Nicolas Couellan

Probabilistic Robustness Estimates for Deep Neural Networks

In the case of deep dense neural networks, under random noise attacks, we propose to study the probability that the output of the network deviates from its nominal value by a given threshold. We derive a simple concentration inequality for the propagation of the input uncertainty through the network using the Cramer-Chernoff method and estimates of the local variation of the neural network mapping computed at the training points. We further discuss and exploit the resulting condition on the network to regularize the loss function during training. Finally, we assess the proposed tail probability estimate empirically on three public regression datasets and show that the observed robustness is very well estimated by the proposed method.

Introduction

Deep neural networks have been found to be very sensitive to data uncertainties [START_REF] Fawzi | The robustness of deep networks: A geometrical perspective[END_REF][START_REF] Szegedy | International Conference on learning representations (ICLR)[END_REF] to the point that a whole research community is now addressing the so-called network attacks. Attacks may be random, when data are corrupted by some random noise or may be adversarial, when the noise is specifically designed to alter the network output [START_REF] Szegedy | International Conference on learning representations (ICLR)[END_REF]. In this article, we will focus on the random case. Most data are usually uncertain, either because the data are related to naturally noisy phenomenon and we only have access to some of its statistics or because assessing devices to do not have sufficient accuracy to record precisely the data. Most people have addressed the problem of robustness to bounded input perturbations through the use of regularization techniques [START_REF] Gouk | Regularisation of neural networks by enforcing lipschitz continuity[END_REF][START_REF] Oberman | Lipschitz regularized deep neural networks converge and generalize[END_REF][START_REF] Virmaux | Lipschitz regularity of deep neural networks: analysis and efficient estimation[END_REF][START_REF] Finlay | Improved robustness to adversarial examples using lipschitz regularization of the loss[END_REF]. The 1 ENAC, Université de Toulouse, Toulouse, France 2 Institut de Mathématiques de Toulouse UMR 5219, Université de Toulouse UPS IMT, Toulouse, France. Correspondence to: Nicolas Couellan <nicolas.couellan@recherche.enac.fr>.

Presented at the ICML 2020 Workshop on Uncertainty and Robustness in Deep Learning. Copyright 2020 by the author(s). main idea is to consider the neural network as a Lipschitz map between the input and output data. The Lipschitz constant of the network is then estimated or upper bounded by the norm of the layer-by-layer weights product. This estimates the expansion or contraction capability of the network and is then used to regularize the loss during training. Often, there is a price to pay: the expressivity of the network may be reduced, especially if the weights are too constrained or constrained layer by layer instead of constrained accross layers (Couellan, 2019). Such strategies are enforcing robustness but do not provide guarantees or estimates on the level of robustness that has been achieved. In the case of adversarial perturbation, some authors have proposed methods for certifying robustness [START_REF] Kolter | Provable defenses against adversarial examples via the convex outer adversarial polytope[END_REF][START_REF] Boopathy | Cnn-cert: An efficient framework for certifying robustness of convolutional neural networks[END_REF]. Recently, a probabilistic approach has also been proposed in the case of random noise for convolutional neural networks [START_REF] Weng | Verifying robustness of neural networks with a probabilistic approach[END_REF]. The authors derive probabilistic bounds based on the idea that the output of the network can be lower and upper bounded by two linear functions. The work proposed here is along the same line but distinct in several aspects. It combines upper bounds on tail probabilities calculated by deriving a specific Cramer-Chernoff concentration inequality for the propagation of uncertainty through the network with a network sensitivity estimate based on a network gradient calculation with respect to the inputs. The network gradient is computed by automatic differentiation and estimates the local variation of the output with respect to the input of the network. The estimation is carried out and averaged over the complete training set. A maximum component-wise gradient variation is also calculated in order to give probabilistic certificates rather than estimates. The certificates can be used in place of estimates whenever guaranteed upper bounds are needed, however they are often not as accurate since they are based on variation bounds rather than averages. We then discuss the use of the derived bounds and estimates in the design of deep robust neural networks and conduct experiments in order to assess the robustness probabilitic estimates for various regularization strategies.

Probabilistic certificates of robustness

Consider feed-forward fully connected neural networks that we represent as a successive composition of linear weighted combination of functions such that x l = f l ((W l ) x l-1 +b l ) for l = 1, . . . , L, where x l-1 ∈ R n l-1 is the input of the l-th layer, the function f l is the L f -Lipschitz continuous activation function at layer l, and W l ∈ R n l-1 ×n l and b l ∈ R n l are the weight matrix and bias vector between layer l -1 and l that define our model parameter θ = {W l , b l } L l=1 that we want to estimate during training. The network can be seen as the mapping g θ : x 0 → g θ (x 0 ) = x L . The training phase of the network can be written as the minimization of the empirical loss L(x, y, θ) = 1 n n i=1 l θ (g θ (x i ), y i ) where l θ is a measure of discrepancy between the network output and the desired output. Assume now that we only have access to noisy observations x i of the input sample. However, we know that these observations are drawn from a distribution D with finite support. We first consider the special case where the functions f l are linear or piece-wise linear (this includes the case of ReLu activation functions) and then extend the analysis to more general functions. We first consider the one layer simple case where the outputs of the network y = x L (with L = 1 in this case) depends linearly of the inputs x = x 0 ∈ R n as follows: w x + b where w = W 1 is the single layer weights vector and b = b 1 is the layer bias. We assume that our input observations are corrupted by some additive noise such that ∼ D and ∀i = 1, . . . , n, we have i ∈ [-γ, γ] with γ < +∞. Our objective is to ensure the following property:

P ∼D ( y -y ≤ Γ) ≥ 1 -α (1)
where y = w (x + ) + b, Γ is the allowed output uncertainty and 1 -α is some predefined level of confidence.

In the following proposition, we give a condition on w that will ensure, with probability greater than 1 -α, that the output uncertainty remains below Γ.

Proposition 2.1 If the network inputs are subject to an additive uncertainty where ∀i = 1, . . . , n, i ∼ D and supp(D) ∈ [-γ, γ] (γ < +∞), then for a given α ∈ [0, 1] and a given output uncertainty level Γ, the following condition holds: If the layer weights vector w satisfies ww F ≤

Γ 2 γ 2 √ 2 log(1/α) , then P ∼D ( y -y ≤ Γ) ≥ 1 -α.
(the proof of Proposition 2.1 is provided in Appendix A). We now address the case where the network is composed of several layers l = 1, . . . , L with linear or piece-wise linear activation functions. Let W = L l=1 W l . Property (1) should now relate to the output of the last layer as follows:

P ∼D x L -x L ≤ Γ ≥ 1 -α (2)
where x L is the output of the layer L when a noisy input x 0 = x 0 + is propagated through the network and is the additive input noise such that ∼ D and ∀i = 1, . . . , n, we have i ∈ [-γ, γ] (γ < +∞). With this setting, we can now state and prove the following:

Proposition 2.2 If the network inputs are subject to an additive uncertainty where ∀i = 1, . . . , n, i ∼ D and supp(D) ∈ [-γ, γ] (γ < +∞), then for a given α ∈ [0, 1] and a given output uncertainty level Γ, the following condition holds: If the layer weights vector w satisfies

WW F ≤ Γ 2 γ 2 √ 2 log(1/α) then P ∼D x L -x L ≤ Γ ≥ 1 -α.
(the proof of Proposition 2.2 is provided in Appendix A). The bound derived above relies on the following inequality (see ( 8) in Appendix A for multilayer architectures):

P ∼D x L -x L ≤ Γ ≥ P ∼D tr WW ≤ Γ 2
(3) The Chernoff bound proposed above may be tight with respect to the right hand side of the above inequality, however, with respect to the left hand side, (3) is not tight in general as the deep complex layer structure of the neural network generates a highly non linear behavior. The linear upper bound is often of poor quality. To address this issue, we propose an alternative to estimate the variation of the neural network response. Without loss of generality, in the following, we will consider that the network output is 1-dimensional. Since may be considered small with respect to the magnitude of the network inputs, we are interested in the local behavior of the network output that we will approximate as follows:

x L -x L = F (x + ) -F (x) ∇ x F (x) (4)
where

F (x) = f L ((W L ) f L-1 (. . . f 0 ((W 0 ) x + b 0 )) + b L-2 ) + b L-1 ) and ∇ x F is the gradient of F with respect to the network input x.
The linear approximation ( 4) is only local but its advantage is that it can easily be evaluated at many x values. Indeed, while visiting all input vectors x during training, this information is usually available at a very low extra computational cost through automatic differentiation in most training computer algorithms and packages for neural networks (such as TensorFlow [START_REF] Abadi | TensorFlow: Large-scale machine learning on heterogeneous systems[END_REF] and Pytorch [START_REF] Paszke | Automatic differentiation in pytorch[END_REF]). Therefore, from local estimates at various training points x 0 i for i = 1, . . . , n, we calculate two n 0 -dimensional vectors of network variation estimates

∇ x F such that ∇ x F k = sign ∇ x F (x 0 i k ) k × v k and ∇ x F = 1 n n i=1 ∇ x F (x 0 i ) where v k = max i∈{1...,n} ∇ x F (x 0 i ) k and i k = argmax i∈{1...,n} ∇ x F (x 0 i ) k .
The quantity ∇ x F accounts for the maximum variation of the network response in every component direction of the network input encountered during the training and gives therefore a tighter linear upper bound than W when input data are part of the training set. The quantity ∇ x F does not provide a linear upper bound but estimates an average linear behavior of the network response that can be used in practice to estimate the required Γ value to reduce

P ∼D x L -x L ≥ Γ below α.
The larger is the training dataset, the higher is the quality of these estimates. Replacing W by these quantities in (12), we derive the following robustness bound for the network:

∇ x F ∇ x F F Γ 2 γ 2 2 log(1/α) . ( 5
)
and the following estimate of Γ to achieve a robustness confidence level of 1 -α:

Γ γ 2 log(1/α) ∇ x F ∇ x F 2 F 1 4 . (6)

Controlling the bound during training

In this section, we are interesting in exploiting the bounds derived above during the process of training the neural network. The main idea would be to ensure that optimal weights after training are satisfying the bound constraint (5). Naturally, this could be formulated as a constrained optimization training problem. Stochastic projected gradient techniques [START_REF] Nedic | On stochastic subgradient mirrordescent algorithm with weighted averaging[END_REF][START_REF] Lacoste-Julien | A simpler approach to obtaining an o(1/t) convergence rate for the projected stochastic subgradient method[END_REF] could be used to solve such a problem. However, in the general case, the projection operator for such constraint is not simple and would require important computational effort. Therefore, instead of ensuring the constraint, we propose to regularize the loss function during training by adding a penalization term as follows:

min θ=(W,b) 1 λ L(x, y, θ) + Q W 2 F
where λ is a positive parameter, L is a loss function (mean squared error for example) and Q W F is the Frobenius norm of a matrix Q W that could be chosen as ∇ x F ∇ x F , ∇ x F ∇ x F or WW , depending on which bound from above we want to exploit. Regularization is a common practice in machine learning [START_REF] Bishop | Pattern Recognition and Machine Learning (Information Science and Statistics)[END_REF][START_REF] Goodfellow | Deep Learning[END_REF] and is usually proposed to avoid overfitting and increase model generalization. The connection between generalization and robustness to input uncertainty in machine learning models has been established in several studies (Xu et al., 2009;[START_REF] Staib | Distributionally robust optimization and generalization in kernel methods[END_REF]. Intuitively, the Q W 2 F regularization term acts as a special weight contraction and it is natural to consider alternative possibilities to reduce the magnitude of the network weights. One alternative is the squared spectral norm of W (largest eigenvalue of Q W ) that would also account for the maximum absolute contraction of a vector when multiplied by Q W . Finally, in (Couellan, 2019), the product L l=1 W l which is an upper bound of the Lipschitz constant of the network has also been proposed as regularization that promotes robustness. It accounts for the overall Lipschitz regularity of the network and acts also as an overall control on the contraction power of the network by coupling layers and allowing some weights to grow for some layers as long as in other layers others weights are getting smaller to compensate. When Q W = WW , its Frobenius norm and the Lipschitz constant gradient can be explicitely derived and integrated into the backpropagation scheme and chain rule of gradients in order to optimize the augmented loss during the training phase. However, for the spectral norm, approximation methods are necessary and gradient will have to be computed using numerical differentiation techniques. In Appendix B, we discuss several available approximation methods and in the next section, we propose to carry out experiments with these various regularization strategies and evaluate their respective impact on the robustness properties of the network.

Experiments

In order to assess the quality of the calculated bound, experiments are conducted on public datasets. We focus on deep neural network regression tasks (linear output activation) and the BOSTON [START_REF] Harrison | Hedonic prices and the demand for clean air[END_REF], DIABETES [START_REF] Tibshirani | Least angle regression[END_REF] and CALIFORNIA [START_REF] Pace | Sparse spatial autoregressions[END_REF] datasets. The neural network and its training and testing are implemented in the python [START_REF] Team | Python: A dynamic, open source programming language, python software foundation[END_REF] environment using the keras [START_REF] Chollet | [END_REF] library and Tensorflow [START_REF] Abadi | TensorFlow: Large-scale machine learning on heterogeneous systems[END_REF] backend. The neural network architecture is composed of 4 dense hidden layers with 50 ReLu activations neuronal units and one dense linear output layer. All results that are presented below are average results from 10 independent runs that are carried out after random shuffling and random splitting of datasets. All dataset samples are scaled so that they lie in [-1, 1]. All neural network training procedures are executed with the ADAM stochastic optimization algorithm with default parameters as given in [START_REF] Kingma | A method for stochastic optimization[END_REF]. Additionnal details about the datasets dimensions and training parameters are given in Table 1 of Appendix C. All comparison results provided below are referring to training with mean squared loss and the following regularization schemes as described in section 3:

no reg: without regularization -Lipschitz reg: L l=1 W l regularizer as described in (Couellan, 2019).

-Gradient reg: Q W 2 regularizer as described above and where

Q W = ∇ x F ∇ x F .
-MaxEig reg: λ max (Q W ) 2 regularizer.

In order to estimate the probability P ∼D ( y -y ≤ Γ), the following procedure is applied. For each test sample from the validation set, we generate random vectors j with j ∈ {1, . . . , 10} and calculate the following probability estimate:

1 10 × T T i=1 10 j=1 1 { y (i) -y (i) j ≤Γ} ( j ) (7)
where T is the number of samples in the testing set, y (i) is the desired output for the i-th testing sample and y

(i) j
is the output of the network calculated via a forward pass through the network for the input vector x (i) + j . In all experiments, we have supp(D) = [-γ, γ] with various levels of γ. Results are reported in the figures of Appendix D where γ is referred as gamma and Γ as Gamma. Figure 1(right) reports these observed probabilities together with the estimated tail probabilities given by exp{-Γ 4 /(2γ 4 ∇ x F ∇ x F 2 F )} for various values of Γ, while Figure 1(left) reports the corresponding mean validation error achieved during the training process. The probability level 1 -α (with α = 0.05) is also marked with a blue dashed on each plot on the right. Figure 2 provides further details about the magnitude of the norm of the network gradient ∇ x F and λ max (Q W ) 2 (re-scaled by a factor 10 to ease the reading of the plot) for each dataset and the four regularization strategies. Finally, Figure 3 provides, for each dataset and each regularization strategies a comparison of the Γ values achieved to reach a 1 -α probability level. Three values are reported each time,

Γ(max) = (2 log(1/α)) 1/4 ∇ x F ∇ x F F , Γ(mean) = (2 log(1/α)) 1/4 ∇ x F ∇ x F F
and the Γ value observed so that the probability given in ( 7) reaches a level 1 -α. We see in Figure 1 that, for the three datasets and for a probability of 0.95, the calculated Γ value (x axis) obtained by the expression of the exponential tail probability, provides a very good estimate of the Γ value given by the observed probability (probability that the output deviates from its nominal value by more than Γ). This validates experimentally, at least for these datasets, the relevance of the estimate given in (6). For the BOSTON dataset, all regularizing strategies give similar Γ values whereas for the DIABETES and CALIFORNIA, the Γ values are more sensitive to the type of regularization employed. However, surprisingly, no general rule can be given from these results. It is difficult to say which regularizer performs best. It is dataset dependent. The left hand side plots of Figure 1, representing the validation mean absolute error, show that similar training performance were achieved by the four regularizing methods and do not provide further explanation of this phenomenon. We believe, without providing any evidence of it, that the high nonlinearity of the neural network error surface may explain it. Indeed, after training, the optimization algorithm has reached a local minimum where the loss value may not have decreased sufficiently to really express the regularization power of the regularizer. This depends on the geometry of the error surface that is greatly dependent on the input data. On Figure 2, the norm of the network gradient tends to be slightly smaller for the Gradient reg strategy. This would confirm that regularizing by the network gradient would help in achieving better robustness. Additionally, the figure also shows that the maximum eigenvalue regularization is not correlated to the network gradient norm and may not be a suitable alternative for robustness purposes. The Γ value comparison in Figure 3 confirms that the Γ estimates calculated by the proposed method are very closed to the observed values. This is true for all datasets and regularizing strategies. Furthermore, the Γ upper bound values Γ(max), are loose as expected in Section 2 but provide certificates for robustness. These certificates follow the same pattern as the norm of the network gradient in Figure 2, which was also expected since their expression in (5) are directly dependent. Therefore, as for the network gradient, we observed that these certificates are better (tighter but still quite loose) for the network gradient regularizing strategy.

Conclusions

In this study, we have proposed analytical probabilistic estimates (and certificates) for deep dense neural networks. The idea combines tail probability bound calculation using the Cramer-Chernoff scheme and the estimation of the network local variation. The network gradient computation is using the automatic differentiation procedure available in many neural network training packages and carried out only at the training samples which does not require much extra computational cost. Experiments with this method has been conducted on public datasets and has shown that the robustness estimates are very good compared to the observed network robustness. Further analysis on these datasets show that the quality of the estimates is not really impacted by the regularization strategy, however, the network gradient regularization tends to generate slightly more robust network architectures. , then P ∼D ( y -y ≤ Γ) ≥ 1 -α.

Proof: We start by observing that in general we have

P ∼D ( y -y ≤ Γ) ≥ P ∼D tr w (w ) ≤ Γ 2 , ( 8
)
where tr(A) defines the trace of a matrix A. Note that in the particular case where all activation functions are linear, equality is achieved. Therefore inequality (1) may be satisfied by ensuring that

P ∼D tr(w (w ) ) ≥ Γ 2 ≤ α, or P ∼D tr( ww ) ≥ Γ 2 ≤ α.
We now state and prove the following Lemma which provides a simple concentration inequality for the above probability.

Lemma A.2 For any random matrix M ∈ R n×n of the form M = vv where v is a random vector such that for all i in {1, . . . , n}, v i are all independent, have finite support included in [-δ, δ] and E(v i ) = 0, we have

∀Q ∈ R n×n , ∀t > 0, P (tr(M Q) ≥ t) ≤ e - t 2 2β 2 δ 4 Q 2 F . (9) where β = min{ρ > 0 s.t. (ρz) 2 2 -log(cosh(z)) ≥ 0}.
Proof: The proof is based on the Cramer-Chernoff method [START_REF] Boucheron | Concentration Inequalities, a nonasymptotic theory of independence[END_REF] to bound the tail probability of the random variable tr(M Q). Applying Markov inequality to the left hand side of (9), we have: (10) Observe now that since for all i = j, we have that cov(m ii , m jj )=0 and since

P (tr(M Q) ≥ t) ≤ E(tr(M Q)) t and for any p ∈ R + , since m ij = v i v j = m ji , P (tr(M Q) ≥ t) ≤ e -pt
E(v i ) = E(v j ) = 0 that cov(m ii , m ij ) = cov(v i v i , v i v j ) = 0.
Therefore, from (10), we have

P (tr(M Q) ≥ t) ≤ e -pt n i,j=1
E (e pmij qji ) .

Let now ψ tr(M Q) be the moment generating function of tr(M Q), its Cramer transform obtained by Fenchel-Legendre duality is

ψ * tr(M Q) (t) = sup p≥0 (pt -ψ tr(M Q) (z)).
Note that m ij ∈ [-δ 2 , δ 2 ] and that by convexity of the exponential function, we can bound the moment generating function of m ij as follows:

∀z ∈ R, E(e zmij ) ≤ 1 2 e -zδ 2 + 1 2 e zδ 2 = cosh(zδ 2 ).
Therefore, around zero, we can write

log(E(zm ij )) ≤ log(cosh(zδ 2 )) ≤ (βzδ 2 ) 2 2
where β is a coefficient in (0, 1] that tightens the bound as much as possible and can be defined as β = min{ρ > 0 s.t. (ρz) 2

2

-log(cosh(z)) ≥ 0}. Replacing z by pq ji with p > 0, we can derive the following upper bound for the Cramer transform of ψ * tr(M Q) :

∀t ∈ R, ψ * tr(M Q) (t) ≤ min p>0    -pt + n i,j=1 (pq ji βδ 2 ) 2 2    .
The minimum in the right hand side of the expression above is reached at p * = t β 2 δ 4 Q 2 F and therefore, applying Cramer inequality, we finally get

log(P (tr(M Q) ≥ t)) ≤ - t 2 2β 2 δ 4 Q 2 F , which completes the proof of the lemma. Note that ∀(i, j) ∈ R n×n , we have ( ) ij ∈ [-γ 2 , γ 2 ].
Hence, applying Lemma A.2 to bound P ∼D tr( ww ) ≥ Γ 2 will lead to

P ∼D tr( ww ) ≥ Γ 2 ≤ e - Γ 4 2β 2 γ 4 ww 2 F ≤ e - Γ 4 2γ 4 ww 2 F , since β ∈ (0, 1].
In order to remain below the level α, we then need that

- Γ 4 2γ 4 ww 2 F ≤ log(α). (11)
This can also be written as ww

≤ Γ 2 γ 2 √ 2 log(1/α) ,
L l=1 W l L l=1 W l F ≤ Γ 2 γ 2 √ 2 log(1/α) then P ∼D x L -x L ≤ Γ ≥ 1 -α.
Proof: Propagating forward the input uncertainties through the network, we can write:

x L -x L = (W L ) L-1 = (W L ) (W L-1 ) L-2 . . . (W 1 ) = L l=1 W l
where l is the propagated noise from layer 1 to layer l. Furthermore, we have Therefore, applying again Lemma A.2 to upper bound the probability in (2), the following condition on the layer weights matrices to ensure property (2) is directly obtained:

L l=1 W l L l=1 W l F ≤ Γ 2
γ 2 2 log(1/α) . (12)

B. Spectral norm approximation

Among available approximation methods, the power iteration algorithm [START_REF] Allaire | Numerical Analysis and Optimization[END_REF], or preferably the Lanczos algorithm [START_REF] Lanczos | An iteration method for the solution of the eigenvalue problem of linear differential and integral operators[END_REF] since Q W is symmetric, is well suited for the purpose. Note that there is no real need to approximate λ max (Q W ) (the largest eigenvalue of Q W ) with great accuracy as it is only used to as a regularization function to guide the optimization process towards optimal regions where the spectral norm is reduced. Therefore, an alternative approximation technique is to use an upper bound of λ max (Q W ). As Q W is positive definite, we propose to use the Dembo's upper bound [START_REF] Dembo | Bounds on the extreme eigenvalues of positivedefinite toeplitz matrices[END_REF] defined as follows:

Let A n ∈ R n×n be an Hermitian positive definite matrix and let λ ( n) 1 ≤ . . . λ ( n) n be the eigenvalues of A n . The matrix A n can be written as

A n = A n

C. Tables

D. Figures

  Xu, H., Caramanis, C., and Mannor, S. Robustness and regularization of support vector machines. J. Mach.Learn. Res., 10:1485-1510, 2009. If the network inputs are subject to an additive uncertainty where ∀i = 1, . . . , n, i ∼ D and supp(D) ∈ [-γ, γ] (γ < +∞), then for a given α ∈ [0, 1] and a given output uncertainty level Γ, the following condition holds: If the layer weights vector w satisfies ww F ≤

  E e p tr(M Q) ≤ e -pt E e p n i,j=1 mij qji leading toP (tr(M Q) ≥ t)

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Mean absolute validation error profiles during training (left) & Comparison of P ∼D ( y -y ≤ Γ) and exponential tail bound for various levels of Γ (right)

Table 1 .

 1 Dataset information and experimental setup

		#	#	test/train	batch	#	learning
	Dataset	inputs	samples	ratio	size	epochs	rate
	BOSTON	13	606	0.2	50	100	0.001
	DIABETES	10	442	0.2	200	30	0.001
	CALIFORNIA	8	20640	0.4	600	30	0.001


Jonathan Dézécot
email: jonathan.dezecot@univ-lemans.fr

Nathalie Fleck
email: nathalie.fleck@univ-lemans.fr

AN EXPLORATION OF THE PERCEPTION OF BRAND MANAGEMENT AND MARKETING IN CRAFTSMANSHIP: STRUCTURED ABSTRACT

INTRODUCTION

Craftsmanship plays a substantial role in the economic environment, with more than 300 billion euros in sales and more than three million employees in France for instance. It has, however been little studied in marketing, where research has mostly focused on consumer goods or services provided by industrial firms. Such firms are often distinct from craft enterprises in that they usually have a dedicated marketing department to work on brand identity (Kapferer 1996). In the present study, we specifically consider independent artisans, whose products can be found in local shops. Recent research on the concept of the human brand can help us understand the artisan as a brand (Thomson 2006, Moulard andal. 2015). A professional human brand can be measured by two dimensionsexpertise and creativity (Parmentier and Fischer 2010) -, both of which are essential for the full development of a human brand (Arnould and Dion 2013). These two dimensions also characterize the artisan: He can be evaluated according to his skills based on his knowledge of the craft, know-how and expertise (Simeoni 1999; Amans, Bravo and Loup 2012) and according to his artistic qualities in relation to the creativity he reveals in his handmade work (Loup and Rakotovahiny 2011).

In the eyes of consumers, craftsmanship is often associated with notions of authenticity, expertise and proximity, whereas industrial brands are associated with elaborate marketing and less authenticity, resulting in a process of commodification (Camus 2001). Interestingly, in some sectors such as luxury or food, major brands claim values similar to those of craftsmanship. Given the usual opposition between craft and industrial brands, can this be reversed? Would an artisan who decides to apply marketing methods and tools used by industrial brands contradict the values of craftsmanship and consumers' perception of it? An interesting parallel may be drawn with artisanal and local products. A study on regional products shows that the marketing of local products is rejected because there is "too much cognitive dissonance between the image of local product and the underlying marketing approach" (Fort and Fort 2006). Specifically, local products and marketing do not match up because consumers are attached to the authenticity of these products and any changes may have a negative impact on their perception.

The aim of this study is to understand whether it is appropriate for an artisan to adopt a marketing approach. Is it risky for his image and might he then lose the values of authenticity and proximity usually associated with craftsmanship? This paper intends examining this question by means of a qualitative exploratory study. A focus group was formed among consumers to explore people's reactions when artisans adopt marketing methods. The main results of this analysis are then presented and discussed.

METHODOLOGY

The topic is relatively new and our study aims to understand in depth what artisans represent for consumers and whether such representations are compatible with brand management and marketing development strategies. For this exploratory study, a focus group was formed with consumers who regularly buy in artisanal shops. This methodology allows us to obtain the in-depth views of the respondents on the basis of their personal experience. It goes further in the exploration of the topic than individual interviews would permit, due to the group dynamics (Festinger and Katz 1974;Gibbs 1997). A convenience sample of six consumers was selected on the basis of the criterion of regularly purchasing from artisanal firms in the food industry. We initially limited ourselves to this sub-sector because it involves day-to-day activities and offers an easy comparison between small artisanal businesses and large industrial firms.

The interview lasted 1h30 with a guideline around three themes: (1) the shopping experience in artisanal outlets and the reasons for this choice, (2) the role of the craftsman in this choice, and (3) reactions to the artisan's development strategies, based on four scenarios (moving to a larger shop, creating a second point of sale in another city, creating an e-commerce website with home delivery, and increasing the number of employees accompanied by the artisan's reorganization of his working practice). The participants' discourse were then transcribed and analyzed thematically.

RESULTS AND DISCUSSION

Analysis of transcripts shows that the various benefits traditionally associated with brands are also associated with the artisan, who may be considered as a brand. Furthermore, the analysis also reveal the boundaries in terms of developing marketing of the business, beyond which the artisan may lose these benefits.

'Artisan-brands' share benefits with traditional brands

The participants' discourses show that the various benefits for consumers associated with major brands, be they functional, experiential or symbolic (Keller 1993), also apply in the case of artisans, as brand's functions do. The functional benefits of artisan-brands correspond to intrinsic values of artisanal products and are strongly linked to the local situation. First, product quality justifies making artisanal food purchases, as was evident from the comments of all the participants during the interviews.

"Yes, what is expected is product quality above all. I was sure of the product quality of my artisan" (Emmanuel); "The artisan is a guarantee of product quality" (Florent).

The second utilitarian benefit is the convenience arising from the geographical proximity of the artisan, which is a primary criterion for consumers in choosing a retail store for their food purchases (Colin et al. 2012).

"The practical side is very important, I alternate between different artisans in my town of residence and those close to my work" (Brice); "I go where it's most convenient" (Florent).

However, product quality always predominates in the choice and is the overriding pre-condition, as Brice and Emmanuel emphasize.

"I think convenience is very important. I shop nearby close and if it's good, whether for bread or meat, I go back. However, if I am disappointed by the quality, I'll go further afield" (Brice), "I go to the same artisan primarily for the quality of the product, and then for the convenience " (Emmanuel).

Experiential benefits of the artisan brand are linked to the consumption of the products and the pleasure derived from buying them. Firstly, consumers acknowledge the pleasure linked to the quality of artisanal products and they clearly distinguish between industrial and artisanal products.

"I go to supermarkets for a lot of products but generally to the baker for bread, as the baker's bread is better than the supermarket's" (Emmanuel); "I buy bread in supermarkets only when I cannot go to my artisan and I see the difference in quality" (Brice).

Furthermore, going to artisanal shops allows consumers to participate in a consumption experience (Holbrook and Hirschman 1982), with the pleasure of talking to the person who manufactures the products and establishing a relationship with him.

"We have developed a cordial relationship" (Florent); "A friendly relationship has been created" (Emmanuel).

As regards the symbolic benefits, buying an artisan-brand can be a means of fulfilling the need for personal expression, social approval, or more generally relating to others (Keller 1993). Artisanal products are viewed as authentic quality products, either homemade or locally produced. These products may be chosen on a more occasional basis, for example to please guests invited to one's home. According to Fuchs, van Osselaer and Schreier (2015), hand-made products generate an emotional dimension in the producer-consumer relationship and this affect is transmitted if a hand-made product is given to a friend or relative. Many studies on food consumption show that consumers are increasingly turning to local products (Fort and Fort 2006;Gabriel and Urien 2006;Merle and Piotrowski 2012). Doing so answers to the need for assurance with regard to food consumption (Gallen 2001), as well as the search for authenticity and nostalgia (Fort and Fort 2006), characteristics that enhance the symbolic aspect of artisanal products.

Consumers' reactions to the development of marketing strategies by artisans

Respondents were presented with four scenarios of development strategies by an artisanal business. The initial situation was the same for the four scenarios, that is, a traditional artisanal butcher, with a small shop in a village, working with his wife, one employee and an apprentice. The scenarios were: (1) moving to a larger shop in a big city, (2) opening a second shop in a different location, (3) setting up an ecommerce website with a home delivery service, and (4) increasing the number of employees, with work reorganization by the artisan, who becomes less present in the store and takes on more administrative tasks. These scenarios highlighted what consumers view as the artisan's business and reveal a line that if crossed can lead to a loss of authenticity.

In consumers' imagination, the artisan is associated with a small shop. Enlarging the sales area is generally rather poorly perceived by respondents. "It's psychological, it's preferable to go to the same small store with the same quality" (Sylvie); "The image of the artisan is that he must be in a cozy shop where the saleswoman is smiling, where the product is good. He will lose artisanal status compared to his original situation" (Emmanuel); "It's puzzling" (Brice).

This situation raises worries about the role of the artisan in his business. "Enlargement will mean the artisan won't be able to manage everything, so it'll be much more like a production line than artisanal work" (Pascale).

This undermines the trust in the artisan and even the perceived quality of the products.

"If the existing organization stays the same, they couldn't produce everything in an artisanal way for supplying a large store. I'd trust it less" (Florent); "I'd tend to say that the quality may decrease because the artisan cannot be everywhere" (Marion).

Filser (1998) developed the theoretical concept of critical size for retailers, which states that below a certain size, a retailer cannot survive in the market. For artisans, the reverse seems to be the case, because an artisan has to sell his products in a small shop to so as ensure the hedonic character of visiting the store. Small size is essential for the perceived image of an artisanal business. The artisan's activity has to remain on a human scale, otherwise its artisanal values may vanish and the perceived quality of its products decline. Being able to visit or at least see artisanal workshops reassures consumers about the retention of artisanal values (Bergadaà 2008), in the same way that the presence of labels of origin on industrial products confers credibility (Larceneux 2003).

Alternatively the artisan may further modernize his business. Opening an e-commerce site tends to be seen as a vector of modernity that does not affect the values and perception of the artisan.

"It doesn't shock me. He will always keep his artisanal values" (Emmanuel); "To have a modern artisan is pretty good, it won't change the quality of the products, it's just that he'll sell his products to new customers" (Brice).

The above is true provided that product quality remains unchanged, otherwise trust in the artisan is undermined.

"These are the same products, so it doesn't change anything" (Pascale); "If the products are handmade, the same as in the store, and transport is good in terms of hygiene and freshness with his name stamped on packages, it's ok" (Emmanuel). But this department mustn't become too large: "I ask myself about the links between food production, the store, and the internet business" (Florent).

The legitimacy of the artisan is a guarantee of quality. With regard to gastronomy, customers will primarily choose a restaurant for the chef, not for the setting (Dion and Boissieu 2013). The main reason is that such chefs have considerable charismatic legitimacy. Because of that legitimacy, a chef who has several restaurants does not have to be present at all them. Similar considerations were observed in our study when consumers were faced with an artisan who created a second point of sale. The purpose was to see if they would retain the same level of trust if the artisan was no longer present in the store where they shopped. The results show that this is the case, provided that the product quality is maintained. Moreover, consumers are in no doubt about this, as confirmed by spontaneous allusions to top chefs.

"I wouldn't mind, I'd still be confident if the quality is the same" (Emmanuel); "For example, if Ducasse opens a restaurant next door to me, I'd go there because I know that the quality will be good even if he's not there himself " (Emmanuel); "Yes, it's the same thing on a smaller scale with the artisan known in his town or county" (Florent).

The physical presence of the artisan in his outlet is not essential for consumers and the development of his business may not have any impact on the perceived quality of the products. If this observation were to be verified by further research, this could prove to be an valuable development lever for artisans.

CONCLUSIONS AND IMPLICATIONS FOR THEORY AND PRACTICE

The primary goal of this paper was to explore the idea that an artisan could be considered as a brand, which is a new topic in marketing. Mobilizing the emerging literature on the human brand, we found relevant similarities among the different human brands studied (celebrities, athletes, chefs, film directors) and artisans. Like any professional human brand, the identity of artisans is based on expertise and creativity (Parmentier and Fischer 2010; Arnould and Dion 2013) and their legitimacy rests on traditional legitimacy, through their know-how and expertise, and on charismatic legitimacy, through their leadership (Dion and Boissieu 2013), because the artisan heads and embodies his business (Loup 2010; Dion and Boissieu 2013). Moreover, like some human brands (Moulard et al. 2015), artisanship is also characterized by its authenticity and its attachment (Thomson 2006). A key concern of this paper was to understand if it may be legitimate for an artisan to adopt a marketing approach without risking his artisanal values. An initial qualified answer can readily be advanced. Some types of marketing approach may be perceived positively by consumers (such as an e-commerce website), whereas others may induce negative perceptions (such as a larger shop).

Research on the legitimacy of brands has been mobilized to understand consumer reactions to artisans who undertake marketing approaches such as an e-commerce site. In addition to our theoretical findings, there are clear practical implications. Our results suggest that the artisan can use marketing without losing artisanal values. Nevertheless, he must be careful to retain many artisanal aspects of his business, such as limiting the size of his store and maintaining the quality of his hand-made products.