Sébastien Razakarivony
email: sebastien.razakarivony@sagem.com

Frédéric Jurie
email: frederic.jurie@unicaen.fr

Small Target Detection combining Foreground and Background Manifolds

This paper focuses on the detection of small objects ( e.g. vehicules in aerial images) on complex backgrounds ( e.g. natural backgrounds). A key contribution of the paper is to show that, in such situations, learning a target model and a background model separately is better than training a unique discriminative model. This contrasts with standard object detection approaches for which objects vs. background classifiers use the same types of visual features/models for both. The second contribution lies in the use of manifold learning approaches to build these models. The proposed detection algorithm is validated on the publicly available OIRDS dataset, on which we obtain state-ofthe-art results.

Introduction

Contrasting with most of the recent papers on object detection -which address the detection of daily life objects in high quality images [START_REF] Everingham | The pascal voc challenge[END_REF][START_REF] Felzenszwalb | Object detection with discriminatively trained part based models[END_REF] -this paper focuses on the detection of small rigid targets (such as vehicles), in any arbitrary position, on complex textured backgrounds (see Fig. 1) . The task is made even more difficult by the fact that some objects can be camouflaged, and because it is often difficult to have large training sets as getting images of the desired targets in real condition is usually costly. Finally, object's context in image (i.e. the pixels surrounding the object) is not strongly correlated with the object itself.

State-of-the-art methods for object detection relyin general -on the use of a discriminative classifier trained to learn class boundaries in the representation space. One typical example is the well known Dalal and Triggs's person detector [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], which represents images with HOG features and classifies person vs. background bounding boxes with SVM classifiers [START_REF] Cortes | Support-vector networks[END_REF].

However, this type of approaches does not seem to be relevant to the detection of small targets on complex backgrounds. First, if the background is rich and the number of (positive) training images is limited, learning reliable discriminative features without over-fitting can not be done without strong regularization, which contrast with the need of having an accurate model of the targets. Second, targets and backgrounds have so different visual properties that it is hard to believe that the same models/features can be adapted to both. Based on these observations, we propose a detection algorithm using two distinct models, one for the background and another for the target, combined to score the candidate windows.

Manifolds are good candidates to model accurately small targets. If a target size is e.g. 40 × 40 pix-Figure 1. Typical images from the OIRDS dataset. Small size vehicles have any orientations. Shadows, highlights and complex textured background make the task very challenging. els its visual appearance lies into a 1, 600-d space despite the fact that only a small number of parameters (among them: the pose, the illumination, etc.) are sufficient to explain its appearance. Manifolds are precisely adapted to represent high dimensional subspaces that can be generated from a space of fewer dimensions. Supporting this assumption, the work of Zhang [START_REF] Zhang | Parametric manifold of an object under different viewing directions[END_REF] shows that images of 3D objects seen from different view points can be represented as points on a low-dimensional manifold. On the other hand, backgrounds do not require (and can not) be modeled as accurately as targets. Regarding their modeling, we follow the work of [START_REF] Carvalho | A weighted image reconstruction based on pca for pedestrian detection[END_REF] and use a PCA based manifold model. Finally, target and background models are combined within a probabilistic framework.

The rest of the paper is as follows: we first introduce some related works, present our approach, and, finally, give an experimental validation on vehicle detection from aerial images, showing our approach constantly outperforms state-of-the-art object detectors.

Related works

Object detection is a very active area of computer vision [START_REF] Everingham | The pascal voc challenge[END_REF]. Most of the recent approaches use the sliding windows framework, proposing powerful descriptors [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], new kernels [START_REF] Vedaldi | Sparse kernel approximations for efficient classification and detection[END_REF], efficient pruning strategies [START_REF] Lampert | Beyond sliding windows: Object localization by efficient subwindow search[END_REF] or new object models [START_REF] Felzenszwalb | Object detection with discriminatively trained part based models[END_REF]. However, none of these recent works are really adapted to the detection of small objets.

Regarding the detection of small objects, state of the art approaches are usually based on saliency detection, the objects to be detected being defined as the regions of the image which do not have the same statistics as the background [START_REF] Seo | Visual saliency for automatic target detection, boundary detection, and image quality assessment[END_REF]. Among the rare papers which tried to explicitly model the targets, we can mention the work of [START_REF] Munder | An experiment study on pedestrian classification[END_REF], which -in addition of introducing a new dataset of 36 × 18 pixels pedestrian imageshave shown that good performance can be obtained by combining standard features such as Haar wavelets or HOG features with SVM/boosting classifiers [START_REF] Enzweiler | Monocular pedestrian detection: Survey and exp[END_REF]. [START_REF] Kembhavi | Vehicle detection using partial least squares[END_REF] presents interesting vehicle detection results by using large and rich set of application-specific image descriptors. Unfortunately their experiments are not reproducible (protocols not given in the paper). Manifold learning has already been used by several authors to address detection tasks. In [START_REF] Pentland | Viewbased and modular eigenspaces for face recognition[END_REF], Pentland et al. introduced the well known eigenfaces, using Principal Component Analysis to build linear face manifolds used for face detection. It has also been applied later to hand detection in [START_REF] Moghaddam | Probabilistic visual learning for object representation[END_REF]. In the same spirit, [START_REF] Carvalho | A weighted image reconstruction based on pca for pedestrian detection[END_REF] uses PCA for object detection, by modeling background and objects as linear manifolds. Interesting results are reported on good quality car and pedestrian images, for high dimensional manifolds. In [START_REF] Feraud | A fast and accurate face detector based on neural networks[END_REF], the authors used autoencoders to build face manifolds for face detection. However, this approach is limited by the lack of background model.

Our approach builds on these recent works by using the best current image features within a manifold learning framework. The contribution of the paper lies in the combination of two types of manifolds, namely autoencoders for the targets and linear manifolds for the backgrounds. As far as we know, this is the first time such a model is proposed.

Our approach

Our approach builds on the standard sliding window framework (e.g. [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]), which consists in classifying densely extracted image sub-windows as foreground or background regions, and applying a non-maximum suppression post processing stage. The contribution of the paper lies in the model used in the scoring function.

As said before, we use two (probabilistic) distinct models, one for backgrounds the other for objects, the score of a candidate window being computed as their log-likelihood:

S(X) = log p obj (X l = obj|X s ) p back (X l = back|X s ) (1)
where X l is the (unknown) class of the window (X l ∈ {obj, back}), X s is the signature of the window (i.e. any visual descriptor such as a HOG descriptor). Probabilities p obj and p back are given by the object and background models respectively. Please note that, as we have two distinct models, p obj = 1p back (which contrasts with standard approaches using a single model).

Both classes (i.e. objects and background) are modeled by manifolds learned during a training stage. If X s,t ∈ H denotes the training signatures 1 representative of a class (H is the signature space), building a Riemanian manifold M representative of these signatures is equivalent to finding a function f , such as 1 we use terms signatures and visual features indistinctively.

(following the Nash embedding theorem):

∀X s,t ∈ M, ∃!Y ∈ R n , Y = f (X s,t ) (2)
f is called the embedding of M, and is an isometric function. Obviously, if X s lies on the manifold, f -1 • f (X s ) = X s . f -1 projects any point of the input space onto the manifold M . By denoting P M = f -1 • f , we can define the distance to the manifold by:

D M (X s ) = |X s -P M (X s )| (3)
where |y| represent the Euclidian norm of y. Finally, we use this distance to derive the probability for a signature X s to be generated by the manifold M:

p(X s ∈ M|X s ) = α exp - D M (X s ) 2 σ 2 M ( 4
)
where α is a normalization factor and σ 2 M a parameter of the model. In practice, as scores are given by a likelihood ratio (eq. ( 1)) and as we are only interested in ranking candidate windows, the normalization factor can be ignored. The only reamining parameter is the object/background ratio of σ 2 M , estimated by crossvalidation.

Object manifolds. Object manifolds are given by autoencoders [START_REF] Kramer | Nonlinear principal component analysis using autoassociative neural networks[END_REF]. Indeed, in addition of being reported as being efficient for several computer vision tasks, they make the computation of f and f -1 possible, which is not the case of most of the manifold models (such as ISOMAP [START_REF] Tenenbaum | A Global Geometric Framework for Nonlinear Dimensionality Reduction[END_REF], or LLE [START_REF] Saul | Think globally, fit locally: unsupervised learning of low dimensional manifolds[END_REF]). Furthermore, they allow to build very expressive models whose complexity can be adapted by varying their number of layers (3 in our case) and hidden neurons (fixed by cross-validation in our experiments). We train our autoencoders by minimizing the reconstruction error of training examples; i.e. Error = Xs,t∈train (X s,tg • f (X s,t )) 2 , where f is the function connecting the input to the central layer of the autoencoder, and g the function connecting the central layer to the output. g • f is then equivalent to the previously seen f -1 • f . In the context of manifold learning, the network is usually used to learn f and f only, providing an embedding of the data [START_REF] Hinton | Reducing the Dimensionality of Data with Neural Networks[END_REF]. In contrast, we keep the full network, which gives us the projection P M (X s ) we are looking for. In practice, we use sigmoid activation functions and train autoencoders, after doing a contrastive divergence initialization [START_REF] Hinton | Training products of experts by minimizing contrastive divergence[END_REF], with a standard back-propagation algorithm. Contrastive divergence is the key to good results, as it helps the neural network to focus on data that were given (instead of the identity function). In practice, to learn the manifold, we take a representative set of training windows, compute their signatures and optimize autoencoder parameters as explained above.

Background manifold. Our hypothesis is that linear models such as the PCA is best suited to model backgrounds. A signature X s can be written as X s = β i * P C i where β is the representation of X s in the PCA basis (P C I are the principal component). We can then project X s into a N-dimensional subspace using the N first principal components.

X s = i=1:N β i P C i + j=N +1:M β j P C j = P (X s ) + P (X s ) (5)
P (X) is the projection of X on the manifold while P (X) is the projection on the space orthogonal to manifold. Interestingly, | P (X)| is the distance to the manifold, which is proportional to the mean square reconstruction error. In our experiments, we randomly sample background windows from training images, compute their signatures and find the best basis by doing a SVD decomposition of their covariance matrix.

Image features and non-maximal suppression. Our algorithm can use any type of image features. In our experiments we have used 3 different ones: (a) raw pixel intensities (b) gradient maps (c) HOG features. Regarding non-maximal suppression, we use a simple and efficient iterative strategy consisting in keeping only the windows which have the maximum score over a disk (which radius is half the window width).

Experimental results

Dataset and protocol. We validated our approach on the OIRDS dataset [START_REF] Colder | Overhead imagery research data set: an annotated data library and tools to aid in the development of computer vision algorithms[END_REF], which is one of the rare publicly available dataset for Automatic Vehicle Detection with aerial images.

OIRDS contains a compilation of around 1,000 aerial images from different sources (e.g. USGS and VIVID), with about 1800 targets. Shadows, specularities, occlusions, as well as the large intra-class variability (e.g. regular cars, pickups, mini-vans, etc.) make this dataset very challenging. The dataset is provided with rich annotations: distance from the camera to the ground, target size (in pixels), bounding boxes, percentage of occlusion, type, etc. are given for each vehicle. Fig. 1 shows typical images from this dataset. The dataset is split in 10 folds and we evaluate the performance using a 10 fold cross validation procedure, by reporting the mean average precision (we use the experimental protocol of [START_REF] Everingham | The pascal voc challenge[END_REF]).

As we are primarily interested in knowing the performance of our detector for small targets detection, images were downscaled to produce a dataset in which targets are not bigger than 40 × 40 pixels.

Finally, as (unfortunately) no reproducible results have been published so far on this dataset (nor on any publicly available dataset for small target detection) we compare the performance of our algorithm with [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], known to get state of the art results on such tasks (comparisons with the part-based model of [START_REF] Felzenszwalb | Object detection with discriminatively trained part based models[END_REF] would not make sense because of target sizes). In addition, we have also implemented a generative model based on a Gaussian mixture model, which is reference for generative models. gradients, supposed to be more robust to illumination changes and (3) HOG features, considered to be the best choice for this task. In practice, raw pixel intensities are computed as the mean of the different color channels (OIRDS images are in color). Gradient images are computed by a Sobel filter. Finally, HOG31 is an histogram of oriented gradient, with 8 pixels overlap cells of 16 × 16 pixels. They contain a 9 bins histogram of unsigned orientations concatenated with a 18 bins histogram of signed orientations.

The linear SVM classifier is taken from the svmlight library [START_REF] Joachims | Making large-scale support vector machine learning practical[END_REF]. The Gaussian mixture models were learnt with the Expectation Maximization algorithm and include 3 gaussians components. The manifold dimensionality of backgrounds models is of 40, 10 and 16 for intensity, gradient and HOG signatures respectively. Autoencoders have 3 layers and have respectively 35, 8 and 10 inner nodes for intensity, gradient and HOG signatures. All these values were determined by preliminary experiments and were kept fixed for all the presented experiments.

Visualizing autoencoder and PCA reconstructions.

Fig. 2(b) shows some vehicle appearances our autoencoder can generate once trained. The model has learnt rotated appearance of cars. Fig. 3 shows 4 candidate windows (1st row), their projection on the background manifold (obtained by PCA, 2nd row), as well as their projection on the car manifold given by an autoencoder (last row). As it can be noticed, target images are better reconstructed by the target model than by the background one, and viceversa.

Quantitative results. We have experimented with 5 different detectors. The first one (so called AE-PCA) is ours, using an autoencoder to model targets and a PCA based manifold for backgrounds. [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] (so called HOG-SVM). In addition, we have also experimented with three other detectors, one using PCA for target and background (PCA-PCA), another one using gaussian mixture model, here again for both target and background (GM-GM), and a last one using an autoencoder for both as well (AE-AE). For these 5 detectors, we report the mean average precision over the 10 folds of the OIRDS datasets in Table 1.

The main conclusion we can draw from these results is that the proposed approach (the AE-PCA detector) outperforms any other detector, for any type of feature. The best results are obtained with HOG31 signatures. We also observe that Gaussian mixture models do not perform well in any case. Indeed, we have noticed that the GM model tends to be specialized to a few images, showing that EM gets stuck in local minima. From these results, we can also conclude that the HOG-SVM detector is outperformed -when using gradient and gray level signatures -by the PCA-PCA detector. HOG-SVM is however better than PCA-PCA with HOG31 features. In addition, we can also observe that using two autoencoders (ob-jects+backgrounds) does not give better results, as the background autoencoder fails to capture the diversity of backgrounds.

We kept aside a dozen images that were more difficult because of their large size for additional experiments (using the previously learnt classifiers). Results are given Table 2. The performance is not as good as on the regular OIRDS images, as images are much larger while not containing more targets. Nevertheless, our AE-PCA detector clearly outperforms the HOG-SVM detector.

Conclusions

This paper proposes a detection algorithm based manifold learning in which targets and background are model by distinct and adapted models. The object is accurately modeled by the mean of an autoencoder learned off-line. In addition, background is modeled by a PCA based linear manifold. We have experimentally validated our approach on a publicly available vehicle dataset, and show results that outperform state-of-theart algorithms.

  Implementation details. Training data are obtained by cropping positive examples, which gives a total of about 3800 positive examples per fold, and 13000 negative windows randomly sampled from the background (no overlap with objects). As the step size of our sliding window is of 8 pixels, when we crop positive images for training we add a random shift up to 4 pixels to make the model more tolerant to small shifts. Some typical positive training examples are given Fig. 2(a). In addition, the training set is extended by adding positive examples obtained by flipping up/down/left and right and by rotating the initial training set. Regarding image signatures, we experimented with three different signatures: (1) normalized raw level intensities, often used for target detection (2) image

  (a) Typical training images with shadows and specular spots. (b) Vehicle manifolds learnt by our autoencoder.

Figure 2 .

 2 Figure 2. Sample training images and autoencoder's manifold.

Figure 3 .

 3 Figure 3. Four candidate windows and their reconstruction by the PCA background manifold (2nd row), and by the target manifold learnt by an autoencoder (last row).

Table 1 .

 1 The second is Mean Average Precision on OIRDS.

		Intensity Gradient HOG31
	GMM-GMM	8.3%	21,3%	17.7%
	HOG-SVM [3]	10.5%	35.2%	46.8%
	PCA-PCA	35.0%	37.9%	42.5%
	AE-AE	35.3%	33.5%	47.5%
	AE-PCA (ours)	35.5%	39.9%	48.9%
		Intensity Gradient HOG31
	HOG-SVM [3]	1.5%	12.1%	12.6%
	AE-PCA (ours)	3.3%	16.4%	17.1%

Table 2 .

 2 Mean Av. Precision on large images. one of the state of the art approaches for detection, namely the Dalal and Triggs's detector

Acknowledgments. This work was supported by ANRT through the CIFRE sponsorship No 2011/0850 and by SAGEM-SAFRAN group.


Thomas Pradeau
email: thomas.pradeau@enpc.fr

F R Éd

AND Éric Meunier
email: frederic.meunier@enpc.fr

Rachana Gupta
email: rachanagupta07@gmail.com

BOUNDING THE PRICE OF ANARCHY FOR GAMES WITH PLAYER-SPECIFIC COST FUNCTIONS

Keywords: atomic splittable congestion game, price of anarchy, multiflow, player-specific cost function

)), we prove a general bound on the price of anarchy for games with player-specific cost functions. This bound generalizes some of their results, especially the bound they obtain for the affine case. However our bound still requires some dependence between the cost functions of the players. In the general case, we prove that the price of anarchy is unbounded, by exhibiting an example with affine cost functions and only two players.

1. Introduction 1.1. Motivation. In many economic fields, companies share common resources while being non coordinated. These resources are often owned by agents that are paid for the access they provide to these companies. These resources are for instance machines in a flexible manufacturing environment, means of transportation in a freight context, or arcs in a telecommunication network. An increase of demand for a resource often leads to an increase of its cost, because the fees increase, or because delay is created. This increase of the cost is seen as a congestion on the resource.

Taking a game-theoretical point of view, Cominetti, Correa, and Stier-Moses [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF] studied the efficiency loss implied by the lack of coordination. The companies become players in a congestion game and the resources become arcs of a network. If there is no conditional use between the resources, the parallel-link network often models correctly the interaction between the players, see for instance Orda, Rom, and Shimkin [START_REF] Orda | Competitive Routing in Multi-User Communication Networks[END_REF], Hayrapetyan, Tardos, and Wexler [START_REF] Hayrapetyan | The effect of collusion in congestion games[END_REF], Wan [START_REF] Wan | Coalitions in nonatomic network congestion games[END_REF]. However, in many situations, several resources have to be chosen simultaneously by each company, but not all subsets of resources are possible. Modeling the possible subsets as routes in a network is a way to cover these situations and makes sense in a freight transportation context for instance. All possible congestion games cannot be modeled with a network, but this representation is helpful and the results often extend to general congestion games without additional work (this is the case for the game we deal with in the present work). To model the congestion, each arc is endowed with a cost function.

A useful notion to quantify the loss of efficiency is the price of anarchy, introduced by Koutsoupias and Papadimitriou [START_REF] Koutsoupias | Worst-case equilibria[END_REF]. The price of anarchy is the worst-case ratio between the social cost at equilibrium and the best possible social cost. Its interest has lead to a considerable amount of work since the seminal paper of Roughgarden and Tardos [START_REF] Roughgarden | How bad is the selfish routing ?[END_REF]. Cominetti, Correa, and Stier-Moses [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF] are able to prove general upper bounds for the price of anarchy of atomic splittable games, which are valid in a large set of situations. These results have been extended by Harks [START_REF] Harks | Stackelberg strategies and collusion in network games with splittable flow[END_REF] and Roughgarden and Schoppmann [START_REF] Roughgarden | Local smoothness and the price of anarchy in atomic splittable congestion games[END_REF].

One of the main assumptions in these papers is that, for each arc of the network, the players all have the same cost function. However, depending on their size, the nature of the goods they carry, or other features, the companies are not equally affected by the congestion. Thus, allowing the companies to have their own cost functions makes sense. To the best of our knowledge, this assumption has not been relaxed in the context of the computation of the price of anarchy, except in Gairing, Monien, and Tiemann [START_REF] Gairing | Routing (un-) splittable flow in games with player-specific linear latency functions[END_REF] for different models (atomic unsplittable and nonatomic games). In this paper, we allow player-specific cost functions and extend some of the results of Cominetti, Correa, and Stier-Moses [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF] in this more general setting.

Model and main results

2.1. Model. The description of the game we deal with in this paper goes as follows. We are given a directed network D = (V, A) and K players identified with the integers 1, . . . , K. Each player k ∈ [K] has to send d k units of flow, its demand, from an origin s k ∈ V to a destination t k ∈ V in this network. A strategy for player k is therefore an s k -t k flow x k ∈ R A + of value d k . Such a flow is an element of

F k = y ∈ R A + : a∈δ + (s k ) y a - a∈δ -(s k ) y a = d k and a∈δ + (v) y a = a∈δ -(v) y a , ∀v ∈ V \ {s k , t k }
and is referred as a feasible flow for player k. Throughout the paper, we consider the decisions of the players as a multiflow x = (x 1 , . . . ,

x K ) ∈ R A×K + . It is feasible if it is an element of F 1 × • • • × F K .
Each player k has his own vector of cost functions c k = (c k a ) a∈A where for each arc a the cost function c k a (•) is a R + → R + function. We assume given a set of allowable cost functions C in which c = (c 1 , . . . , c K ) is taken. Since each player has his own vector of cost functions, the game is said to be with player-specific cost functions. We will sometime say that it is a multiclass game. On the contrary, if c 1 = • • • = c K for every c ∈ C, we speak of a single-class game.

We denote the total flow on an arc a by x a = k∈[K] x k a . The cost experienced by a player k is

C k ( x) = a∈A x k a c k a (x a ).
The goal of this player consists in sending its d k units of flows while minimizing this cost.

A feasible multiflow x N E = (x N E,1 , . . . , x N E,K ) is a Nash equilibrium if for each player k, we have

(1) C k ( x N E ) = min y∈F k C k (y, x N E,-k ),
where (y, x N E,-k ) = (x N E,1 , . . . , x N E,k-1 , y, x N E,k+1 , . . . , x N E,K ).

This kind of games belongs to the class of atomic splittable network congestion games. Atomic, because each player has nonnegligible impact. Splittable, because the goods are seen as a flow whose support is not necessarily a unique route. For the game we consider in this paper, a Nash equilibrium always exists [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF][START_REF] Orda | Competitive Routing in Multi-User Communication Networks[END_REF]. However, it may not be unique and even the total flows on the arcs may vary among the multiple Nash equilibria [START_REF] Richman | Topological Uniqueness of the Nash Equilibrium for Selfish Routing with Atomic Users[END_REF][START_REF] Bhaskar | Equilibria of atomic flow games are not unique[END_REF].

The social cost of a multiflow is defined as

C( x) = k∈[K] C k ( x) = k∈[K] a∈A x k a c k a (x a ).
A multiflow of minimal social cost is a social optimum.

Main results.

An instance of the game is defined by the network, the set of players with their origin-destination pairs, their demands, and their cost functions. We denote by NE(I) the set of Nash equilibria for an instance I and by x OP T (I) a feasible multiflow achieving the minimal social cost. This paper is focused on the price of anarchy (PoA) of the game. Given a set of allowable instances I, the price of anarchy is

PoA = sup I∈I sup x∈NE(I) C( x) C( x OP T (I))
.

Theorem 1. Consider an atomic splittable network congestion game with player-specific cost functions. Suppose that the available cost functions in C are differentiable, nonnegative, increasing, and convex, and define

λ(C) = sup a∈A, k,ℓ∈[K], c∈C, x∈R + x(c ℓ a ) ′ (x) c k a (x) . If λ(C) < 3, we have PoA ≤ 1 1 -λ(C)/3 .
This theorem is also valid if we are in the single-class case, in which case λ(C) coincides with γ(C) -1 in Cominetti et al. [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF].

The upper bound on the price of anarchy given in Theorem 1 can be improved with a more complicated formula, see Proposition 5 below. A corollary is then the following proposition, which is the generalization of Proposition 3.5 in Cominetti et al. [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF] to the multiclass case.

Proposition 1. Consider an atomic splittable network congestion game with player-specific cost functions. Suppose that the allowable cost functions are affine of the form c k a (x) = p k a x + q k a with p k a > 0, q k a ≥ 0, and define ∆ = sup

a∈A sup ℓ∈[K] p ℓ a inf k∈[K] p k a . If ∆ < 3, we have PoA ≤ 3∆(K -1) + 4 ∆(3 -∆)(K -1) + 4
.

Proposition 1 uses the parameter ∆ with the same definition and with a similar purpose as in Gairing et al. [START_REF] Gairing | Routing (un-) splittable flow in games with player-specific linear latency functions[END_REF]. This parameter is independent of the constant terms of the functions. This bound is valid for ∆ < 3, i.e. when the marginal cost caused by the congestion does not vary too much among the players. In particular, it handles cases where the marginal impact of the congestion for any player is at most twice the one of the others. In practice, for example in the transportation context, these situations are likely to happen. Note that in the single-class case, we have ∆ = 1, and our proposition coincides with the aforementioned Proposition 3.5 of Cominetti et al. [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF] giving a bound of 3K+1 2K+2 . If the cost functions are affine, Proposition 1 shows that the price of anarchy is bounded when ∆ < 3. However, according to the next proposition, it is unbounded in general, even when the cost functions are affine. Proposition 2. For any M > 0, there is an instance of an atomic splittable network congestion game with player-specific affine cost functions, with two players, and with

PoA > M.
This result contrasts with the single-class case. Indeed, in this case it has been showed that the price of anarchy is bounded for polynomial cost functions of degree d: Harks [START_REF] Harks | Stackelberg strategies and collusion in network games with splittable flow[END_REF], followed by Roughgarden and Schoppmann [START_REF] Roughgarden | Local smoothness and the price of anarchy in atomic splittable congestion games[END_REF] found the closed-form upper bound of 1+ √ d+1 2

d+1

. Bounds also exist for different games that are nonatomic games [START_REF] Roughgarden | Bounding the inefficiency of equilibria in nonatomic congestion games[END_REF] and atomic unsplittable games [START_REF] Aland | Exact price of anarchy for polynomial congestion games[END_REF].

The remaining of the paper is organized as follows. Section 3 gives a general but not explicit bound on the price of anarchy, from which we deduce Theorem 1 and Proposition 1. Section 4 presents the proof of Proposition 2. In Section 5 we discuss the results and some open questions.

3. The price of anarchy for general cost functions 3.1. Preliminary remarks. Throughout the paper, the components of an x ∈ R K + are denoted x k . Given a K-tuple of cost functions c = (c 1 , . . . , c K ), we define c k : R K + → R + by

c k (x) = ∂ ∂x k x k c k (x) where x = ℓ∈[K]
x ℓ .

We have then

c k (x) = x k c k ′ (x) + c k (x).
Note that the c k in Cominetti et al. [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF] coincides with our c k .

The following proposition is standard in this context and is obtained by writing the optimality conditions of Equation ( 1).

Proposition 3. The multiflow x N E is a Nash equilibrium if and only if, for all k ∈ [K], it satisfies (2) a∈A c k a (x N E a )(y k a -x N E,k a
) ≥ 0, for any feasible flow y k for player k,

where x N E a = (x N E,1 a , . . . , x N E,K a ) ∈ R K + .

3.2.

A general bound. Following Cominetti et al. [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF], for a K-tuple of cost functions c = (c 1 , . . . , c K ), we define

β(c) = sup x,y∈R K + k∈[K] c k (x) -c k (y) y k + c k (x) -c k (x) x k k∈[K] x k c k (x)
,

where x stands for k∈ [K] x k and y for k∈[K] y k . We assume 0/0 = 0. Note that β(c) ≥ 0, since the function we want to maximize is zero when x = y. We also define β(C) = sup c∈C, a∈A β(c a ), where c a = (c 1 a , . . . , c K a ). The following proposition gives a general bound, yet nonexplicit, on the price of anarchy. For sake of simplicity, we assume that (1 -β(C)) -1 = +∞ for β(C) ≥ 1. Proposition 4. Let x N E be a Nash equilibrium and x OP T be a social optimum. Then

C( x N E ) ≤ 1 1 -β(C) C( x OP T ).
This proposition coincides with Proposition 3.2 of Cominetti et al. [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF] in the single-class case. They refer to Roughgarden [START_REF] Roughgarden | Selfish routing with atomic players[END_REF] and use ideas of Correa et al. [START_REF] Correa | Selfish routing in capacitated networks[END_REF]. The proof is routine.

Proof of Proposition 4. We have

C( x N E ) = k∈[K] a∈A c k a (x N E a ) -c k a (x N E a ) x N E,k a + c k a (x N E a )x N E,k a ≤ k∈[K] a∈A c k a (x N E a ) -c k a (x N E a ) x N E,k a + c k a (x N E a )y k a ≤ a∈A   β(c a ) k∈[K] x N E,k a c k a (x N E a )   + C( y) ≤ β(C)C( x N E ) + C( y)
where we use Equation ( 2) to get the first inequality and the definition of β(•) to get the second inequality.

We finish by taking y = x OP T .

3.3.

Computation of the bound. We give now an explicit upper bound on β(c).

Proposition 5. Consider a K-tuple of cost functions c = (c 1 , . . . , c K ) and define

δ(c) = sup k,ℓ∈[K], x∈R + (c ℓ ) ′ (x) (c k ) ′ (x) and λ(c) = sup k,ℓ∈[K], x∈R + x(c ℓ ) ′ (x) c k (x) .
If each c k is differentiable, nonnegative, increasing, and convex, then the following inequality holds

(3) β(c) ≤ λ(c) 3 1 1 + 4 3 1 δ(c)(K-1)
.

Before proving this proposition, let us note that the special case K = 1 gives β(c) = 0 and a price of anarchy of 1 as expected: if there is only one player, Nash equilibrium and social optimum coincide.

Proof of Proposition 5. Here and throughout the proof, c k ′ stands for the derivative of c k . We have thus

β(c) = sup x,y∈R K + f (x, y) k∈[K] x k c k (x) , where f (x, y) = k∈[K] x k c k ′ (x) + c k (x) -c k (y) y k -x k 2 c k ′ (x) .
The proof goes as follows. We will first find an upper bound on f (x, y) depending only on x, Equation ( 6) below. Then, using the quantities δ(c) and λ(c) defined in the statement of the proposition, we will get an upper bound on β(c) expressed as a supremum of a concave function on a convex domain, Equation ( 7), which will lead by straightforward computations to the desired expression.

We compute now an upper bound on f (x, y) depending only on x. We have

(4) f (x, y) ≤ sup y∈R + ,ℓ∈[K] x ℓ c ℓ ′ (x) + c ℓ (x) -c ℓ (y) y - k∈[K] x k 2 c k ′ (x),
since for a y ∈ R K + with fixed sum y = k∈[K] y k , the first sum in the definition of f can be made maximum by putting all the weight on a single term.

The map g

: y → x ℓ c ℓ ′ (x) + c ℓ (x) -c ℓ (y) y is concave on R + . Its derivative is g ′ (y) = x ℓ c ℓ ′ (x) + c ℓ (x) -c ℓ (y) -yc ℓ ′ (y). Let y ⋆ ∈ R + such that g ′ (y ⋆ ) = 0. We have (5) y ⋆ c ℓ ′ (y ⋆ ) + c ℓ (y ⋆ ) = x ℓ c ℓ ′ (x) + c ℓ (x)
and thus, since

x ℓ ≤ x, x ℓ c ℓ ′ (x ℓ ) + c ℓ (x ℓ ) ≤ y ⋆ c ℓ ′ (y ⋆ ) + c ℓ (y ⋆ ) ≤ xc ℓ ′ (x) + c ℓ (x).
Since the map u → uc ℓ ′ (u) + c ℓ (u) is nondecreasing, these inequalities imply

x ℓ ≤ y ⋆ ≤ x.
Hence

g(y ⋆ ) = x ℓ c ℓ ′ (x) + c ℓ (x) -c ℓ (y ⋆ ) y ⋆ ≤ (x ℓ + x -y ⋆ )c ℓ ′ (x)y ⋆ ≤ x + x ℓ 2 2 c ℓ ′ (x),
where the first inequality is a consequence of the convexity of c ℓ (•) and where the second is obtained via direct calculations. Using this bound in Equation ( 4), we get an upper bound that does not depend on y and which is valid for all x ∈ R K + :

f (x, y) ≤ sup ℓ∈[K] x + x ℓ 2 2 c ℓ ′ (x) - k∈[K] x k 2 c k ′ (x) ≤ sup ℓ∈[K] x 2 c ℓ ′ (x) 4   1 + 2 x ℓ x -3 x ℓ x 2 - 4 δ(c) k =ℓ x k x 2   . (6)
Using Equation [START_REF] Harks | Stackelberg strategies and collusion in network games with splittable flow[END_REF] in the definition of β(c) and with the help of the parameter λ(c) defined in the statement of the proposition, we get

β(c) ≤ λ(c) 4 sup x∈R K + ,ℓ∈[K]   1 + 2 x ℓ x -3 x ℓ x 2 - 4 δ(c) k =ℓ x k x 2   .
Without loss of generality we can assume that the maximum is attained with ℓ = 1. This inequality can thus be rewritten as [START_REF] Hayrapetyan | The effect of collusion in congestion games[END_REF] β(c) ≤ λ(c)

4 sup z∈△ 1 + 2z 1 -3 z 1 2 - 4 δ(c) K k=2 z k 2 ,
where

△ is the (K -1)-dimensional simplex {z ∈ R K + : k∈[K] z k = 1}.
The value of right-hand side can be obtained by maximizing a concave function on a convex domain. We compute z ⋆ ∈ △ maximizing its value:

z ⋆,1 = 1 + θ(c) 3 + θ(c) and z ⋆,2 = • • • = z ⋆,K = 1 -z ⋆,1 K -1 , where θ(c) = 4 δ(c)(K-1)
. Inequality (7) becomes then

β(c) ≤ λ(c) 4 1 + 2z ⋆,1 -3(z ⋆,1 ) 2 -θ(c)(1 -z ⋆,1 ) 2 ,
which leads to the upper bound given in the statement of the proposition.

Note that the proof of Proposition 5 does not seem to be adaptable to deal with a parameter ω analogue to the one of Harks [START_REF] Harks | Stackelberg strategies and collusion in network games with splittable flow[END_REF].

We explain now how to deduce Theorem 1 and Proposition 1 from Proposition 5.

Proof of Theorem 1. We have λ(C) larger that any λ(c), and we use 1

1 + 4 3 1 δ(c)(K-1) ≤ 1.
Proof of Proposition 1. In this special case, we have ∆ larger than any δ(c), and we have moreover δ(c) = λ(c) for all c. A straightforward calculation leads to the desired formula.

The price of anarchy with affine cost functions is unbounded

In this section, we prove Proposition 2 by exhibiting an instance of the game with affine cost functions giving a price of anarchy which can be made arbitrarily large.

Consider the network with one origin-destination pair and two parallel arcs a and b. There are two players. The first player has a total demand of M and his cost functions are c 1 a (x) = x and c 1 b (x) = x + 2M . The second player has a total demand of 1 and his cost functions are c 2 a (x) = 2M 2 x + 1 and c 2 b (x) = M 3 x. Denote by I(M ) this instance. Since the network has parallel arcs, the Nash equilibrium is unique [START_REF] Orda | Competitive Routing in Multi-User Communication Networks[END_REF]. We prove that it is reached with the multiflow x where player 1 puts all his demand on the arc a and player 2 puts all his demand on the arc b. Indeed, we have in this case, for any y 1 ∈ F 1 and y 2 ∈ F 2 ,

c 1 a (x a )(y 1 a -x 1 a ) + c 1 b (x b )(y 1 b -x 1 b ) = 2M (y 1 a -M ) + (2M + 1)(M -y 1 a ) = M -y 1 a ≥ 0 and c 2 a (x a )(y 2 a -x 2 a ) + c 2 b (x b )(y 2 b -x 2 b ) = (2M 3 + 1)(1 -y 2 b ) + 2M 3 (y 2 b -1) = 1 -y 2 b ≥ 0. Proposition 3 gives then that x is a Nash equilibrium. The social cost at equilibrium is C( x) = M 2 + M 3 .
Consider now the multiflow z where player 1 puts all his demand on the arc b and player 2 puts all his demand on the arc a. These flows are feasible and give a social cost C( z) = 5M 2 + 1. We have then

PoA(I(M )) ≥ C( x) C( z) = M 3 + M 2 5M 2 + 1 .
Since lim M →+∞ PoA(I(M )) = +∞, we get the result.

Proof of Proposition 2. The instance I(6M ) gives a price of anarchy greater than M .

Discussion and open questions

The bound of Proposition 5 makes sense only when λ(c) -3 < 4 δ(c)(K-1) , since otherwise it is larger than 1. When λ(c) > 3, this condition is met only when

K < 1 + 4 δ(c)(λ(c) -3)
.

In other words, there is a whole range of cost functions and numbers of players for which we are unable to provide any concrete bounds. It would be interesting to extend the bound to a larger set of instances. Proposition 2 shows that the price of anarchy is unbounded as soon as the set of affine cost functions is included in the set of allowable cost functions. Its proof needs that the demand of one player become infinitely larger than the demands of the others. The question whether the price of anarchy is bounded with affine cost functions, when for instance the quantity Another result from Cominetti et al. [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF] that can be extended to the multiclass case is their Proposition 3.7 stating that the social cost at equilibrium is bounded by the optimal cost of the game where the demands are multiplied by 1 + β(C). More precisely, we can adapt their proof to find the following proposition. Proposition 6. Consider an atomic splittable network congestion game with player-specific cost functions. Suppose that the available cost functions in C are differentiable, nonnegative, increasing, and convex. Consider an instance I with an equilibrium multiflow x N E (I) and the instance αI where all demands are multiplied by α ≥ 1, with an optimal multiflow x OP T (αI). Suppose that β(C) < 1, we have

C( x N E (I)) ≤ 1 α -β(C)
C( x OP T (αI)).

Proof. The proof is almost the same as in Cominetti et al. [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF]. For the ease of reading, we denote

x N E = x N E (I). Let y be a flow feasible for the instance αI, then Another possible further development would be to compare the game studied in this paper with the nonatomic case. Cominetti et al. [START_REF] Cominetti | The impact of oligopolistic competition in networks[END_REF] proved that when all players have same cost functions, same demand, and same origin-destination pair, the social cost at equilibrium of the atomic game is bounded by the one of the corresponding nonatomic game. In particular, the price of anarchy of the atomic game is bounded by the one of the nonatomic game. The key point in their context is that the atomic game is potential, which is unlikely to be the case here. Whether this bound holds with player-specific cost functions is an open question.

  ] d k remains bounded for each player j, remains an open question.

  c k a (x N E a ) -c k a (x N E a ) x N E,k a + c k a (x N E a ) c k a (x N E a ) -c k a (x N E a ) x N E,k a + c k a (x N E a )y k a ≤ β(C)C( x N E ) + C( y)where we use Equation (2) with y k a α to get the first inequality, the fact that α ≥ 1 and c k a (x N E a )c k a (x N E a ) ≤ 0 to get the second inequality, and the definition of β(•) to get the last inequality. We finish by taking y = x OP T (αI). This proposition extends Proposition 4 which deals with the case α = 1. In particular we have C( x N E (I)) ≤ C( x OP T ((1 + β(C))I)).


Peggy Candas
email: peggy.candas@unistra.fr

Analyzing student practice in language resource centres of the Louis Pasteur University in Strasbourg : is planning really central to selfdirected learning?

Keywords: learner autonomy, self-directed learning, planning, monitoring, language learning, activity, language resource centre, student, university

The Language Resource Centres (LRCs) of the Louis Pasteur University 1 (LPU) in Strasbourg were created to promote both language learning and the development of learner autonomy among university students. Teacher support in the LRCs is based on the prevailing principles found in the literature on self-directed learning. But little is in fact known about how learner autonomy actually develops and a few authors have started to challenge the assumption that planning is central to self-directed learning. Our doctoral research set out to comprehensively study how university students learning a foreign language in an LRC proceed when encouraged to self-direct learning. The corpus of data was obtained using a methodology combining direct observation of individual learners' work sessions and subsequent interviews. The confrontation of the data with literature on autonomy led us to identify four lines of analysis, according to which student activity was then qualitatively analyzed. This article will focus on one of them: if and how the learners controlled or monitored their learning process. Although our results remain to be confirmed on larger numbers of learners studied for a longer time span, they open challenging areas of investigation into the nature of the processes involved in selfdirected learning.

Introduction

In France as well as in Europe considerable energy has been devoted to discussing and promoting the development of autonomy in learners, young and old, in particular in the field of language teaching and learning (e.g. [START_REF] Little | Learner Autonomy -1 : Definitions, Issues and Problems[END_REF][START_REF] Albero | Autoformation et enseignement supérieur[END_REF][START_REF] Carré | L'apprenance: Vers un nouveau rapport au savoir[END_REF][START_REF] Dumazedier | Penser l'autoformation: Société d'aujourd'hui et pratiques d'autoformation[END_REF]). The authors of numerous books and articles (e.g. [START_REF] Carré | L'autoformation dans la formation professionnelle[END_REF][START_REF] Carré | L'autoformation: Psychopédagogie, ingénierie, sociologie[END_REF]Courtois and Prévost 1998;[START_REF] Foucher | L'autoformation dans l'enseignement supérieur: Apports européens et nord-américains pour l'an 2000[END_REF][START_REF] Carré | La formation autodirigée: Aspects psychologiques et pédagogiques[END_REF] attempt to operationalize this notion so as to guide teachers and trainers in their daily practice. However, a close look at the existing literature reveals that what [START_REF] Little | Learner Autonomy -1 : Definitions, Issues and Problems[END_REF] more precisely calls 'learner autonomy' is rather vaguely defined. Although it is not often explicitly stated in the French and European literature on learner autonomy, most of the practical advice given to educators to develop learner autonomy rests on research on self-directed learning (e.g. [START_REF] Knowles | The Modern Practice of Adult Education: Andragogy vs Pedagogy[END_REF][START_REF] Tough | The Adult's Learning Project : A Fresh Approach to Theory and Practice in Adult Learning[END_REF]), a more pragmatic notion which was originally developed, under different names, by north-American research in the adult education field (cf. Tremblay and Eneau 2006, 76-79) and then imported into French and European research. As far as this very problematic terminological confusion is concerned, Holec's work stands as an exception, in that he clearly distinguished autonomy, defined as an ability, a power to do something, from self-direction in learning, which he described as an attitude, a way of doing something, thereby using the above-mentioned ability (personal translation, Holec 1981b, 21).Therefore, for this author, the word autonomous is a term which refers to a particular characteristic of the learner: an autonomous learner is a learner who knows how to learn. In no way can it be used to describe the way learning is conducted, contrary to self-direction in learning, which, he argues, leads to self-directed learning. When turning to North-American research on autonomy and self-directed learning, similar terminological difficulties can be found. According to Tremblay (2003, 59), the notion of selfdirected learning remains ambiguous, definitions being unstable and sometimes even contradictory. For Benson (2001, 34), further confusion around the terms autonomy and self-directed learning, and their distinction, arises from the fact that they are often used differently in the fields of adult education and language learning. As a necessary basis for our doctoral research, we chose to keep Benson's definitions as reference definitions as we found they had reached the widest consensus:

'Perhaps the most important distinction to be made in the field of language learning is between autonomy as an attribute of the learner and self-directed learning as a particular mode of learning in which the learner makes the important decisions about content, methods and evaluation. Autonomy can be considered as something that learners possess to various degrees. Self-directed learning can be considered as something that learners are able to do more or less effectively, according to the degree that they possess this capacity'. (Benson 2001, 34) Since the 1970s and the publication of numerous articles on the organisational dimension of adult self-directed learning, especially in north-American research, the idea that planning is central to self-directed language learning has prevailed:

'In previous research in self-directed learning, [START_REF] Tough | The Adult's Learning Project : A Fresh Approach to Theory and Practice in Adult Learning[END_REF], [START_REF] Peters | Adult learning projects: a study of adult learning in urban and rural Tennessee[END_REF], [START_REF] Penland | Self-initiated learning[END_REF][START_REF] Hiemstra | The older adult and learning[END_REF] have reported or implied the existence of detailed preplanning of self-directed learning by adults. Specifically, Tough (1971, 116-117) identified thirteen steps in planning that the learner 'sometimes takes in a self-planned project'. These steps include: identifying in detail what is to be learned, resources and methods for learning, time devoted to the effort, where to learn, pace of learning, deadline for completion, etc. Tough suggests that these actions may take place so rapidly that the learner is not even aware of them but that they occur nonetheless.' (Spear and Mocker 1984, 2) This widely accepted principle in the field is of course of the utmost importance for teachers and trainers and largely influences the way they try to foster the process of autonomy development in learners. Researchers therefore tried to determine the kind of institutional training which could be offered to learners who wanted to acquire or develop abilities to self-direct their projects (Benson 2001, 80-84). Research on learning strategies [START_REF] Wenden | Literature review: the process of intervention[END_REF][START_REF] Oxford | Language Learning Strategies: what every Teacher should Know[END_REF][START_REF] O'malley | Learning Strategies in Second Language Acquisition[END_REF], which set out to identify types of attitudes and strategies used by efficient learners and train less efficient learners to use them, turned out to be a major source of inspiration.

The Language Resource Centres (LRC) of the Louis Pasteur University (LPU) in Strasbourg were created to promote both language learning and the development of learner autonomy among university students not majoring in languages. A detailed historical account of how these centres were designed after several years of work and experimentation by a team of researchers and teachers in both the fields of educational sciences and language learning can be found elsewhere [START_REF] Poteaux | Nouveaux dispositifs, nouvelles dispositions[END_REF][START_REF] Poteaux | L'autoformation à l'université : de quelques éléments dialectiques[END_REF]. However, it is important to note that the pedagogical principles implemented in those centres largely derive from the French and north-American research on selfdirected learning. Practically speaking, there are seven language resource centres (replacing traditional classrooms) where students come to learn languages with their teachers. A great number of resources, very diverse, are available for language learning and teachers are always present. Although not majoring in languages, the students have compulsory credits for at least one language. The development of learner autonomy while learning one or several languages is set as an explicit goal and teachers act both as language experts and counsellors [START_REF] Gremmo | Conseiller n'est pas enseigner: le rôle du conseiller dans l'entretien de conseil[END_REF] who discuss needs, goals and learning trajectories with learners. Needless to say that the teachers also design and/or implement reflexive tools, such as the European Language Portfolio 2 , for this purpose. Through frequent teacher-student interaction, students are encouraged to constantly reflect on their work, i.e. set learning objectives and choose adequate resources accordingly, define practical modalities and learning strategies, and self-evaluate their progress (Holec 1981a[START_REF] Holec | L'autonomie de l'apprenant en langues vivantes -Learner autonomy in modern languages[END_REF]. They are also evaluated at the end of the semester on their capacity to discuss their learning programme critically, assess their progress and define general directions for the compulsory language module they will enroll in the following semester.

A preliminary study to our doctoral research [START_REF] Candas | Les usages des étudiants dans les centres de ressources de langues de l'Université Louis Pasteur : construction raisonnée d'un parcours ou itinéraire fortuit ? Paper presented at the Troisième Colloque Mondial sur l'Autoformation[END_REF], conducted in the LPU centres, showed an important gap between teachers' intentions and student practice. Indeed, although teachers encouraged learners to define learning objectives, a programme and steps, the choices students made appeared to be linked to a large extent to other parameters like circumstances -or chance -or resources present in the environment. This finding seemed to challenge the assumption that planning is central to self-directed language learning, although the university context led us to remain very cautious. Following further inquiry in the adult education field, we found out that Danis andTremblay (1985, 1987) were actually very critical of the otherwise largelyaccepted principles prevailing in the field of adult education practice and had conducted a critical review of prevailing adult learning principles [START_REF] Danis | Principes d'apprentissage des adultes et autodidaxie[END_REF]:

'Adult education practice tends to be influenced by the particular sets of principles prevailing in the field [START_REF] Fox | Current action principles and concepts from research and theory in adult learning and development[END_REF]). These principles, in turn, reflect the underlying explicit or implicit assumptions [START_REF] Brundage | Adult learning principles and their application to program planning[END_REF] regarding how adults learn and how they should be taught. (...) a critical review of the prevailing principles may prove to be a most useful prerequisite to their implementation, inasmuch as some of these assertions and recommendations [START_REF] Parsons | Adults learn differently than children: an examination of an old basic assumption[END_REF][START_REF] Pratt | Andragogical assumptions: some counter intuitive logic[END_REF][START_REF] Zemke | 30 things we know for sure about adult learning[END_REF]) may be found to be either partly or completely erroneous.' (Danis and Tremblay 1987, 4) As far as the organisational dimension of self-directed learning is concerned, and more precisely the issue of planning, Eneau (2000, 41-45), Benson (2001, 77-78) and Tremblay (2003, 150) all refer to what they consider to be an important study in the field, carried out by [START_REF] Spear | The organizing circumstance : environmental determinants in self-directed learning[END_REF]:

'Spear and Mocker's findings are of interest because they call into question the assumption that autonomous learning is necessarily planned learning. ' (Benson 2001, 78) All this led us to embark on a comprehensive research project on self-directed learning in an academic context, in the sense that we aimed, not at validating or invalidating pre-established hypotheses, but, on the contrary, at comprehending the observed phenomena and contributing to their description in order to suggest possible explanations or a model. In the frame of this article, it is not possible to adequately detail the whole of our doctoral research; therefore we will focus on the chosen design and methodology of the study and present the results for one line of analysis only.

Design of the study

The aim of our doctoral research was to comprehensively study how university students in the LPU language resource centres proceeded when asked to endorse responsibility so as to learn a language, encouraged and helped in this design by teachers. This obviously meant gathering data on individual learners' activity when learning. [START_REF] Vermersch | L'entretien d'explicitation[END_REF] showed that, when asked to explain in great detail how they proceeded for a particular task in their specialized field, the description that experts made of the sequence of actions they thought they performed did not coincide with what they really did in an effective situation. This led us to decide against collecting discourse material as the primary source of data and to prefer direct observation, on the basis that it would enable us to obtain detailed and situated descriptions of learners' activity in the language resource centres -activity being understood in a broad sense as choice, action, interaction, etc. However, preliminary tests made it quite clear that learners' activity could not be fully apprehended through direct observation only, since part of the activity was not visible. What is more, we subscribe to the definition of 'activity' given in [START_REF] Astolfi | La Saveur des Savoirs: Disciplines et plaisir d'apprendre[END_REF], where it is described as three-fold and comprising the level of operation (the material dimension of activity, the how), the level of action (the cognitive dimension, the why, the aim), and the level of activity (the intentional dimension, the for what, the motive). This model is inspired from [START_REF] Leontiev | Activity, Consciousness, Personality[END_REF] and implies that what a person's actions mean to him/her is essential to fully comprehend a particular sequence of events. The chosen methodology for data collection therefore combined observation sessions of individual learners working in the language resource centres with directive interviews, immediately following observation sessions (the sessions were therefore still fresh in the learners' minds).

Participants

Teachers working in the Louis Pasteur University (LPU) language resource centres were informed by mail about the study. They were told that the researcher was looking for volunteers among first-year students who had enrolled for an English compulsory module and that the protocol included the observation of several working sessions in the LPU centre for each volunteer, as well as students' interviews. One male and three female teachers responded positively and accepted that the researcher come in one or several of their classes to try to recruit one volunteer in each of the classes. In their groups of students, five female and three male students accepted to participate, knowing four of their working sessions would be observed by the researcher and that there would be a twenty-minute interview with the researcher (and without his/her teacher) after each of the observed session. Each student was guaranteed that his/her name would not appear in the data and signed an informed consent. The aim of the study was presented in general terms to all participants:

'learning more about how students work in the language resource centres'.

First-year students in their second semester of study at university were targeted because it was hypothesized that this was a relevant period to observe learner autonomy developing, since students had recently left high school and entered university. Moreover, as it is deemed a crucial time for students in the process of familiarisation with the language resource centres, LPU language teachers are known to concentrate a lot of their efforts as counsellors on first-year students. Because the researcher is proficient in English, and not in the other languages offered in the centres (German, Spanish), and had to be able to understand precisely the observed student's actions and interactions (e.g. what he/she chose to do, how he/she was going about it, what he/she said to other students or the teacher, what he/she was told), participants had to study this particular language in the centres to be eligible for the study. No other criterion was imposed. If French is also taught in the centres, foreign students participating in study abroad and exchange programmes such as Erasmus programmes and studying French in the LRCs as part of their curriculum were voluntarily excluded from the study because they constitute a very special population inasmuch as they are immersed all day long in situations where the language they are trying to learn is spoken or used, contrary to most 'more traditional' students coming to the centres. The researcher made sure that the group of students volunteering went to different language resource centres and were majoring in diverse disciplines. The eight participants came to be included in the study quite randomly, although it must be said that, while most of them volunteered when the teacher asked the whole group of students in the class if someone accepted to participate in the study, some were among two or three students to which their English teacher asked directly if they wanted to participate.

Data collection

For each of the eight students included in the study, a series of four observation sessions was conducted. Each observation session lasted an hour and a half to two hours, depending on the student's timetable and on his/her wishes. Indeed, the official work sessions included in the students' timetable last two hours, but each student was free to interrupt the session or continue to work; however, the researcher asked him/her to stop working half an hour before leaving the centre so as to have enough time for the interview.

During the session, the researcher, carrying only a clipboard, sheets of paper, a pen and a highlighter, was sitting behind the observed student and slightly on his/her right or left, but at quite a short distance so as to be able to see precisely what actions the student was performing and what he/she was looking at, as well as hear what he/she was saying or was told. If the observed student went anywhere in the centre, the researcher followed after a few seconds, standing at a reasonable distance. The participant and the researcher did not talk or interact in any way, except at the beginning of the session when they greeted one another and also sometimes when the student felt he/she had justify an action, e.g. when doing something personal or going to sit somewhere else in the centre. The researcher had also informed the LRC teachers that she would prefer not being addressed during the observation sessions.

Globally, the researcher's request was respected.

All this was negotiated with the observed students individually when the researcher met them for the first time: the researcher explained how she intended to proceed and why, insisting that she would try not to disturb the student's work, encouraged the student to give his/her opinion or impressions, discussed slight modifications which could be agreed upon, and tried to obtain explicit consent on the procedure. During the interviews following the observation sessions, the researcher regularly asked the student for feedback on how she conducted her observations, trying to make sure that the student still felt satisfied by the way in which the researcher was carrying out the established contract and was not judging her too intrusive.

During each observation session, detailed notes were taken by the researcher on the student's actions and interactions, without any grid in order not to limit observation to pre-established categories. Indeed, the aim was to study the activity of students in the LRCs in its most general sense, and not only in the restrictive meaning of intellectual work. The researcher therefore recorded as many actions as possible so as to obtain a written observation report as faithful to what she had seen as possible.

Of course, recording every single action is utterly impossible and the researcher's gaze acts as a filter, undeniably. Moreover, the speed with which successive actions are undertaken can be amazing, as in the case of one of our participants when he surfed the internet. However, the researcher did her best to record all practical actions and interactions with other students and teachers, without neglecting more subtle activity such as pauses. To facilitate the subsequent interview, any action which might require an explanation was immediately highlighted by the researcher in her notes. On the same day as the observation session, the detailed observation report was typed by the researcher on her computer; the rough notes were nevertheless carefully kept. At the end of each observed session and after each time asking for the student's explicit consent, photocopies of the student worksheet and of all work documents used or produced by the student during that session were made. The researcher sometimes had to insist because the students felt some work documents such as drafts were not interesting enough (there were no cases where the students refused because they thought a document was too personal).

The subsequent interviews were done in French, which was either the mother tongue of the student or a language he/she mastered well enough to be adequately understood by the researcher. They were directive interviews, in which the researcher, using her notes, narrated almost everything she had seen the student doing in chronological order to him/her, asking very vague and open questions to obtain details without inducing a particular answer. People who are asked to talk about their activity performed in a singular situation at one given time easily wander away from particular actions and tend to discuss classes of events (i.e. what they generally do when they are in such a situation) rather situated events (i.e. what they did in this particular situation at that precise moment) [START_REF] Vermersch | L'entretien d'explicitation[END_REF], so the researcher sometimes had to interrupt and redirect the interviewee when he/she strayed from situated actions. The interviews lasted twenty to forty minutes. With the student's consent, they were recorded on an audio tape and later fully transcribed.

At the end of the study, each participant accepted a final individual interview in French with the researcher, in which the learner's work during the two semesters spent in the LRC was discussed. They also filled in several questionnaires aiming to collect personal and biographical information. These enabled the researcher to gain more general knowledge about them.

Data analysis

Through empirical inquiry into the data after it had been transcribed, ideas found in the literature read by the researcher in diverse theoretical and research fields including biology, neurosciences, educational sciences were repeatedly confronted with the data available for each participant in the study. The researcher's aim was to try to identify common elements to the corpus of data and to literature, which might help to make some progress in our understanding of how learners proceed when self-directing language learning. This led to the identification of four main lines of analysis, a process we will detail for one of them.

We have already briefly mentioned the study led by [START_REF] Spear | The organizing circumstance : environmental determinants in self-directed learning[END_REF].

Their preliminary findings show that adults who successfully self-directed a project to learn something did not necessarily plan their learning very precisely at the beginning of the project. On the contrary, in their data the authors found only rare instances of planning, which turned out to be also very vague. As the learning process nevertheless appeared deliberate and logical, they set out to identify the factors structuring selfdirected projects. Their final results showed that the participants in the study had not tried to find a great number and variety of resources so as to choose the best ones for their learning project, but had in fact used a very limited number of resources, sometimes even only one, readily available in their immediate environment or found by chance. They had always chosen one of the first ones that suited the project. Moreover, all that was necessary to the learning process -structure, method, resources, learning conditions -was most of the time given or dictated by circumstances. The learning process progressed, not in a linear way, but from one learning episode to another, and it was only retrospectively that it gave the impression of being linear and directed towards a long-term objective. The authors coined the term 'organizing circumstance' to define these opportunities which the adult learners had seized: '(...) the researchers derived the concept of the Organizing Circumstance which postulates that self-directed learners, rather than preplanning their learning projects, tend to select a course from limited alternatives which occur fortuitously within their environment, and which structures their learning projects.' (4) Spear and Mocker's study suggests that specific circumstances at a precise moment in the life of a learner play an essential role in structuring, organizing, one might even say limiting, the learning process -more than the learner's own will and ability to preplan and monitor it. Danis andTremblay (1985, 1987) reached similar conclusions. In this study, these researchers critically reviewed prevailing adult learning principles in the adult education field by comparing the learning principles suggested in the works of experts in this field (called 'the authors' in the quotations) with principles inferred from the analysis of the descriptions of their own learning experiences made by successful selftaught adult learners. Danis and Tremblay (1985, 438) cautiously remark, in their conclusion, that extrapolations from self-taught adult learners to adult learners in general should be validated, but the fact is that, as Benson reminds (2001, 76-77), part of the research on adult self-directed learning (e.g. [START_REF] Gibbons | Toward a theory of self-directed learning: a study of experts without formal training[END_REF][START_REF] Brookfield | Independent adult learning[END_REF]) has long tried to identify recurrent traits or attitudes in self-taught adults, usually defined as learners who became experts in a domain for which they did not receive formal training. If Danis and Tremblay's study validated the principle according to which 'the learning process is activated and controlled by the learners themselves' (personal translation, Danis and Tremblay 1985, 427), the learning process did not seem to be composed of a succession of learning activities following one another in a sequential and unidirectional order, geared towards a precise goal, with carefully predetermined steps. Therefore, the results contradict the assumption that linear preplanning is central to self-directed learning. Yet, it was unclear what kind of control or monitoring the learners exerted:

'Based on a linear, cyclical interpretation of the adult learning process (...) the authors mostly interpret the learners' activities in terms of management (...). Self-taught adults do not describe their own learning processes from a managerial point of view; they do seem to follow linear, predetermined steps related to the planning and evaluation of their learning activities. Yet, they seem to exert control over their whole learning endeavor : Throughout their learning process, self-taught adults assume the monitoring of their own learning, even when consulting external agents or when participating in formal educational activities. Little is known to date about the type of monitoring which is involved here.' (Danis and Tremblay 1987, 6) The study does give some insights into what actually structured the learning process of these learners: '(...) the heuristic learning process seems to prevail: Self-taught adults proceed in a heuristic manner, within a learning approach which they organise around intentions, redefine and specify without following any predetermined patterns. In this context, the impact of random events upon the entire learning process also stands out: Self-taught adults, in order to learn, take advantage of any opportunities that random events may offer them. ' (6) In other words, the self-taught adults described their learning process in terms of very large and flexible goals, which they rarely chose at the beginning of the learning project, but that they adapted as they progressed according to their preferences, personal needs and according to random events, or 'organizing circumstances' as Spear and Mocker call them. Other similar results in Danis and Tremblay's study (1985) conflict with generally-accepted principles in the adult education field : the learners included in the study did not use a unique combination of learning methods that they thought was the best way to learn, but made use of a variety of learning modes and resources, turning from one to the other according to situations or learning events (432); to decide whether they had succeeded or not, these learners did not seek some kind of formal assessment, but considered the practical results they obtained or the inner feeling they had of having succeeded (430-431).

The learning contexts in the studies led by Spear and Mocker and by Danis and Tremblay are very different from that of our university setting. However, we follow [START_REF] Little | Learner Autonomy -1 : Definitions, Issues and Problems[END_REF] who sees learning autonomy 'as a psychological necessity':

'For it appears that effective and worthwhile learning may actually depend on the extent to which learners achieve autonomy' ( 14).

Against the background of this thesis, learner autonomy being seen as fundamental to learning, we postulate that processes are similar in any type of learning, be it at home, at school or anywhere else. As one of the main lines of analysis in our doctoral research, we therefore set out to determine if and how students in our university language resource centres controlled or monitored their learning process. Very briefly, the three other lines of analysis retained were the following: to what extent learners explored the resources available in the LRCs; whether they developed learning routines; how they managed institutional constraints.

The corpus of data was then qualitatively analysed along the four main lines presented above. The procedure will be outlined here. One table was created per subject, showing all the data for the four working sessions, in order to apprehend observation data and interview data together. In the table, the different actions performed by the learner according to the observation report appeared in one column, while in the next column the quotations from the learner's discourse directly linked to each action were copy-pasted. In these tables, the researcher systematically identified all the parts of the observation data and interview data which she would consider for the four different lines of analysis: an item might therefore appear as an instance of exploration, control of the learning process by the learner, routine or constraint, or several of those. For practical purposes, four different colours were used in all tables, each corresponding to a line of analysis. The eight case studies were first analyzed as singular cases; then they were compared, in search of common and recurrent features, to try to come up with plausible hypotheses concerning the nature of processes involved in self-directed language learning in a university setting.

Results and discussion

The final results obtained for one line of analysis will be presented here and discussed.

The participants in the study were never paralyzed or lost: they always chose a resource and worked quite a lot with it. Most students appeared to spend very little time choosing resources, after exploring only a very small part of what was available in the centre (eg. reading the list of films in the resource catalogue; taking a brief look at available magazines on the shelves; spotting a pile of boardgames to choose from).

When they chose to work with a resource, the learners rarely had a learning goal in mind: they did not often know or could not say why they had chosen to work with a certain resource, or if they had an aim, it was very general (e.g. learn more English vocabulary, understand spoken English better). They frequently indicated they had chosen this or that type of resource or resource because they liked it, which confirms the predominance of the emotional dimension in the learning process [START_REF] Berthoz | La Décision[END_REF][START_REF] Damasio | Descartes' Error: Emotion, Reason, and the Humain Brain[END_REF][START_REF] Candas | Autonomie de l'apprenant et dimensions affectives de l'autoformation. In Enjeux et dilemmes de l'autonomie[END_REF]. For instance, student Et10 liked playing a particular boardgame in French and therefore wanted to try playing in English; student Et6 loved this or that film. During the interviews and when the researcher insisted, some students justified some of their choices, using a cluster of more or less vague criteria having to do with what they thought of the linguistic value of the resource (because it was familiar or thanks to what teachers had said about it), what they knew about the process of learning a language, past learning experiences and more down-to-earth parameters such as what they felt like doing that day and how tired they were. However, it did not seem that these criteria had been rationally and consciously mobilized at the precise moment of the choice. It is only after having worked for quite a long period of time (eg. a whole work session) with a resource that some students said they had decided that they would go on with it or choose one of the same type: for example, student Et10, who, during her session, watched a cartoon in English which she had seen in French and almost knew by heart, stated in the interview that it was a more useful exercise to choose a video one knew well because of the work it had enabled her to do, comparing play-on-words in both languages.

Unfortunately, the limited number of observation sessions did not enable us to confirm that these resources were then regularly made use of, except in one case study where student 9 used three times the same book and audio cassette on general scientific vocabulary because he felt it enabled him to progress as far as his listening skills and understanding of scientific vocabulary were concerned. In our data as in [START_REF] Danis | Principes d'apprentissage des adultes et autodidaxie[END_REF], learners appear to assess the efficiency of a particular learning episode thanks to practical results or their inner feeling of having benefited from it.

In our study, we also analyzed how the participants proceeded with the chosen resources. For that, we distinguished two types of resources: traditional methods or lessons composed of a succession of standard exercises and resources where learners had more freedom to decide how to proceed. The first kind turned to be very interesting, particularly when several students worked with a similar resource. We will give one example with a dictation CD-Rom in which the learner's task is to transcribe an audio file verbatim, sentence after sentence, the software providing help by giving hints as to meaning and uncovering words at the learner's request. Student Et8 listened to every sentence a fixed and limited number of times, using hints but spending little time on them, before uncovering all the unheard words; she argued she did not want to spend too much time on each sentence. On the contrary, students Et9 and Et10 listened to each sentence an amazing number of times, before using a single hint and listening again numerous times, and uncovering one unheard word only after having tried everything else; they argued it was more beneficial that way. Student Et8 executed automatic actions without any reflection on the chosen method as far as learning is concerned, contrary to the two other students. For the second type of resources, the choice of the task or tasks by the learner revealed multiple influences, among which the exam modalities played a predominant role (students had to show a file with all their written productions to the teacher assessing their work), as well as teachers' general recommendations and past learning experiences at school which both placed high value on the same kinds of tasks (e.g. writing down news words and their translation, writing a summary). When asked why they were doing a summary of an article or a film, student Et5 answered that it meant she would have done something, while student Et4 first said it was for her, then corrected herself and admitted that she did not know, before finally talking about this task being a habit, something she always did (interestingly enough, in the next interview, she spontaneously told the researcher that she now thought she did summaries because they were an exam requirement). Institutional constraints and school routines seem to leave little room for more personal reflection and informed decisions, students being logically ready to do whatever is needed to pass their exams. Considering all the learning tasks chosen by the participants altogether, our data showed a very poor variety. However, our data also includes two clear instances where simply describing and explaining actions to the researcher induced learner reflection on a previous choice, leading the student to re-evaluate it in relation to an objective clarified out loud: for example, student Et6 systematically put English subtitles when watching videos, but, after being questioned during one of the interviews on this choice, ended up saying that, although she had not thought about it, it would be interesting to try another video without subtitles so as to train to better understand spoken English. We argue that, this finding, unexpectedly obtained thanks to the chosen methodology of the study, has important consequences for the nature of teacher support when wishing to promote the development process of learner autonomy: indeed, more than giving practical advice or encouraging and training learners to determine precise learning objectives and steps, it might be useful to act as the mirror of their actions and reflect back to them what they are doing and how they are proceeding and to enable them to talk about it, in the hope that explaining and justifying their choices might make them aware of incoherencies and envisage necessary changes. This interviewing technique of course reminds us of 'the counselling sessions' and its protagonist, 'the counsellor', whose role has been theorized by [START_REF] Gremmo | Conseiller n'est pas enseigner: le rôle du conseiller dans l'entretien de conseil[END_REF].

Conclusion

To sum up, what our analysis shows is that the learning paths of the participants in the study are structured by large and flexible goals [START_REF] Danis | Principes d'apprentissage des adultes et autodidaxie[END_REF], or, more adequately, by a sort of 'fleeting intentionality', in which what one likes and how one feels also play a part, and by the resources readily available in the environment [START_REF] Spear | The organizing circumstance : environmental determinants in self-directed learning[END_REF]. We chose to refer to the way in which these learners control or monitor their learning process as 'loose piloting', where institutional constraints and habits or routines play a predominant role, rendering teacher support essential to trigger reflective thinking in learners and enable them to make more personal choices. The learners' focus is clearly on doing things and any inflection in the learning path or modes is decided upon as they progress, much more than things are planned from the beginning onwards. It must be said that the participants in the study admitted working mainly to validate their compulsory language credits, although all said they liked or found the English language useful, and that they did not report having a precise personal learning project as far as the English language was concerned. And of course one might argue that these eight learners are not autonomous at all. Our hypothesis, on the contrary, is that these students are autonomous learners at different degrees. Therefore, we argue that our study contradicts the widely-accepted assumption that planning (setting learning objectives, defining a programme and linear steps, choosing relevant learning methods and modes and assessing results in relation to pre-defined objectives) is central to self-directed learning. Although our results remain to be confirmed on larger numbers of learners studied for a longer time span, they open challenging areas of investigation into the nature of the processes involved in self-directed learning.

These results were obtained thanks to an original methodology, combining direct observation of the activity of individual learners during a series of work sessions in a language resource centre and subsequent directive interviews.

Preliminary analyses showed that data resting solely on observation were not sufficient to formulate strong hypotheses on the nature of the processes taking place.

Indeed, two students, apparently doing the same task on a similar resource, were in fact proceeding very differently. It is true that very fine observation of a series of actions when carrying out a task might enable the researcher to infer plausible hypotheses, but, to reach a conclusion, the students' answers in the interviews are needed. Our study also confirmed, after [START_REF] Vermersch | L'entretien d'explicitation[END_REF], that interviewees tended to discuss classes of events rather situated events, thus requiring that the interviewer firmly directs the interview. So, to have access to reliable data, observing effective work sessions is essential. To conclude, the exposed methodology, though complex and long to put into practice, can be deemed relevant for the study of the processes involved in self-directed learning in that it gives access to extensive and detailed information on these processes, which are not accessible by means of more traditional types of data collection.

  I am deeply indebted to Jérôme Eneau, doctor in educational sciences and senior lecturer at The University of Rennes, France, for providing several major references without which my work would not have been what it is. Notes 1) The Louis Pasteur University became part of The University of Strasbourg on January 1 st 2009 when the three universities of Strasbourg (Marc Bloch University, Robert Schumann University and Louis Pasteur University) merged into a single university. 2) I am referring here to the version of the European Language Portfolio (ELP) which has been specifically designed for use in universities across Europe. It was accredited by the Council of Europe's Validation Committee at the request of CercleS (Confédération européenne des centres de langues de l'enseignement supérieur). Peggy Candas, now doctor in educational sciences and a member of the LISEC (Laboratoire Interuniversitaire des Sciences de l'Education et de la Communication, The University of Strasbourg), defended her PhD at the University of Strasbourg in February 2009. She is also currently working in the language resource centres of the University of Strasbourg, as an English teacher and counsellor.

	Notes on contributor

Acknowledgements