Gwenaël Le Rhun
email: gwenael.le-rhun@cea.fr

Franklin Pavageau

Christel Dieppedale

New challenges for piezoelectric MEMS

Lead zirconate titanate, Pb(Zr,Ti)O3 (PZT), has been the most studied piezoelectric thin film for MEMS actuators the last two decades due to its high transverse piezoelectric coefficient d31. The PZT technology is now available in several MEMS foundries and PZT thin film actuators are already integrated in commercialized devices like inkjet printheads, micro-speakers and autofocus camera. Emerging applications such as micro-mirror for LIDAR, haptics for humanmachine interface, micro-pumps, as well as PMUT for fingerprint, medical probe or gesture recognition are also based on thin film PZT actuators. The potential of use of PiezoMEMS is thus expanding, but it may require to functionalize non silicon substrate (Glass, Flex, Metal,…), large surface (smart windows), curved surface (haptics). In some cases also, transparency is requested.

 or transparent PZT membranes [3].


Keywords: silent lms, gender roles, role reversal, servicemen, female empowerment

"Men are disarmed, women, indispensable. " This quote from Joanna Bourke's article on gender roles in The Cambridge History of the First World War enlightens the confusing and challenging relationships established during wartime. The two American short lms, Love and Duty (1916) and A.W. O.L (1919), could offer a relevant illustration of the new social organization deriving from the con ict and heralding women's agency. If none of these lms has been directed by a woman and seem to deal with women during the Great War, they inadvertently do so despite their male-centered plot illustrating the misadventures of American soldiers when trained or demobilized. Spoo ng in ated heroism in romantic melodramas, Love and Duty is a one-reel comedy from the Plump and Runt series produced Blackhawk Films in Jacksonville, Florida. As Plump (Oliver Hardy) and Runt (Billy Ruge) presumably serve as reservists, they are trained in a bucolic camp even if America has not entered the war yet. This slapstick piece may exemplify the paci st stance adopted by Hollywood in the early years of the Great War, as Americans dreaded taking part in an international con ict which they considered alien to their interests. A.W.O.L (All Wrong Old Ladiebbuck) is a one-reel propaganda cartoon commissioned by the U.S. Army Signal Corps at the very end of the war to encourage the soldiers to wait patiently for the general demobilization after the 1918 Armistice. It was designed by Charles Bowers, one of the pioneers of animation lms, to warn the impatient soldiers too eager to get back home of the risks of going A.W.O.L, absent without of cial leave. Both lms reveal the presence of women around servicemen and how the war paradoxically separated and brought together combatants and non-combatants. These lms presents a non-scary and distant vision of the war -an anticipation of the con ict and an illustration of its aftermath, picturing not so much the ac- Love and Duty (1916) Love and Duty (1916) Love and Duty offers a grotesque gallery of un t servicemen, as the regiment on screen seems to gather the rejects of the US army. Plump and Runt, the title characters, are too stout or too short to t the standards of military manliness introduced by recruiters and military boards. Lieutenant Runt tries to ease his frustration by bullying Private Plump with nonsensical and useless exercises and maneuvers which cause the poor lad to gesticulate in an absurd manner and sweat like a pig as he juggles with his rif e. Runt spents his time yelling stupid orders at Plump who cannot do as told and almost faints with exhaustion. Along with this antithetical pair, other peculiar soldiers form the battalion: a group of idle simpletons and an over-aged bugler. Instead of getting ready for war, they just chat with the local women, relax in the countryside, and watch passively Plump being harassed by Runt. They look like a regiment of outcasts who may have been gathered because of the call for preparedness of some Americans who considered the US would ght the war sooner or later.Yet the war don't seem to be on the mind of these soldiers and they show no sense of professionalism or military duty. The of cer in charge, Colonel Tracy (Burt Tracy), is busier trying to convince his daughter (Florence McLaughlin) not to fall for Plump than training properly his men, and the soldiers are mainly interested in wooing women, especially Plump who only cares about escaping to meet his sweetheart (Ray Godfrey). A.W.O.L. opens on the conversation of a group of soldiers who are desperate to leave Europe and want to return home. As the ghting is over, they are bored to death, complaining about the slowness of demobilization and waiting idle. One of them is more vocal about his discontent while the others are coping with the situation and he decides to escape the barracks. Even if they are still wearing their uniforms, they are not of any use to their country because of their apathy. Because of his impatience, the noncompliant private decides to run away with a lady of pleasure who shows up outside the camp to invite young men for a joy ride. The cartoon presents US soldiers as dispirited and even de ant to Uncle Sam's orders. The men are indeed disarmed, physically and morally, in these lms because they are not granted any strength or will power which could enable them to match the archetypal standards of the ideal warrior. Before and after the war their lack of motivation to serve their country or obey the rules prevents them from becoming heroes in these two short programs. As a result, we may wonder if women become heroines despites themselves in Love and Duty and A.W.O.L.?

"AWOL" (1919 silent lm short) "AWOL" (1919 silent lm short)

Female agency is a common trait in both productions. In Love and Duty, women are granted the power to galvanize the regiment, give men orders and most importantly to decide for themselves. Plump's sweetheart keeps the morale of the troops high as she entertains the men with her cheeky behavior and she is not scared to defy Runt by throwing a stone at him to make him stop molesting Plump. When she meets with her beau, she is the one bringing owers to him, thus reversing the codes of romance by leading the seduction instead of being courted. She is the one kissing Plump and, when Runt sends a soldier to interrupt their tender embrace, she knocks the soldier off with her bouquet in order to continue doing what seems more important to her. The regiment watches the scene and laughs out loud, enjoying this impudent display of de ance. The sweetheart has a rival in the camp: the Colonel's daughter. The latter is also in love with Plump, to her father's and Runt's dismay because the two men planned on marrying her to the tiny Lieutenant. Challenging the authority of her father, she refuses to change her mind and disparages Runt's exuberant courtship. Instead, she prefers chasing Plump around the camp, also transforming the traditional codes of seduction as she openly shows her desires and behaves like a suitor. When she discovers she is not the target of Plump's affection, her features are distorted with jealousy and fury. Her power over men is fueled by her rage against the couple and she starts ordering soldiers around in order to separate the lovebirds. The Colonel's daughter uses Runt to have him send Plump to the guard house and she commands the soldiers to expel the sweetheart from the military camp: "Take her off". Despite her condition as a woman, she is given some of cial power over the servicemen and bosses them around as she pleases. At her command, the soldiers even point their ri es at Plump's lady friend when they oust her out of the camp. While her father is never shown leading his men, the daughter behaves as if https://artswarandpeace.univ-paris-diderot.fr/2022/07/10/film-review-fearing-empowered-womanhood-during-world-war-one/ 4/6

she was the real commanding of cer of the camp and soldiers salute her with deference. Despite her misuse of the power transferred to her thanks to her father's rank, she is presented as the one ruling the regiment. Her ruthless decision is not challenged by men but by the sweetheart who works out a clever plan to get Plump out of his prison cell. Only another woman is capable of countering the supremacy of the Colonel's daughter and save her captive lover by pretending his baby needs to visit him and bringing tools in the cell. The comical exaggeration may show the ladies as a dictator in petticoats and a cunning intriguer, yet it cannot be denied that they are conferred a form of superiorityeven open control in the case of the daughter -over men in a military environment.

In A.W.O.L, the portrayal of women is far less attering because the female character, "Miss AWOL"as she introduces herself, is dangerous, a loose woman who lures the poor soldiers to their doom. However, she embodies the liberty and authority women gained during the war. With her short bobbed hair and her sophisticated out t and make up, she may be a apper challenging traditional gendered behavior. Seeking fun right after the war, she shows up at the window of the barracks and calls out soldiers to join her for a joy ride in her fancy car, a hardly covered allusion to sexual intercourses. The cartoon criticizes brazen women who chased young men in uniforms but it also reveals new trends giving women more opportunities to become autonomous. Miss AWOL initiates the encounter with the soldier who does not behave like an audacious man but rather like an embarrassed maiden when he is approached by the undaunted seducer. Uncomfortable and impressed, he makes funny faces and wipes his sweaty palms on his trousers while she seems perfectly at ease and tries to convince him by shooting: "Oh do, come on, we'll have a bully time". He obeys her command to loosen up and have fun and jumps into the back seat of her car. His position at the rear of the car is signi cant of his subordinate role while the woman takes on the leading role and decides for what they should do and where they should go. He depends on her for getting away from his military barracks and he is submitted to her whims. The joy ride ends up being less joyful than expected because of her reckless driving and her desire to maintain him at her disposal: when he tries to move to the passenger seat in the front he pushed to the back seat, reminded where he should stand. The car almost falls from a cliff and he is completely panicked. After they manage to escape the peril, the apper orders him to repair a at tire but he makes a fool of himself as he has no talent as a mechanic and make the tire explode. He is knocked out by the explosion but she insists on continuing the crazy ride until they get arrested by a police of cer and are taken to Judge Gloom. The soldier pays the ne but his lady companion refuses to stop and abducts him for a new ride before she nds another admirer she fancies more. Miss AWOL appears to be authoritative and quite manipulative and the soldiers, probably bewitched by her interest in them, are presented as submissive and quite impotent, disarmed because of the trauma of the war and overwhelmed by the social transformations generated by the con ict.

The cartoon offers a negative portrayal of women after the war changed their looks and behaviors because they had been called to endorse more active roles in society as bread-winners, political protesters, helpers in the combats zones and even ghters sometimes. Despite the cynical denunciation of the wildness of Miss AWOL, the spectator is presented a shift of power and how women became indispensable because they were turned into decision-makers while men seemed overtaken by events. In the cartoon, the female character is obviously a temptress driving the soldier away from his military duty and causing his fellow soldiers to turn their backs on him, but the moral lesson of the Love and Duty and A.W.O.L are rather unknown World War One lms and have been often left aside by scholarly works analyzing anti-war lms or propaganda productions. Both the comedy and the cautionary cartoon are worth of interest for their comparison of male distress and female empowerment in a context of exacerbated military masculinity. Even when the lms disapprove of the attitudes and choices of women, they interpret a period of transition during which women took charge of their own destiny and of men's fate. Love and Duty announces already the challenges to male command witnessed in the post-war world presented in A.W.O.L., in which new gender relationships are tensed and often contested. After the war, most men longed for a return to the normality they once knew -in which they held the upper hand. These lms show that that mythical normality is a remain of the past and that women cannot resume a life guided by the by-gone principles motherhood and home-life after they have been given emotional, sexual, professional and political independence. LES COMMENTAIRES SONT FERMÉS.

[1]

  EMPOWERED WOMANHOOD DURING WORLD WAR ONE? 10 JUILLET 2022 | ARTS WARPEACE | UN COMMENTAIRE Arts of War and Peace tual military ghting but what Ford Maddox Ford called "life as a perpetual sex-battle" (The Good Soldier: A Tale of Passion, 1915) in which women have the upper hand over men, thus turning males into the weaker sex.

  Fearing empowered Womanhood during World War One? | Arts of War and Peace https://artswarandpeace.univ-paris-diderot.fr/2022/07/10/film-review-fearing-empowered-womanhood-during-world-war-one/ 3/6

Films

  A.W.O.L. Directed by Charles Bowers. United States, 1919.Love and Duty. Directed by Will Louis. United Joanna Bourke, "Gender roles in killing zones", in Jay Winter (ed.), The Cambridge History of the First World War: Volume III, Civil Society(Cambridge: Cambridge University Press, 2014), 153. Clémentine Tholas is an associate professor of American Studies at Sorbonne Nouvelle University. She is a silent lm scholar, with a special interest in World War One lms. She co-edited the volumes Humor, Entertainment and Popular Culture during WWI (NY: Palgrave Macmillan, 2015), and New Perspectives on the War Film (NY: Palgrave Macmillan, 2019). She also produced a pedagogical documentary on sexual violence and harassment in French higher education, entitled Briser le silence des amphis (2022). UNE RÉFLEXION SUR « FILM REVIEW: FEARING EMPOWERED WOMANHOOD DURING WORLD WAR ONE? » Ping : Women's War Writing | Arts of War and Peace

  lm incriminates the private who ends up locked in the guard house while his brothers in arms are sent back to the U.S. and Miss AWOL happily runs free with another boyfriend.

	05/12/2022 13:04	Film Review: Fearing empowered Womanhood during World War One? | Arts of War and Peace

https://artswarandpeace.univ-paris-diderot.fr/2022/07/10/film-review-fearing-empowered-womanhood-during-world-war-one/ 5/6

empowered-womanhood-during-world-war-one/ 1/6 Love and Duty (1916) and A.W.O.L. (1919


Anthony Bardou
email: anthony.bardou@ens-lyon.fr

Thomas Begin
email: thomas.begin@ens-lyon.fr

INSPIRE: Distributed Bayesian Optimization for ImproviNg SPatIal REuse in Dense WLANs

Keywords: Networks → Wireless local area networks, Network performance analysis, • Theory of computation → Bayesian analysis, Reinforcement learning, Gaussian processes, Distributed algorithms Machine Learning, Gaussian Process, Spatial Reuse, IEEE 802.11, Power Control

WLANs, which have overtaken wired networks to become the primary means of connecting devices to the Internet, are prone to performance issues due to the scarcity of space in the radio spectrum. As a response, IEEE 802.11ax and subsequent amendments aim at increasing the spatial reuse of a radio channel by allowing the dynamic update of two key parameters in wireless transmission: the transmission power (TX_POWER) and the sensitivity threshold (OBSS_PD). In this paper, we present INSPIRE, a distributed online learning solution performing local Bayesian optimizations based on Gaussian processes to improve the spatial reuse in WLANs. INSPIRE makes no explicit assumptions about the topology of WLANs and favors altruistic behaviors of the access points, leading them to find adequate configurations of their TX_POWER and OBSS_PD parameters for the "greater good" of the WLANs. We demonstrate the superiority of INSPIRE over other state-of-the-art strategies using the ns-3 simulator and two examples inspired by real-life deployments of dense WLANs. Our results show that, in only a few seconds, INSPIRE is able to drastically increase the quality of service of operational WLANs by improving their fairness and throughput.

CCS CONCEPTS

•

INTRODUCTION

Since their introduction in the late 1990s, WLANs (Wireless Local Area Networks) have rapidly overtaken wired networks to become the primary means of connecting devices to the Internet. According to Cisco [START_REF]Cisco visual networking index: Forecast and trends, 2017-2022 white paper[END_REF], they will account for 57% of the Internet traffic in 2022, compared to 22% and 21% for mobile and wired networks, respectively. The current WLAN architecture is defined by the IEEE standard 802.11 (commercially known as Wi-Fi). APs (Access Points) are the centerpiece of this setup, serving as relays for wireless devices; we refer to the latter as STAs (Stations) throughout this paper. Typically, each AP is equipped with a wired interface giving access to the LAN and then the Internet as well as a wireless interface providing connectivity to nearby STAs through radio communications.

Space on the radio spectrum is a scarce resource as it is often shared by multiple WLANs. The radio bands used by the IEEE 802.11 standard (currently 2.4 and 5 GHz, soon to be joined by 6 GHz) are divided into channels. Different APs can then be assigned to different, orthogonal channels enabling the APs to transmit at the same time without interfering with each other. Equally important thanks to the limited radio range of radio waves, APs configured on the same radio channel can transmit concurrently provided that they are sufficiently far away from each other. This ability was central to the success of WLANs and it is commonly known as the spatial reuse of radio channels.

However, the spatial reuse of radio channels as performed by today's WLANs may be reaching its limit. This is particularly true in places where WLAN deployments are very dense, such as offices, shopping malls and train stations. This is because, in these areas the distance between APs is small, so that an AP is more likely to be blocked by the transmissions of one or several nearby APs operating on the same channel. This will in turn take a hefty toll on the WLANs' performance.

A solution to this issue can be found in the 2021 amendment to 802.11 known as 802.11ax [START_REF]IEEE Standard for Information Technology-Telecommunications and Information Exchange between Systems Local and Metropolitan Area Networks-Specific Requirements Part 11: Wireless LAN[END_REF], which enables the dynamic configuration of two key parameters at each AP: TX_PWR and OBSS_PD. The former parameter specifies the power level (in dBm) at which the AP transmits its data. The latter parameter defines the sensitivity threshold (in dBm). If the energy received is below this level, this indicates to the AP that the radio channel is clear and thus available for transmission. Otherwise, the AP must defer its transmissions. While prior amendments to 802.11 held these TX_PWR and OBSS_PD parameters constant (typically 20 dBm and -82 dBm respectively), 802.11ax has made them dynamic with their values spanning from 1 to 21 dBm for the former and from -82 to -62 dBm for the latter. Adjusting the configurations of TX_PWR and OBSS_PD can help overcome the limitations of spatial reuse in dense environments by allowing APs that are close to each other to transmit on the same channel. Figure 1 depicts a simple example of two APs operating on the same radio channel and illustrates how different configurations of the TX_PWR and OBSS_PD parameters can lead to different performance.

Despite the potential of 802.11ax to improve the spatial reuse of radio channels, finding an adequate configuration of TX_PWR and OBSS_PD for the APs in a WLAN is a complex problem. First, an adequate configuration is very topology-specific. In other words, knowing a suitable configuration for a given scenario is of no value for another scenario. Second, a distributed solution would be more appreciated than a centralized solution. Not only does this avoid the search in an otherwise very high dimensional space but this (a) With the default configuration of TX_PWR and OBSS_PD, the two APs are within each other's detection range so that they cannot transmit simultaneously.

(b) The value of TX_PWR is reduced on both AP so that they do not belong to each other's detection range. Under this configuration, concurrent transmissions from the two AP may occur at the same time.

Figure 1: Adequately configuring the TX_PWR parameter of APs can significantly improve the spatial reuse of radio channels in WLANs. Note that concurrent transmissions of the two APs could also be attained by increasing OBSS_PD at each AP. While similar, reducing TX_PWR and increasing OBSS_PD may affect the WLANs' performance differently (see Table 2 of [START_REF] Wilhelmi | Potential and pitfalls of Multi-Armed Bandits for decentralized Spatial Reuse in WLANs[END_REF] for more details).

avoids the assumption of having a centralized entity (e.g., a controller) deciding the configurations of all APs. This assumption is acceptable if all the interfering APs belong to the same WLAN but unrealistic if they belong to concurrent WLANs. Third, forecasting the performance of WLANs with an analytical model that can subsequently help establish "optimal" configurations is difficult. The degree of details in the models will be either too coarse and thus inapplicable, or adequate but unscalable when scenarios involve multiple APs and STAs. A solution to this is that APs can apply new configurations of their parameters, measure the effect of these changes on their performance, and exchange their experience with surrounding APs. This paves the way for the use of online and reinforcement learning techniques in a distributed manner.

In a previous work [START_REF] Bardou | Improving the Spatial Reuse in IEEE 802.11ax WLANs: A Multi-Armed Bandit Approach[END_REF], we introduced a centralized solution based on the sampling of Gaussian mixtures and Thompson sampling to improve the spatial reuse of WLANs by adequately configuring TX_PWR and OBSS_PD parameters at each AP. In this paper, we present a distinct contribution named INSPIRE, which is a distributed online learning solution performing local Bayesian optimizations based on GPs (Gaussian Processes) to address the issue of improving the spatial reuse of WLANs. INSPIRE makes no explicit assumptions about the topology of WLANs or the radio environments and thus can apply to any WLANs. Additionally, it can operate even when the APs to be configured belong to different concurrent WLANs. The contributions of this paper are as follows:

• We demonstrate the ability of GPs at approximating the reward function, which reflects the performance of WLANs, and at exploring efficient AP configurations; • We establish the superiority of a divide-and-conquer approach to handle the complex problem of setting the TX_PWR and OBSS_PD parameters at each AP; • We introduce INSPIRE a distributed solution that lets the APs of concurrent WLANs automatically adapt their internal parameters' setting in their own interest as well as in the interest of obtaining a more efficient spatial reuse of radio channels;

• We evaluate the efficiency of INSPIRE on real-life inspired case studies using a detailed network discrete-event simulator and compare its performance with several state-of-the-art solutions (centralized or distributed).

The remainder of this paper is organized as follows. The next section discusses the related work. Section 3 describes the proposed strategy and the main theoretical results obtained to address the issue of spatial reuse of a radio channel in WLANs. Its performance are evaluated in Section 4.2. Section 5 concludes this paper.

RELATED WORK

The release of the IEEE 802.11ax amendment [START_REF]IEEE Standard for Information Technology-Telecommunications and Information Exchange between Systems -Local and Metropolitan Area Networks-Specific Requirements -Part 11: Wireless LAN[END_REF] in late 2021 marks a new era for the spatial reuse of radio channels of WLANs: Nodes can dynamically adjust their transmission power (TX_PWR) and sensitivity threshold (OBSS_PD) parameters. For a detailed explanation of how this new feature is implemented, we refer the interested reader to [START_REF] Wilhelmi | Spatial Reuse in IEEE 802.11ax[END_REF], which also provides simple scenarios to illustrate its potential benefits.

Years before IEEE released the 802.11ax amendment, the idea of dynamically updating TX_PWR and OBSS_PD has been explored by some researchers. The pioneering work of [START_REF] Zhu | Adapting physical carrier sensing to maximize spatial reuse in 802.11 mesh networks[END_REF] presents an analytical model that, based on the current radio channel conditions, dynamically configures OBSS_PD on each node of a Wi-Fi-based mesh network. Concurrently, [START_REF] Kim | SP-TPC: a selfprotective energy efficient communication strategy for IEEE 802.11 WLANs[END_REF] established that adapting TX_PWR can lead to increased throughput and reduced energy consumption. More recently, in 2020, [START_REF] Qiu | Joint Access Point Placement and Power-Channel-Resource-Unit Assignment for 802.11ax-Based Dense WiFi with QoS Requirements[END_REF] casts the issues of positioning the APs of a WLAN and choosing their TX_PWR as an optimization problem. The authors provides a solution to this problem that delivers a static configuration of TX_PWR for a WLAN. But their solution does no account for the number of STAs nor the type of traffic in the WLAN.

The difficulty of accurately modeling the dependency between the configuration parameters of a large WLAN and its performance is a strong hurdle to the development of spatial reuse strategies based on analytical models. As a result, most of the proposed strategies are data-driven. Adaptive by construction, they seem to constitute promising candidates in the search for configurations that improve the spatial reuse of a radio channel and the performance of WLANs.

Machine learning (ML) techniques are natural candidates for addressing problems requiring a data-driven approach, and the spatial reuse problem is no exception. [START_REF] Ak | FSC: Two-Scale AI-Driven Fair Sensitivity Control for 802.11ax Networks[END_REF] addresses the problem of configuring TX_PWR and OBSS_PD with a two-scale solution using artificial neural networks (ANN). In their strategy, STAs and APs first adjust their value of OBSS_PD to minimize interference. Then, an ANN, which was trained offline through simulation, is used to increase the fairness between STAs in terms of attained throughput. However, given the vast diversity of WLAN topologies, the offline learning of the ANN appears as a clear limitation to the generalization of this strategy. An online learning procedure is proposed by [START_REF] Bardou | Improving the Spatial Reuse in IEEE 802.11ax WLANs: A Multi-Armed Bandit Approach[END_REF], which uses reinforcement learning and more precisely the Multi-Armed Bandit (MAB) framework to find the optimal configuration of TX_PWR and OBSS_PD in a WLAN. The approach comprises two agents with one sampling promising configurations through a multivariate normal distribution, and the other identifying the best configuration among those already sampled with Thompson sampling and Normal-Gamma priors. These two ML solutions [START_REF] Ak | FSC: Two-Scale AI-Driven Fair Sensitivity Control for 802.11ax Networks[END_REF][START_REF] Bardou | Improving the Spatial Reuse in IEEE 802.11ax WLANs: A Multi-Armed Bandit Approach[END_REF] were tested on the network simulator ns-3 and lead to significant WLAN improvements. However, in order to perform their optimization, they both assume the presence of a central controller that has access and control over all the APs in the WLANs. By construction, these approaches are centralized, and hence cannot be applied to cases where concurrent WLANs managed by different owners interfere with others.

Distributed approaches are undisputedly better fit than centralized approaches to handle cases with a set of concurrent WLANs. [START_REF] Afaqui | Evaluation of dynamic sensitivity control algorithm for IEEE 802.11ax[END_REF] introduces a distributed algorithm named Dynamic Sensitive Control which is run on every STAs of a WLAN. In short, each STA tries to dynamically reduce its value of OBSS_PD to favor concurrent transmissions while keeping it high enough to ensure a high quality signal reception. Similarly, [START_REF] Lee | LSR: Link-aware Spatial Reuse in IEEE 802.11ax WLANs[END_REF] proposes Link-aware Spatial Reuse (LSR), a distributed algorithm designed for the APs. In LSR, each AP chooses another AP, which is allowed to transmit concurrently, and then prescribes a value of TX_PWR for the selected AP. These two algorithms rely on a single measurement metric reflecting the quality of the received signal, namely the Received Signal Strength, to choose the nodes configuration. More recently, strategies using distributed MAB approaches have been proposed [START_REF] Wilhelmi | Potential and pitfalls of Multi-Armed Bandits for decentralized Spatial Reuse in WLANs[END_REF][START_REF] Wilhelmi | Collaborative Spatial Reuse in wireless networks via selfish Multi-Armed Bandits[END_REF]. They both use Thompson sampling with Gaussian priors to find the best couple of TX_PWR and OBSS_PD at each AP. In [START_REF] Wilhelmi | Collaborative Spatial Reuse in wireless networks via selfish Multi-Armed Bandits[END_REF], each AP seeks to maximize the throughput of its associated STAs. On the other hand, in [START_REF] Wilhelmi | Potential and pitfalls of Multi-Armed Bandits for decentralized Spatial Reuse in WLANs[END_REF], the authors assume that every AP has access to the performance of all other APs in the WLAN; then each AP attempts to maximize a global reward that takes into account the performance of all the other nodes. Both strategies [START_REF] Wilhelmi | Potential and pitfalls of Multi-Armed Bandits for decentralized Spatial Reuse in WLANs[END_REF][START_REF] Wilhelmi | Collaborative Spatial Reuse in wireless networks via selfish Multi-Armed Bandits[END_REF] solutions were evaluated in a self-made simulator with simple random scenarios.

Table 1 summarizes the main characteristics of the data-driven strategies discussed above. It shows that, out of the six considered strategies, two (namely [START_REF] Afaqui | Evaluation of dynamic sensitivity control algorithm for IEEE 802.11ax[END_REF][START_REF] Ak | FSC: Two-Scale AI-Driven Fair Sensitivity Control for 802.11ax Networks[END_REF]) only focus on the configuration of the OBSS_PD parameter (keeping the TX_PWR parameter fixed). To help in the comparison of the different strategies, we introduce two concepts: "pull area" and "push area". The pull area indicates the area on which each node is assumed to obtain information (this typically includes parameter configurations and performance measurements). Depending on the strategy being considered, the pull area can include just the node itself, the surrounding nodes, or the whole set of nodes in the WLANs. The push area designates the area which each AP can influence typically through the prescription of parameter configurations. In the case of centralized strategies (e.g., [START_REF] Ak | FSC: Two-Scale AI-Driven Fair Sensitivity Control for 802.11ax Networks[END_REF][START_REF] Bardou | Improving the Spatial Reuse in IEEE 802.11ax WLANs: A Multi-Armed Bandit Approach[END_REF]), the pull and push areas naturally cover the whole set of APs. We distinguish partially distributed strategies (e.g., [START_REF] Wilhelmi | Potential and pitfalls of Multi-Armed Bandits for decentralized Spatial Reuse in WLANs[END_REF]) wherein either the pull or push area includes the whole set of APs with fully distributed strategies (e.g., [START_REF] Afaqui | Evaluation of dynamic sensitivity control algorithm for IEEE 802.11ax[END_REF][START_REF] Lee | LSR: Link-aware Spatial Reuse in IEEE 802.11ax WLANs[END_REF][START_REF] Wilhelmi | Collaborative Spatial Reuse in wireless networks via selfish Multi-Armed Bandits[END_REF]) in which both the pull and push areas differ from the whole set of APs. We observe in Table 1 that only three out of the six state-of-the-art strategies can be considered as fully distributed. The last four columns of Table 1 pertain to the performance evaluation used to validate each of these strategies. It appears that most strategies were evaluated without considering the dynamical selection of the Modulation Coding Scheme (MCS) for the speed of the wireless links, nor bidirectional (with upstream and downstream) traffic. This can be seen as a strong limitation since this overlooks some associated trade-offs. For instance, increasing the value of TX_PWR certainly enables the communication to operate with a faster data rate (larger MCS), but at the same time, it increases the level of interference with surrounding APs. Additionally, most strategies were evaluated on relatively simple scenarios (with a few APs and a limited number of radio channels), often using a self-made network simulator.

In this paper, we propose a fully distributed strategy to address the problem of the spatial reuse of radio channels in WLANs. The proposed strategy can be applied to any arrangement of WLANs and its novelties are mostly twofold. First, to the best of our knowledge, it is the first strategy making use of Gaussian Processes to explore promising WLAN configurations in the quest of discovering the optimal one. Gaussian processes are recognized tools to deal with the exploration vs. exploitation dilemma (see [START_REF] Ray | On kernelized multi-armed bandits[END_REF][START_REF] Srinivas | Gaussian process optimization in the bandit setting: No regret and experimental design[END_REF]) which is at the center of the spatial reuse problem. Second, unlike the existing fully distributed strategies, INSPIRE allows each AP to account for its surroundings thanks to pull and push areas broader than a single node. Through the use of a simple consensus method, APs of the WLANs achieve to behave altruistically selecting configurations for the "greater good" of the WLANs. We also introduce realistic scenarios, inspired by real-life WLANs, with dynamic MCS and bidirectionnal traffic, to evaluate and compare the efficiency of all the considered strategies.

PROPOSED SOLUTION 3.1 WLANs under study

Let W denote the set of concurrent WLANs under study, each of which being comprised of one or more APs. We let 𝑉 be the set of APs in W that operate on the radio channel of interest. We denote by 𝑁 the number of APs, by 𝑠 𝑖 the set of STAs associated with AP 𝑖 and by 𝑆 the total number of STAs in the considered radio channel of W. Thus, we have: 𝑆 = 𝑁 𝑖=1 |𝑠 𝑖 |. Finally, we use N 𝑖 to designate the set of APs that are within the communication range of AP 𝑖 (when every AP is under the default configuration of the TX_PWR and OBSS_PD parameters). Note that AP 𝑖 itself belongs to N 𝑖 . We refer to the APs in N 𝑖 as the surroundings of AP 𝑖.

We make no assumptions on W, including on the specific arrangement of its APs and STAs, other than the three detailed below. [START_REF] Afaqui | Evaluation of dynamic sensitivity control algorithm for IEEE 802.11ax[END_REF] Associated STAs Associated STAs Up Self-made 100/3 WCNC'21 [START_REF] Lee | LSR: Link-aware Spatial Reuse in IEEE 802.11ax WLANs[END_REF] AP itself AP itself Down ns-3 6/1 Globecom'20 [START_REF] Ak | FSC: Two-Scale AI-Driven Fair Sensitivity Control for 802.11ax Networks[END_REF] All APs All APs Up/Down ns-3 3/1 ADHOC'19 [START_REF] Wilhelmi | Collaborative Spatial Reuse in wireless networks via selfish Multi-Armed Bandits[END_REF] AP itself AP itself Down Self-made 8/1 JNCA'19 [START_REF] Wilhelmi | Potential and pitfalls of Multi-Armed Bandits for decentralized Spatial Reuse in WLANs[END_REF] All APs AP itself Down Self-made 8/1 MSWiM'21 [START_REF] Bardou | Improving the Spatial Reuse in IEEE 802.11ax WLANs: A Multi-Armed Bandit Approach[END_REF] All First, we assume that every AP 𝑖 is able to exchange control frames (possibly through its beacon frames) with its surrounding APs (i.e., the ones in N 𝑖 ). By the same token, we suppose that at least one AP 𝑖 has another AP in its communication range (i.e., ∃𝑖 ∈ 1, 𝑁 , N 𝑖 ≠ {𝑖}), otherwise the spatial reuse of the radio channel would already be at its apex.

Second, we assume that the 𝑁 APs have their TX_PWR and OBSS_PD parameters configurable (as it is the case since the introduction of the 802.11ax amendment). We use 𝑥 𝑡 𝑖 to denote the configuration of AP 𝑖 with regards to its two TX_PWR and OBSS_PD parameters at time 𝑡. Analogously, 𝑥 𝑡 represents the configuration of the 𝑁 APs from W at time 𝑡. Thus, we have:

𝑥 𝑡 𝑖 ∈ 𝐶 = -82, -62 × 1, 21 dBm and 𝑥 𝑡 ∈ 𝐶 𝑁 .
Lastly, we assume that each AP in W can periodically run performance tests and obtain, in return, the mean throughput attained by each of its STAs over a short time interval 𝛥𝑡. More formally, we use the vector 𝑇 𝑡 ∈ R +𝑆 to denote the throughput attained by the 𝑆 STAs of W given the WLAN configuration 𝑥 𝑡 at time 𝑡. Throughout this paper, we sometimes refer to 𝑇 𝑡 as 𝑇 (𝑥 𝑡 ) to explicitly show the dependency between the STAs' throughputs and APs' configurations.

In this work, we seek to discover an adequate configuration 𝑥 * of the 𝑁 APs composing W that improves the collective experience of the 𝑆 STAs through a better reuse of their radio channel. We address this problem as a reinforcement learning task in which, at regular time intervals 𝑡, the APs collect measurements 𝑇 𝑡 associated to their current configuration 𝑥 𝑡 , and need to decide their next configuration 𝑥 𝑡 +1 . The obstacles towards that objective are mostly threefold. (i) We need to define a meaningful objective function that APs will attempt to optimize collectively; (ii) We are facing the well-known exploration vs. exploitation dilemma since the search for an adequate configuration of the WLANs should be as seamless as possible (without disrupting the STAs). This leads us to cast the problem as a MAB problem where the arms are the WLANs' configurations. Following the MAB terminology, we refer to the objective function as the reward function; (iii) We are looking for a strategy that can be applied in a distributed way since it would be in general unrealistic to assume that (concurrent) APs have a fine knowledge beyond their surroundings.

Reward function

We need to define a reward function 𝑅 that appraises the "goodness" of a configuration 𝑥 with regards to the WLANs performance. Because multiple criteria may be considered in the definition of 𝑅, there is no universal definition. However, assessing the quality of a configuration 𝑥 can be derived from the STAs throughputs 𝑇 (𝑥) obtained with APs configured with 𝑥. Among all easily computable reward functions, 𝑅(𝑥) = 𝑇 𝑖 ∈𝑇 (𝑥) 𝑇 𝑖 is called the proportional fairness (PF) and provides a convenient trade-off between fairness and cumulated throughput. However, PF is often criticized for its high variability, since 𝜕𝑅 𝜕𝑇 𝑖 = 𝑇 𝑗 ∈𝑇 (𝑥),𝑗≠𝑖 𝑇 𝑗 . To overcome this drawback, we consider the logarithm of PF. This lowers its variability, which becomes: 𝜕𝑅 𝜕𝑇 𝑖 = 1 𝑇 𝑖 (note that 𝑇 𝑖 is typically much larger than 1). This also emphasizes the contribution of STAs with low throughputs in the computation of 𝑅 and provides a pleasant closed-form to optimize. For an arbitrary set of APs 𝑋 , we can define 𝑅 𝑋 = log 𝑖 ∈𝑋 𝑗 ∈𝑠 𝑖 𝑇 𝑗 (𝑥). Then, our global reward function 𝑅 is:

𝑅(𝑥) = 𝑅 𝑉 (𝑥) = log 𝑖 ∈𝑉 𝑗 ∈𝑠 𝑖 𝑇 𝑗 (𝑥) = ∑ 𝑖 ∈𝑉 𝑗 ∈𝑠 𝑖 log𝑇 𝑗 (𝑥) (1)
However, to compute Equation 1, an AP must have a complete knowledge of the performance attained by the STAs of all APs or, at least, be able to communicate with all the APs in W. This is in contradiction with our assumption that APs only have a partial knowledge of W, limited to their surrounding APs. To design a reward function compatible with the distributed case, we proceed as follows. Each AP 𝑖 applies Equation 1 but restricted to the set of its associated STAs and obtains in return a "selfish" reward denoted by 𝑅 {𝑖 } . Previous work [START_REF] Wilhelmi | Collaborative Spatial Reuse in wireless networks via selfish Multi-Armed Bandits[END_REF] have showed that considering such selfish rewards may have a positive but limited impact on the WLANs performance. Therefore, we introduce a more altruistic reward, denoted by 𝑅 𝑖 that accounts not only for the "selfish" reward of AP 𝑖 (i.e., 𝑅 {𝑖 } ) but also for the rewards of the surrounding APs (i.e., the ones in N 𝑖 ). The "altruistic" local reward of AP 𝑖 is computed as:

𝑅 𝑖 (𝑥) = ∑ 𝑗 ∈N 𝑖 𝑅 { 𝑗 } (𝑥) |N 𝑗 | (2)
Note that 𝑅 𝑖 defined with Equation 2 ensures that a configuration 𝑥 maximizing the local rewards is also a maximum for the global reward 𝑅 since we have: 𝑁 𝑖=1 𝑅 𝑖 (𝑥) = 𝑅(𝑥).

Proof. This is a straightforward property since

𝑁 ∑ 𝑖=1 𝑅 𝑖 (𝑥) = 𝑁 ∑ 𝑖=1 ∑ 𝑗 ∈N 𝑖 𝑅 { 𝑗 } (𝑥) |N 𝑗 | (3)
By noticing that 𝑖 ∈ N 𝑗 ⇐⇒ 𝑗 ∈ N 𝑖 , we can permute the indices and use the definition of 𝑅 { 𝑗 } :

𝑁 ∑ 𝑗=1 𝑅 { 𝑗 } (𝑥) |N 𝑗 | ∑ 𝑖 ∈N 𝑗 1 = 𝑁 ∑ 𝑗=1 ∑ 𝑘 ∈𝑠 𝑗 log𝑇 𝑘 (𝑥) = 𝑅(𝑥) (4)
□

Local reward maximization

For the sake of clarity and since all variables in this section are relative to an AP 𝑖, we often omit the subscript 𝑖 in the notations. Now that each AP 𝑖 has its own local reward function, we need a model of the knowledge of AP 𝑖 about 𝑅 𝑖 in order to find its argmax. We represent the beliefs of AP 𝑖 about 𝑅 𝑖 by defining a prior distribution on the reward functional space with a Gaussian Process (GP).

Gaussian process. In our case, a GP can be defined as a collection of random variables indexed by configurations of APs in N 𝑖 :

{𝑌 𝑐 ; 𝑐 ∈ 𝐶 | N 𝑖 | } such that every finite collection 𝑌 𝑐 1 , • • • , 𝑌 𝑐 𝑛 ∼ N (𝜇, 𝛴).
Without loss of generality, we assume the GP to have zero mean so that it is entirely determined by its covariance function

𝛴 : 𝐶 | N 𝑘 | × 𝐶 | N 𝑘 | → R + .
As shown by [START_REF] Christopher | Gaussian Processes for Regression[END_REF], GPs can be used as priors on a function space. We use 𝑋 𝑡 to denote the 𝑡 × 2|N 𝑘 | features matrix gathering the tested configurations 𝑥 1 , • • • , 𝑥 𝑡 𝑇 and 𝑌 𝑡 to denote the 𝑡 × 1 label vector gathering the corresponding local reward values 𝑅 𝑖 (𝑥 1 ), • • • , 𝑅 𝑖 (𝑥 𝑡 ) 𝑇 . Given 𝑋 𝑡 and 𝑌 𝑡 , we can infer the distribution of the reward value for an arbitrary configuration 𝑥, 𝑅 𝑖 (𝑥), from Bayes' theorem: 𝑅 𝑖 (𝑥)|𝑋 𝑡 , 𝑌 𝑡 ∼ N 𝜇 (𝑥), 𝜎 2 (𝑥) with 𝜇 (𝑥) and 𝜎 2 (𝑥) defined in Equations 5 and 6, respectively.

𝜇 (𝑥) = 𝛴 (𝑥, 𝑋 𝑡 )𝛴 (𝑋 𝑡 , 𝑋 𝑡 ) -1 𝑌 𝑡 (5) 𝜎 2 (𝑥) = 𝛴 (𝑥, 𝑥) -𝛴 (𝑥, 𝑋 𝑡 )𝛴 (𝑋 𝑡 , 𝑋 𝑡 ) -1 𝛴 (𝑋 𝑡 , 𝑥) (6)
Since GPs can be used as a prior on a functional space, they are useful to solve regression problems as well as maximization tasks. In our case, the AP 𝑖 uses a GP to model 𝑅 𝑖 and to assist the exploration of promising configurations of the APs in N 𝑖 , maximizing 𝑅 𝑖 in a Bayesian way.

Choosing the covariance function 𝛴 is a critical step when designing a GP as it determines some key features such as its isotropy and smoothness. Since the reward function, which quantifies the quality of spatial reuse in N 𝑖 , is likely to exhibit threshold effects, we choose a covariance function that decreases rapidly as the distance between two considered configurations increases. Thus, the regularity constraint is not too restrictive on the modeled function. Because we have no incentive to prefer any particular direction over another, we let the covariance function 𝛴 (𝑥, 𝑥 ′ ) depend only on ||𝑥 -𝑥 ′ || to ensure the isotropy of the GP. This leads us to use a Matérn kernel [START_REF] Marc | Classes of kernels for machine learning: a statistics perspective[END_REF] with parameter 𝜈 = 3 2 , which is defined as

𝛴 (𝑥, 𝑥 ′ ) = 𝑠 2 1 + √ 3||𝑥 -𝑥 ′ || 𝜌 𝑒 - √ 3||𝑥 -𝑥 ′ || 𝜌 (7)
where 𝑠 2 and 𝜌 are two hyperparameters whose values are approximated by maximizing the likelihood of 𝑌 𝑡 (which is Gaussian) during the learning process.

As discussed before, each AP 𝑖 faces the exploitation vs exploration dilemma in its attempt to find the optimal configuration. A common way in the MAB framework to appraise a given strategy 𝜋 is then to consider the cumulative regret 𝛤 (𝜋). In our problem, 𝛤 (𝜋) is expressed with Equation 8for an episode of 𝐷 steps, since it is expressed as the cumulative sum of the differences between the best reward that the AP can get and 𝑅 𝑖 (𝜋 (𝑡)), which is the actual reward obtained at time 𝑡 for the strategy 𝜋.

𝛤 (𝜋) = 𝐷 ∑ 𝑡 =1 max 𝑥 ∈𝐶 |N 𝑖 | 𝑅 𝑖 (𝑥) -𝑅 𝑖 (𝜋 (𝑡)) (8)
Minimizing the cumulative regret with GP models is usually done by defining a strategy 𝜋 from the maximization of an acquisition function 𝐴: 𝜋 (𝑡) = arg max 𝑥 ∈𝐶 |N 𝑖 | 𝐴 𝑡 (𝑥). However, this assumes that our search space 𝐶 |N 𝑖 | is continuous. Since each AP 𝑖 deals with discrete configurations of APs in N 𝑖 , we systematically round the recommendation of the GP to the nearest valid WLAN configuration. Many acquisition functions exist, such as Knowledge Gradient (KG) [START_REF] Gupta | Bayesian look ahead one-stage sampling allocations for selection of the best population[END_REF], GP-UCB [START_REF] Srinivas | Information-Theoretic Regret Bounds for Gaussian Process Optimization in the Bandit Setting[END_REF] or the Expected Improvement (EI) [START_REF] Jones | Efficient Global Optimization of Expensive Black-Box Functions[END_REF]. We choose EI over KG (whose computational cost can rapidly become prohibitive) and GP-UCB (which was found to be less efficient on our examples). The EI acquisition function is expressed as 𝐴 𝑡 (𝑥) = E (𝜇 𝑡 +1 (𝑥)max 1≤𝑘 ≤𝑡 𝑅 𝑖 (𝑥 𝑘 )) + given that 𝑋 𝑡 +1 = (𝑋 𝑡 , 𝑥), 𝑌 𝑡 +1 = (𝑌 𝑡 , 𝑅 𝑖 (𝑥)). Since 𝑅 𝑖 (𝑥) ∼ N (𝜇 (𝑥), 𝜎 2 (𝑥)), we can derive a convenient closed-form for EI, as shown in Equation 9.

𝐸𝐼 (𝑥) = (𝜇 (𝑥) -𝑅 * 𝑖,𝑡 )𝛷 (𝑍 ) + 𝜎 (𝑥)𝜙 (𝑍 )

with 𝑅 * 𝑖,𝑡 = max 1≤𝑘 ≤𝑡 𝑅 𝑖 (𝑥 𝑘 ), 𝑍 = 𝜇 (𝑥)-𝑅 * 𝑖,𝑡 𝜎 (𝑥) , 𝛷 and 𝜙 being respectively the CDF and the PDF of a standard Gaussian distribution.

Then, the AP 𝑖 can try to maximize Equation 9 by differentiating it and performing a gradient ascent. By applying its strategy 𝜋 𝑖 (𝑡) = arg max 𝑥 ∈𝐶 |N 𝑖 | 𝐸𝐼 (𝑥) and classical gradient ascent techniques on Equation 9, AP 𝑖 provides promising configurations for its surrounding APs in N 𝑖 .

Aggregation of local prescriptions

In the previous sections, we have described how each AP 𝑖 computes its local reward and relies on its model GP 𝑖 to explore promising configurations for the APs in N 𝑖 .

However, more coordination between APs is required. By construction, the collection F = (N 𝑘 ) 1≤𝑘 ≤𝑁 is a cover of the set of APs in W but not a partition. In fact, if F had only null intersections (i.e., ∀𝑗, 𝑘, N 𝑗 ∩ N 𝑘 = ∅), then the spatial reuse of the radio channel would already be at its apex and there is no need for improvement. Figure 2 illustrates an example with 5 APs in which the collection F = (N 1 , N 2 , N 3 , N 4 , N 5 ) exhibits multiple non-null intersections. As a result, most APs will receive a set of different prescriptions for the configuration of their TX_PWR and OBSS_PD parameters at their next iteration. For instance, AP 1 will receive prescriptions from APs 2 and 4 in addition to its own prescription. Since APs can only test one configuration at a time, one of those prescriptions must be chosen, or preferably, a consensus between them must be made.

In general, maximizing local rewards is very likely to lead to a sub-optimal situation since, for non-linear optimization problems, individual interests are often not aligned with the global interest (e.g., the famous Tragedy of the Commons [START_REF] Hardin | The Tragedy of the Commons[END_REF]). Without more information on the relation between the configuration of the APs and the measured throughputs of STAs, it seems difficult to provide an expression for the argmax of the global reward function 𝑅 given the argmax of the local reward functions 𝑅 𝑖 . However, recall that 𝑁 𝑖=1 𝑅 𝑖 (𝑥) = 𝑅(𝑥). We can leverage this property to provide guarantees. Theorem 1. Let {𝑅 𝑖 } 1≤𝑖 ≤𝑁 be a set of Lipschitzian functions, 𝑤 ∈ [0, 1] 𝑁 , ||𝑤 || 1 = 1 be a weight vector and {𝑥 𝑖 𝑗 } 𝑖 ∈N 𝑗 be the prescriptions received by the AP 𝑗, with 𝑥 𝑖 = arg max 𝑥 ∈𝐶 |N 𝑖 | 𝑅 𝑖 (𝑥), then the weighted marginal median of the received sets of prescriptions is a maximin optimum x of 𝑁 𝑖=1 𝑅 𝑖 ( xN 𝑖 ):

x 𝑗 = 𝑚𝑒𝑑 { 𝑥 𝑖 𝑗 , 𝑤 𝑖 } 𝑖 ∈N 𝑗 (
𝛹 ( x) = 𝜆 𝐷 ∑ 𝑑=1 𝑁 ∑ 𝑗=1 ∑ 𝑖 ∈N 𝑗 𝑤 𝑖 |𝑥 𝑖 𝑗,𝑑 -x 𝑗,𝑑 | = 𝜆 𝐷 ∑ 𝑑=1 𝑁 ∑ 𝑗=1 ∑ 𝑖 ∈N 𝑗 𝑥 𝑖 𝑗,𝑑 < x 𝑗,𝑑 𝑤 𝑖 ( x 𝑗,𝑑 -𝑥 𝑖 𝑗,𝑑 ) + ∑ 𝑖 ∈N 𝑗 𝑥 𝑖 𝑗,𝑑 ≥ x 𝑗,𝑑 𝑤 𝑖 (𝑥 𝑖 𝑗,𝑑 -x 𝑗,𝑑 ) (12)
We want to minimize 𝛹 ( x). This is equivalent to finding x so that ∇𝛹 is 0, which can be written as, ∀( 𝑗, 𝑑) ∈ 1, 𝑁 × 1, 𝐷 :

𝜕𝛹 𝜕 x 𝑗,𝑑 = 𝜆 ∑ 𝑖 ∈N 𝑗 𝑥 𝑖 𝑗,𝑑 < x 𝑗,𝑑 𝑤 𝑖 - ∑ 𝑖 ∈N 𝑗 𝑥 𝑖 𝑗,𝑑 ≥ x 𝑗,𝑑 𝑤 𝑖 = 0 ( 13
)
The only value of x 𝑗,𝑑 which ensures this relation is the weighted median of the sample {𝑥 𝑖 𝑗,𝑑 } 𝑖 ∈N 𝑗 , provided that the weights can cancel out. It remains to verify that this critical point is a minimum by considering the coefficients of the Hessian matrix. We can rewrite the partial derivative of 𝛹:

𝜕𝛹 𝜕 x 𝑗,𝑑 = -𝜆 ∑ 𝑖 ∈N 𝑗 𝑤 𝑖 𝑥 𝑖 𝑗,𝑑 -x 𝑗,𝑑 = -𝜆 ∑ 𝑖 ∈N 𝑗 𝑤 𝑖 2𝐻 1 2 𝑥 𝑖 𝑗,𝑑 -x 𝑗,𝑑 -1 (14)
with 𝐻 1 2 the Heaviside function. Then, we obtain the Hessian matrix coefficients:

𝜕𝛹 𝜕 x 𝑗,𝑑 𝜕 x 𝑗 ′ ,𝑑 ′ = 0 if ( 𝑗, 𝑑) ≠ ( 𝑗 ′ , 𝑑 ′ ), 2𝜆 𝑖 ∈N 𝑗 𝑤 𝑖 𝛿 𝑥 𝑖 𝑗,𝑑 -x 𝑗,𝑑
otherwise.

(15) with 𝛿 the Dirac impulsion. These coefficients define a positive definite matrix if x 𝑗,𝑑 ∈ {𝑥 𝑖 𝑗 } 𝑖 ∈N 𝑗 , which is necessarily the case if x 𝑗,𝑑 is a weighted median of the sample. However, we must not create new data: if two elements are eligible to be the weighted median, we must select one of them and not taking the average between the two.

Since the weighted marginal median of the prescriptions minimizes the upper bound of 𝑅 * -R, it is a maximin optimum for R = 𝑁 𝑖=1 𝑤 𝑖 𝑅 𝑖 ( xN 𝑖 ). □

Since, by definition of the local reward functions, 𝑁 𝑖=1 𝑅 𝑖 (𝑥) = 𝑅(𝑥) (see Eq. 4), we can apply Theorem 1 with, ∀𝑖 ∈ 1, 𝑁 , 𝑤 𝑖 = 1 𝑁 and state that taking the marginal median of the prescriptions is a good way to reach high values of the global reward 𝑅.

Algorithm and complexity

Compute the consensus x𝑡+1

𝑖 with Equation 107:

Test 𝑥 𝑡 +1 𝑖 for 𝛥𝑡 seconds and compute its selfish reward 𝑅 {𝑖 } with Equation 1 applied only to AP 𝑖 Contrary to what one might think, the most resource intensive operation in Algorithm 1 is not the inversion of the 𝛴 (𝑋 𝑡 , 𝑋 𝑡 ) matrix. Since, at step 𝑡, we already know the Cholesky decomposition of 𝛴 (𝑋 𝑡 -1 , 𝑋 𝑡 -1 ) = 𝐿𝐿 𝑇 , it is very easy to compute the Cholesky decomposition of 𝛴 (𝑋 𝑡 , 𝑋 𝑡 ). The most resource-intensive operation is the maximization of the acquisition function through gradient ascent. Since this requires computing many matrix-vector multiplications for at most 𝑚 steps, the computational complexity of Algorithm 1 at time 𝑡 is 𝑂 𝑚𝑡 2 .

It is worth noting that the dimensionality of the problem (i.e. dim(𝐶 | N 𝑖 | ) = |N 𝑘 | dim 𝐶) does not appear in the expression of the asymptotic computational complexity of INSPIRE. This interesting property results from the use of a kernel function by GPs to compare WLANs configurations. This gives INSPIRE the ability to handle arbitrarily dense WLANs, or to optimize more parameters than just TX_PWR and OBSS_PD, without taking a hefty toll on its execution time. In fact, the real burden to the execution time of INSPIRE is 𝑡. This compels us to bound the size of 𝑋 𝑡 and to find a balance between the amount of collected data on the WLANs' performance and configuration and a quick execution time. We keep the possibility of approximation methods to reduce the computational complexity of INSPIRE for future works. As for now, we recommend using windowing methods (such as a moving window) to bound the size of 𝑋 𝑡 and so the computational complexity of INSPIRE.

PERFORMANCE EVALUATION 4.1 Experimental settings

To evaluate the ability of INSPIRE at improving the spatial reuse of a radio channel through the configuration of the TX_PWR and OBSS_PD parameters, we consider two distinct scenarios.

The first scenario is inspired by the WLAN deployment of Cisco in their offices in San Francisco. In [START_REF]High Density Wi-Fi Deployments[END_REF], Cisco provides the location of 60 APs that together deliver wireless connectivity to their employees on a floor. To account for the WLANs' activity from other floors, we consider a three-floor building and we replicate on each floor the same arrangement of APs as in Cisco's offices. This leads us to a total number of 180 APs spanned over three floors. Assuming 18 independent radio channels, we run a radio channel allocation algorithm to determine the radio channel used by each AP. For our first scenario, we consider the subgraph resulting from the channel allocation with the highest density. We use T1 to refer to this topology (i.e., arrangements of APs and STAs), which is illustrated in Figure 3a. T1 exhibits a total of 10 APs and we associate a number of 5 STAs per AP.

The second scenario addresses the case of many single-AP WLANs deployed and operated independently in a relatively limited area. This is typically the case in housing units where each apartment is equipped with its own AP so that the APs are often only a few meters away from a number of others. More specifically, we consider a nine-story building with 216 apartments of 25 m2 each. We randomly position an AP within each apartment as well as 4 STAs per AP. Then, similarly to the first scenario, we apply a radio channel allocation algorithm given a total of 18 radio channels, to obtain the topology of interest denoted by T2. Note that T2 consists of 14 APs and 56 STAs. Figure 3b depicts the topology T2.

For each scenario, we consider heavily loaded conditions. APs attempt to transmit frames to each of their associated STAs at a rate of 50 Mbps while the latter attempt to send their frames to the AP at a lower rate of 3.33 Mbps. These assumptions are in line with the downstream traffic largely exceeding the upstream traffic in WLANs. Given the speed of wireless links in 802.11ax, the buffers of the APs will always be full of frames waiting to be sent. More generally, considering APs in saturation represents undoubtedly the most difficult case when dealing with the spatial reuse of a radio channel. Therefore, if INSPIRE manages to significantly improve the WLANs' performance under these circumstances, then it can only do better under normal conditions.

To better appraise the quality of INSPIRE, we also consider a control strategy as well as several state-of-the-art solutions, which were discussed in Section 2 and briefly summarized here:

• DEFAULT: Every AP keeps its default configuration for the TX_PWR and OBSS_PD parameters (i.e., (-82, 20) dBm); • WCNC'15: Each AP implements a simple distributed algorithm to dynamically update its OBSS_PD parameter [START_REF] Afaqui | Evaluation of dynamic sensitivity control algorithm for IEEE 802.11ax[END_REF]; • JNCA'19: Each AP solves a MAB problem using Thompson sampling to dynamically update their TX_PWR and OBSS_PD parameters [START_REF] Wilhelmi | Potential and pitfalls of Multi-Armed Bandits for decentralized Spatial Reuse in WLANs[END_REF]; • MSWiM'21: Similar to JNCA'19, except that the sampling of new configurations is performed through a multivariate Gaussian mixture, and that the solution is centralized [START_REF] Bardou | Improving the Spatial Reuse in IEEE 802.11ax WLANs: A Multi-Armed Bandit Approach[END_REF].

We implemented INSPIRE (based on the open-source Gaussian process library LibGP [START_REF] Blum | LibGP. GitHub[END_REF]) as well as the four strategies described above in the open-source network simulator ns-3 [16]. ns-3 is a wellestablished realistic discrete-event simulator that implements most of the network protocols involved in the WLANs communication from the Physical up to the Application layer. We report in Table 2 the simulation parameters used in the rest of this section. Unlike previous works (e.g., [3-5, 20, 22])) with the exception of [START_REF] Lee | LSR: Link-aware Spatial Reuse in IEEE 802.11ax WLANs[END_REF], our simulations incorporate the mechanism of rate adaptation that let APs and STAs dynamically vary the speed of their wireless links (through the use of different Modulation Coding Scheme (MCS)) in response to the quality of the received signal. This is particularly important for the sake of our study since changing the value of TX_PWR necessarily affects the quality of the received signal and thus the MCS. Since our simulated WLANs take place in buildings, we choose an appropriate path loss by combining the models ItuR1238 and InternalWallsLoss, both implemented by ns-3. With these propagation models, the signal is decreased by an additional attenuation coefficient each time it goes through a floor or a wall. The attenuation coefficients are respectively -4 dBm (which is the default value in ItuR1238) and -8 dBm.

We instrumented ns-3 to collect and compute a number of performance metrics. At the end of each iteration, the quality of the spatial reuse is assessed with Equation 1, although distributed strategies may internally use the local reward defined in Equation 2. Then, we compute the classical metric to analyze the efficiency of a strategy at dealing with a MAB problem: (i) The cumulative regret (with Equation 8 using a normalized version of the global reward in Equation 1). We also collect the following performance metrics to reflect the effect of each strategy on the behavior of the WLANs and of their STAs: (ii) The number of starving STAs, which we define as STAs experiencing a very low throughput (namely, less than 10% of their attainable throughput) and (iii) The cumulated throughput, which simply sums all STAs' throughput. Each simulation lasts 30 seconds and we replicated them independently 22 times to obtain and visualize their first, second, and third quartiles. When the quartiles of a performance metric vary too much within a single simulation, we apply an exponential moving average (with 𝛼 = 0.04) to extract the underlying trends of the quartiles sequences. The metrics are collected throughout the whole duration of the simulation. At the end of each iteration, we compute all the performance metrics and then we refer to the current strategy to decide what will be the next configuration of the WLANs. Since an iteration lasts 𝛥𝑡 = 75 ms and a simulation lasts 30 seconds, the quality of each solution is assessed over 400 iterations.

Numerical results

Figure 4 illustrates the performance metrics delivered by the ns-3 simulator for each strategy in the case of topology T1. The cumulative regret, represented in Figure 4a, indicates which strategy has performed the best at any time of the simulation. INSPIRE is found to be the most efficient strategy, reducing the cumulative regret by 70% more than DEFAULT and by over 50% than WCNC'15, which happens to be the most efficient state-of-the-art strategy. We now look at the other performance metrics to better understand how much INSPIRE is able to improve the behavior of the WLAN and of its STAs. Taking DEFAULT as baseline, Figure 4b shows that INSPIRE reduces the number of STAs in starvation by 80% while Figure 4c demonstrate that our proposed solution manages to increase the cumulated throughput (+400%).

We now turn to the case of topology T2. First, we observe in Figure 5a that among the four considered strategies, INSPIRE is the one that manages to decrease the most the cumulative regret with a decline of about 36% compared to the DEFAULT configuration at the end of the simulation. The proposed solution also outperforms MSWiM'21, which is found to be the best state-of-the-art strategy on this topology, by a margin of 14%. Looking at the performance of WLANs and of their STAs, Figure 5b shows that INSPIRE is able to limit the number of STAs starving from throughput by a degree of 36% when compared to the DEFAULT configuration. Similarly, the cumulated throughput of STAs have their value increased by 28% and nearly doubled with INSPIRE (see Figure 5c).

Overall, through the study of topologies T1 and T2, INSPIRE demonstrates its superiority over the other state-of-the-art strategies. The significant improvements brought by our proposed solution on all performance metrics are permanently obtained after 100 iterations only (7.5 seconds). In other words, in less than 10 seconds, INSPIRE manages to significantly improve the behavior of the WLANs and of the associated STAs thanks to a better spatial reuse of the radio channel. This efficiency in searching and finding an adequate configuration of the TX_PWR and OBSS_PD parameters at each AP of the WLANs mostly results from the distributed, altruistic use of Gaussian processes which we further discuss in the next section.

CONCLUSIONS

In this work, we have presented INSPIRE, a reinforcement learning method to improve the spatial reuse of radio channels in WLANs by configuring two parameters of APs: the transmission power (TX_PWR) and the sensitivity threshold (OBSS_PD), that can be dynamically configured with the latest Wi-Fi amendments. To address the difficult problem of sharing efficiently and fairly the resource of a radio channel, INSPIRE works as a distributed solution where each AP solves a local Multi-Armed Bandit problem with the help of information and actions limited to its surrounding APs (i.e. within its communication range). The development of the solution includes (i) an intuitive quantification (based on STAs throughputs) of the "goodness" of a configuration of TX_PWR and OBSS_PD for concurrent APs in WLANs, both at local and global scales, (ii) the use of an acquisition function and Gaussian processes to find local configurations that maximize approximations of local reward functions, and (iii) an altruistic behavior facilitated by prescriptions to surrounding APs along with a consensus method which aggregates the prescriptions of surrounding APs for the "greater good" of the WLANs.

INSPIRE has been evaluated and compared with other state-ofthe-art strategies addressing the same problem, using the opensource network simulator ns-3 that implements all the layers of the network stack. The different strategies were compared on two examples inspired by real-life deployments of dense WLANs in both professional and domestic environments. INSPIRE was found to outperform other state-of-the-art strategies by significantly reducing the number of STAs in starvation and increasing the cumulated throughput of the WLANs in only a few seconds.

As future work, we plan to assess the quality of INSPIRE for a specific class of WLANs where STAs are mobile (e.g., customers in a shopping mall). Another natural follow-up would be to experiment INSPIRE with real material on a testbed.

Figure 2 :

 2 Figure 2: A WLAN represented by a graph with APs depicted as labelled triangles and STAs as black dots. An edge exists between two APs when they are in the communication range of each other. We use different colors to illustrate the surroundings of each AP in F .

10 )

 10 Proof. Let 𝑥 𝑖 be the prescription of AP 𝑖, with 𝑥 𝑖 𝑗 the prescription of AP 𝑖 for AP 𝑗 and 𝑤 ∈ [0, 1] 𝑁 a normalized weight vector (||𝑤 || 1 = 1). Consider the quantities 𝑅 * = 𝑁 𝑖=1 𝑤 𝑖 𝑅 𝑖 (𝑥 𝑖 ) and R = 𝑁 𝑖=1 𝑤 𝑖 𝑅 𝑖 ( xN 𝑖 ) for a given consensus x ∈ 𝐶 𝑁 . If 𝑥 𝑖 = arg max 𝑥 ∈𝐶 |N 𝑖 | 𝑅 𝑖 (𝑥), then it follows that ∀ x ∈ 𝐶 𝑁 , 𝑅 * -R > 0. We want to minimize this difference. If all the functions 𝑅 𝑖 are 𝜆-Lipschitzian, we have: 𝑅 * -R = 𝑁 ∑ 𝑖=1 𝑤 𝑖 𝑅 𝑖 (𝑥 𝑖 ) -𝑁 ∑ 𝑖=1 𝑤 𝑖 𝑅 𝑖 ( xN 𝑖 ) = 𝑁 ∑ 𝑖=1 𝑤 𝑖 (𝑅 𝑖 (𝑥 𝑖 ) -𝑅 𝑖 ( xN 𝑖 )) ≤ 𝜆 𝑁 ∑ 𝑖=1 𝑤 𝑖 ||𝑥 𝑖 -xN 𝑖 || 1 x 𝑗,𝑑 | = 𝛹 ( x) (11) by expliciting the 𝐿 1 norm over the 𝐷 dimensions of 𝐶. By rearranging the indices and splitting the absolute values, we have:

8 : 11 :

 811 Broadcast 𝑅 {𝑖 } , |N 𝑖 | and x𝑡+1 𝑖 to APs in N 𝑖 9: Receive 𝑅 { 𝑗 } , |N 𝑗 | and x𝑡+1 𝑗 from AP 𝑗, 𝑗 ≠ 𝑖, 𝑗 ∈ N 𝑖 10: Compute the local reward 𝑅 𝑖 with Equation 2 and the local configuration x𝑡+1 N 𝑖 Add the pattern x𝑡+1 N 𝑖 , 𝑅 𝑖 to GP 𝑘 12: end while Algorithm 1 summarizes the main steps of our proposed strategy INSPIRE run on each AP of the WLANs.

Figure 3 :

 3 Figure 3: The two considered topologies. APs are shown as red triangles, connected by a arrow if they lie in each other's communication range. Associated STAs are shown as dots colored according to their throughputs: warm, yellowish colors indicate that the STA has enough throughput most of the time while, on the contrary, cool, blueish colors indicate that the STA has mostly not enough throughput under the default configuration of 802.11: 20 dBm for TX_PWR and -82 dBm for OBSS_PD.

Figure 4 :

 4 Figure 4: Performance metrics delivered on topology T1 by each strategy.

Figure 5 :

 5 Figure 5: Performance metrics delivered on topology T2 by each strategy.

Table 1 :

 1 Comparison of the state-of-the-art data-driven strategies. The last column refers to the size of the scenarios involved in the performance evaluation of the strategy. For instance, 216/18 means the evaluation comprises 216 APs distributed over 18 radio channels.

	Proposed	Tuning of	Degree of centralization	Dynamic	Traffic	Simulator APs /
	solution	TX_PWR	Pull area	Push area	MCS	Up/Down	channels
	WCNC'15

  Algorithm 1 INSPIRE run at each AP 𝑖 Input: subset N 𝑖 of APs 1: Initialize the Gaussian Process GP 𝑖 2: while true do 3: Find a prescription 𝑥 𝑖 = arg max 𝑥 ∈𝐶 |N 𝑖 | 𝐸𝐼 𝑡 𝑖 (𝑥) by gradient ascent 4:Broadcast 𝑥 𝑖 to APs in N 𝑖

	5:	Receive the prescriptions 𝑥

𝑗

𝑖 from AP 𝑗, 𝑗 ≠ 𝑖, 𝑗 ∈ N 𝑖 6:

Table 2 :

 2 ns-3 parameters.

	Parameter	Value
	ns-3 version	3.31
	Number of repetitions	22
	Simulation duration	30 s
	Duration of an iteration (𝛥𝑡) 75 ms
	Packet size	1,464 bytes
	Downlink traffic	50.0 Mbps
	Uplink traffic	3.33 Mbps
	Channel size	20 MHz
	Frequency band	5 GHz
	A-MDPU Aggregation	4
	Path loss	HybridBuildings (ItuR1238 +
		InternalWallsLoss)
	Wi-Fi Manager	IdealWifiManager

ACKNOWLEDGMENTS

This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).