Nabil Chamroune

Florence Delange

Nathalie Caillault

Fabrice Morvan

Yongfeng Lu

Akira Kawasaki

Jean-François Silvain

Jean-François Silvain

Dr Jean-Francois Silvain
email: jean-francois.silvain@icmcb.cnrs.fr

Synergetic Effect of Discontinuous Carbon Fibers and Graphite Flakes on Thermo-Mechanical Properties of Aluminum Matrix Composites Fabricated by Solid-Liquid Phase Sintering

Keywords: Metal-matrix composites (MMCs), Discontinuous reinforcement, Powder processing, Thermal properties

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

Introduction

Metal matrix composite materials (MMC) are currently studied and have several promising applications. One of them concerns their use as heat sinks in microelectronic industries where two conditions are required: a thermal conductivity (TC) as high as possible in order to evacuate the heat generated by the silicon chip and therefore to increase the reliability of the electronic device, and a coefficient of thermal expansion (CTE) close to the CTE of ceramic substrate (2 to 8×10 -6 /K) and the electronic chip (2 to 4×10 -6 /K) [1][START_REF] Zweben | Annual IEEE Semiconductor Thermal Measurement and Management Symposium[END_REF][START_REF] Rohatgi | [END_REF][4]. Currently Copper (Cu) and Aluminum (Al) heat sinks are used because of their high TC (400 W/mK for Cu and 220 W/mK for Al). However, the large CTE mismatch with the ceramic substrate (CTE of Cu equal to 17×10 -6 /K and CTE of Al equal to 24×10 - 6 /K) generates thermo-mechanical stresses at the heat sink-ceramic interfaces which can generate cracks inside the solder joint of inside the ceramic substrate and then catastrophic failure of the electronic device.

Compared to the others matrices (Copper or Silver), Aluminum has a lower TC and a higher CTE. However, due to his low density, 2.7 g/cm 3 , and low price, Al has a high specific thermal conductivity (TC divided by density) and low cost which is a great advantage in terms of the fabrication of mobile electronic devices for automobile or aeronautic industries [4].

Among carbon reinforcements, diamonds exhibit outstanding isotropic thermal properties (TC of ~2000 W/m.K and CTE close to 2.5×10 -6 /K) [5][6][7]. However, the applications are limited due to its high price and poor machinability. Pitch-based carbon fibers (FC) are in contrast less expensive and have a good machinability but a lower and anisotropic thermal properties (longitudinal direction: TC of 100-1000 W/m.K depending on graphitization degree and CTE of -1 10 -6 /K ; transverse direction: TC of 5-10 W/m.K and CTE of 12×10 -6 /K) [8,[START_REF] Pierson | Handbook of Carbon, Graphite, Diamond and Fullerenes[END_REF]. Graphite flakes (GF) have significant attention for thermal management applications due to their superior thermal properties, low cost and ease of machining. Indeed it has been reported that GF exhibit a CTE of -1×10 -6 /K and a TC higher than 1000 W/m.K in the plane of graphite flake [START_REF] Chen | [END_REF]11]. A TC of 1600-2000 W/m.K has been reported for highly oriented pyrolytic graphite (HOPG) [12,13]. Out of plane, whatever the graphite type, a TC of 5-10 W/m.K and a CTE of 28×10 -6 /K has been reported [11,13,14].

In functional metal matrix composite (MMC), extended works have been reported on reactivity between matrix and reinforcement [15,16], dispersion and orientation of reinforcement [17,18], fabrication process [4] (sintering [START_REF] Chen | [END_REF]18], squeeze casting [START_REF] Chen | [END_REF]20] and pressure infiltration process [21]). However, few attempts in solid-liquid phase sintering [22] have been carried out especially with a mixture of GF and CF reinforcements. Pietro et al. [23] and Oddone et al. [24] reported composites with a mixture of GF and CF, fabricated by gas pressure infiltration and spark plasma sintering, respectively. They focused their studies on sintering parameters (heating rate, current flow and sintering temperature) and the nature of the metal. However, no understanding on the behavior of the thermal properties of Metal/(GF+CF) compared to the Metal/GF and Metal/CF composite materials was reported.

Several studies reported the fabrication of metal/GF composite materials with a significant increase of TC along the GF principal axis (in-plane TC) and a decrease of TC perpendicular to the GF principal axis (through-plane TC) [25,26]. However, concerning the CTE of MMC, studies reported a slight decrease of the in-plane CTE and, surprisingly, a significant decrease of through-plane CTE [26,27,28,29]. In Cu/50vol.%GF systems, throughplane CTE of 2.5×10 -6 /K [26], 1.9×10 -6 /K [27] and -1×10 -6 /K [28] was reported, which is very close to CTE of silicon (2.8×10 -6 /K). In Al/50vol.%GF systems, a through-plane CTE of -10×10 -6 /K and an in-plane TC of 330 W/m.K were reported [29]. However, this CTE presents a significant CTE mismatch with the ceramic substrate.

Therefore, in order to obtain a CTE close to 3×10 -6 /K in Al/GF system, the vol.% of GF must be decreased which will, consequently, decreases the TC of Al/GF composite [29].

This investigation reports a way to control the CTE of Al/C composite by adding small quantity of CF to Al/GF composites in order to obtain a CTE compatible to the ceramic in the in-plane or the through-plane direction.

Al/Carbon composites materials were fabricated by solid-liquid phase sintering. The quantity of Al-12wt%Si alloy, in liquid state during the sintering process, was optimized for a total carbon content of 50 vol.% in order to obtain fully dense (Al+Al-Si)/Carbon materials. Different GF/CF ratios was used in order to investigate the effect of the solid-liquid phase sintering and the carbon composition on the TC and the CTE of the (Al+Al-Si)/(GF+CF) composite materials. Al/C composite materials were characterized by scanning electron microscopy (SEM), energy-dispersive X-ray microscopy (EDS) and X-ray tomography in order to understand the behavior of thermomechanical properties of Al/(GF+CF) composite materials.

Experimental procedure

Materials

Gas-atomized Al spherical powder (ULTD0065, Hermillon powders, France, Fig. 1a) and Al-12wt%Si spherical powder (16-0006G, Toyal, Japan, Fig. 1b) with an average diameter of 10 µm and 8 µm, respectively, were used as the starting metal powder. Graphite Flakes (GF; Yanxin-Graphite Co., Ltd, Fig. 1c), with an average size and thickness of 550 µm and 30 µm respectively, and Carbon Fibers (CF, K223HM, CTN DIALEAD TM /BOBBINS, Sumitomo Corporation Europe Limited, Fig. 1d) with an average length and diameter of 250 µm and 10 µm respectively, were added to metal powder. The four components were mixed in Turbula mixer (Turbula Shaker/Mixer Model T2C, Germany) for 2 hours to obtain a homogeneous (Al+AlSi)/(GF+FC) distribution. mm 3 ) were fabricated by hot pressing for 30 min at 600°C under a uniaxial compressive stress of 60 MPa and primary vacuum. The temperature of the steel mold is measure with a thermocouple place inside the mold and close to the composite powder. Heating time is close to 10 min and cooling time to 60 min. Differential thermal analysis of Al-Si alloy powder has shown that the melting temperature is 578°C. For these processing conditions, Al-Si alloy is in liquid state during the densification of the materials as illustrated in Figure 2. The volume fraction of GF and CF in the Al/GF and Al/CF composites respectively, was fixed at 10, 20, 30, 40, and 50%. Concerning Al/(GF+CF) composite materials, the composition of the carbon mixture (vol.%GF + vol.%CF) was fixed at (48+2), (46+4), (44+6), (42+8), (40+10), (30+20), (20+30) and (10+40) for a total carbon volume fraction of 50 vol.%. The hot-pressing temperature was monitored via a K-type thermocouple located 2 mm from the sample in the steel mold. Considering that the GF and carbon fibers [8,18] tend to be aligned perpendicularly to the uniaxial compressive stress, the cylindrical Al/GF composites were machined to Ø6 x 5 mm 3 in order to measure TC and CTE in the in-plane and through-plane direction.

Characterization

Microstructural characterization of the Al/GF composite was carried out via scanning electron microscopy (SEM, Tescan, VEGA© II SBH). Elemental analysis of the Al/C composites was performed through energy dispersive X-ray spectroscopy (EDS, Apollo X, EDAX Inc.). In order to reveal the microstructure, Al/GF composites were prepared using mechanical polishing with polishing cloth. The polished samples were placed in etchant aqueous solution of sodium hydroxide. X-Ray tomography was carried out via micro-scanner (General Electrics V/Tome/x s). The acquisition resolution is equal to 2.8 µm 3 . The voltage and rotation resolution are equal to 180 kV and 0.1°, respectively. The dimension of the sample is Ø6 x 5 mm 3 and 2.5 x 2.5 x 5 mm 3 . Tomography data were treated with ImageJ software.

The CTE of the Al/C composite materials was measured with a dilatometer (NETZSCH DIL 402, PC) in two thermal cycles from 40°C to 300°C with a heating/cooling rates of 2°C/min, under argon gas flow. The technical mean values of CTE were recorded between 50°C and 150°C. Each measure point is an average of 3 samples with a standard deviation close to 3%.

Heat capacity was measured by differential Scanning Calorimetry at 70°C (DSC 8000 Pyris Diamond PERKIN ELMER) for graphite flake, carbon fiber, Al-Si and Al powders. The TC of Al/GF composites was calculated using the following equation TC = α x ρ x Cp where α is the thermal diffusivity of Al/GF composites measured via the laser flash method (NETZSCH LFA 45, MicroFlash) at 70°C. Each measure point is an average of 3 samples with a standard deviation close to 5%. The thermal diffusivity was measured in the in-plane and transverse direction. Cp and ρ are the heat capacity and the measured density of the Al/GF composites, respectively. Cp was calculated from heat capacity of each component by a rule of mixture. Considering that the composite materials only contain closed porosity (ranging from 0.2 to 2.5%), the density of the Al/C composites was measured using the Archimedes principle. The theoretical density of the composites was calculated using a rule of mixture.

Results and discussion

Microstructure analysis

Figure 3 shows the in-plane view SEM micrographs of the Al/50vol.%GF (Fig. 3a), Al/vol.%(40GF+10CF) (Fig. 3b), and Al/50vol.%CF (Fig. 3c) composite materials. Dark contrasts are associated to graphite flake (black rectangle) and carbon fiber (black circle) and gray one refers to Al matrix. At that scale of resolution, identical microstructure are found for the Al(Al-Si) matrix. For Al/50vol.%GF and Al/vol.%(40GF+10CF) composites, graphite flakes are preferentially oriented in a plane perpendicular to the pressure direction. In the case of Al/50vol.%(GF+CF), the presence of CF does not affect the orientation of GF compared to Al/GF composite materials. In Al/50vol%CF composites, carbon fibers are also oriented in the plane perpendicular to the pressure direction but randomly oriented in this plan. Moreover, because of the high-volume fraction of carbon reinforcement (50 vol.%), GF-GF (Fig. 4a), GF-CF (Fig. 3b) and CF-CF (Fig. 3c) contact points are observed in all of the Al/C composites materials. and Al/50vol.%CF (g, h, i, and k) composite materials. The carbon network of Al/50vol.%GF, Al/vol.%(40GF+10CF) and Al/50vol.%CF composites were reconstructed in a cubic block of 1.84 mm 3 . Yellow arrows show the pressure direction (Z axis) applied during the sintering process. These reconstructions clearly show different microstructure observed in (xz) and (xy) plan which is characteristic of a transversally isotropic material. In the (xy) plan, the top surface of the GF can be observed (Fig. 4c, 4f and4j) whereas the CF are randomly oriented (Fig. 4f, 4i and4k). In the (xz) plan, the section of GF and CF can be observed. These observations imply that the pressure applied during the hot pressing provides an orientation of the carbon reinforcement perpendicular to the pressure axis. It has to be noticed that in some area, carbon reinforcement is still disoriented [18]. Moreover, the carbon network of Al/vol.%(40GF+10CF), Al/50vol.%CF were reconstructed in a square of 1.1×1.1 mm 2 and a thickness of 30 µm (Fig. 4j and4k). This allows highlighting the formation of a continuous 2D carbon networks composed by GF and CF oriented randomly in this same plan. However, by comparing Al/vol.%(40GF+10CF) and Al/50vol.%CF, which have the same carbon content, it can be seen an as dense carbon fiber network in Al/vol.%(40GF+10CF) than Al/50vol.%CF. Indeed, in Al/50vol.%CF, the carbon fibers are dispersed in 50 vol.% of Al whiles in Al/vol.%(40GF+10CF), the CF are dispersed in lower volume (50 vol.% of Al). The dense carbon fiber network observed in Al/vol.%(40GF+10CF) will be used in order to explain further physical properties of the Al/(GF+CF) composite materials.

Figure 5 shows an energy dispersive X-ray spectroscopy (EDS) scan realized on (Al+Al-Si)/50vol.%CF composite material in order to highlight the distribution of the silicon inside the Al matrix. The amount of Al-12wt%Si were optimized for a GF content of 50 vol.% in order to achieve a fully dense composite materials and to show its effect on the in-plane TC. Table 1 shows the evolution of the measured relative density of the (Al+Al-Si)/50vol.%GF composite materials with the volume percentage of Al-12w%Si liquid phase ranging from 0 to 15 vol.% with 2.5 vol.% step increase. It is well known that the relative density of composites decreases when the reinforcement content increases [16,18]. Indeed, in addition to the porosity, which is inherent to the powder metallurgy (PM) route, the decrease of the relative density may be correlated with the Al-GF interfacial nano-voids [26] and graphite-graphite contact points (Fig. 3 and4) which appear significantly from a carbon volume fraction close to 30 vol.%GF [START_REF] Chen | [END_REF]18]. Al/50vol.%GF composite fabricated by conventional PM route shows a relative density of 97.5%. When the volume percentage of liquid phase increases, the relative density increases to reach a value of 98.5% for a liquid content of 15 vol.%. This behavior is explained by the fact that the liquid phase formed during the sintering step, reduces the pores and infiltrates the GF/GF contact point where no sintering can occurs. Therefore, the liquid phase sintering helps to increase the relative density of Al/C composite materials with high carbon content. With this fabrication process, MMC, with carbon volume fraction up to 70%, have been successfully fabricated with a porosity level smaller than 3%. Moreover, Table 1 shows that the evolution of the relative density can be correlated with the evolution of the TC.

In the in-plane direction, the graphite flakes are oriented parallel to the direction of thermal conduction measurement. Al/50vol.%GF composite fabricated by conventional PM route shows an in-plane TC of 400 W/m.K. From 0 to 5 vol.% of liquid phase, the in-plane TC increases from 400 to 425 W/m.K. Then, from 5 to 10 vol.% of liquid phase, the in-plane TC reaches values close to 425 W/m.K, and then decreases at 409 W/m.K for higher Al-Si content. It has also to be noticed that, whatever the Al-Si content, the in-plane TC of (Al+Al-Si)/GF composites are always higher than the Al/50vol.%GF fabricated by conventional PM route. This behavior can be explained by two factors: 1) the increase of relative density with the liquid phase content and 2) the lower intrinsic TC of Al-12wt%Si alloy compared to the pure Al (experimental values of 163 W/m.K and 217 W/m.K, respectively). Indeed, the increase of TC from 0 to 10 vol.% of liquid phase is correlated to the increase of the relative density of (Al+Al-Si)/GF composite materials. Then, from 10 to 15 vol.% of Al-Si, even if the relative density increases slightly, the important quantity and the intrinsic TC of Al-Si alloy, results in a decrease of the in-plane TC of (Al+Al-Si)/50vol.%GF composite materials. Hence, a volume percentage of 10% were selected in order to fabricate (Al90vol.%+Al-Si10vol.%)/50vol.%Carbon composites, where the carbon is a mixture of graphite flakes and carbon fibers.

Thermal conductivity of Al/GF and Al/CF composite materials

Al/GF and Al/CF composites were fabricated in order to evaluate the individual effect of the GF and CF on the thermal properties of Al/C composite materials. Figure 6 shows the evolution, of the in-plane and through-plane TC of Al/GF and Al/CF composites with different carbon volume fraction. First of all, Fig. 6 shows a TC of 218 W/m.K for the sintered Al powder. (TC of 239 W/m.K is reported for high purity aluminum at 70°C [30]). The difference between sintered and high purity aluminum materials (foundry route) is mainly attributed to the residual porosity and the amorphous aluminum oxide presents on the surface of each aluminum grain when using PM route.

In the in-plane direction, the graphite flakes are preferentially oriented parallel to the direction of thermal conduction measurement. Then, the TC increases with the graphite content to reach a value of 400 W/m.K. This behavior is in accordance to the intrinsic in-plane TC of the GF (close to 1000 W/m.K [START_REF] Chen | [END_REF][11]). The TC of the Al/CF composites decreases slightly from 218 to 216 W/m.K from 0 to 30 vol.% of CF and then reach a value of 197 W/m.K from 30 to 50 vol.% of CF. This slight decrease of TC may be explained by the fact that, due to the random distribution of the CF in the plane of measurement, the supposed intrinsic TC of the CF in the plane, 280 W/m.K [16] is an average of the along (550 W/m.K; manufacturer data) and perpendicular to the CF axis (5-10 W/m.K). Thermal resistance at the Al-C interface and the decrease of the relative density with the increase of carbon content can explain the slight decrease of in-plane TC of Al/CF composite materials.

In the through-plane direction (dotted line), the GF and CF are preferentially oriented perpendicularly to the direction of thermal conduction measurement. Then, the TC decreases with the carbon content to reach a value of 87 and 46 W/m.K for Al/GF and Al/CF composites, respectively. This decrease is in accordance to the intrinsic through-plane TC of the GF and CF (5-10 W/m.K [START_REF] Chen | [END_REF][11]). However, the behavior of through-plane TC with the carbon content differs between the two reinforcements. Indeed, in the Al/GF composites, the variation of the TC with the GF content has a behavior of an inverse function: a drastic decrease of TC from 0 to 10 vol.% of GF (218 to 123 W/m.K: 43% of decrease) and a slower decrease of TC from 10 to 50 vol.% of GF (123 to 46 W/m.K: 62% of decrease). In contrast, the TC of Al/CF composites decreases linearly with CF content and stay higher than Al/GF for 50vol.% of carbon. This difference of behavior can be explained by the difference of morphology of the reinforcement. Indeed, the planar morphology and larger size (average size and thickness of 550 µm and 30 µm, respectively) than CF (average length and diameter of 250 µm and 10 µm, respectively) allow the GF to slow the thermal flow more efficiently than CF in the through-plane direction. Moreover, the orientation of the reinforcement, in a plane perpendicular to the pressure direction, which is induced by the hot-pressing process, is easily achieved for the GF than for the CF (Fig. 5a). Figure 7 shows the evolution of the CT of Al/(GF+CF) and (Al90vol.%+Al-Si10vol.%)/(GF+CF) composites with a total carbon content of 50 vol.%, in the in-plane (Fig. 7a and7c) and through-plane (Fig. 7b) directions. For comparison purpose, the CT of Al/GF composite materials was added in Fig. 7a (dotted line). It can be shown that the in-plane TC of the Al/(GF+CF) composites is lower than the Al/GF composites with the same volume content of GF. It should be noted that the gap between the two sets of materials decreases when the GF/CF ratio increases.

Indeed, as shown in Fig. 6, the decrease of the in-plane TC of Al/CF is negligible between 0 to 30 vol% of CF which explains the weak gap between Al/(GF+CF) and Al/GF from a carbon content of (30 vol.%GF + 20 vol.%CF). For higher CF content a decrease of the in-plane TC of Al/(GF+CF), compared to Al/GF with the same GF content, is observed. Fig. 7b shows the through-plane TC of Al/(GF+CF) composites materials. It can be seen an almost linear behavior of the TC of Al/(GF+CF) composite materials.

The Al/(GF+CF) composite materials fabricated by conventional PM route (black triangle) were compared to the equivalent composites fabricated by solid-liquid phase sintering (red triangle). It can be seen that solid-liquid phase sintering provides an increase of the in-plane TC (Fig. 7a and7c). As explained in section 4.2, the liquid phase sintering tends to increase the relative density of Al/C composite materials with high carbon content by infiltrating the pores and the carbon-carbon contact points where no sintering can occur. Concerning Al/CF composites, a decrease of the in-plane CTE is observed and reaches a value of 4.6×10 -6 /K for a CF content of 50 vol.%. This result is in accordance with the intrinsic in-plane CTE of CF (-1×10 -6 /K). However, the through-plane CTE of Al/CF composites show an important increase between 0 and 20 vol.% of CF, before to reach a maximum value of 47×10 -6 /K from 20 to 40 vol.% and finally, a decrease at 50 vol.% CF. Unlike the evolution of the in-plane CTE, this behavior is in contradiction with the intrinsic through-plane CTE of CF (12-15×10 -6 /K [8]). Considering the fact that the CFs restrain the dilatation of Al in the in-plane direction leads to a compressive strain in the same direction and then a decrease of the in-plane CTE [31]. This in-plane compressive strain induces an increase of through-plane CTE because of the positive Poisson ratio of Al and CF (0.345 [25] and 0.2 [32], respectively).

Concerning the Al/GF composites, the CTE differ radically in the two directions of measurement. In the in-plane direction, the CTE decreases slightly to a value of 21.8×10 -6 /K for 50 vol.% of GF. In the through-plane direction, the CTE decreases drastically and reaches a negative value of -10.5×10 -6 /K for 50 vol.% of GF. These counterintuitive behaviors were reported in several studies for Al/GF and Cu/GF systems [26][27][28][29]. In order to understand these behaviors, Firkowska et al. [27] modeled the mechanical and thermal interplay of graphite and Copper within elasticity theory and considered a sandwich-like structure of graphite and Copper. They found that the temperaturedependent in-plane strain in the graphite embedded in a matrix copper results in a negative through-plane CTE (α33) of -26×10 -6 /K calculated by equation ( 1 We propose here to use the formulation (1) of Firkowska et al. [27] to explain the behavior of through-plane CTE of Al/GF composite materials and explain the important decrease of the CTE compared to Cu/GF system. The inplane lattice constant of the GF follows the CTE of the composites which causes an in-plane strain ε11= ε22 in the GF that varies with temperature as shown on equation [START_REF] Zweben | Annual IEEE Semiconductor Thermal Measurement and Management Symposium[END_REF]. In Al/GF system, an in-plane CTE of 21.8×10 -6 /K for 50 vol.% of GF was obtained, which is higher than in plane CTE of Cu/50vol.%GF reported in previous studies (14×10 -6 /K [26], 12×10 -6 /K [27] and 13.6×10 -6 /K [28]). This difference is due to the higher intrinsic CTE of Aluminum compared to Copper (24×10 -6 /K and 17×10 -6 /K, respectively [1]). Then the higher in-plane CTE difference of Al/GF and GF (αgr,1 = -1×10 -6 /K) induce a higher in-plane strain in the GF. For an Al/GF system Δαx is equal to 22.8×10 -6 /K which is higher than Cu/GF system. In a second step, intrinsic strain, induced by the CTE difference of the different constituent, has to be consider. Indeed, after the consolidation step, when the temperature decreases to room temperature, the reinforcement and the matrix shrink at different rates, which induce a residual in-plane strain 𝜀 11 0 equal to ~-10 -3 for Cu/GF system [27]. Considering that CTEAl is ~1.5 higher than CTECu, it can be reasonably assumed that the in-plane residual 𝜀 11 0 of Al/GF system is also 1.5 times higher than 𝜀 11 0 for Cu/GF system. By using the formulation (1), a value of -54×10 -6 /K was calculated for the through-plane CTE of GF which is quite different from the intrinsic CTE value of the through-plane GF (28×10 -6 /K). This calculated value will be used to calculate theoretical values of CTE for the Al/GF composite materials.

Schapery model [32] was chosen to predict the CTE of the Al/GF system with anisotropic properties [33] and then compared with the obtained experimental values:

𝛼 𝑐,2 = 𝑉 𝑚 (1 + 𝑣 𝑚 )𝛼 𝑚 + 𝑉 𝑓 (1 + 𝑣 𝑓,2 )𝛼 𝑓,2 -𝛼 𝑐,1 𝑣 𝑐,1 (3)
In these equations, α, V, E and v are the CTE, the volume fraction, the Young's modulus and the Poisson's ratio, respectively. 1 and 2 refer to in-plane and through-plane direction, respectively. The subscripts c, m, and f refer to composite, matrix and reinforcement, respectively. The material parameters for the calculations are given in Table 2 and the through-plane CTE of GF used in this model is the value calculated with the Firkowska formulation [27]. In-plane GF -1 1109 0.12 [34] Through-plane GF -58* 38.7 0.01 [27]*; [34] Al 25 69 0.345 [25] *Value calculated with Firkowska formulation [27] Figure 8 shows a comparison of the theoretical values of the through-plane CTE of Al/GF composite materials (blue dashed line), calculated using Schapery model (3) and Firkowska formulation (1) with the experimental values for fabricated Al/GF composites materials. The Schapery prediction is in fairly good agreement with the experimental CTE of Al/GF composite materials in the through-plane direction. We can therefore argue that the Firkowska formulation, given for Cu based materials, can be transposed for an Al/GF system and allows to demonstrate that the higher mismatch between the matrix and the GF for a Al/GF system compared to a Cu/GF system, induce a higher in-plane strain on the GF and resulting in a through-plane CTE for Al/GF composites significantly lower than for Cu/GF composites.

Thermal expansion of Al/(GF+CF) and (Al+Al-Si)/(GF+CF) composite materials

Figure 10 shows the evolution of the CTE of Al/(GF+CF) and (Al90vol.%+Al-Si10vol.%)/(GF+CF) composites with a total carbon volume percentage of 50%, in the in-plane (Fig. 10a andc) and through-plane (Fig. 10b andc) directions. For comparison purpose, the CTE of Al/CF and Al/GF composites were added to this figure (dotted line). For pure GF reinforcement, in the in-plane direction, the CTE of the composites decreases slightly to reach value of 21.8×10 -6 /K for 50 vol.% of GF. However, with a mix carbon composition (CF + GF), a huge decrease for a CF vol. % up to 10% (9×10 -6 /K) is observed (a detail of this variation is given in Fig. 10c) follow by a slower decrease to reach the CTE value of the Al/50vol.%CF material (4.6×10 -6 /K).

Considering that GF alone has no significant effect on the in-plane CTE, Al/(40vol.%GF + 10vol.%CF) can be compared to Al/10vol.%CF composite material. It can be show that for the same content of CF, the CTE decreases from 18.9×10 -6 /K to 9.2×10 -6 /K. Favier et al. [35] reported that the geometric percolation of cylindrical reinforcement, with an aspect ratio of 25, is close to 3 vol.% and 18 vol.% for a 3D and 2D random orientation, respectively. However, the geometric percolation should be distinguished from the mechanical percolation. Indeed, for electrical system, the geometric percolation is identical with the electric percolation. However, for thermomechanical properties, the geometric percolation is not enough to ensure the rigidity of the whole system. Then, the mechanical percolation, which is higher than the geometric percolation, should be considered. The mechanical percolation of the CF in Al/CF composite materials have already been associated to the formation of rigid skeleton of reinforcement, from a CF volume fraction close to 30 vol.% [8]. This behavior may explain the slope change of the in-plane CTE observed from 30 vol.% of CF (Fig. 10a). The addition of graphite to the CF provides the formation of rigid CF skeleton with only 10 vol.% of CF. Indeed, as it was demonstrated by X-Ray tomography, the CF network in Al/10vol.% is not enough dense to build a rigid skeleton, and thus the mechanical percolation is not reached. The huge decrease of CTE for the CF + GF reinforcement can therefore be attributed to the formation of a rigid CF skeleton in-between the network of the graphite flakes. Moreover, the fact that CF are interleaved between GF may prevent the disorientation of CF out of (xy) plane, as observed in Al/10vol.%CF composite material (Fig. 4h).

In the through-plane direction, the CTE increases significantly from a value of -10.5×10 -6 /K for 50 vol.% of GF to a value of 37.6×10 -6 /K. Indeed, for a carbon composition of (40 vol.%GF + 10 vol.%CF), an important increase of the CTE is observed and reach a value of 17.6×10 -6 /K (a detail of this variation is given in Fig. 9d). Then, when the CF/GF ratio increases, the CTE increases slowly to tend to the CTE of Al/50vol%CF (37.8×10 -6 /K). Therefore, the addition of CF to the GF provides an important CTE increase of Al/(GF+CF) composite material due to the individual effect of the CF in the through-plane direction showed in Fig. 10b (circle mark and dotted line). The fact that the through-plane CTE decreases in Al/GF composites and increases in Al/CF composites explains the behavior of Al/(GF+CF) composites materials. was observed between the Al particles, at the Al/C interface and at the carbon-carbon contact points (Fig. 5). The presence of the Al-Si liquid phase at the C-C contact points allows soldering the carbon reinforcement in this zone and therefore to consolidate the carbon rigid network observed by x-ray tomography (Fig. 4). The formation of this carbon rigid network, induced by the liquid phase, can explain the decrease of the in-plane CTE of Al/(GF+CF) composite materials. The fact that the solid-liquid phase sintering generates a decrease of the in-plane CTE and an increase of the through-plane CTE (in average), compared to the PM route shows that the physical properties of the Al based composites in two direction are linked. This behavior is in accordance with the Firkowska formulation (eq. 1). Indeed, the reduction of the in-plane CTE of Al/50%vol.GF composites with the solid-liquid phase sintering (from 21.8×10 -6 /K to 19×10 -6 /K) induces a decrease of Δαx (eq. 2) and in-plane strain ε11= ε22. Then, the through-plane CTE of GF embedded in the matrix (α33 eq. ( 1)) will increase. This explains, therefore, the increase of the through-plane CTE of Al/GF composites with the solid-liquid phase sintering (from -10.5×10 -6 /K to -4.1×10 -6 /K). With the addition of small amount of CF, the CTE value of (Al+Al-Si)/C composite fabricated by solid-liquid phase sintering can be controlled in order to obtain a CTE compatible to the ceramic domain (2×10 - 6 /K to 8×10 -6 /K) in the in-plane direction or through-plane direction (Fig. 9a and9d) without affecting significantly the in-plane TC of the composite materials (Fig. 7c).

Specific thermal conductivity of Al/(GF+CF) composite materials

Figure 11 shows the specific TC of Metal/C (with a reinforcement content close to 50 vol.%) composites materials in function of CTE reported in the literature. Thanks to the low density of Al/50vol.%C (2.44 g/cm 3 ) and a high TC (411 W/m.K), our MMC shows a high specific TC even higher than Cu/GF composite materials. Moreover Al/(GF+CF) shows a low CTE which make these composite materials compatible with electronic applications. In the ceramic domain, only Mg-Ca/GF composite shows a higher specific TC than our work thanks to the lightweight of the matrix.

Figure 11: Specific TC of (Al90vol.%+Al-Si10vol.%)/(GF+CF) (red square) compared to the Metal/C composite materials reported in the literature

Conclusion

The addition of 50 vol.% of GF into the Al matrix improves remarkably the in-plane TC (400 W/m.K). However, these materials show a through-plane CTE (-10.5×10 /K, respectively). The GF, in addition to increase the TC of the matrix, allows to decrease the mechanical percolation threshold of carbon fibers and then, provide the formation of a continuous and rigid (GF+CF) network that enhance the decrease of in-plane CTE.

In the through-plane direction, it has been shown that Al/(40vol.%GF+10vol.%CF) composite exhibits a CTE of 17.6×10 -6 /K which is significantly higher than an Al/40%GF (-6.4×10 -6 /K) composite and lower than Al/10vol.%CF (33×10 -6 /K) composite. Recognizing that CF allows to increase the through-plane CTE, the GF/CF ratio was optimized for a total reinforcement of 50 vol% in order to control the value of the CTE without decrease significantly the in-plane TC of the composite materials.

Solid-liquid phase sintering was used to fabricate Al/C composites. The amount of liquid phase alloy was optimized for a carbon content of 50 vol.%. The addition of 10 vol.% of Al-12wt%Si alloy to the matrix of Al/C composite materials induces:

 The increase of the relative densification of composite materials for reinforcement content of 50 vol.%.

 The increase of the in-plane TC of Al/GF composite from 400 to 428 W/m.K.

 The decrease of the in-plane CTE and the increase of through-plane CTE of Al/C composites materials.

 The presence of liquid phase in the carbon-carbon contact points leads to the formation of a solid CF+GF network.

Finally, two sets of (Al90vol.%+Al-Si10vol.%)/50%Carbone composite materials have been developed and optimized:

 High in-plane TC (363 W/m.K) and low in-plane CTE (6.3×10 -6 /K) In-plane GF -1 1109 0.12 [34] Through-plane GF -58* 38.7 0.01 [27]*; [34] Al 25 69 0.345 [25] *Value calculated with Firkowska formulation [27]

Figure 1 :

 1 Figure 1: SEM micrographs of starting materials: (a) Al powder, (b) Al-Si powder, (c) Graphite flakes (d) Carbon fibers

Figure 2 :

 2 Figure 2: Illustration of the solid-liquid phase sintering process. The initial powder mixture (Al+Al-Si+C) is sintered under 60 MPa at 600°C which result in the fusion of the Al-12wt%Si alloy.

Figure 3 :

 3 Figure 3: SEM micrographs (in-plane view) of (a) Al/50vol%GF, (b) Al/(40vol.%GF+10vol.%CF) and (c) Al/50vol%CF composite materials

Figure 4 :

 4 Figure 4: X-Ray tomography analysis of: (a), (b), and (c) Al/50vol.%GF; (d), (e), (f,) and (j) and Al/(40vol.%GF+10vol.%CF); (g), (h,) (i) and (k) Al/50vol.%CF composite materials. (Yellow arrows show the pressure direction)

Fig. 5 .

 5 b shows the microstructure of Al/CF composites fabricated solid-liquid phase sintering respectively. It shows the presence of white area in the (Al+Al-Si)/50vol.%CF, more precisely in the Al matrix, Al-CF interface and CF/CF contact points. These white parts were not observed in composites fabricated by conventional PM route. The spot 1 and 2 correspond to the aluminum matrix and carbon fiber, respectively. The spots scan 3, 4 and 5 reveals the presence of aluminum and silicon in the white area, which corresponds to the Al-12wt%Si phase. Then, EDS scan confirms the infiltration of the liquid phase in the porosity between Al particles, Al-CF interphase and CF/CF contact point during the sintering process.

Figure 5 :

 5 Figure 5: EDS spot scan analysis of (Al+Al-Si)/50vol.%CF (a and b) composite material

Figure 6 :

 6 Figure 6: Comparison of experimental TC of Al/GF (red square) and Al/CF (black circle) composite materials in the through-plane (dotted line) and in-plane (line) direction (standard deviation 5%)

Figure 7 :

 7 Figure 7: Experimental CT of (Al90vol.%+Al-Si10vol.%)/(GF+CF) (red triangle) and Al/(GF+CF) (black triangle) composite materials in (a and c) the in-plane direction and (b) through-plane (b) direction (standard deviation 5%)

Figure 8 : 3 . 5 .Figure 9

 8359 Figure 8: TEM micrographs of a) Al/CF composite material and b) al-Al-Si/CF composite material

  ) 𝛼 33 = 𝑑𝜀 33 𝑑𝑇 = 𝛼 𝑔𝑟,3 + 𝑣 2𝐷 ∆𝛼 𝑋 -𝑑𝑣 2𝐷 𝑑𝑇 𝜀 11 0 where ε33 and T are the through-plane strain of the graphite and the temperature, respectively. αgr,3 is the intrinsic through-plane CTE of graphite (28×10 -6 /K). v2D and 𝑑𝑣 2𝐷 𝑑𝑇 = -4.3 × 10 -2 /𝐾 are the 2D equivalent Poisson ratio of graphite and its temperature derivative. 𝜀 11 0 ~-10 -3 is the estimated residual in-plane strain of graphite in Copper matrix. is the temperature derivative of the in-plane strain ε11= ε22 in graphite and it is calculated using equation (2) 𝑑𝜀 11 𝑑𝑇 = ∆𝛼 𝑋 = 𝛼 𝑋 -𝛼 𝑔𝑟,1 where αx and αgr,3 are the in-plane CTE of Cu/50vol%GF and GF, respectively. The equation (1) shows that the change of CTE value (from 28×10 -6 /K to -26×10 -6 /K) is due to the large Poisson ratio of graphite (v2D) and its strong temperature dependence.

Figure 9 :

 9 Figure 9: Comparison of experimental CTE of Al/GF (red square) and Al/CF (black circle) composite materials in the through-plane (dotted line) and in-plane (line) direction. Theoretical predictions by the Schapery model (blue dashed line) for Al/GF system in the through-plane direction (standard deviation 3%)

Figure 10 :Figure 10

 1010 Figure 10: Experimental CTE of (Al90vol.%+Al-Si10vol.%)/(GF+CF) (red triangle) and Al/(GF+CF) (black triangle) composite materials in (a and c) the in-plane direction and (b and c) the through-plane direction (standard deviation 5%)

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 12345 Figure 1: SEM micrographs of starting materials: (a) Al powder, (b) Al-Si powder, (c) Graphite flakes (d) Carbon fibers

Figure 6 :Figure 7 :

 67 Figure 6: Comparison of experimental TC of Al/GF (red square) and Al/CF (black circle) composite materials in the through-plane (dotted line) and in-plane (line) direction

Figure 8 :Figure 9 :Figure 10 :Figure 11 :

 891011 Figure 8: TEM micrographs of a) Al/CF composite material and b) al-Al-Si/CF composite material

Table 1 :

 1 Influence of the Al-12wt%Si liquid content (vol.%) on the relative density (%) and the in-plane TC

	(W/m.K)
	Liquid phase content (vol.%)	0	2.5	5	7.5	10	12.5	15
	Relative density (%)	97.5	97.9	98.1	98.3	98.2	98.3	98.5
	In-plane TC (W/m.K)	400	404	425	423	428	414	409

Table 2 :

 2 Materials parameters for calculating the theoretical CTE for Al/GF composite materials

	Material	α (10 -6 /K)	E (GPa)	v	Ref.

  -6 /K) and in-plane CTE (21.8×10 -6 /K) incompatible with ceramic domain. Carbon fibers were added to Al/GF composite in order to increase the through-plane CTE and to decrease the in-plane CTE of Al/GF composites materials. It has been shown that by addition of (40vol.%GF+10vol.%CF) to the Al matrix, induces a decrease of the in-plane CTE to 9.2×10 -6 /K which is significantly lower than an Al/10vol.%CF and Al/40vol.%GF composite materials (18.9×10 -6 /K and 22.9×10-6

Table 1 :

 1 Influence of the Al-12wt%Si liquid content (vol.%) on the relative density (%) and the in-plane TC

	(W/m.K)
	Liquid phase content (vol.%)	0	2.5	5	7.5	10	12.5	15
	Relative density (%)	97.5	97.9	98.1	98.3	98.2	98.3	98.5
	In-plane TC (W/m.K)	400	404	425	423	428	414	409

Table 2 :

 2 Materials parameters for calculating the theoretical CTE for Al/GF composite materials

	Material	α (10 -6 /K)	E (GPa)	v	Ref.

Acknowledgement

The authors thank the University of Bordeaux (2015-FD-24) for financial support.

 High in-plane TC (411 W/m.K) and low through-plane CTE (2.4×10 -6 /K) Therefore, with their remarkable specific TC (0.168 W.m 2 /Kg.K), the composite materials developed in this study are applicable as a lightweight heat sink in microelectronic industries.


M Vernet

S Ducrot

L Leibnitz

& Chaix

Vision, Attention and Reading : Identifying preschool children at risk for reading difficulties

Introduction Conclusion and perspectives

Using a longitudinal study, the aim was to determine whether it is possible to predict the future reading skills in kindergarten children from oculomotor skills. (Richman and Garzia, 1987 ;Richman, 2015) Aim of the study

Method Results

DEM-test

MATERIAL

DEM-test (Richman and Garzia, 1987 ;Richman, 2015) ❖ Oculomotor skills and naming number skills Alouette-R (Lefavrais, 1967(Lefavrais, , 2005) ) ❖ Reading skills Speeded naming subtest -NEPSY-II (Korkman, Kirk and Kemp, 2012) ❖ Naming skills Alouette-R (Lefavrais, 1967)

PROCEDURE

Oculomotor skills in reading

To read, eyes have to make saccades and fixations to overcome its anatomo-functional constraints (i.e., decreased visual acuity with eccentricity and crowding effect; Leibnitz & al., 2016). Attention plays a crucial role in overcoming these constraints by : Rayner, 1979;Optimal Viewing Position;O'Regan & al., 1984) ❖ Decrease in the number of regression saccade. This improvement in oculomotor capacity is related to an increase of the accuracy and the speed of reading. Therefore, the oculomotor skills directly affect the reading skills.

What about learning ?

It is well known that reading instruction requires efficient phonological skills. However, many studies have shown the major involvement of visual-attentional and perceptual-motor skills in reading skills. Longitudinal studies have shown the involvement of these skills as necessary preliteracy skills from kindergarten on the future reading skills of these children (Franceschini & al., 2012;Bellocchi & al., 2017).

Setting

  PARTICIPANTS 30 children from two different French public schools→ Normal or corrected-to-normal vision

  OF THE 1ST GRADEEarly identification and care for future reading difficulties in preschool children

❖

  Making the letters prominent in foveal vision ❖ Attenuating information in parafoveal vision ❖ Participating in ocular guidance Speeded naming -NEPSY-II (Korkman, Kirk and Kemp, 2012) DEM-test Children had to name the numbers, first on the vertical boards and then on the horizontal number board. Mesures Vertical time (VT) Horizontal time (HT) Errors (E) Ratio (R) → HT/VT DEM-test Children had to name the numbers, first on the vertical boards and then on the horizontal number board. Speeded naming Participants had to name the shape, color and size of items and from 7 years old, children also had to name a sheets of numbers and letters presented in lines. With the training, oculomotor skills become more efficient : ❖ Increase in the landing position accuracy of the saccade towards the left-center of the word (Preferred Viewing Location;

  up oculomotor training at the school to promote learning to read Development of remediation tools for reading difficulties 6th conference of the ARN -Neuroeducation: a new science for school? -June 07

LINK BETWEEN OCULOMOTOR SKILLS IN KINDERGARTEN AND READING SKILLS IN GRADE 1 CASE STUDY

* : p<.05 ; ** : p<.01 ; *** : p<.001

Results of linear regression analysis

Reading accuracy according to the DEM-test measurements 1 The Developmental Test of Visual Perception -Second Edition (Hammill, Pearson & Voress, 1993) : it measures visual perception and visual-motor integration skills. We identified 4 children with difficulties with at least one DEM-test result below the -1.65 standard deviation threshold. We examined their profile more precisely from DEM-test, Alouette-R, speeded naming test and DTVP-2 data.

GVP = General Visual

1

The reading accuracy deficit of children 1 and 2 is related to ➢ Oculomotor deficit outcomes.

➢ Other measures are not affected for any children.

 Oculomotor skills influence reading accuracy. Therefore, the reading difficulties can be predicted by the DEM-test.

-6,00 -5,00

-4,00 -3,00 -2,00 -1,00


Jérôme Bolte
email: jerome.bolte@ut-capitole.fr

Lilian Glaudin
email: lilian@glaudin.net

Edouard Pauwels
email: edouard.pauwels@irit.fr

Mathieu Serrurier
email: mathieu.serrurier@irit.fr

A Hölderian backtracking method for min-max and min-min problems

We present a new algorithm to solve min-max or min-min problems out of the convex world. We use rigidity assumptions, ubiquitous in learning, making our method applicable to many optimization problems. Our approach takes advantage of hidden regularity properties and allows us to devise a simple algorithm of ridge type. An original feature of our method is to come with automatic step size adaptation which departs from the usual overly cautious backtracking methods. In a general framework, we provide convergence theoretical guarantees and rates. We apply our findings on simple GAN problems obtaining promising numerical results.

Introduction

Adversarial learning, introduced in [START_REF] Goodfellow | Generative adversarial nets[END_REF], see also [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF], calls for the development of algorithms addressing large scale, smooth problems of the type

min x∈R d max y∈Y L(x, y), (1.1)
where Y is a constraint set, and L is a given cost function. This structure happens to be ubiquitous in optimization and game theory, but generally under assumptions that are not those met in learning. In optimization it stems from the Lagrangian approach and duality theory, see e.g., [START_REF] Boyd | Convex Optimization[END_REF][START_REF] Bertsekas | Constrained optimization and Lagrange multiplier methods[END_REF][START_REF] Combettes | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF], while in game theory it comes from zero-sum 2-players games, see e.g., [START_REF] Von | Zur theorie der gesellschaftsspiele[END_REF][START_REF] Von | On Rings of Operators. Reduction Theory[END_REF][START_REF] Laraki | Mathematical foundations of game theory[END_REF]. Dynamics for addressing (1.1) have thus naturally two types. They may be built on strategic considerations, so that algorithms correspond to a sequence of actions chosen by antagonistic players, see [START_REF] Laraki | Mathematical foundations of game theory[END_REF] and references therein. In general these methods are not favorable to optimization because the contradictory interests of players induce oscillations and slowness in the identification of optimal strategies. Optimization algorithms seem more interesting for our purposes because they focus on the final result, i.e., finding an optimal choice x, regardless of the adversarial strategy issues. In that respect, there are two possibilities: the variational inequality approach which treat minimization and maximization variables on an equal footing, see e.g. [START_REF] Korpelevich | An Extragradient Method for Finding Saddle Points and for Other Problems[END_REF][START_REF] Nemirovski | Prox-Method with Rate of Convergence o(1/t) ) for Variational Inequalities with Lipschitz Continuous Monotone Operators and Smooth Convex-Concave Saddle Point Problems[END_REF][START_REF] Combettes | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF] or [START_REF] Mertikopoulos | Optimistic mirror descent in saddle-point problems: Going the extra(-gradient) mile[END_REF][START_REF] Hsieh | On the convergence of single-call stochastic extra-gradient methods[END_REF][START_REF] Gauthier Gidel | A variational inequality perspective on generative adversarial networks[END_REF] in learning.

On the other hand, some methods break this symmetry, as primal or augmented Lagrangian methods. In those, a large number of explicit steps, implicit steps, or global minimization are performed on one variable while the other is updated much more cautiously in an explicit incremental way, see e.g., [START_REF] Bertsekas | Constrained optimization and Lagrange multiplier methods[END_REF][START_REF] Sabach | Lagrangian methods for composite optimization[END_REF].

Our work is written in this spirit: we assume that the under-max argument is tractable with a good precision, and we construct our algorithm on the following model:

y n = argmax y∈Y L(x n , y), x n+1 = x n -γ n ∇ x L(x n , y n ), γ n > 0, n ≥ 0.
As explained above, the rationale is not new 2 , and is akin to many methods in the literature on learning where the global optimization is performed approximately by multiple gradient steps [START_REF] Maher Nouiehed | Solving a class of non-convex min-max games using iterative first order methods[END_REF][START_REF] Maher Nouiehed | Solving a class of non-convex min-max games using iterative first order methods[END_REF] or by clever revision steps, as in the "follow the ridge" method, see [START_REF] Wang | Towards A Unified Min-Max Framework for Adversarial Exploration and Robustness[END_REF].

Backtrack Hölder What is new then? The surprising fact is that we can provide theoretical grounds to devise large steps and thus obtain aggressive learning rates with few assumptions. This is done by exploiting some hidden properties of the value function g = max y L(•, y) under widespread rigidity assumptions. Let us sketch the ideas of our approach. First, under a uniqueness assumption on the maximizer, our method appears to be a gradient method on the value function for "player I" ( "the generator" of GANs)

x n+1 = x n -γ n ∇g(x n ).
Secondly, we use the fact that g has a locally Hölderian gradient 3 whenever L is semialgebraic or analytic-like, a situation which covers most of the problems met in practice. With such observations, we may then develop automatic learning rate strategies and a diagonal backtracking method, that we call "Backtrack Hölder methods for min-max".

Contributions

• We provide new algorithms whose steps are tuned automatically: Backtrack Hölder gradient and Backtrack Hölder for min-max methods.

• Our algorithms are shown to perform with O( -[2+c] ) rate, where c is a cost incurred by the diagonal backtracking process (which is negligible in practice), and to provide general convergence guarantees to points (x * , y * ) satisfying y * = argmax y L(x * , y) and ∇ x L(x * , y * ) = 0. This is done within a fairly general framework, since L is merely assumed semi-algebraic while the "best response" of player II is only required to be singled-valued.

• A byproduct of our work is a global convergence result for Hölder methods, which were earlier investigated in the literature [START_REF] Berger | On the Quality of First-Order Approximation of Functions with Hölder Continuous Gradient[END_REF][START_REF] Geovani | Tensor Methods for Minimizing Functions with Hölder Continuous Higher-Order Derivatives[END_REF][START_REF] Nesterov | Universal gradient methods for convex optimization problems[END_REF][START_REF] Yashtini | On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients[END_REF].

• Our work is theoretical in essence. It is merely a first step towards more involved research, regarding the effect of nonsmoothness or stochastic subsampling. We propose however numerical experiments on learning problems. First on the "Sinkhorn GANs", [START_REF] Genevay | Learning generative models with sinkhorn divergences[END_REF][START_REF] Genevay | Gan and vae from an optimal transport point of view[END_REF], which rely on optimal transport losses regularized through the addition of an entropic term, and second on Wasserstein GANs [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF] which are a natural extensions of GANs [START_REF] Goodfellow | Generative adversarial nets[END_REF].

Gradient algorithms à la Hölder

Our results often use semi-algebraic assumptions which are pervasive in optimization and machine learning, see e.g. [START_REF] Castera | An inertial newton algorithm for deep learning[END_REF] and references therein.

Our method and proofs are presented in view of solving min-max problems, but the techniques are identical for the min-min case. R d , R d are endowed with their Euclidean structure.

Framework: a single valued best response and semi-algebraicity

Let Y ⊂ R d be a nonempty closed semi-algebraic set, see Definition A.1 in Appendix. The first part of the assumption is satisfied whenever L(x, •) is strictly concave, see e.g. [START_REF] Lin | On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems[END_REF]. Note also that if L(x, •) is concave, as in a dual optimal transport formulation, some regularization techniques can be used to obtain uniqueness and preserve semi-algebraicity, see e.g., [START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transport[END_REF]. As for the H2 continuity assumption, it is much less stringent than it may look: Combining these assumptions with Tarski-Seidenberg theorem and the properties of semialgebraic functions [START_REF] Bochnak | Géométrie algébrique réelle[END_REF], we obtain the following.

Properties of the value function and its best response

Assumption 2.1 (Standing assumptions) L is a C 1 semi-algebraic function on R d × R d such that (x, y) → ∇ x L(x, y) is jointly continuous. Furthermore, for any compact sets K 1 ⊂ R d and K 2 ⊂ R d , there exist β 1 , β 2 ∈ ]0, +∞[ such that, ∀x 1 , x 2 ∈ K 1 , ∀y 1 , y 2 ∈ K 2 , ∇ x L(x 1 , y 1 ) -∇ x L(x 2 , y 2 ) ≤ β 1 x 1 -x 2 + β 2 y 1 -y 2 . ( 2
Proposition 2.
= R + , L(x, y) = xy -1 3 y 3 , then g(x) = max y∈R + L(x, y) = 2 3 x 3/2 with ∇g(x) = √ x.
Comments and rationale of the method At this stage the principles of our strategy can be made precise. We deal with problems which are dissymetric in structure: the argmax is easily computable or approximable while the block involving the minimizing variable is difficult to handle. This suggests to proceed as follows: one computes a best response mapping (even approximately), the gradient of the value function becomes accessible via formula (i) in Proposition 2.4, and thus a descent step can be taken. The questions are: which steps are acceptable? Can they be tuned automatically? This is the object of the next sections.

Gradient descent for nonconvex functions with globally Hölderian gradient

The results of this section are self contained. We consider first the ideal case of a gradient method on a globally Hölder function with known constants, see e.g. [START_REF] Nesterov | Universal gradient methods for convex optimization problems[END_REF][START_REF] Yashtini | On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients[END_REF]. We study Algorithm 2, previously presented in [START_REF] Yashtini | On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients[END_REF] for which we prove sequential convergence.

Assumption 2.6 (Global Hölder regularity)

f : R d → R is C 1 , semi-algebraic and ∀x 1 , x 2 ∈ R d , ∇f (x 1 ) -∇f (x 2 ) ≤ β x 1 -x 2 ν , with β > 0, ν ∈]0, 1]. (2.2)
Algorithm 1: Hölder gradient method Input: ν ∈ ]0, 1], β ∈ ]0, +∞[ and γ ∈]0, (ν + 1)/β[ Initialization :

x 0 ∈ R d for n = 0, 1, . . . do γ n (x n ) = γ ν+1 β 1/ν-1 ∇f (x n ) 1/ν-1 x n+1 = x n -γ n (x n )∇f (x n ) Proposition 2.
(i) the sequence (f (x n )) n∈N is nonincreasing and converges, (ii) the sequence (x n ) n∈N converges to a critical point x * ∈ R d of f , i.e., ∇f (x * ) = 0, (iii) for every n ∈ N 4 , min 0≤k≤n ∇f (x k ) 1 ν +1 ≤   f (x 0 ) -f (x * ) γ -γ ν+1 β ν+1 ν β ν + 1 1-ν ν   1 n + 1 = O 1 n . Choosing γ = ν+1 β 1 ν+1 1/ν , we obtain min 0≤k≤n ∇f (x k ) 1 ν +1 ≤ (f (x) -f (x * ))β 1/ν (ν + 1) ν(n + 1
) .

The "Backtrack Hölder" gradient algorithm and diagonal backtracking

In practice, the constants are unknown and the Hölderian properties are merely local. The algorithm we present now (Algorithm 2), is in the spirit of the classical backtracking method, see e.g., [START_REF] Bertsekas | Constrained optimization and Lagrange multiplier methods[END_REF]. The major difference is that we devise a diagonal backtracking, to detect both constants β, ν at once in a single searching pass.

Assumption 2.8 f : R d → R is a C 1 semi-algebraic function such that ∇f is locally Hölder.

In the following algorithm, α, γ > 0 are step length parameters, δ > 0 is a sufficient-decrease threshold and ρ > 0 balances the search between the unknown exponent ν and the unknown multiplicative constant β, see Assumption 2.6.

The following theorem provides convergence guarantees under local Hölder continuity (Assumption 2.8 for Algorithm 2).

Theorem 2.9 (Convergence of Backtrack Hölder for nonconvex functions) Under Assumption 2.8, consider a bounded sequence (x n ) n∈N generated by Algorithm 2. Then the following hold:

(i) (γ n ) n∈N is well defined, (ii) the sequence (f (x n )) n∈N is nonincreasing and converges, (iii) there exists x * ∈ R d such that x n → x * and ∇f (x * ) = 0, Algorithm 2: Backtrack Hölder gradient method Input: δ, α ∈ ]0, 1[ and γ, ρ ∈ ]0, +∞[ Initialization : x 0 ∈ R d , k -1 = 0 for n = 0, 1, . . . do k = k n-1 γ n (x n ) = α k min{1, ∇f (x n ) ρk }γ while f (x n -γ n (x n )∇f (x n )) > f (x n ) -δγ n (x n ) ∇f (x n ) 2 do k = k + 1 γ n (x n ) = α k min{1, ∇f (x n ) ρk }γ k n = k x n+1 = x n -γ n (x n )∇f (x n ) (iv) the while loop has a uniform finite bound k := sup n∈N k n < +∞. Moreover min 0≤i≤n ∇f (x i ) = O 1 n 1 2+ρ k , (v) suppose moreover that there exist β ∈ ]0, +∞[ and ν ∈]0, 1] such that ∇f is globally (β, ν) Hölder. Then sup n∈N k n ≤ 1 + 1 ν max    log (1-δ)(ν+1) γ ν β log(α) , 1 -ν ρ    . (2.3)

Remark 2.10 (Diagonal backtracking alternatives and comments)

In the previous theorem, we ask (k n ) n∈N to be a nondecreasing sequence and (2.3) is actually a bound on the total number of additional calls to the function in the while loop. In practice, this approach might be too conservative and other strategies may provide much more aggressive steps at the cost of additional calls to the function. We will use two variations to update (k n ) n∈N :

• Initialize k to 0 for fine tuning to the price of longer inner loops (see Algorithm 8).

• For some iterations, decrease the value of k by 1 (see Algorithm 5 for example).

The parameters α, γ, δ, and ρ are made to tune finely the number of iterations spent on estimating local Hölder constants.

In Theorem 2.9(v), we need to suppose that the gradient is globally Hölder contrary to Assumption 2.8.

Backtrack Hölder for min-max problems

Gathering the results provided in Section 2.3, we provide here our main algorithm (Algorithm 3).

Several comments are in order:

-The above contains an inner loop whose overhead cost becomes negligible as n → ∞, this allows one for automatic step size tuning. The form of Algorithm 3 is slightly different from Algorithm 2 to avoid duplicate calls to the max-oracle required both to compute gradients and evaluate functions.

-As described in Remark 2.10, the backtracking strategy is one among others and it is adaptable to different settings. In this min-max case, the cost of the max-oracle may have some impact: either it is costly and extra-flexibility is needed or it is cheap and it can be kept as is. Two examples are provided in Sections 4.1 and 4.2.

-A direct modification of the above method, provides also an algorithm for min

Initialization : x 0 ∈ R d , y 0 = arg max z∈Y L(x 0 , z), k -1 = 0 for n = 0, 1, . . . do k = k n-1 γ n (x n ) = γα k min{1, ∇ x L(x n , y n ) ρk } x = x n -γ n (x n )∇ x L(x n , y n ) y = arg max z∈Y L(x, z) while L(x, y) > L(x n , y n ) -δγ n (x n ) ∇ x L(x n , y n ) 2 do k = k + 1 γ n (x n ) = γα k min{1, ∇ x L(x n , y n ) ρk } x = x n -γ n (x n )∇ x L(x n , y n ) y = arg max z∈X L(x, z) k n = k, x n+1 = x, y n+1 = y.
-Algorithm 3 is a model algorithm corresponding to a monotone backtracking approach (i.e., the sequence (k n ) is nondecreasing), but many other variants are possible, see Appendix D. Algorithms 5 and 6 are for the min-min problem, with a non monotone backtracking and the same guarantees. A heuristic version is also considered: it is Algorithm 8 where an approximation of the argmax is used.

To benchmark our algorithms, we compare them to Algorithms 4, 6, and 7 in Appendix D, with constant but finely tuned step sizes or with Armijo search. (iii) Suppose that there exist β ∈ ]0, +∞[ and ν ∈]0, 1] such that ∇g is (β, ν) Hölder everywhere. Then the cost of the while loop is bounded by

sup n∈N k n ≤ 1 + 1 ν max    log (1-δ)(ν+1) γ ν β log(α) , 1 -ν ρ    . (3.2)
Remark 3.2 In [18, Proposition 2], the authors mention an algorithm akin to what we proposed, but without backtracking. They insist on the fact that if one had access to the maxoracle, then one would be able to implement a gradient descent by using "sufficiently small updates". Our theoretical results are an answer to this comment as we offer a quantitative characterization of how small the step should be, as well as a backtracking estimation technique.

Numerical experiments

We compare our method with constant step size algorithm and Armijo backtracking for the Generative Adversarial Network (GAN) problem, first using Sinkhorn divergences and second considering Wasserstein adversarial networks. Data lie in R d = R 2 , the sample size is N = 1024 and we consider x 1 , . . . , x N ∈ R d a fixed sample from a distribution P d , which is a Gaussian mixture, see Figure 1, and z 1 , . . . , z N ∈ Z a fixed sample from latent distribution We consider as generator G, a dense neural network with three hidden layers containing respectively 64, 32, 16 neurons with a ReLU activation between each layer. We write G : Z × Θ G → X , with inputs in Z and parameters θ G ∈ Θ G where Θ G = R q with q the total number of parameters of the network (2834 in our case).

P z = U ([0, 1] × [0, 1]), uniform on Z, where Z = [0, 1] × [0, 1].

Sinkhorn GAN

We first consider training generative network using Sinkhorn divergences as proposed in [START_REF] Genevay | Learning generative models with sinkhorn divergences[END_REF]. This is a min-min problem which satisfies Assumption 2.2 (see also the remark in Equation 3.1). Sinkhorn algorithm [START_REF] Sinkhorn | A relationship between arbitrary positive matrices and doubly stochastic matrices[END_REF][START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transport[END_REF] allows us to compute a very precise approximation of the min-oracle required by our algorithm, we use it as an exact estimate. Note that the transport plan for the Sinkhorn divergence is regularized by an entropy term whence the inner minimization problem has a unique solution and the corresponding p is continuous. This is a perfect example to illustrate our ideas. Consider the following probability measures

μ = 1 N N i=1 δ xi , (empirical target distribution), µ(θ G ) = 1 N N i=1 δ G(zi,θ G ) ,
(empirical generator distribution).

We then define the Sinkhorn divergence between these two distributions.

W (μ, µ(θ G )) = min

P ∈R N ×N +    Tr(P C(θ G ) T ) + N i,j=1 P ij log(P ij ) ; P 1 N = 1 N , P T 1 N = 1 N    where > 0 is a regularization parameter, C(θ G ) = [ G(z i , θ G ) -x j ] i,j ∈ R N ×N
is the pairwise distance matrix between target observations (x i ) 1≤i≤N and generated observations (G(z i , θ G )) 1≤i≤N . Here Tr is the trace, and 1 N is the all-ones vector. The optimum is unique thanks to the entropic regularization and the optimal transportation plan P can be efficiently estimated with an arbitrary precision by Sinkhorn algorithm [START_REF] Sinkhorn | A relationship between arbitrary positive matrices and doubly stochastic matrices[END_REF][START_REF] Cuturi | Sinkhorn distances: Lightspeed computation of optimal transport[END_REF].

Training our generative network amounts to solving the following min-min problem

min θ G W (μ, µ(θ G )).

Remark 4.1 (Global subanalyticity ensures Łojasiewicz inequality)

The cost function of the Sinkhorn GAN problem is not semi-algebraic due to log. However we never use the logarithm in a neighborhood of 0 during the optimization process because of its infinite slope. Hence the loss can actually be seen as globally subanalytic. Whence p, g are globally subanalytic and the Łojasiewicz inequality as well as Hölder properties still hold, see [START_REF] Bochnak | Géométrie algébrique réelle[END_REF][START_REF] Attouch | Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the kurdyka-łojasiewicz inequality[END_REF][START_REF] Bolte | Nonconvex lagrangian-based optimization: monitoring schemes and global convergence[END_REF] for more on this.

Algorithmic strategies for Sinkhorn GAN

The monotone diagonal backtracking is too conservative for this case, so we use a variant described in Algorithm 5 instead. At each step, the idea is to try to decrease k of 1 whenever possible, keeping some sufficient-decrease property valid. Otherwise k is increased as in the monotone method, until sufficient decrease of the value is ensured. This approach is particularly adapted to the Sinkhorn case, because estimating the best response is cheap.

Note that, to propose a fair comparison and keep the same complexity between algorithms, we count each call to the min-oracle, both in the outer and in the inner while loop, as an iteration step. The parameters used in this experiment for Algorithm 5 are γ = 1, α = 0.5, δ = 0.25, δ + = 0.95, and ρ = 0.5. We compare with Algorithm 4 presented in Appendix D, which is a constant step size variant, we try with different step size parameters γ ∈ {0.01, 0.05, 0.1}. We compare with the standard Armijo backtracking algorithm (see Algorithm 6 in Appendix D) which uses a similar approach as in Algorithm 5 to tune the step size parameter γ n , but does not take advantage of the Hölder property. All algorithms are initialized randomly with the same seed. We observe on the left part of Figure 2 that both Hölder and Armijo backtracking provide decreasing sequence and avoid oscillations. Both algorithms converge faster than the constant step size variant. Furthermore, since our algorithm can take into account the norm of the gradient, the number of intern loop is smaller and that explain why the Non Monotone Hölder backtracking is faster.

Wasserstein GAN

We treat the Wasserstein GAN (WGAN) heuristically with an approximation of the max-oracle and use Algorithm 8 in Appendix D which matches this setting.

Consider a second neural network, called discriminator, D : R d × Θ D → R with inputs in R d and parameters θ D ∈ Θ D whose architecture is the same as G (i.e., Θ D = Θ G ) but with a fullsort activation between each layer, see [START_REF] Anil | Sorting out Lipschitz function approximation[END_REF]. We consider the following problem

min θ G max θ D n i=1 D(x i , θ D ) - n j=1 D(G(z j , θ G ), θ D ).
In order to implement the analogy with Kantorovitch duality in the context of GANs [START_REF] Arjovsky | Wasserstein generative adversarial networks[END_REF], one has to ensure that the discriminator D is 1-Lipschitz, when seen as a function of its input neurons. This is enforced using a specific architecture for the discrimintator network D. We use Bjork orthonormalization and fullsort activation functions [START_REF] Anil | Sorting out Lipschitz function approximation[END_REF] which ensure that the network is 1-Lipschitz without any restriction on its weight parameters θ D .

For this problem, we use Algorithm 8, which is a heuristic modification of our method designed to deal with the inner max. Both the argmax and max are indeed approximated by gradient ascent. Algorithm 8 then implements the same bactracking idea which is evaluated on the same to benchmark as in the previous section. Extra discussions are provided in the Appendix. Doing so, the extra-cost induced by the while loop becomes negligible and we can find the optimal value of k by exhaustive search. For this reason, in this heuristic context, Hölder backtracking schemes have very little advantage compared to Armijo and we do not report comparison. Detailed investigations for large scale networks is postponed to future research. Since GAN's training is delicate in practice [START_REF] Gulrajani | Improved training of wasserstein GANs[END_REF], we provide comparison with many step size choices for the constant step size algorithm. Algorithm 7 that the difference between Armijo method and ours As for Backtrack Hölder min-max, we use parameters γ = 1, δ = 0.75, α = 0.75, and ρ = 0.20 for the Hölder backtracking algorithm and constant step size parameter γ ∈ {0.001, 0.01, 0.05, 0.1, 0.5} for the constant step size variant. All algorithms are initialized randomly with the same seed.

Figure 2 displays our results on the right. The optimal loss equals 0. One observes that constant large steps are extremely oscillatory while small steps are stable but extremely slow. Backtrack Hölder takes the best of the two world, oscillates much less and stabilizes closer to the optimal loss value compared to constant step size variants.

Broader impact

The authors think that this work is essentially theoretical and that this section does not apply.

Acknowledgements.

A Extra-material

Definition A.1 (Semi-algebraic sets and functions) (i) A subset S of R m is a real semi-algebraic set if there exist r and s two positive integers, and, for every k ∈ {1, . . . , r}, l ∈ {1, . . . , s}, two real polynomial functions P kl , Q kl : R m → R such that

S = r k=1 s l=1 {x ∈ R m : P kl (x) < 0, Q kl (x) = 0} (ii) A function f : A ⊂ R m → R n is semi-algebraic if its graph (x, λ) ∈ A × R n f (x) = λ is a semi-algebraic subset of R m+n .
For illustrations of this notion in large scale optimization and machine learning we refer to [START_REF] Attouch | Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the kurdyka-łojasiewicz inequality[END_REF][START_REF] Castera | An inertial newton algorithm for deep learning[END_REF].

One will also find in this work references, definitions and examples of globally subanalytic sets that are necessary for our proofs on Sinkhorn GANs.

B Proofs

Proof of Proposition 2.3 Let us proceed with the case when Y is compact; the other case is similar. Let (x n ) n∈N be a sequence such that x n → x * ∈ R d . We need to prove that p(x n ) → p(x * ). For every n ∈ N, set y n = p(x n ) and since Y is compact, let y * ∈ Y a cluster point of (y n ) n∈N . Since g and L are continuous we have g(

x n ) → g(x * ) and L(x n k , y n k ) → L(x * , y * ). Since p(x n k ) = y n k one has L(x n k , y n k ) ≥ L(x, y n k ) for all x in R d .
Thus at the limit L(x * , y * ) ≤ L(x, y * ) for all x in R d . This implies that L(x * , y * ) = g(x * ), and so, by uniquennes of the argmax, p(x * ) = y * . Whence p is continuous.

Proof of Proposition 2.4 (i)

This is a consequence of [START_REF] Rockafellar | Variational Analysis[END_REF]Theorem 10.31].

Proof of Proposition 2.4 (ii)

According to the definition of a semi-agebraic function, we need to prove that their graph is semi-algebraic.

For g: the

epi g = (x, ξ) ∈ R d × R g(x) ≤ ξ = (x, ξ) ∈ R d × R (∀y ∈ Y) L(x, y) ≤ ξ and its complement set is (x, ξ) ∈ R d × R (∃ y ∈ Y) L(x, y) > ξ which is the projection of (x, ξ, y) ∈ R d × R × R d L(x, y) > ξ R d × R × Y.
As a conclusion it is semi-algebraic by Tarski-Seidenberg principle. The same being true for the hypograph, g is semi-algebraic.

For p: graph p = (x, y) ∈ R d × Y (∀y ∈ Y) L(x, y) ≥ L(x, y ) . Then graph p is defined from a first-order formula and the conclusion follows from [13, Theorem 2.6].

Proof of Proposition 2.4 (iii)

Using Assumption 2.2 H2 and (ii), p is continuous and semi-algebraic so using Proposition C.1, p is locally Hölder. Similarly Assumption 2.2 H2, (i) and (ii) ensure that ∇g is also continuous ans semi-algebraic and the result follows again from Proposition C.1.

Proof of Proposition 2.7

Let n ∈ N and set d n = ∇f (x n ) . For the clarity of the proof, the dependence of γ n in x n is dropped.

Lemma C.3 with U = R d provides f (x n+1 ) ≤ f (x n ) + d n , x n+1 -x n + β ν + 1 x n+1 -x n ν+1 ≤ f (x n ) - 1 γ n x n+1 -x n 2 + β ν + 1 x n+1 -x n ν+1 = f (x n ) - 1 γ n x n+1 -x n 2 - βγ n ν + 1 x n+1 -x n ν+1 . (B.1)
By definition of γ n we have

γ 1/ν n = γ ν + 1 β 1/ν-1 x n+1 -x n 1/ν-1
and thus

γ n = γ ν ν + 1 β 1-ν x n+1 -x n 1-ν . Set δ = 1 -γ ν β ν+1 ν
. Since γ < ν+1 β by hypothesis in Algorithm 1, we have δ > 0 and we deduce from (B.1) that

f (x n+1 ) ≤ f (x n ) - 1 γ n x n+1 -x n 2 -γ ν β ν + 1 ν x n+1 -x n 2 = f (x n ) - 1 -γ ν β ν+1 ν γ n x n+1 -x n 2 = f (x n ) - δ γ n x n+1 -x n 2 . (B.2)
Hence (f (x n )) n∈N is nonincreasing and since (x n ) n∈N is bounded, (f (x n )) n∈N is also bounded and converges, this proves (i). Since, for all n, ∇f (x n ) = x n+1 -x n /γ n and (x n ) n∈N bounded we can apply Theorem C.5 and obtain that x n → x * ∈ R d . Finally, it follows from (B.2) that x n+1 -x n → 0 and that x n+1 -

x n = γ n ∇f (x n ) = γ ν+1 β 1/ν-1 ∇f (x n ) 1/ν → 0.
Hence x * is a critical point, which proves (ii).

For n fixed, we have

δ γ n x n+1 -x n 2 = δγ n ∇f (x n ) 2 = δγ ν + 1 β 1 ν -1 ∇f (x n ) 1 ν +1 .
Then it follows from (B.2) that ∇f (x n )

1 ν +1 ≤ 1 δγ β ν + 1 1-ν ν (f (x n ) -f (x n+1 )) (B.3) whence (n + 1) min k=0,...,n ∇f (x k ) 1 ν +1 ≤ n k=0 ∇f (x k ) 1 ν +1 ≤ 1 γ -γ ν+1 β ν+1 ν β ν + 1 1-ν ν (f (x 0 ) -f (x * )). Choosing γ = ν+1 β 1 ν+1 1/ν , we obtain γ -γ ν+1 β ν + 1 ν = ν + 1 β 1 ν + 1 1/ν - ν + 1 β 1 ν + 1 1/ν ν+1 β ν + 1 ν = ν + 1 β 1 ν + 1 1/ν 1 - 1 1 + ν = ν + 1 β 1 ν + 1 1/ν ν 1 + ν .
from which we deduce

(n + 1) min 0≤k≤n ∇f (x k ) 1 ν +1 ≤ (f (x) -f (x * ))(ν + 1) ν β ν+1 1-ν ν ν+1 β 1 ν+1 1/ν = (f (x) -f (x * ))β 1/ν ν + 1 ν ,
which proves (iii).

Proof of Theorem 2.9 (i) : For every n ∈ N, by using Taylor expansion, the test implies

(∀γ ∈ ]0, +∞[) -γ ∇f (x n ) 2 + o(γ) ≤ -δγ ∇f (x n ) 2 , thus (∀γ ∈ ]0, +∞[) o(γ) γ ≤ (1 -δ) ∇f (x n ) 2
which is satisfied for γ small. (ii): It follows from Algorithm 2 that for every n ∈ N,

f (x n+1 ) ≤ f (x n ) - δ γ n x n+1 -x n 2 (B.4)
so the descent property holds.

(iii): One has ∇f (x n ) ≤ 1 γn x n+1 -x n . Since (x n ) n∈N is bounded, we conclude by Theorem C.5(ii) that there exists x * ∈ R d such that x n → x * . This follows directly from (ii) and (B.4).

(iv): Since f is locally Hölder, there exist U ⊂ R d , a convex neighborhood of x * , ν ∈ ]0, 1], and β ∈ ]0, +∞[ such that ∇f is (β, ν) Hölder on U and (x n ) n≥N remains in U for N sufficiently large.

Fix any

K ∈ N such that K ≥ max    log (1-δ)(ν+1) γ ν β log(α)ν , 1 -ν ρν    (B.5)
then we also have

α K ≤ 1 γ (1 -δ)(ν + 1) β 1/ν and ρK ≥ 1 ν -1.
We deduce that for any x ∈ U such that x -λ∇f (x) ∈ U ,

λ ν := α Kν min{1, ∇f (x) ρKν }γ ν ≤ (1 -δ)(ν + 1) β min{1, ∇f (x) ρKν } ≤ (1 -δ)(ν + 1) β min{1, ∇f (x) 1-ν }. (B.6)
We derive from Lemma C.3 and (B.6) that for any

x ∈ U f (x -λ∇f (x)) ≤ f (x) -λ ∇f (x) 2 + β ν + 1 λ ν+1 ∇f (x) ν+1 ≤ f (x) -λ ∇f (x) 2 + βλ ν + 1 (1 -δ)(ν + 1) β min{1, ∇f (x) 1-ν } ∇f (x) ν+1 . Since min{1, ∇f (x) 1-ν } ∇f (x) ν+1 ≤ ∇f (x) 2 , we have for all x ∈ U such that x - λ∇f (x) ∈ U , f (x -λ∇f (x)) ≤ f (x) -λ ∇f (x) 2 + (1 -δ)λ ∇f (x) 2 = f (x) -δλ ∇f (x) 2 . (B.7)
Fix any N 0 ∈ N large enough such that x N ∈ U for all N ≥ N 0 . Suppose that K = k N0 satisfies (B.5), then for all N ≥ N 0 we may consider equation (B.7) with x = x N , λ = γ N , noting that x N +1 = x N -γ N ∇f (x N ) ∈ U . This is exactly the negation of the condition to enter the while loop of Algorithm 2. Hence, by a simple recursion, the algorithm never enters the while loop after step N 0 and we have k N = k N0 for all N ≥ N 0 . On the other hand, if k N0 does not satisfy (B.5), then since k is incremented by 1 at each execution of the while loop, using the fact that (B.5) implies (B.7), it must hold that

k N ≤ 1 + max    log (1-δ)(ν+1) γ ν β log(α)ν , 1 -ν ρν    .
In all cases, we have using monotonicity of k N in N that for all N ∈ N,

k N ≤ 1 + max    k N0 , log (1-δ)(ν+1) γ ν β log(α)ν , 1 -ν ρν    , (B.8) hence (k n ) n∈N is bounded.
Now we use (B.4) and (ii) which ensures that

x n+1 -x n 2 γ n = γ n ∇f (x n ) 2
is summable and thus tends to 0 as n → ∞, whence, either (γ n ) n∈N or ( ∇f (x n ) 2 ) n∈N tends to zero. Using the fact that (k n ) n∈N is bounded, in any case we have, ∇f (x n ) → 0 as n → ∞.

It follows for n large enough that ∇f (x n ) ≤ 1, from the while loop condition and the fact that k := sup n∈N k n < +∞, that

δα k ∇f (x n ) 2+ρ kγ ≤ δα kn ∇f (x n ) 2+ρkn γ = δγ n (x n ) ∇f (x n ) 2 ≤ f (x n ) -f (x n+1 ).
Using the convergence of (f (x n )) n∈N , and by summing the previous equation and taking the minimum, we obtain that min 0≤i≤n ∇f

(x i ) 2+ρ k = O(1/n).
(v): The result follows from (B.8) with k N0 = 0, since in this case the same reasoning can be applied for all N ∈ N with U = R d .

Proof of Theorem 3.1 Recall that g(•) = max y∈Y L(•, y) and p(•) = argmax y∈Y L(•, y). It follows from Proposition 2.4 that for every n ∈ N, ∇ x L(x n , y n ) = ∇g(x n ). We derive from Proposition (iii) that ∇g is locally Hölder. It turns out that Algorithm 3 applied to L is the same as Algorithm 2 applied to g. Thus Theorem 2.9 ensures the convergence of (x n ) n∈N to a critical point x * ∈ R d of g. Furthermore, it follows from the continuity of p that y n → y * = p(x * ). We conclude that (x n , y n ) n∈N converges to a critical point of L, satisfying y * = argmax y∈Y L(x * , y). Finally, since for every n ∈ N, ∇g(x n ) = ∇ x L(x n , y n ), we conclude by Theorem 2.9(v).

C Lemmas

Proposition C.1 (Continuity and semi-algebraicity implies Hölder continuity) [START_REF] Bochnak | Géométrie algébrique réelle[END_REF] Let f : R d → R d be a semi-algebraic continuous function, then f is locally Hölder, i.e., for all compact set K, ∃ β ∈ ]0, +∞[ , ∃ ν ∈ ]0, 1] , ∀x, y ∈ K, f (x) -f (y) ≤ β x -y ν .

We recall below the Łojasiewicz inequality, see e.g [START_REF] Kurdyka | On gradients of functions definable in o-minimal structures[END_REF] and references therein. Remark C.6 (Convergence and semi-algebraicity) (a) Note that when f is semi-algebraic, we have in fact an alternative, for any sequence:

• either x n → ∞

• or (x n ) n∈N converges to a critical point x * .

Indeed if we are not in the diverging case, there is a cluster point x * which must be a critical point. Whence we are in the situation of (ii) above. (b) If f is, in addition, coercive, i.e., lim x →+∞ f (x) = +∞, each Hölder gradient sequence converges to a critical point since the first alternative is not possible because (f (x k )) k∈N is non increasing so that (x k ) k∈N is bounded.

D Numerical Experiments : Complements

In practice, it can be difficult to calculate the argmax (or the argmin) or to perform rigorously the internal while loop, we propose two algorithms to simplify this implementation aspect. We also present the constant step size algorithm that we use to assess the efficiency of our method.

. 1 )Assumption 2 . 2 (

 122 Borrowing the terminology from game theory, one defines the value function as g(•) = max y∈Y L(•, y) and the best response mapping p(•) = argmax y∈Y L(•, y) for x ∈ R d . Well posedness) H1. p(x) is nonempty and single valued for every x ∈ R d , H2. p is continuous.

Algorithm 3 :

 3 x∈R d min y∈Y L(x, y). (3.1) Monotone Backtrack Hölder for min-max Input: δ, α ∈ ]0, 1[ and γ, ρ ∈ ]0, +∞[

Theorem 3 . 1 (

 31 Backtrack Hölder for min-max) Under Assumptions 2.1 and 2.2, consider the sequences (x n ) n∈N and (y n ) n∈N generated by Algorithm 3. Suppose that (x n ) n∈N is bounded. Then (i) The while loop has a uniform bound, i.e., sup n∈N k n < +∞.(ii) (x n ) n∈N converges to x * in R d and (y n ) n∈N converges to y * ∈ Y, with ∇ x L(x * , y * ) = 0 and y * = argmax y∈Y L(x * , y).

Figure 1 :

 1 Figure 1: Data distribution x 1 , . . . , x N

Figure 2 :

 2 Figure 2: Left: Sinkhorn loss with respect to number of Sinkhorn max-oracle evaluation for different gradient step rules. The x axis accounts for all oracle calls, not only the ones used to actually perform gradient steps. Right: WGAN loss with respect to iteration number for different gradient step rules.

Definition C. 2 (

 2 Łojasiewicz inequality) A differentiable function f : R n → R has the Łojasiewicz property at x * ∈ R n if there exist η, C ∈ ]0, +∞[ and θ ∈ ]0, 1[ such that for all x ∈ B(x * , η), the following inequality is satisfiedC|f (x) -f (x * )| θ ≤ ∇f (x) .In this case the set B(x * , η) is called a Łojasiewicz ball.Proof.(i): By assumption, we have x N ∈ B(x * , η/2) and|f (x N ) -f (x * )| 1-θ < δC(1 -θ)η/2. It follows from Lemma C.4 that δC(1 -θ) x N +1 -x N ≤ (f (x N ) -f (x * )) 1-θ -(f (x N +1 ) -f (x * )) 1-θ .Let us prove by strong induction that for every n ≥ N , x n ∈ B(x * , η). Assume n ≥ N + 1 and suppose that for every integerN ≤ k ≤ n -1, x k ∈ B(x * , η). Lemma C.4 yields (∀k ∈ [N, . . . , n -1]) δC(1 -θ) x k+1 -x k ≤ (f (x k ) -f (x * )) 1-θ -(f (x k+1 ) -f (x * )) 1-θ .By summing we haveδC(1 -θ) n-1 k=N x k+1 -x k ≤ (f (x N ) -f (x * )) 1-θ -(f (x n ) -f (x * )) 1-θ ≤ (f (x N ) -f (x * )) 1-θ < δC(1 -θ)η/2. (C.1) Since x n -x * ≤ n-1 k=N x k+1 -x k + x N -x * < η/2 + η/2 = ηwe have x n ∈ B(x * , η). We have proved that for every n ≥ N , x n ∈ B(x * , η).(ii): Since (f (x n )) n∈N is nonincreasing by [a], we deduce that f (x n ) → f (x * ) and f (x n ) ≥ f (x * ). Since f is semi-algebraic, fhas the Łojasiewicz property at x * [23]. Hence, let us define θ, C, and η as in Definition C.2 relative to x * . Since x * is a cluster point and f (x n ) → f (x * ), there exists N as in (i) above. For every integer n ≥ N , it follows from (C.1) that δC(1 -θ) n-1 k=N x k+1 -x k ≤ (f (x N ) -f (x * )) 1-θ ≤ (f (x 0 ) -f (x * )) 1-θ < +∞ hence the serie converges, increments are summable and (x k ) k∈N converges to x * .

3 (Continuity of

	p) Suppose that Assumption 2.1 and Assumption 2.2-H1 are
	satisfied, and that either Y is compact, or p is bounded on bounded sets. Then the best response
	p is a continuous function, that is Assumption 2.2-H2 is fulfilled.

Proposition 2.4 (Properties of p and

	g) Suppose that Assumption 2.1 and Assumption 2.2
	are satisfied. Then
	(i) g is differentiable and for all x ∈ R d , ∇g(x) = ∇ x L(x, p(x)),
	(ii) both the value function g and the best response p are semi-algebraic,
	(iii) the gradient of the value function, ∇g, is locally Hölder.
	Remark 2.5 Consider L(x, y) = xy, Y = [-1, 1], one sees that g(x) = max y∈[-1,1] L(x, y) =
	|x| while p(x) = sign x if x = 0 and p(0) = [-1, 1]. This shows that Assumption 2.2 is
	a necessary assumption for g to be differentiable. One cannot hope in general for ∇g
	to be locally Lipschitz continuous. For instance set Y

7 (Convergence of the Hölder gradient method for nonconvex functions)

	Under Assumption 2.6, consider a bounded sequence (x n ) n∈N generated by Algorithm 1. Then
	the following hold:

  The authors acknowledge the support of ANR-3IA Artificial and Natural Intelligence Toulouse Institute. JB and EP thank Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant numbers FA9550-19-1-7026, FA9550-18-1-0226, and ANR MasDol. JB acknowledges the support of ANR Chess, grant ANR-17-EURE-0010 and ANR OMS.

It can be traced back to the origin of augmented Lagrangian methods, see e.g.,[START_REF] Tyrrell | Proximal subgradients, marginal values, and augmented lagrangians in nonconvex optimization[END_REF]

Recall that G : R d → R d is locally Hölderian if for all bounded subset V ⊂ R d , there exists β and ν positive such that G(x) -G(y) ≤ β x -y ν , whenever x, y ∈ V .

This result is essentially present in[START_REF] Yashtini | On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients[END_REF]

Lemma C.3 (Hölder Descent Lemma)

 [38, Lemma 1]Let U ⊂ X be a nonempty convex set, let f : R d → R be a C 1 function, let ν ∈]0, 1], and let β ∈ ]0, +∞[. Suppose that ∀(x, y) ∈ U 2 , ∇f (x) -∇f (y) ≤ β x -y ν .

Then ∀(x, y) ∈ U 2 , f (x) ≤ f (y) + ∇f (x), x -y + β ν + 1 y -x ν+1 .

Lemma C.4 (Controlled descent) Let δ, θ ∈ ]0, 1[, let C ∈ ]0, +∞[ and let x, y, x * ∈ R d Suppose that the following hold:

Proof. First, if y = x, then the inequality holds trivially. Second, if f (x) = f (x * ), then by the first two items, y = x and the inequality holds also. Second, also. Hence we may suppose that f (x) -f (x * ) > 0 and y = x. We have

By concavity of s → s 1-θ , we have

which concludes the proof.

We slightly adapt the recipe from [START_REF] Bolte | Proximal alternating linearized minimization for nonconvex and nonsmooth problems[END_REF] and add a trap argument from [START_REF] Attouch | Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods[END_REF].

Theorem C.5 (Recipe for convergence [START_REF] Bolte | Proximal alternating linearized minimization for nonconvex and nonsmooth problems[END_REF] and the trapping phenomenon) Let

Then the following results hold:

We also present an Armijo search process for this problem in Algorithm 6. It has a structure similar to the "Non Monotone Hölder Backtrack" but with a much less clever update for γ n .

Algorithm 6: Non Monotone Armijo for min-max

D.2 Wasserstein GAN

As explained in Section 4.2, this problem does not formally match our setting. In particular, the argmax cannot be computed fast, so we use a gradient ascent to provide an approximation expressed by using the sign ≈. We also provide a constant step size method (Algorithm 7) to benchmark our algorithm. Algorithm 7: Heuristic gradient method for min-max with constant step size

Besides, since the max is not easily accessible, we modify the while loop by using y n instead of the exact argmax to validate the sufficient decrease. This approach gives Algorithm 8.