Sylvain Maugeais

Non parametric harmonization of a tabla's membrane

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The tabla, along with other Indian percussion instruments like pakhawaj and mridangam, is known amongst membranophones for its harmonic quality (cf. [START_REF] Raman | Indian musical drums[END_REF]) and resonance (cf. [START_REF] Roda | Resounding Objects: Musical Materialities and the Making of Banarasi Tablas[END_REF], chapter 6.3). These specificities are mainly due to the syahi, which is an extra mass added at the center of the membrane. Although it used to be made of clay that had to be applied each time the instrument was used (cf. [START_REF] Bharata-Muni | The Natyasastra. Bibliotheca Indica[END_REF] chapter 33, verse 25-26), it is now made either of single use dough (bass side of pakhawaj of mridangam) or syahi : "a paste of starch, gum, iron oxide, charcoal, or other materials." (cf. [START_REF] Fletcher | The physics of musical instruments[END_REF] chapter 18.5) (for tablas). The process of either applying the dough (cf. [START_REF] Bharata-Muni | The Natyasastra. Bibliotheca Indica[END_REF]) or the syahi (cf. [START_REF] Roda | Resounding Objects: Musical Materialities and the Making of Banarasi Tablas[END_REF]) is quite complex. The latter requires the application of different layers of syahi paste, drying after each step and rubbing with a stone. These different steps ensure there is a reticulum of cracks forming in the syahi (see fig. [START_REF] Antunes | Harmonic configurations of nonhomogeneous membranes[END_REF]), which reduces the stiffness of the added mass (cf. [START_REF] Roda | Resounding Objects: Musical Materialities and the Making of Banarasi Tablas[END_REF], chapter 5.11). The shape of the syahi has to be very precise to achieve harmonicity. Leaving aside the sustain, various authors have tried to describe the its geometry by numerically optimising the inharmonicity: from [START_REF] Ramakrishna | Vibration of indian musical drums regarded as composite membranes[END_REF] (refined by [START_REF] Gaudet | The evolution of harmonic indian musical drums: A mathematical perspective[END_REF]) viewing it as a composite membrane, [START_REF] Sathej | the eigenspectra of indian muscial drums[END_REF] and [START_REF] Samejima | Vibration analysis of a musical drum head under nonuniform density and tension using a spectral method[END_REF] in a similar way but with a smooth approximation of the composite density, to [START_REF] Maugeais | How to apply a plaster a drum make it harmonic[END_REF] and [START_REF] Antunes | Harmonic configurations of nonhomogeneous membranes[END_REF] who used non paramet-ric optimisation. All these articles found inhomegeneous membranes whose first in vacuo eigenfrequencies are almost harmonic, and whose shape is very similar to that of an actual physical tabla. On the other hand, following [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] who studied air loading on timpani membrane, [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF] applied the same optimisation procedure as [START_REF] Ramakrishna | Vibration of indian musical drums regarded as composite membranes[END_REF] on composite membranes but taking into account the air loading. They too obtained an almost harmonic set of frequencies. As a byproduct, they also got decay times. Unfortunately, these are sometimes unrealistic mainly because their model does not take into account the viscoelastic losses, which can be incorporated along the line of [START_REF] Gallardo | Sound model of an orchestral kettledrum considering viscoelastic effects[END_REF] who studied it for the kettledrum. The present article expands these results by applying a non parametric optimisation (like [START_REF] Antunes | Harmonic configurations of nonhomogeneous membranes[END_REF]), taking into account air loading (like [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF]) and viscoelastic losses (like [START_REF] Gallardo | Sound model of an orchestral kettledrum considering viscoelastic effects[END_REF]), and imposing conditions on the density ensuring the playability of the tabla (like [START_REF] Maugeais | How to apply a plaster a drum make it harmonic[END_REF]).

Following [START_REF] Henrique | Optimal design and physical modelling of mallet percussion instruments[END_REF] in their work on optimisation of musical instruments, the finite element method has been chosen to compute the eigenfrequencies relative to general densities. Together with boundary element method and optimisation, these techniques form the technical core of the present work. The remainder of the article is organised as follows: in section 2, the model used for the tabla and air loading is described and the modes in vacuo are computed using the finite element method (FEM). In section 3, the matrices involved in the boundary element method (BEM) are computed, first outside the kettle, and then inside. In section 4, the solutions of the complete system, membrane together with air loading, is computed, and are compared with the results of the bibliography. Finally, in section 5, the actual optimisation is carried out and discussed. The article concludes with open questions and future directions.

Problem formulation

Throughout the text, the only concern is that of the right hand tabla. The left hand one, although it has also been studied in other works (eg. [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF]), is much more difficult to model as the player's wrist is usually used during playing by resting at different positions and applying different pressures while fingers strike the membrane ( [START_REF] Kippen | The tabla of Lucknow : a cultural analysis of a musical tradition[END_REF] chapter

Model

Following [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF], the tabla is modeled as a rigid cylinder of radius a and height L, capped at the top by a membrane Σ and at the bottom by a rigid disk. An infinite rigid plane flange enclosing the membrane in the plane of the membrane is assumed (see ibid. concerning the validity of this assumptions). Let us write x = (r, θ, z) for the cylindrical coordinates, the membrane Σ being positioned in the z = 0 plane and the center of the membrane at r = 0.

Following [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] (and [START_REF] Gallardo | Sound model of an orchestral kettledrum considering viscoelastic effects[END_REF] for the viscoelastic term), the displacement η of the membrane is governed by the equation

σ ∂ 2 η ∂t 2 = T ∆ η + ν ∂η ∂t + p in z=0 -p out z=0 (1)
where σ(x) is the (non uniform) density of the membrane at the point x, T is the membrane tension by unit length, ν is the viscoelastic damping coefficient, p in and p out are the acoustic pressure fields inside and outside the kettle (i.e. above the plane z = 0). Assuming the membrane is fixed at the rim yields η(x, t) = 0 for all t and x ∈ ∂Σ

As the kettle acts as a baffle, a Neuman condition is used on the fixed borders of the kettle

∂p in ∂ρ ρ=a = ∂p in ∂z z=-L = 0 for all t. (3)
In the same way, the assumption that the membrane is surrounded by an infinite plane flange implies ∂p out ∂ρ ρ>a,z=0 = 0 for all t.

Denote by η (resp. p in , p out ) the temporal Fourier transform of η (resp. p in , p out ) with variable ω. The global acoustic pressure field p composed of p in and p out satisfies the wave equation

∆ + ω 2 c 2 a p = 0 (5)
where c a denotes the speed of sound in air.

Denote by G ω in and G ω out the Green functions for equation [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] for the inside and outside domains. Then the pressure fields can be computed (cf. [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF], eqs. ( 21) and ( 22))

p in (x) = Σ G ω in (x|x ′ ) ∂ p in ∂z ′ z ′ =0 dx ′ (6)
p out (x) = Σ G ω out (x|x ′ ) ∂ p out ∂z ′ z ′ =0 dx ′ . ( 7
)
The compatibility of the speeds of air and membrane at z = 0, together with linearized fluid dynamics, imply

∂ p in ∂z z=0 = ∂ p out ∂z z=0 = ω 2 ρ 0 η, (8)
with ρ 0 denoting the density of air at equilibrium. Finally the equation verified by η is given by

-ω 2 σ η = T ∆ ( η -jων η) + ω 2 ρ 0 Σ G ω in (x|x ′ ) z ′ =0 η(x ′ )dx ′ - Σ G ω out (x|x ′ ) z ′ =0 η(x ′ )dx ′ . (9)

Description of the elements and solutions in vacuo

The membrane is meshed uniformly with triangles and a spectral element method of order n is used. Let S be the set of shape functions defined on the triangulation using Legendre-Gauss-Lobatto points that are not on the boundary of ∂Σ. Define a hermitian product on the Sobolev space

H 1 0 (Σ) by ⟨f, g⟩ = Σ f (x)g(x)dx. ( 10
)
For the density function σ on Σ and tension parameter T , define the matrices

A σ = (⟨σφ, ψ⟩) φ,ψ∈S (11)
B = T (⟨grad φ, grad ψ⟩) φ,ψ∈S . (12)
The computation of the scalar products for computing the matrices A σ and B are performed using the collocation method. The lossless in vacuo equation implied by ( 9) is given by

-ω 2 A σ Φ = BΦ (13)
and its solutions ω, Φ are obtained through the computation of eigen-pairs, and are Galerkin weak approximations of in vacuo frequencies and modes of the membrane with density σ.

According to [START_REF] Babuška | Eigenvalue problems[END_REF] (see also [START_REF] Marburg | Six boundary elements per wavelength: is that enough?[END_REF] for a general discussion on the size of the mesh), the error term for ω or Φ can be bounded for small enough eigenvalues by h n ω 2 , where h denotes the mesh-size, which is the maximum diameter of the triangles of the mesh.

The ωs are counted with multiplicities and sorted by their magnitude as a set ω i i∈I where I denotes the set of eigen-pairs of [START_REF] Kippen | The tabla of Lucknow : a cultural analysis of a musical tradition[END_REF], and the corresponding eigenvector is denoted by Φ i .

The error on frequencies in cent for different meshes and different orders are given on figure [START_REF] Bharata-Muni | The Natyasastra. Bibliotheca Indica[END_REF] for the first 120 modes (extra logarithmic scale on the cents is used to allow the comparison between high and low order). This number of modes is necessary to ensure a good precision during the next step of the computation (cf. section 4.1). As expected, the error increases with frequency (cf. [START_REF] Babuška | Eigenvalue problems[END_REF]) and should be minimal for the first ones. The poor precision on the first eigenvalues is explained by the approximation of the geometry: a lesser number of points in the mesh implies a worse approximation of the circle.

As the number of modes is important for the precision required in section 4.1, and the speed of computation is not relevant, order 3 and 1096 points are used in the rest of the article.

The numerical method was implemented in Python from scratch on a desktop computer without using any finite element package.

Boundary element method

Contrary to [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF], [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF] and [START_REF] Gallardo | Sound model of an orchestral kettledrum considering viscoelastic effects[END_REF], the integrals in ( 9) are not computed "explicitly" because the density of the membrane is not constant, so the projection onto the eigenfunctions of the radial component of the membrane equation cannot be readily applied. Instead, the Boundary Element Method (BEM) is used (cf. for example [START_REF] Hall | The Boundary Element Method[END_REF] or [START_REF] Kirkup | The boundary element method in acoustics: A survey[END_REF]), together with a spectral element methods (SEM) of order 3. All the integrals involving only regular functions are computed using collocation methods.

Computation of the outside Green function

According to [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF], equation [START_REF] Samejima | Vibration analysis of a musical drum head under nonuniform density and tension using a spectral method[END_REF], G out can be expressed as

G ω out (x|x ′ ) = -1 4π e j ω ca ∥x-x ′ ∥ ∥x -x ′ ∥ + e j ω ca ∥x * -x ′ ∥ ∥x * -x ′ ∥ (14)
with x * = (r, θ, -z) if x = (r, θ, z). In particular, when r, r ′ ∈ Σ we get

G ω out (r|r ′ ) = -2 4π e j ω ca ∥r-r ′ ∥ ∥r -r ′ ∥ . ( 15
)
Define the matrix G out (ω) whose entries are

(G ω out ) φ,ψ = -1 4π Σ φ(r) Σ ψ(r ′ ) e j ω ca ∥r-r ′ ∥ ∥r -r ′ ∥ dr ′ dr. (16)
for any two φ, ψ ∈ S shape functions.

Denote by κ

ψ (r) = Σ ψ(r ′ ) e j ω ca ∥r-r ′ ∥ ∥r -r ′ ∥ dr ′ so that (G ω out ) φ,ψ = -1 2π Σ φ(r)κ ψ (r)dr (17)
For each s ∈ S, denote by r s its node point and w s its weight in the Lobatto quadrature. As κ φ is regular, (G ω out ) φ,ψ is approximated by

(G ω out ) φ,ψ ∼ = -1 4π s∈S w s φ(r s )κ ψ (r s ) = w φ κ ψ (r φ ) (18)
To compute κ ψ , the usual technique is used (cf. [START_REF] Costabel | Principles of boundary element methods[END_REF], 6.1): separate the singular part and compute it analytically, the rest of the integral is computed numerically. For this

write e j ω ca ∥rφ-r ′ ∥ ∥rφ-r ′ ∥ = 1 ∥rφ-r ′ ∥ -E(∥r φ -r∥ ′ , ω), with E is a regular function. Then κ ψ (P φ ) = Σ ψ(r ′ ) 1 ∥r φ -r ′ ∥ dr ′ κ * φ,ψ + Σ ψ(r ′ )E(∥r φ -r ′ ∥, ω)dr ′
The first integral κ * φ,ψ can be computed explicitly using a polar parametrization of triangles, together with antiderivatives of trigonometric polynomials. As for the second, E being regular, it can be approximated using the collocation method

Σ ψ(r ′ )E φ (r ′ , ω)dr ′ ∼ = w ψ E(∥r φ -r ψ ∥, ω). Finally (G ω out ) φ,ψ ∼ = -1 2π w φ κ * φ,ψ + w ψ E(∥r φ -r ψ ∥, ω)
which is an approximation of order n thanks to Lobatto's quadrature.

Finally, define the matrices H and E(ω) by

H φ,ψ = w φ κ * φ,ψ (19)
E φ,ψ (ω) = w φ w ψ E(∥r φ -r ψ ∥, ω) (20)
so that

G ω out = -1 2π (H + E(ω)) (21)

Computation of the Green function inside the shell

Let (Ψ n ) n∈N be a complete system of eigenvectors for the Laplace operator on Σ with Neuman boundary conditions, and denote by ϖ n its eigenvalue.This data can be computed using the finite element method as above changing the boundary condition, the weak approximation now taking place in the Sobolev space H 1 (Σ) instead of H 1 0 (Σ). Figure 4 gives the error in the computations of the ϖ n for different meshes and different orders. As for the Dirichlet condition, order 3 with 1096 points give very good results. A computation following [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] equation [START_REF] Roda | Resounding Objects: Musical Materialities and the Making of Banarasi Tablas[END_REF] gives an explicit form for the Green function inside the shell:

G ω in (r|r ′ ) = - n Ψ n (r)Ψ n (r ′ ) cotan(µ n L) µ n (22)
with

µ n = ϖ 2 n -ω 2 c 2 a .
To compute the weak approximation matrix of G ω in at entries φ, ψ ∈ S remark that

(G ω in ) φ,ψ = Σ Σ G ω in (r|r ′ )φ(r)ψ(r ′ )drdr ′ , (23)
so that

(G ω in ) φ,ψ = - n cotan(µ n L) µ n ⟨Ψ n , φ⟩⟨Ψ n , ψ⟩ (24)

Projection onto the space of in vacuo modes

To keep track of the right symmetries when looking for eigenmodes, as well as reducing the dimension of the search space, it is convenient to use a projection of the equations onto the space of eigenmodes in vacuo

H vac = vect (Φ i ) 0≤i≤ℓ .
It is a good candidate for the approximation of the exact solution u thanks to Sturm-Liouville's theory. Restricting to this space is the equivalent of the truncation of the infinite series done in [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] for equations ( 38) and (46). For any matrix N among A, B, G ω in , G ω out , define its restriction to the space H vac by

N i,j = t Φ i NΦ j . (25)
for all 0 ≤ i, j ≤ ℓ.

The projection of the fundamental equation ( 9) can then be written as

-ω 2 A σ M = (1-jων) BM +ω 2 ρ 0 G ω in -G ω out M (26)
In general, the results depend on the value of ℓ, a bigger value of ℓ giving more precise results. We found that a typical value of ℓ = 80 is a good choice for the computations to ensure a good precision for the first 15 modes (cf. section 5). This high value of ℓ is the main reason why order 3 methods were chosen (see figures 3 and4).

Solutions of the complete system

The last step to find the eigenmodes of the loaded non uniform membrane is to solve equation (26) which is non polynomial in ω because of the presence of this variable in the matrices G ω in and G ω out . This is achieved following the same "fixed point" strategy as in [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF]. It should be noted however that this method does not always give a solution (see ibid. section IV) and is also confronted to the well known problem of spurious eigenvalues and nonuniqueness of solutions (see [START_REF] Chen | Recent development of dual bem in acoustic problems[END_REF]). Although this is a real problem for bigger drums, we found that all the solutions given by this method for the tabla are of the same order as that of a physical tabla, owing to the relatively small size of the membrane and the reduced air loading compared to that of the timpani.

Fixed point algorithm

The method chosen here follows only the "simple case" of [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] section IV: start with an eigensolution in vacuo (ω 0 , M 0 ), where M 0 denotes the coordinates of the solution in the basis of the Φ i s. In particular, M 0 is part of the canonical basis:

M 0 = t (0, • • • , 0, m 1 , 0, • • • , 0).
Denote by m the non zero coordinate. Then solve recursively the eigenvalue problem

-ζ 2 A σ Λ = (1 -jω n ν) B + 2ω 2 n ρ 0 G ωn in -G ωn out depends only on ωn Λ (27)
with indeterminates ζ and Λ, where the previous guess ω n has been replaced in the Green functions and the viscoelastic losses.

In general, this system has ℓ + 1 eigensolutions. To ensure that the chosen solution has the same kind of symmetries as the initial guess M 0 in terms of nodal cirlces, the one that has the largest relative m-th coordinate (hence closest to M 0 ) is picked. This is the crucial step and requires that the perturbation is not too strong ( [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] section IV). This hypothesis not only allows the fixed point method to converge, but also gives realistic results for the tabla. The fixed point method is described precisely below (cf. algorithm 1).

Algorithm 1: Fixed point algorithm Data: m < ℓ Result: solution of (26)

ϖ 0 = ω m do S ←-{(ζ s , M s )} = set of normalized eigen-solutions of (27) i ←- argmax{m-th coordinate of|M s |, (ζ s , M s ) ∈ S} ϖ n+1 ←-ζ i n ←-n + 1 while |ϖ n+1 -ϖ n | > ε
The output of this algorithm is compared on figure 5 for the uniform membrane of a timpani to the results of [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] table I, for different meshes and different orders. The modes (0, 1) and (0, 2) where removed because the algorithm does not converge in that case (or sometimes converges to a spurious mode) which is in agreement with the discussion of [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF], section IV. Indeed, these two modes are the most perturbed ones, and the real solutions are far from the modes in vacuo. However, for the other modes, the results are within the margin of error as the results in op. cit. are given with a precision of only 10 cents. The results obtained in this figure confirm that a high order scheme is necessary, together with a fine mesh.

Figure 5: Relative error in cents for the timpani compared to [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF], table I, for different uniform meshes and different orders. Modes (0, 1) and (0, 2) where removed because of convergence problems.

Comparison with known results

Case of tabla

The results obtained by the fixed point algorithm for different uniform meshes and different orders are compared to those of [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF], table II for the in vacuo modes (see figure 6) and for modes loaded by air (see figure 7). As for the uniform membrane detailed on figure 5, the results are within the margin of error, as the results in [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF] Table II are given only with a precision of 10 cents. However, the effect of high order is less clear on these figures than for figure 5. This is due to the discontinuity of the composite membranes which induces less regularity on the FEM problem and a slower convergence. The modes in vacuo and loaded are well approximated at order 2 and 3 for a uniform mesh of 1096 points.

Regularisation of the composite membrane

To get a clearer result, the method is tested on a regularisation of the composite density and compared to the computation with the same algorithm but with degree 3 and finer mesh (1096 points). The results can be seen on figure 8 for different mesh sizes and are compatible with convergence of the method.

On the necessity of the viscoelastic losses

Because the only dissipative mechanism used in the models of [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] and [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF] is the sound radiation, both articles give unrealistic values for τ 60 's of some modes (cf. [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF], section IV and [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF], section III.C). To get more realistic decay times, a straightforward improvement is to add viscoelastic losses, as in [START_REF] Gallardo | Sound model of an orchestral kettledrum considering viscoelastic effects[END_REF] who studied the timpani model of [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF] together with such an added term. From here onwards, we consider the same value as [START_REF] Gallardo | Sound model of an orchestral kettledrum considering viscoelastic effects[END_REF] for the viscoelastic coefficient: ν = 0.6 × 10 -6 s.

Figure 6: Relative error in cents for inharmonicities of the modes in vacuo relative to mode (0, 2), compared to [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF], table II, for different uniform meshes and different orders. Mode (0, 2) is removed as it is taken as reference Figure 7: Relative error in cents for inharmonicities of the modes in vacuo relative to mode (0, 2) compared to [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF], table II, for four different uniform meshes and different orders. Mode (0, 2) is removed as it is taken as reference Figure 8: Relative error in cents for the eigenfrequencies of the loaded modes for a regularisation of the composite density of [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF], table II, for four different uniform meshes. The comparison is made with order 3 and 1096 points. The top right corner shows the regularisation used Figure 9: Comparison between the τ 60 obtained by [START_REF] Christian | Effects of air loading on timpani membrane vibrations[END_REF], table III (black curve for the computation, blue curve for the experiment) and the FEM/BEM method (green curve).

The comparison of τ 60 's obtained from the FEM/BEM method exposed in section 3 taking the viscoelastic losses into account are presented on figure 9. As pointed out in [START_REF] Gallardo | Sound model of an orchestral kettledrum considering viscoelastic effects[END_REF], the addition of the viscoelastic losses give more realistic values for all the modes and the remaining differences could come from the model of the room acoustics (cf. [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF]).

Optimisation

The goal of this section is to give the optimisation tools necessary to find a solution to the tuning process. The method chosen here relies on the gradient algorithm as a "natural" way to go toward a minimum. This method is not only suitable because of the big dimension space in which the optimisation process takes place (the dimension of the space is given by the number of forms in the FEM method, which is here about 10 000) but also because it gives a natural, local direction to search for the minimum.

Computation of gradients

The goal of this section is to study the dependence of solutions of the system

-σω 2 η = (1 -jνω)T ∆ η + G ω * η (28)
under small deformations of the functional parameter σ For this, consider the perturbed system

-(σ + ε σ)(ω + ε ω) 2 ( η + ε η) = (1 -jν(ω + ε ω))T ∆( η + ε η)+ ρ 0 (ω + ε ω) 2 G ω+ε ω * ( η + ε η) (29)
where ε denotes a "small" parameter.

Taking the Taylor expansion in ε and keeping only the part in ε give

-σω 2 η + 2σω ω η -jν ωT ∆ η + ρ 0 ω 2ωG ω + ω 2 ∂G ω ∂ω η = σω 2 η + T (1 -jνω)∆ η + ρ 0 ω 2 G ω η. ( 30
)
To simplify this equation, take the scalar product ⟨., η⟩ with the complex conjugate η of η of the whole equation, and use the hermitian symmetries of the operators ∆, G ω * and ∂Gω ∂ω * . The fact that (ω, η) is a solution of (9) yields

σω 2 η + 2σω ω η -jν ωT ∆ η+ ρ 0 ω 2ωG ω + ω 2 ∂G ω ∂ω * η, η = 0. ( 31
)
Thereby giving

grad σ ω = -ω 2 2ω⟨σ η, η⟩ -jνT ∆ η, η + ρ 0 ω 2G ω + ω ∂Gω ∂ω * η, η η 2 (32)
Any perturbation of σ along a direction ε σ can then be computed as

ω(σ + ε σ) = ω(σ) + ε⟨ σ, grad σ ω⟩ (33)
Note that this is true for both real and imaginary parts, so that it is possible to see the tendency of both frequency and decay time in a particular direction. This is what is done on figures 10 (for the frequency) and 11 (for decay time) taking for σ the uniform density σ U = 0.245, for σ a regular function that cancels on the rim

σ(ρ, θ) = σ U (1 -ρ/a) 2
and a not so small parameter δ ∈ [-0.3, 0.3] so that the evolution along δ σ can be compared with the derivative computed using the gradient. Note that δ is also chosen so that the density remains positive. For both figures, the colored curves represent the frequency or decay time for the different modes computed with de FEM/BEM method (using a 1096 points mesh and order 3, together with ℓ = 80) and the dashed lines represent the derivatives computed with formula of the gradient (formulas (32) and (33)).

Evolution of both frequencies and decay times are very well predicted by the gradient alone.

Definition of inharmonicity

As was already noticed by Raman in [START_REF] Raman | Indian musical drums[END_REF], the first nine modes of a tabla are in an almost harmonic progression. The different modes, defined by their nodes, are described on figure 12 together with the approximate ratio to the mode (1, 1) which serves as a reference. It should be noted that this description of the modes hides the fact that most of them are actually degenerate: for symmetry reasons, each mode (α, β) is actually doubled if α > 0. However, among the twin modes, only the mode shapes differ (by a rotation) and not the frequency. The inharmonicity function is therefore be defined as

I(σ) = m∈modes γ m 2ℜ(ω m (σ)) p m ℜ(ω (1,1) (σ)) -1 2 (34)
where γ m designates a weight affected to mode m. The choice of these weights is actually crucial as they favor which modes are to be optimised first. For example, [START_REF] Sathej | the eigenspectra of indian muscial drums[END_REF] chose γ m = p 2 m which gives a relatively strong weight to high frequency modes. This choice leads in our case to the gradient algorithm to not converge.

Another interesting choice is the one by [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF], equation (24), which is closely related to the uniform case but taking into account psychoacoustic phenomena. The present article follows [START_REF] Gaudet | The evolution of harmonic indian musical drums: A mathematical perspective[END_REF] (with a different choice of target ratios p (0,3) ) and [START_REF] Maugeais | How to apply a plaster a drum make it harmonic[END_REF], giving the same weight to all modes γ m = 1. This choice makes the gradient algorithm converge toward a physically realistic solution (cf. figure 17).

Monotonification

As can be seen on figure 13, the gradient of the inharmonicity is neither of constant sign, nor monotonic (as a function of r), and does not even have a precise circular symmetry. All these characteristics are however expected from the density of the syahi. The non constancy of the sign is problematic to build the syahi, as it would necessitate to remove some matter, even maybe to the point of having negative density at some places: the syahi is usually built by adding small layers of mass. The non circularity of the gradient comes from the fact that only a finite set of modes taken into account for the calculation of inharmonicity, together with the choice of the limits in the definition of inharmonicity and the precision of the computations. This "problem" is more of a numerical nature, but is relatively natural for the "monotonification" process described hereafter.

Finally, even for densities with circular symmetry, an optimal solution could have non playable shapes (for example those proposed in [START_REF] Antunes | Harmonic configurations of nonhomogeneous membranes[END_REF] Figure 7) which could be Figure 14: Function σ 0 -δgrad σ I(σ 0 ) as a multivalued function of r (in black) together with its monotonification σ 0 -δgrad σ I(σ 0 )

♮ (in red).

harmful to the player. Moreover, owing to the current building process of the syahi (cf. [START_REF] Roda | Resounding Objects: Musical Materialities and the Making of Banarasi Tablas[END_REF] or [START_REF] Courtney | Manufacture and repair of tabla[END_REF]), which is done by adding small concentric layers of paste of decreasing radius, letting them dry and rubbing them with a stone to produce the characteristic cracks (cf. figure 1), one hypothesis that seems reasonable, and which is found in actual physical tablas, is to suppose that the density is monotonic as a function of r.

A simple way to achieve all three hypotheses is to use a process called here "monotonification" which is done in three steps: for any real valued function λ on Σ

• transform λ into a non negative function,

• take the maximum of λ for every fixed radius r, thus ensuring the circular symmetry,

• take the cumulative maximum, thus ensuring the monotonicity in r.

These steps can be summarized in the formula

λ ♮ (r) = max r ′ ≥r max |r|=r ′ max(λ(r), 0) (35)
giving a new function which depends only on r, is non negative and decreasing. On figure 14, the monotonified version of

σ 0 -δgrad σ I(σ 0 )
♮ is given for a uniform density σ = 0.245 and a parameter δ given by the first step of the gradient algorithm (see algorithm 2). It should be noted that σ 0 -δgrad σ I(σ 0 ) is drawn only as a function of r, which is why it appears as a multivalued function.

Gradient algorithm

The algorithm used in this article is the usual gradient descent algorithm. Although there are more effective algorithms to find a global minimum, this one is very suitable for our application as the dimension of the search space is quite big (about 10 000) and it gives a result in only a few steps (about 30 steps for the order 3 FEM/BEM method with a mesh of 1096 points). The algorithm used for optimisation is detailed hereafter in algorithm 2.

Algorithm 2: Gradient algorithm with monotonification of σ Data: σ Result: locally optimal density n ←-0

σ 0 ←-σ I 0 ←-inharmonicity(σ 0 ) do grad = gradient of inharmonicity(σ n ) δ ←--I n /⟨grad, grad⟩ σ n+1 ←-σ n + δ.grad ♮ I n+1 ←-inharmonicity(σ n+1 ) n ←-n + 1 while I n > I n-1
The only significative difference with the usual gradient algorithm is the use of monotonification in the step

σ n+1 ←-σ n + δ.grad ♮ .
It should be noted that another choice is possible by applying the monotonification on the gradient, rather than on the density. This latter choice has the effect of producing a series of increasing densities : σ n+1 ≥ σ n , which may be closer to the reality of construction of the syahi, but give results that are further from the optimal.

Results of optimisation

The results of the gradient algorithm are given on figures 15 to 18 for an initial uniform density σ 0 = 0.245kg/m 2 as in [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF], Table II. The parameters used for the tabla are therefore T = 1822N/m, a = 0.05m and L = 0.1m. Denoting by σ T the optimal density obtained in loc. cit., we get I(σ T ) = 0.00229 with formula 34, which is different from the inharmonicity computed in [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF] because of the choice of weights. It should be noted that, although σ T was obtained for another choice of weights, it is very close to the optimal inharmonicity, emphasizing the robustness of the composite density. The decrease of inharmonicity is clearly seen on figure 15 and the algorithm stops after 36 steps for an optimal density σ opt for which I(σ opt ) = 0.00019. The fact that I(σ T ) > I(σ opt ) is not a surprise as σ T verifies all the hypotheses (symmetry, monotonicity) that are imposed upon the search space, which therefore include that of [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF]. The evolution of the frequency of each mode, normalised by the frequency of mode (1, 1), can be seen on figure [START_REF] Maugeais | How to apply a plaster a drum make it harmonic[END_REF]. It seems that the higher the frequency, the slower the convergence. This is readily explained by the choice of weights that is made, which favours the low frequency. The evolution of the density is displayed in figure 17, together with the composite optimal σ T in red for reference. The proximity of σ T and σ opt is striking, emphasizing the fact that seeing the syahi as a composite membrane is a very good approximation. It should be noted that from the very beginning of the optimisation, the rightmost point where σ n > σ 0 shifts to the left during optimisation. This is due to the choice of taking the monotonification on σ, rather than on grad σ I. It allows the optimisation process to get a smaller support for the syahi. A third ramp can be seen on the last obtained densities around r = 0.01m. This suggests that to improve even further the composite membrane of [START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF], another ring should be added (see [START_REF] Vautour | Quasiharmonic circular membranes with a central disc or an isolated ring of different density[END_REF] for a study in this direction without the air loading). The evolution of decay time for each mode can be seen on figure 18. During all the optimisation process, the decay times are increasing, giving for the optimal density a sustain that is up to 10 times greater for σ opt than for σ 0 , in particular for the low frequency modes. This is compatible with the knowledge of tabla makers, that adding smaller central circle adds to the resonance (cf. [START_REF] Roda | Resounding Objects: Musical Materialities and the Making of Banarasi Tablas[END_REF], section 6.3).

(s) [0, 1] [1, 1] [2, 1] [0, 2] [3, 1] [1, 2] [3, 1] [1, 2] [0, 3]
Sounds obtained at each step of the simulation are available online at http://perso.univ-lemans.fr/ ~smauge/tabla/

Conclusion

A non parametric optimisation procedure is proposed to produce a non homogeneous membrane loaded by air whose first partials are harmonic. Together with mild hypotheses ensuring the playability of the membrane, this procedure gives a realistic density that is close to that of an actual physical tabla. As a byproduct, it gives an explicit formula for the changes in frequencies and decay times of eigenmodes with respect to an infinitesimal variation of the density. Moreover, being based on finite elements and boundary elements methods, it can readily be applied to non circular geometries. The method developped here gives a density close to those obtained in previous studies, highlighting the good aproximation given by a composite membrane. The main novelty of the present article lies in the numerical scheme, which uses a non parametric optimisation together with mild hypotheses that are inherent to the manufacture of the syahi : circular symmetry and radial monotonicity. As such, it could be used in more complex settings where no obvious solution such as the composite one is known. As for the previously developed methods, the optimisation is done only for the first five frequencies (equivalent to 15 eigenmodes, some of them being degenerate). This is justified as these are the only prevalent frequencies observed in the spectrum of the tabla. It was conjectured by Raman in [START_REF] Raman | Indian musical drums[END_REF] that the reason for this limited spectrum is the presence of a second, annular shaped membrane that rests on the main one. We hope that the tools developed in the present article will help investigate this conjecture. One of the main drawback of the method is the assumption that the stiffness is negligible. This would require a precise investigation as the added mass is important and the material appears to be quite rigid. As stated in the introduction, the usual explanation for this low stiffness is the reticulum of cracks which, to our knowledge, has not been studied in depth and would deserve more attention.

Figure 1 :

 1 Figure 1: Close up of the syahi. The reticulum of cracks and different layers can be seen

Σ

  Surface of the drumhead F Infinite flange a Radius of the drumhead (in m) = 0.05m for a tabla L Height of the kettledrum (in m) = 0.1m for a tabla x = (ρ, θ, z) Point in space, cylindrical coordinates r = (ρ, θ) Point in the membrane, polar coordinates σ Surface density of the drumhead (in kg/m 2 ) = 0.245kg/m 2 for a uniform tabla membrane η Transverse displacement of the drumhead (in m) T Membrane tension (in N/m) = 1822N/m for a tabla ν Viscoelastic coefficient of the drumhead (in s) = 0.6 × 10 -6 s p in Pressure inside the kettledrum pout Pressure above the flange F ρ 0 Air density = 1.20kg/m 3 ca Speed of sound = 344m/s

Figure 2 :

 2 Figure 2: Model of the tabla

Figure 3 :

 3 Figure 3: Relative error in cents for the modes in vacuo for a uniform circular membrane compared to theoretical values, for different uniform meshes and different orders.

Figure 4 :

 4 Figure 4: Relative error in cents for the eigenvalues of the Laplace operator with zero Neuman boundary condition, for different uniform meshes and different orders.

Figure 10 :

 10 Figure 10: Evolution of frequencies along the line σ U +δ σ together with the derivative computed with the real part of formula (32).

Figure 11 :Figure 12 :

 1112 Figure 11: Evolution of τ 60 along the line σ U +δ σ together with the derivative computed with the imaginary part of formula (32)

Figure 13 :

 13 Figure 13: Gradient of inharmonicity computed for a uniform density

Figure 15 :Figure 16 : 2 )Figure 17 :

 1516217 Figure15: Evolution of inharmonicity during optimisation (blue curve). Inharmonicity of the optimal obtained in[START_REF] Tiwari | Effects of air loading on the acoustics of an indian musical drum[END_REF] is printed for reference (dotted red curve)

Figure 18 :

 18 Figure 18: Evolution of τ 60 of each mode during optimisation

Table 1 :

 1 List of symbols


Christophe Reffay

Marie-Laure Betbeder

CSCL -Workshop

Extending validation of tools and analyses in CSCL situations: How to collaborate on interaction analysis?

Christophe Reffay, Marie-Laure Betbeder

Introduction

The Mulce Project aims at developing a server that would allow researchers to share Learning and Teaching Corpora (Letec). Our main goals are: first, to facilitate the work of researchers in the CSCL field by reusing existing corpora instead of creating new experiments, collecting and organizing data, and secondly, to connect these corpora to shared analyses and visualization tools. We also hope that this process of sharing would deepen and widen the validity of research tools and analyses. The limited space of our publications generally does not allow the authors to include their data nor the detailed context of their source experiments, which means that these results cannot be reused nor reproduced.

In the first part of this paper we present some of our work trends related to the workshop topics. In particular, visualization and analysis tools for synchronous and asynchronous interactions. In the second part we discuss the lack of validity for indicators and analysis tools and results in this particular domain. The last part presents the Mulce project which is a tentative to face this stated lack of validity.

Our work on Analysis and visualization tools

In previous research, presented in CSCL2003, we built an automatable computing process based on Social Network Analysis to evaluate the cohesion of a learning group using e-mail and forum messages [START_REF] Reffay | How social network analysis can help to measure cohesion in collaborative distance-learning?[END_REF]. Another work, mainly conducted during the phd thesis of A. Mbala [START_REF] Mbala | Analyse, Conception, spécification et développement d'un système multi-agents pour le soutien des activités en formation à distance[END_REF] and presented in ITS2002 (Intelligent Tutoring System) [START_REF] Mbala | Integration of automatic tools for displaying interaction data in computer environments for distance learning[END_REF], is a multi-agent system that tries to predict if some learners are less and less participating, in order to prevent their abandon. We have also worked on the analysis of data resulting from a tailorable framework to support collective activities in a learning context [START_REF] Betbeder | Symba: a tailorable framework to support collective activities in an learning context[END_REF]. We analysed act's contents and worked out the proportion of acts related to the following four categories: activity achievement, group organization, environment tailoring and socialization.

In collaboration with a research team in linguistics, we are also involved in the field of multimodal interaction analysis [START_REF] Betbeder | Interactions multimodales synchrones issues de formations en ligne : problématiques, méthodologie et analyses[END_REF], combining quantitative (macroscopic level) and content analysis (microscopic level) on micro actions that each participant can realize in an audiographic synchronous collaborative environment (audio talking turns, chat acts, votes, paragraph production in a shared text document, objects in a shared concept map or whiteboard). Our contributions in this field deal with (1) pattern discovering in sequences of such actions [START_REF] Betbeder | Recherche de patterns dans un corpus d'actions multimodales[END_REF], (2) a visualization tool [START_REF] Betbeder | Interactions multimodales synchrones issues de formations en ligne : problématiques, méthodologie et analyses[END_REF] which emphasizes the intertwinement of the actors' synchronous acts. In such environment, actors can interact by using different modalities at the same time. Due to the importance of time and sequence of acts, such phenomena are hardly visible through a database representation.

We are currently working on navigation through corpora. This includes the selection of corpora and visualization of archived interaction acts. Once the corpus has been chosen, a system of requests provides selection of parts of the corpus by considering for example: time restriction, selected communication tools, author or group or string search in interaction contents. The resulting set of acts 1560-4292/08/$17.00 © 2008 -IOS Press and the authors. All rights reserved. can be visualized in the appropriate form, according to the communication tool they are recorded from. This heterogeneous set of acts is organised by our XSD schema, specified by the Mulce project, presented in (Reffay & al., 2008) and available on the Mulce project site1 .

The need of validation

In CSCL domain, it seams to us that the lack of validity of results and tools becomes more and important. We can precise our view on this problem in the following terms:

-Lack of transparency and availability (accessibility for other researchers) of interaction data resulting from an online learning situation. These data are unusable for others. Furthermore, the pedagogical and scientific contexts of these learning situations are rarely published.

-Too many analysis tools are developed by a given team and only tested in a (unspecified) given context. Interaction and research data being unreachable for the rest of the community, the scientific results cannot be replicated or reproduced [START_REF] Rourke | Methodological Issues in the Content Analysis of Computer Conference Transcripts[END_REF][START_REF] Garrison | Revisting methodological issues in the analysis of transcripts: Negotiated coding and reliability[END_REF]. As a consequence, there is no possible comparison between methods or results concerning one set of interaction data and another.

Studying collaborative online learning, in order to either understand this specific type of situated human learning, or evaluate scenario and associated devices, or improve technological environments, requires accessibility to interaction data collected from various actors (e.g. learners, teachers, tutors, etc.) that participate to learning situations. Recent international research publications and scientific events are related to this topic. But interdisciplinary communities involved in this research have not been able to characterise a sharable scientific object according to a comprehensible methodology.

On the one hand, we have partial data, not contextualised with pedagogical and technological learning situation elements, or else raw data that are inextricably tangled in specific software using proprietary formats. A simple collection of student's online interaction data is not a scientific object for the research community focused on online learning. In the language learning field, this idea is emphasised by Kern, Ware and Warshauer as follows:

Researchers must carefully document the relationships among media choice, language usage, and communicative purpose, but they must also attend to the increasingly blurry line separating linguistic interaction and extra linguistic variables.

[…] Studies of linguistic interaction will likely need to account for a host of independent variable: the instructor's role as mediator, facilitator, or teacher; cross-cultural differences in communicative purpose and rhetorical structure; institutional convergence or divergence on defining course goals; and the affective responses of students involved in online language learning projects. [START_REF] Kern | Crossing frontiers: new directions in online pedagogy and research[END_REF]) Our research domain is not only concerned by learning, but more widely by all pedagogical aspects and viewpoints and particularly by teaching. Then, some studies aim at "[…] gather evidence about the effects of instructional conditions of instruction" (Chapelle, 2004: 594). Success in such studies requires gathering the context elements, and in particular those characterising the pedagogical situation. From a methodological point of view, this leads to link up the various data sources in order to create a scientific object, worthy of analysis. This idea is emphasised by the following excerpt concerning interaction in discussion board:

Research in computer mediated communication in education aims at describing complex phenomena by using content analysis methods that flavour partially only some aspects of the communication. The method should be able to consider the discourse as a situated verbal interaction, in its various dimensions: linguistic, situational (in the universe of reference and interaction situation) and hierarchical constraints of the discourse. [START_REF] Henri | L'analyse des forums de discussion Pour sortir de l'impasse[END_REF]) On the other hand, inaccessibility to research data, that is in fact the reality in the quite whole international community, hinders online learning situations from being considered as a scientific object: it impedes verifications, controversy, replication, refinement, multiple analyses, etc. Even if multiple analyses are strongly encouraged by research heads, they are still exceptions that we can illustrate by the studies of Kramsch and Thorne [START_REF] Kramsch | Foreign language learning as global communicative practice[END_REF] and [START_REF] Thorne | Artifacts and cultures-of-use in intercultural communication[END_REF]. Analysing some interaction pieces, extracted from a learning situation informally described in [START_REF] Kern | Literacy and Language Teaching[END_REF], they gave a different interpretation to explain the failure in this interaction between learners of different mother tongues (Kern et al., 2004:p251). Reanalysis can be motivated by various factors like verification (in the previous example), use of alternative content analysis methods (i.e. special issue of Computers & Education (Valke & Martens, 2006)), comparison of results provided by distinct disciplinary approaches [START_REF] Corbel | A method for capitalizing upon and synthesizing analyses of human interactions[END_REF], etc. However, except this contrastive or alternative view, one can consider that analysis approach is by nature a cumulative process, built by distinct research teams, supported by previous analysis, each of these giving its set of annotations. In this sense, the transcription process of audio-oral or multimodal interaction (Vetterr & Chanier, 2006;Ciekanski & Chanier, 2008) is the necessary first step before any other analyses. In the same way, we can consider that chat sessions or forum sets have to be prepared in a target format from row data in the proprietary structures tangled in the various platforms in order to explicit speech turns or messages, speaker/scripter, etc.). This first step, giving the content in a textual and structured form, can be followed by a first level of annotation coming from a conversational analysis, and by a second one concerning a discourse analysis. Such research practices are already well rooted in the NLP (Natural Language Processing) research domain, where, from corpus extracts, distinct researchers can cumulate different level of description/annotation: morphologic, syntactic, anaphoric, etc. [START_REF] Salmon-Alt | Un modèle générique d'organisation des corpus en ligne[END_REF].

A research community becomes mature by sharing (contextualised) resources, tools and practices. Resources and tools are components, aside our publications, of the scientific contribution that our research directions ask us to put in open access (Berlin, 2003;Chanier, 2004:121). A necessary condition to share such resources is open access that should be supported by a set of techniques and protocols (standardisation, interoperability, metadata, etc.). We also need to solve the problems of rights in our research domain and, concerning learning and teaching studies, involving human beings in social experiments; the major ethic question has been quoted by Chapelle in the following terms:

Any discussion of technology in second language research would not be complete without raising the ethical challenges that researchers face in SLA [Second Language Acquisition] research in general and particularly in research involving the collection and archiving of personal performance data that reveal personal attributes (Chapelle, 2004: 599).

Mulce project

In order to widen and strengthen the scientific approach in online learning domain, and specifically analysis of online interaction in learning situations, we launched the MULCE (MULtimodal Corpus Exchange) project [START_REF] Mulce | English version of the MULCE Project homepage[END_REF]. The first step of this project has been to define (and exemplified) the notion of "Learning and Teaching Corpus". Its main objectives are:

To define the data structure of a "Learning and Teaching Corpus", To specify and develop a technical support to effectively share such corpora, integrating OLAC specifications (Open Language Archives Community) and OAI-PMH (Open Archives Initiative's Protocol for Metadata Harvesting), To rebuild 2 of our global corpora (Simuligne and Copeas) to be accessible through out this platform, and documented according specifications of a "Learning and Teaching Corpus".

We propose (1) a formalism to describe learning and teaching corpora and (2) a platform to share them among the research community. The formalism defines the information which can be contained in a corpus and the structure of the data. Through the platform, researchers can share their corpora with the community and access the data shared by other members of the community.To share a corpus, a researcher has to provide metadata describing the corpus' components and upload a file describing each component. While accessing a corpus, an identified researcher is provided with a variety of tools to browse the corpus components, to navigate through the contextualized interaction data, to visualize and to analyze them.

Our major goal is to develop efficient tools and technical environments to help the wide variety of actors involved in online teaching and learning.

Mulce data structure available here: http://mulce.univ-fcomte.fr/metadata/mce-schemas/mce_sid.xsd


Christophe Reffay

Marie-Laure Betbeder

Sharing Corpora, Analysis and Tools for CSCL Interaction Analysis

Sharing Corpora, Analysis and Tools for CSCL Interaction Analysis

Christophe Reffay, Marie-Laure Betbeder, Computer Science Laboratory of the Franche-Comté University, France.

Interaction analysis

In previous research, presented in CSCL2003, we built an automatable computing process based on Social Network Analysis to evaluate the cohesion of a learning group using e-mail and forum messages [START_REF] Reffay | How social network analysis can help to measure cohesion in collaborative distance-learning? In proceeding of CSCL[END_REF]. Another work, mainly conducted during the phd thesis of A. Mbala [START_REF] Mbala | Analyse, Conception, spécification et développement d'un système multi-agents pour le soutien des activités en formation à distance[END_REF] and presented in ITS2002 (Intelligent Tutoring System) [START_REF] Mbala | Integration of automatic tools for displaying interaction data in computer environments for distance learning[END_REF], is a multi-agent system that tries to predict if some learners are less and less participating, in order to prevent their resignation. We have also worked on the analysis of data resulting from a tailorable framework to support collective activities in a learning context [START_REF] Betbeder | Symba: a tailorable framework to support collective activities in an learning context[END_REF]. We analysed the content of the acts and worked out the proportion of acts related to the following four categories: activity achievement, group organization, environment tailoring and socialization.

In collaboration with a research team in linguistics, we are also involved in the field of multimodal interaction analysis [START_REF] Betbeder | Interactions multimodales synchrones issues de formations en ligne : problématiques, méthodologie et analyses[END_REF], combining quantitative (macroscopic level) and content analysis (microscopic level) on micro actions that each participant can realize in an audiographic synchronous collaborative environment (audio talking turns, chat acts, votes, paragraph production in a shared text document, objects in a shared concept map or whiteboard). Our contributions in this field deal with (1) pattern discovering in sequences of such actions [START_REF] Betbeder | Recherche de patterns dans un corpus d'actions multimodales[END_REF], (2) a visualization tool [START_REF] Betbeder | Interactions multimodales synchrones issues de formations en ligne : problématiques, méthodologie et analyses[END_REF] which emphasizes the intertwinement of the actors' synchronous acts. In such an environment, actors can interact by using different modalities at the same time. Due to the importance of time and sequence of acts, such phenomena are hardly visible through a database representation.

We are currently working on navigation through corpora. This includes the selection of corpora and visualization of archived interaction acts and the context of the learning situation. Once the corpus has been chosen, a system of requests provides selection of parts of the corpus by considering for example: time restriction, selected communication tools, author or group or string search in interaction contents. The resulting set of acts can be visualized in the appropriate form, according to the communication tool they are recorded from. This heterogeneous set of acts is organised by our XSD schema, specified by the Mulce project, presented in (Reffay & al., 2008) and available on the Mulce project site1 .

To facilitate analysis on the platform, we also want to propose tools originated from our research team or from partnership. For example we have two running collaborations: the Calico2 project and Tatiana [START_REF] Dyke | Analysing face to face computer-mediated interactions[END_REF]. The Calico project aims at proposing different visualization and analysis tools available as servlets on their platform, specialized in discussion forums. Tatiana is a client application that combines and synchronizes various data sources (i.e. videos, logs, etc.). It includes a replayer and an annotator as well as various visualization facilities typically useful for synchronous interactions (face to face, chat, synchronously shared document editors).

Mulce project

The Mulce3 Project aims at developing a server that would allow researchers to share Learning and Teaching Corpora (Letec). Our main goals are: first, to facilitate the work of researchers in the CSCL field by reusing existing corpora instead of creating new experiments, collecting and organizing data, and secondly, to connect these corpora to shared analyses and visualization tools. We also hope that this process of sharing would deepen and widen the validity of research tools and analyses.

Its main objectives are (1) to define the data structure of a "Learning and Teaching Corpus", (2) to specify and develop a technical support to effectively share such corpora, integrating OLAC specifications (Open Language Archives Community) and OAI-PMH (Open Archives Initiative's Protocol for Metadata Harvesting) and (3) to rebuild 2 of our global corpora (Simuligne and Copeas) to be accessible throughout this platform, and documented according to the specifications of a "Learning and Teaching Corpus".

We propose a formalism to describe learning and teaching corpora and a platform to share them among the research community. The formalism defines the information which can be contained in a corpus and the structure of the data. Through the platform, researchers can share their corpora with the community and access the data shared by other members of the community. To share a corpus, a researcher has to provide metadata describing the corpus' components and upload a file describing each component. While accessing a corpus, an identified researcher is provided with a variety of tools to browse the corpus components, to navigate through the contextualized interaction data, to visualize and to analyze them. Our major goal is to develop efficient tools and technical environments to help the wide variety of actors involved in online teaching and learning.

Mulce data structure available here: http://mulce.univ-fcomte.fr/metadata/mce-schemas/mce_sid.xsd

French national research project coordinated by E. Bruillard (ERTÉ: Technical Research Team in Education) http://calico.inrp.fr/

Multimodal Learning Corpus Exchange (2007-2009). http://mulce.univ-fcomte.fr/axescient.htm#eng


Ousmane Touat

Towards Robust and Bias-free Federated Learning

Keywords: CCS Concepts:, Computing methodologies → Distributed artificial intelligence, • Social and professional topics → User characteristics, • Computer systems organization → Reliability Federated Learning, Bias Mitigation, Robustness, Byzantine Behavior

INTRODUCTION

Federated Learning (FL) is a distributed learning paradigm that allows multiple entities, denoted FL clients, to collaborate to learn a model with the help of a central node called the server, with privacy in mind, as the entities do not share their training data [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF]. The deployment of FL in high-risk sectors such as healthcare has sparked concerns as the models may exhibit bias due to underrepresented minorities in data, leading to poorer performance for specific groups [START_REF] Yahya | FairFed: Enabling Group Fairness in Federated Learning[END_REF]. For example, there have been reports of image classification models used for diagnosing skin cancer showing less accurate results for the person of color [START_REF] Roxana | Disparities in dermatology AI performance on a diverse, curated clinical image set[END_REF]. Furthermore, some FL clients denoted Byzantine client, can significantly impact the FL model training by sending arbitrary updates to the server [START_REF] Blanchard Peva | Machine Learning with Adversaries: Byzantine Tolerant Gradient Descent[END_REF]. Addressing model bias and Byzantine clients in FL systems is critical, and several existing works are addressing such issues separately. For instance, FL bias mitigation methods formulate an optimization approach with bias constraints, requiring FL clients to send additional statistics on their local data [START_REF] Cui | Addressing Algorithmic Disparity and Performance Inconsistency in Federated Learning[END_REF][START_REF] Yahya | FairFed: Enabling Group Fairness in Federated Learning[END_REF][START_REF] Wei | Fairness-Aware Agnostic Federated Learning[END_REF]. On the other hand, robust FL systems usually rely on passive outlier detection mechanisms using robust aggregation or gradient clipping [START_REF] Blanchard Peva | Machine Learning with Adversaries: Byzantine Tolerant Gradient Descent[END_REF][START_REF] Pillutla | Robust Aggregation for Federated Learning[END_REF][START_REF] Yin | Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates[END_REF]. However, incorporating both goals into a single FL system is challenging, as the current objectives are contradictory.

FL bias mitigation requires considering outliers clients for adding representativity in their local data, while robust FL systems may eliminate outlier clients, which can affect model bias for worse [START_REF] Sai | Byzantine-Robust Learning on Heterogeneous Datasets via Bucketing[END_REF][START_REF] Wang | Attack of the Tails: Yes, You Really Can Backdoor Federated Learning[END_REF]. Therefore, Authors' addresses: Ousmane Touat, ousmane.touat@insa-lyon.fr, INSA Lyon -LIRIS, Lyon, France; Sara Bouchenak, sara.bouchenak@ insa-lyon.fr, INSA Lyon -LIRIS, Lyon, France. finding a balance between these two goals in a single FL system is complex and requires careful consideration to avoid compromising one for the other (see Figure 1 and Figure 2). This paper investigates the challenges of building a FL system that combines bias constraints and robustness guarantees. We empirically showcase where classical robust aggregation degrades model bias and hinders bias mitigation. We then discuss some research directions to build a FL system combining robustness and bias mitigation goals.

Client 1

FL Robustness Mechanism

The FL Robustness Mechanism may also filter the client 3 ("Honest but minority") update, loosing interesting data for the FL Bias Mitigation mechanism.

FL Server

Client 2

...

Client 3 Client 4

Honest clients Byzantine

FL Bias Mitigation Mechanism

Fig.

Client 3 Client 4

Honest clients Byzantine

FL Robustness Mechanism

The FL Bias Mitigation Mechanism is directly exposed to the Byzantine influence Fig. 2. Applying bias mitigation first in Byzantine setting does not work

BACKGROUND AND SYSTEM MODEL

In this section, we recall the principles of FL, demographic bias and the standard Byzantine attack model.

Federated Learning

In Federated Learning (FL), we have 𝑛 clients aiming to optimize a shared objective function collectively managed by a central server. The training process progresses through communication rounds, in which the central server sends the current global ML model with parameters 𝜃 to the FL clients. In this setup, the 𝑖-th client has its local data denoted as D 𝑖 and trains a local model denoted as 𝜃 𝑖 . We denote the union of clients' local data D. At the end of the round, the clients send their updated model to the server, which update the new global model by aggregating the clients' updates. The classical approach for FL aggregation is called FedAvg [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF] and consists in averaging the clients' model parameters as follows :

𝜃 = 𝑘=𝑁 ∑ 𝑘=1 |D 𝑘 | |D | .𝜃 𝑘 (1)

Bias in Federated Learning

Bias refers to the property of an classifier model to disparately treat different groups based on sensitive attributes, denoted 𝑆. A sensitive attribute is binary as it partition the dataset into two groups. One widely used metric for measuring bias with respect to sensitive attributes is statistical parity difference (SPD) [START_REF] Dwork | Fairness Through Awareness[END_REF].

With 𝑦 the prediction value from the FL model and 𝑆 the value of the sensitive attribute we define 𝑆𝑃𝐷 𝑆 as follows:

𝑆𝑃𝐷 𝑆 = |𝑃𝑟 (𝑦 = 1|𝑆 = 1) -𝑃𝑟 (𝑦 = 1|𝑆 = 0)| (2)
In this formulation, a value of zero for 𝑆𝑃𝐷 𝑆 indicates that the algorithm is statistically fair concerning the sensitive attribute, as it predicts positive outcomes with the same probability for both groups. Minimizing the value of such metric is relevent in cases where the classifier output must be independant from the sensitive attribute (e.g algorithms with any societal impact which encounters issue of demographic discrimination ).

Byzantine Attack Model in Federated Learning

The presence of malicious parties, known as Byzantine clients, denoted B, can threaten the integrity of the FL process. The fraction of Byzantine clients is given by 𝜌, and the number of Byzantine clients denoted by 𝑞 such that 𝑞 ≤ 𝜌𝑛. Byzantine clients can deviate from the standard protocol and send arbitrary updates to the server, and they may even collude and have knowledge of the states of all other clients. The remaining 𝑛 -𝑞 clients who follow the protocol, denoted as honest clients, are represented by set H . FL systems such as FedAvg are directly vulnerable to Byzantine attacks [START_REF] Blanchard Peva | Machine Learning with Adversaries: Byzantine Tolerant Gradient Descent[END_REF]. For example , we take a FL system of 𝑛 clients, including one Byzantine client. We suppose that each honest client sends an update denoted as 𝜃 𝑖 for 𝑖 ∈ {1, . . . , 𝑛 -1}. The Byzantine client can craft an update

𝜃 𝑏𝑦𝑧 = |D |.𝜃 𝑤𝑡 -1 | D | (D 𝑘 .𝜃 𝑖 + • • • + D 𝑛-1 .𝜃 𝑛-1
) to make the server compute any global model 𝜃 𝑤𝑡 desired. This happen as the server, unaware of the presence of the Byzantine client, aggregates all received updates, producing the following global model 𝜃 𝑤𝑡 as the Byzantine model wanted. Such unwanted behavior harms the FL system greatly, making it unable to converge reliably, motivating the research on robust FL methods.

RELATED WORK

Robustness to Byzantine behavior in Federated Learning

The Byzantine fault tolerance problem is well-known in distributed systems, where malicious nodes may provide incorrect information, leading to incorrect results. In FL, this problem can arise when some participating clients behave maliciously and manipulate the aggregated model parameters. Several Byzantine resilient FL methods have been proposed in the literature using the statistical properties of the client updates.

Krum [START_REF] Blanchard Peva | Machine Learning with Adversaries: Byzantine Tolerant Gradient Descent[END_REF], and by extension Multi-Krum, order the clients' updates using a score stating how close they are to its 𝑛 -𝑓 -2 nearest neighbors using Euclidean distances. RFA [START_REF] Pillutla | Robust Aggregation for Federated Learning[END_REF] computes an approximate value of the geometric median of all client updates using Weiszfeld's algorithm. TrimmedMeans [START_REF] Yin | Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates[END_REF] computes the global model by taking the mean value of each coordinate of the client updates, truncating extreme values. For model poisoning attack settings, NDC [START_REF] Sun | Can You Really Backdoor Federated Learning?[END_REF] adopts a norm-thresholding policy that limits the model's parameter norm by clipping it when it exceeds a fixed threshold. However, such robust aggregation techniques fail when used on non-IID data, cannot withstand attacks, which degrade the global model performance [START_REF] Sai | Byzantine-Robust Learning on Heterogeneous Datasets via Bucketing[END_REF]. For non-IID data, some defense mechanisms have been proposed to mitigate the impact of Byzantine behavior. Recently, new defense mechanisms have tackled the impact of Byzantine behavior in non-IID FL context. FedInv [START_REF] Zhao | FedInv: Byzantine-Robust Federated Learning by Inversing Local Model Updates[END_REF] performs model inversion on the clients' updates to generate dummy datasets and compare clients' Wasserstein distance on the dummy datasets distribution, empirically showing robustness in non-IID FL context. Recently Karimireddy and al. introduced a new bucketing technique where the server randomly partitions client updates into "buckets", providing some guarantee when using this technique alongside the previously mentioned robust aggregation technique in non-IID and Byzantine FL context [START_REF] Sai | Byzantine-Robust Learning on Heterogeneous Datasets via Bucketing[END_REF]. Furthermore, Byzantine resilience can also be addressed by ensuring the integrity of the participating nodes using enclaves such as TEEs [START_REF] Hanieh | Byzantine-Robust and Privacy-Preserving Framework for FedML[END_REF].

Bias Mitigation in Federated Learning

The issue of model bias is well-known in machine learning [START_REF] Ninareh | A Survey on Bias and Fairness in Machine Learning[END_REF] and has also received prime attention in FL. In FL, we distinguish between approaches that mitigate model performance discrepancy between clients [START_REF] Li | Ditto: Fair and Robust Federated Learning Through Personalization[END_REF][START_REF] Song | Profit Allocation for Federated Learning[END_REF] and ones that address model bias between demographic groups in FL, from where our work is positioned. We encounter two approaches to tackle this topic, namely client-side techniques and server-side techniques. The client-side technique aims to apply a local debiasing mechanism, such as data reweighting [START_REF] Abay | Mitigating Bias in Federated Learning[END_REF]. However, such a technique may fail on the global bias objective, especially where the FL setup is highly non-IID [START_REF] Yahya | FairFed: Enabling Group Fairness in Federated Learning[END_REF]. In server-side techniques, most FL methods seeks to solve a global bias-constrained objective, requiring local clients to share information about the local statistic of sensitive attributes [START_REF] Cui | Addressing Algorithmic Disparity and Performance Inconsistency in Federated Learning[END_REF][START_REF] Yahya | FairFed: Enabling Group Fairness in Federated Learning[END_REF][START_REF] Wei | Fairness-Aware Agnostic Federated Learning[END_REF][START_REF] Daniel Yue Zhang | FairFL: A Fair Federated Learning Approach to Reducing Demographic Bias in Privacy-Sensitive Classification Models[END_REF]. AgnosticFair [START_REF] Wei | Fairness-Aware Agnostic Federated Learning[END_REF] uses local information from the client to optimize the data reweighting coefficient sent back to the client to seek its globally constrained objective problem. FairFL [START_REF] Daniel Yue Zhang | FairFL: A Fair Federated Learning Approach to Reducing Demographic Bias in Privacy-Sensitive Classification Models[END_REF] train a client-selection policy function using multi-agent reinforcement learning to maximize a gain function based on how bias mitigation is performed on the global model. FairFed [START_REF] Yahya | FairFed: Enabling Group Fairness in Federated Learning[END_REF] linearly scalarize the inputted local model parameters based on their local bias discrepancy to the global bias metric. FCFL [START_REF] Cui | Addressing Algorithmic Disparity and Performance Inconsistency in Federated Learning[END_REF] also linearly scalarize the gradient updates from the clients to optimize both a bias constraint and consistency in the client's model performance.

Federated Learning with Bias and Robustness Guarantees

While the issues of demographic bias and robustness have been extensively explored in the literature, combining these two notions within a single system needs to be further examined. Ditto [START_REF] Li | Ditto: Fair and Robust Federated Learning Through Personalization[END_REF] confronts the notions of robustness against client-level fairness, showing that these two notions are competing, and implements a FL system that combines these properties using client model personalization. Recently, Singh and al. [START_REF] Singh | Fair detection of poisoning attacks in federated learning on non-i.i.d. data[END_REF] built a robustness mechanism that minimizes the amount of honest "minority" clients detected as malicious ones. Assuming each client represents only one group, the server employs a decentralized micro aggregation algorithm to cluster the clients based on their published sensitive attribute, identifying malicious clients having updates too far from those cluster centroids.

PROBLEM ILLUSTRATION

While the robustness of FL systems is essential, using robust aggregators in FL may impair model bias, particularly in non-IID FL settings. The robustness methods may eliminate outlier clients with different data distribution, suppressing their contribution to the global model. In this section, we will further illustrate this problem and explore its implications for model bias in FL.

Impact of FL Robustness Methods on Bias

We compared the behavior of 4 robust aggregators that are Multi-Krum [START_REF] Blanchard Peva | Machine Learning with Adversaries: Byzantine Tolerant Gradient Descent[END_REF], RFA [START_REF] Pillutla | Robust Aggregation for Federated Learning[END_REF], TrimmedMeans [START_REF] Yin | Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates[END_REF] and NDC [START_REF] Sun | Can You Really Backdoor Federated Learning?[END_REF], on two tabular datasets, MEPS [START_REF] Cohen | Design Strategies and Innovations in the Medical Expenditure Panel Survey[END_REF] and Adult [START_REF] Kohavi | Scaling Up the Accuracy of Naive-Bayes Classifiers: A Decision-Tree Hybrid[END_REF], in non-IID FL settings. We also added FedAvg [START_REF] Yin | Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates[END_REF] as another FL baseline without robustness mechanism for comparison. On MEPS, an hospital expenditure dataset, the target attribute for the binary classification task is the utilization of medical facilities with the sensitive attributes of race and gender. The MEPS dataset is cut into a 4-client FL setup, showing the opposite trend of the three other clients that share similar data distribution, with the minority race becoming the majority. In the Adult dataset, the target attribute is income, and the considered sensitive attributes are gender and age. We use a 10-client FL setup generated with a Dirichlet function [START_REF] Hsu | Measuring the Effects of Non-Identical Data Distribution for Federated Visual Classification[END_REF] on the distribution of the sensitive attribute and target feature combo to ensure heterogeneity between FL clients.

We report the model bias for the baseline FedAvg method and the robust aggregators in Figure 3. We observe that the robust aggregators, including Multi-Krum, RFA, and NDC, increase the model bias compared to the FL baseline on both MEPS (Figure 3(c)) and Adult (Figure 3(a),3(b)). The robust aggregators' behavior in such experiments is concerning as it demonstrates a degradation in the model bias in a setup without any Byzantine attacks. We attribute such behavior to the fact that robust aggregators tend to exclude clients with different data distributions. We observe for example that in the 4-client setup of the MEPS experiment, the client that shows the reverse bias trend is systematically excluded, or have its influence reduced by the robustness techniques. In our case, it corresponds to clients showing opposite bias dispositions, which are necessary to offset the biases from the other FL clients, thus showing the robust aggregators' difficulty working in a heterogeneous context when considering demographic bias.

Impact of FL Robustness Methods on FL Bias Mitigation

We study the behavior of two robust aggregators, Multi-Krum and NDC when used with a state-of-the-art bias mitigation method called FCFL [START_REF] Cui | Addressing Algorithmic Disparity and Performance Inconsistency in Federated Learning[END_REF]. We add the robust method as the first step after getting clients' updates before sending the updates to the FCFL server solver to perform the optimization task. Using NDC with FCFL means the clients' updates are clipped if their norm exceeds a fixed threshold and sent to the server solver while for Multi-Krum it means that clients are selected to be sent to the server solver based on the set of clients with the lowest Krum score. The solver will then work with fewer updates to compute the global model. For all experiments, we used the same hyper-parameters on FCFL, specifically the SPD threshold and the learning rate. As explained earlier, we perform such experiments on the same datasets and sensitive attributes. We report the model bias of both added robust mechanisms against using FCFL alone and the baseline FL method in Figure 4. We observe how the robust mechanism influences the behavior of FCFL. Specifically, on both MEPS (Figure 4(c)) and Adult (Figure 4(a),4(b)), we observed that NDC influenced the bias mitigation process, getting a higher value of SPD than the typical method behavior. For Multi-Krum, we can see its influence on FCFL the most on Adult (Figure 4(a)) and MEPS (Figure 4(c)). We observe that using classical robustness mechanism in conjunction with FCFL worsens the bias mitigation process, further motivating the issue of building a bias-aware FL robustness mechanism. FL Robustness to Byzantine Behavior. The FL system must be robust against the attack of 𝑓 = 𝜌𝑛 Byzantine client, which is achieved when for any 𝜖 > 0, the server can output a global model 𝜃 such that we can bound the diameter of the gradient of the loss function 𝑓 (𝜃 ) by a constant 𝜖. Ideally, the system can distinguish the honest FL clients, thus selecting only the latter for aggregation. We call such aggregation

𝜃 = 1 | H |
𝑗 ∈ H 𝜃 𝑗 the true average. A robust aggregator must output a model 𝜃 close to the true average. Assuming that we can bound the honest client model heterogeneity ∥𝜃 𝑗 -𝜃 𝑖 ∥ 2 < 𝛿 2 for all 𝑖, 𝑗 ∈ H , the output θ from the robust Byzantine aggregator must respect the following constraint [START_REF] Sai | Byzantine-Robust Learning on Heterogeneous Datasets via Bucketing[END_REF] :

E∥ θ -𝜃 ∥ 2 < 𝑘𝜌𝛿 2 (3)
FL Bias Mitigation. The goal is to learn the optimal value of the global model parameters 𝜃 that solve the bias-constrained optimization on the 𝑆𝑃𝐷 𝑆 value, where 𝜖 is the 𝑆𝑃𝐷 𝑆 constraint and 𝑓 𝑘 (𝜃 ) is the loss of the given model 𝜃 over local data of the honest client 𝑘 :

min 𝜃 𝑓 (𝜃 ) = min 𝜃 1 |H | ∑ 𝑘 ∈ H 𝑓 𝑘 (𝜃 ) (4a) s.t |𝑆𝑃𝐷 𝑆 (𝜃 )| ≤ 𝜖 (4b)

WHY HANDLING BIAS AND BYZANTINE BEHAVIOR IN FL IS DIFFICULT

This section shows the challenges of building a FL system combining robustness against Byzantine behavior, and model bias mitigation. We cover the following scenarios: Using classical robust aggregators first and then the FL Bias mitigation process separately (in section 6.1). Using the FL Bias mitigation process first and then the robustness mechanism separately (in section 6.2) and finally combine both robustness and bias mitigation objectives at once (in section 6.3). We provide then interesting research direction to address the cited challenges.

Why Applying a Classical FL Robustness Mechanism Followed by Classical FL Bias Mitigation Does Not Work

Suppose we first apply classical robust aggregation techniques and subsequently perform bias mitigation.

The resulting system may fail to operate effectively in a non-IID environment. With a FL system consisting of 𝑛 clients, with one honest client having a very different data distribution with better representation of the minority group, its contribution to the FL system would then help bring down the global model bias. However, we must expect our FL system to be robust to Byzantine clients by assuming the latter's presence within our system. Let us also suppose the presence of one Byzantine clients in our system. Using classical robust aggregators such as Multi-Krum would mean that one client get eliminated based on how far the model parameters are from all other clients. In section 4, we illustrated how robust aggregators could eliminate the client with different data representativity. Unfortunately, this can lead to the exclusion of honest clients that offer demographic minority representation (see Figure 1), losing crucial data representation that could be used by bias mitigation methods to help reduce the bias between demographic groups.

Observation 1:

Using classical robust aggregators may eliminate honest client, affecting the normal behavior of FL bias mitigation.

6.2 Why Applying Classical FL Bias Mitigation Followed By a Classical FL Robustness

Mechanism Does Not Work

In a FL setup where we suppose the presence of Byzantine clients, we must keep the Byzantine client from being considered by the FL Bias mitigation method as we need to trust their behavior. Suppose a FL system with honest clients and Byzantine clients. At the end of the FL round, each honest client sends regular updates, while the Byzantine sends arbitrary information. In this scenario, the FL bias mitigation component is applied before any robustness mechanism in this scenario. The classical bias mitigation method happening server-side would then get exposed to the Byzantine clients, which can directly harm the model utility (see Figure 2). A tradeoff with model utility is often expected with the bias mitigation methods [START_REF] Cui | Addressing Algorithmic Disparity and Performance Inconsistency in Federated Learning[END_REF][START_REF] Yahya | FairFed: Enabling Group Fairness in Federated Learning[END_REF][START_REF] Wei | Fairness-Aware Agnostic Federated Learning[END_REF]. Adding Byzantine clients would further increase the tradeoff, making some bias-constrained optimization tasks like AgnosticFair [START_REF] Wei | Fairness-Aware Agnostic Federated Learning[END_REF] more difficult. Furthermore, as mentioned in section 3.2, some bias mitigation methods that intervene server-side, such as FairFed [START_REF] Yahya | FairFed: Enabling Group Fairness in Federated Learning[END_REF], require clients to send information on their local bias metrics in order to compute the global model. Byzantines can exploit this mechanism by sending falsified local bias information, which can cause the system to consider their harmful model updates even better.

Observation 2:

Using the classical FL bias mitigation method before any robustness mechanism expose the bias mitigation method to the influence of the Byzantine clients.

On the Impossibility to Handle Bias and Byzantine Behavior at Once in Classical Approaches

Each client sends their locally trained model update to the server during the FL process for optimization. In our case, we assume wanting to maximize the global model utility on a data distribution representing the union of honest client data under bias constraint. At the same time, we want our model to have robustness guarantees against the current Byzantine activity in the system. In order to solve the bias-constrained objective problem on the global data distribution, we must optimize our objective based on the set of models updates coming from the honest clients set, and nothing else as we do not trust the updates coming from the Byzantine clients as mentioned in subsection 6.2. However, such a set of honest client updates is heterogeneous due to outlier clients, which we assume represents additional information on minorities compared to the one found in most updates. In [START_REF] Sai | Byzantine-Robust Learning on Heterogeneous Datasets via Bucketing[END_REF], Karimireddy and al. argue that we can construct a set of model updates including all Byzantines that would pass the robust aggregation part. Our system may consider those Byzantine updates for the server-side optimization part, which brings an error component to the global optimization process that we cannot ignore. The output of such a system combining bias mitigation and robustness is not guaranteed to keep the bias constraint under the global data distribution.

Observation 3:

Using both objectives is impossible in heterogeneous settings using only model updates as information, as the sever cannot distinguish between outlier clients and Byzantine ones.

Discussion and Research Directions

Despite the challenges of combining bias mitigation and robustness to Byzantine behavior methods in FL, interesting research directions remain to explore. We have seen the necessity of providing robustness before any other bias mitigation operation at the server level in subsection 6.2. This requires a new method for detecting Byzantine clients in non-IID environments. First, such a FL system must guarantee the selection of outlier clients when there are no Byzantine attacks on the system. Also, the FL system must reduce the amount of falsely predicted Byzantine clients, as these clients probably contain critical data representativity.

The server can try spotting such "honest and minority" clients among the deemed Byzantine clients by the robust aggregators. To distinguish such clients from the Byzantine ones, we could rely on reputation mechanisms to precisely identify Byzantine clients [START_REF] Xu | A Reputation Mechanism Is All You Need: Collaborative Fairness and Adversarial Robustness in Federated Learning[END_REF]. Furthermore, we saw how most FL bias mitigation approaches require the clients sending additional information on local bias (see subsection 3.2). Then a new solution could have the server also asking its clients to send additional information on their local data distribution and privately compare such information with their sent model updates. The system could then analyze inconsistencies between the information on the data distribution and what can be extracted from the client model update (i.e., using model inversion [START_REF] Zhao | FedInv: Byzantine-Robust Federated Learning by Inversing Local Model Updates[END_REF]), revealing the Byzantine nature of the client.

Then selecting of the "honest and minority" clients could be performed on how interesting their added data representativity for minorities are.

CONCLUSION

FL model bias and robustness against Byzantines have been extensively studied over the last few years.

Addressing both problems within a single FL system, while critically needed, has been elusive. This study investigated the challenges of constructing a FL system with both robustness to Byzantine behavior properties and bias guarantees. By design, a robust FL system degrades inclusiveness, particularly for outlier model behavior, which may have the side effect of setting aside data representing minorities, which is necessary for proper functioning FL bias mitigation techniques. Furthermore, we discuss the limitations of combining classical bias mitigation methods with existing robustness mechanisms. Finally we formulate possible research directions for building robust, bias-free FL.

Fig. 3 .

 3 Fig. 3. Impact of FL robustness methods on Bias

Fig. 4 .

 4 Fig. 4. Interaction between FL bias mitigation and FL robustness mechanisms

ACKNOWLEDGMENTS

This work was partially supported by the ANR French National Research Agency, through SIBIL-Lab LabCom (ANR-17-LCV2-0014) and ByBlos project (ANR-20-CE25-0002-01). Thanks to Paul Wawszczyk, who has also contributed to this work during his internship.