were made and stained by hematoxilin and eosin. For each rat, 10 sub-cortical fields were visualized and analyzed on a Leica DM4000 microscope at a magnification of 200 X. The percentage of injured tubules was blindly analyzed. Tubular injury was assessed in Hematoxylin and eosin stained sections in at least 10 cortical fields per rat. The number of tubules displaying epithelial cell necrosis and detachment, cast formation or tubular dilation was determined. The percentage of injured tubules was calculated against the total number of tubules per field. For CD68 immunohistochemistry studies, 4 m sections from paraffin embedded tissues were de-waxed and probed against CD68 [ED1] (Abcam antibody ab31630). Ten cortical fields (magnification 400X) were randomly recorded from each kidney slide. The number of cells that were positive for CD68 were counted and reported as CD68 positive cells per high power field.

RNA extraction and real time PCR

Total RNA extraction was carried out from kidney cortex by using the TRIZOL reagent (Life Technologies) according to the manufacturer's instructions. The reverse transcription was performed with 1 g of RNA and the Superscript II Reverse transcriptase Kit (Life Technologies, Carlsbad, CA, USA). Transcript levels of genes were analyzed by real-time PCR (fluorescence detection of SYBR green) in an iCycler iQ apparatus (Bio-rad).

The mRNA levels were normalized by the amount of 18S as an endogenous control. The primer sequences of the analyze genes are listed in the supplementary table 1.

Oxyblot

The protein carbonylation levels were assessed with the Oxyblot protein oxidation detection kit, following the manufacturer's instructions (Chemicon, Millipore). For each assay, 10 μg of protein were used. To quantify protein oxidation, we defined an oxidation index as the ratio of the densitometric value of the Oxyblot bands versus -actin levels (Las3000 DarkBox, Fuji Photo Film Europe, GMBH).

Western blot analysis

Total renal proteins were isolated from cortexes from each group and homogenized in 1 % SDS homogenization buffer with protease inhibitor (Roche). Protein samples containing 20 g of total protein were diluted in Laemmli buffer and heated at 95 °C. The proteins were resolved in 4-15% Mini-PROTEAN TGX precast polyacrylamide gels (Bio-rad) and transferred onto polyvinylidinedifluoride membranes (Bio-rad). Membranes were then blocked with 5 % blotting-grade non-fat dry milk. Membranes were then incubated in 0.1 % blotting-grade non-fat dry milk with their respective antibodies. Specific antibodies against ET B (Santa Cruz Biotechnology 1:500), Caspase-3 (Santa Cruz Biotechnology 1:200) or pS1177-eNOS (Abcam 1:5000) were used. After incubation with primary antibody, membranes were washed and incubated with their respective secondary antibody. As a loading control, membranes were incubated) overnight at 4 C with goat anti-actin antibody (Santa Cruz Biotechnology, 1:5000 dilution). Proteins were detected with an enhanced chemiluminescence kit (Bio-rad). For urinary Hsp72 detection by western blot, urine was diluted 1:100 in 0.9% saline solution, and 10 µL of each dilution was loaded and resolved by 8.5% SDS-PAGE electrophoresis and electroblotted, as previously described. Membranes were then blocked with 5% blotting-grade non-fat dry milk and incubated in 0.1% blottinggrade non-fat dry milk with anti Hsp72 antibody (Santa Cruz Biotechnology, 1:10000).

Immunoprecipitation

For IP analysis the tissues from each group were pooled and the proteins were extracted in dimedone lysis buffer. ET B receptor was immune-precipitated using protein A agarose beads (Thermo Scientific) and 4 g of anto ET B antibody (Santa Cruz Biotechnology). The immune precipitated proteins were eluted by boiling in Laemmli buffer, blotted and probed for anti-cysteine sulfenic acid (Millipore) and ET B .

Urinary hydrogen peroxide and nitrates and nitrites excretion

The urinary levels of hydrogen peroxide were detected with the Amplex® Red kit (life technologies) according to the manufacturer's instructions. For the detection of urinary nitrates and nitrites the colorimetric nitric oxide assay kit (Oxford Biomedical Research) was used. The data were normalized by the urinary creatinine.

Statistics

The results are represented as means ± SEM. The statistical differences among the groups were determined by Kruskal-Wallis test using the Dunn's post hoc test for multiple comparisons in the GraphPad Prism 6 software. The p value < 0.05 was defined as statistically significant.

Fig 1 .

 1 Fig 1. Benefit of BR-4628 (10 mg/kg) administration against renal injury induced by IR.

Fig 2 .

 2 Fig 2. Effect of renal IR on ET-1 and ET B receptor. (A) mRNA levels of endothelin-1. (B)

Fig 3 .

 3 Fig 3. The benefit of BR-4628 is prevented by ET B receptor antagonism. (A) Plasma

Fig 4 .

 4 Fig 4. Effect of renal IR on ET B receptor sulfenic acid modification and eNOS

ACKNOWLEDGEMENTS

This work was supported by grants from Institut National de la Santé et de la Recherche Médicale, the "Centre de Recherche Industrielle et Technique", the Fondation de France, the Agence de la Biomédecine (Appel d'offre recherché "greffe" 2014) and a research grant from Bayer (12127a10). J B-C was supported by a postdoctoral fellowship from the French Society of Nephrology (SPF2012FDR_SN_FRM_Barrera Chimal) and the French Foundation for Medical Research (SPF20130526725).

DISCLOSURE

Peter Kolkhof is an employee at BAYER HealthCare Pharmaceuticals.