Marine Agogue

Béatrice Parguel

Do you believe Red Bull gives wings? Do you believe Red Bull gives wings? When implicit theories of creativity impair creative performance

Keywords: creativity, implicit theories, LIToC, stimulating creativity, energy drinks

Everyone seems to have something to say about creativity, thus participating in the reproduction of persistent myths about creativity that may influence creative behavior. This research explores the influence of Laypeople's Implicit Theories of Creativity (LIToC) regarding the drivers of creativity on creative performance, to ascertain whether having strong convictions about the drivers of creativity either enhances or hinders creative productivity when these convictions align with the actual methods of stimulating creativity. An experiment randomly involved 69 subjects who were invited to drink the exact same fruit juice before performing a creative task. In one condition, they were told this was indeed juice, in the other condition they were told that it was mixed with Red Bull. Analyses showed an interaction effect with the subjects' LIToC, such that among subjects displaying strong LIToC, individual creative performance was lower when they perceived the conditions to stimulate creativity were activated, than otherwise. These results suggest that having strong beliefs in the effects of some creativity drivers might then trigger a complacent attitude and reduce the invested effort in generating creative ideas. This research contributes to rethinking how we use specific drivers to stimulate creativity, as strong LIToCs about those drivers may have a counterproductive effect on creative performance.

INTRODUCTION

Creativity stands as a crucial competency for many organizations, empowering them to navigate their surroundings, maintain competitiveness, and embrace novel realities. First and foremost, a strategic approach to creativity management enables the systematic cultivation of novel ideas, ensuring a continuous stream of innovative solutions to challenges [START_REF] Martins | Building organisational culture that stimulates creativity and innovation[END_REF]. Additionally, the systematic management of creative processes facilitates the identification and harnessing of individual and collective talents, maximizing the organization's intellectual capital [START_REF] Wang | Core self-evaluation, individual intellectual capital and employee creativity[END_REF][START_REF] Crema | Managing Intellectual Capital in Italian Manufacturing SMEs[END_REF].

Moreover, by instilling a culture that encourages and channels creativity, organizations can enhance employee engagement, job satisfaction, and overall organizational performance [START_REF] Naranjo-Valencia | Organizational culture and radical innovation: Does innovative behavior mediate this relationship?[END_REF]. Ultimately, the judicious management of creativity is indispensable for organizations aspiring to adapt to changing market demands, sustain long-term growth, and remain at the forefront of their respective industries.

Operating as a socio-cognitive process fostering the development of novel ideas [START_REF] Amabile | Creativity in Context: Update to the Psychology of Creativity[END_REF][START_REF] Mumford | Alternative knowledge structures in creative thought[END_REF], creativity materializes across multiple tiers of the organization: at an individual level, within teams, and at the overarching organizational level [START_REF] Woodman | Toward a Theory of Organizational Creativity[END_REF]. Creativity thus emerges from the dynamic interactions among individuals and their engagement with the various facets of their environment, and specific skills and conditions must be present for creativity to occur [START_REF] Epstein | How is creativity best managed? Some empirical and theoretical guidelines[END_REF]Amabile, 1983). As a result, many professionals and scholars in the realm of creativity have been motivated to explore optimal approaches for managing creativity [START_REF] Epstein | How is creativity best managed? Some empirical and theoretical guidelines[END_REF][START_REF] Mejia | Major and recent trends in creativity research: An overview of the field with the aid of computational methods[END_REF]. Understanding how to intentionally foster individual [START_REF] Egan | Factors influencing individual creativity in the workplace: An examination of quantitative empirical research[END_REF][START_REF] Phelan | Understanding creativity in the workplace: An examination of individual styles and training in relation to creative confidence and creative self-leadership[END_REF] and collective creativity [START_REF] Zhou | Research on workplace creativity: A review and redirection[END_REF][START_REF] Cirella | Organizational variables for developing collective creativity in business: A case from an Italian fashion design company[END_REF], through both direct and subtle means [START_REF] Stieler | Innovation nudging-A novel approach to foster innovation engagement in an incumbent company[END_REF], has become a pivotal focus in managerial strategies. This encompasses a spectrum of strategies, ranging from refining leadership competencies [START_REF] Ollila | Creativity and innovativeness through reflective project leadership[END_REF][START_REF] Bäcklander | Doing complexity leadership theory: How agile coaches at Spotify practise enabling leadership[END_REF] to providing comprehensive employee training [START_REF] Rampa | Developing radical innovation capabilities: Exploring the effects of training employees for creativity and innovation[END_REF][START_REF] Valgeirsdottir | Studying creativity training programs: A methodological analysis[END_REF]. It extends further to the strategic utilization of objects [START_REF] Chen | The influence of objects on creativity[END_REF] and the deliberate design of physical spaces [START_REF] Kristensen | The physical context of creativity[END_REF][START_REF] Haner | Spaces for creativity and innovation in two established organizations[END_REF][START_REF] Perez Mengual | Physical interaction platforms: A taxonomy of spaces for interactive value creation[END_REF][START_REF] Haner | Spaces for creativity and innovation in two established organizations[END_REF], as well as virtual environments [START_REF] Abi Saad | Creativity in virtual teams: Systematic review, synthesis and research agenda[END_REF], all aimed at enhancing and magnifying the creative potential within organizational contexts.

However, individuals involved in creative activities are typically not unaware of the nature of creativity and how it can be nurtured. Everyone tends to formulate their own understanding of creativity, shaping layperson's implicit theories of creativity that often draw from commonly held beliefs that might not necessarily be supported by scientific evidence.

Such beliefs about the nature and process of creativity significantly interact with individuals' behavior. For instance, the belief that creativity can be developed through training and effort leads to improved problem-solving outcomes and more innovative ideas [START_REF] O'connor | Consequences of Beliefs about the Malleability of Creativity[END_REF]. The misconception that creative idea generation declines over time harms creative performance (Lucas & Nordgen, 2020). Or those who think that creativity and mental illness are linked tend to perform better in creative tasks [START_REF] Kaufman | Insane, poetic, lovable: Creativity and endorsement of the "Mad Genius" stereotype. Imagination[END_REF]. Although organizations and managers strive to control and boost creativity, there has been limited focus on investigating the precise influence of LIToCs related to stimulating creativity on actual creative performance. For instance, in the context of alcohol's impact on creativity, there's a common belief that it has a positive effect on creative output. However, while studies have explored cognitive performance under the influence of alcohol (e.g., [START_REF] Benedek | Creativity myths: Prevalence and correlates of misconceptions on creativity[END_REF][START_REF] Lapp | On the Enhancement of Creativity by Alcohol: Pharmacology or Expectation?[END_REF][START_REF] Benson | Attentional and working memory performance following alcohol and energy drink: A randomised, double-blind, placebocontrolled, factorial design laboratory study[END_REF], the specific influence of this belief on creativity has been largely overlooked. In the only study investigating the effect of holding LIToC regarding the stimulation of creativity, the focus is on how it impacts decisions about how to manage organizational creativity, and not on creative performance [START_REF] Baas | Conceiving creativity: The nature and consequences of laypeople's beliefs about the realization of creativity[END_REF].

Such gap in our current understanding of implicit theories of creativity poses an issue because if deeply held beliefs about what spurs creativity interact with real creative performance, attempts to oversee creativity within organizations could yield counterproductive outcomes. Consequently, it is imperative to conduct additional research to ascertain whether having strong convictions about the drivers of creativity either enhances or hinders creative productivity when these convictions align with the actual methods of stimulating creativity. This research thus aims to contribute to a deeper comprehension of when and how creativity management strategies might result in either favorable or adverse outcomes.

THEORETICAL BACKGROUND

Laypeople's Implicit Theories of Creativity (LIToC)

Scholars specializing in creativity often find themselves in situations where they share their research findings with various audiences, such as students, practitioners, and even friends. In these interactions, they frequently encounter individuals who harbor strong preconceptions about what creativity entails and how it operates. Some of these preconceived notions align with the findings of scientific research, influenced in part by popular press portrayals of creativity [START_REF] Rainone | Lost in translation? A review of popular-press claims about organizational creativity and innovation[END_REF], while others delve into more obscure and unconventional interpretations [START_REF] Baas | Conceiving creativity: The nature and consequences of laypeople's beliefs about the realization of creativity[END_REF]. Everyone seems to form their personal conception of creativity, which can take the shape of widely held beliefs that sometimes lack scientific support. For example, half the population believes that "Creative thinking mostly happens in the right hemisphere of the brain" or that "Creative accomplishments are usually the result of a sudden inspiration" (Benedek et al., 2021, p3). These assumptions about creativity can differ significantly across different cultural contexts, [START_REF] Loewenstein | Implicit theories of creative ideas: How culture guides creativity assessments[END_REF], and may also be influenced by the specific professional domains or occupational niches to which individuals belong [START_REF] Malmelin | Rethinking creativity in journalism: Implicit theories of creativity in the Finnish magazine industry[END_REF].

In essence, what becomes evident is that many people tend to tailor their understanding of creativity to align with their personal viewpoints, giving rise to a diverse range of beliefs that can occasionally lack firm grounding, coherence, or even internal consistency. Despite these variations, these individual beliefs continue to play a significant role in shaping the broader lay theories of creativity [START_REF] Ritter | Lay theories of creativity[END_REF].

Implicit theories encompass the convictions that individuals harbor about the characteristics of both humans and nonhumans, as well as broader phenomena on a global scale [START_REF] Furnham | Lay theories of schizophrenia[END_REF][START_REF] Zedelius | The science of lay theories[END_REF] However, when these individuals were prompted to tap into their LIToCssuch as when evaluating profiles of students or employees who could potentially be creativetheir approach seemed to shift towards embracing distinct, more grounded interpretations of creativity [START_REF] Lim | Creativity through a lens of social responsibility: Implicit theories of creativity with Korean samples[END_REF][START_REF] Sternberg | Implicit theories of intelligence, creativity, and wisdom[END_REF][START_REF] Plucker | The Patient is Thriving! Current Issues, Recent Advances, and Future Directions in Creativity Assessment[END_REF].

The literature identifies four categories of LIToC that concern what creativity is (Nature), how creativity occurs (Process), who can be creative (Person) and what can foster creativity (Stimulation) [START_REF] Benedek | Creativity myths: Prevalence and correlates of misconceptions on creativity[END_REF]. As an illustration, [START_REF] Baas | Conceiving creativity: The nature and consequences of laypeople's beliefs about the realization of creativity[END_REF] demonstrate that LIToCs regarding the Process of creativity often do not match self-reported processes leading to actual creative behavior. Typically, people generally believe that creativity occurs unexpectedly while they describe past creative behavior as deliberate and conscious work. and what is not [START_REF] Pichot | The construct validity of creativity: empirical arguments in favor of novelty as the basis for creativity[END_REF][START_REF] Weisberg | From explicit to implicit theories of creativity and back: The relevance of naive criteria in defining creativity[END_REF]. Individuals also hold LIToCs about the creative Person, such as the stereotype of the "eccentric poet" [START_REF] Dumas | The creative stereotype effect[END_REF][START_REF] Fisher | The Designer's Self-Identity-Myths of Creativity and the Management of Teams[END_REF] or of the "mad genius" [START_REF] Kaufman | Insane, poetic, lovable: Creativity and endorsement of the "Mad Genius" stereotype. Imagination[END_REF][START_REF] Prichard | Creative selves? Critically reading 'creativity'in management discourse[END_REF], and use them to discern who they judge as creative (Gralewski & Karwowski, 2016;[START_REF] Paek | Teachers' beliefs of creative children[END_REF][START_REF] Puccio | Implicit theories of creativity: Laypersons' perceptions of the creativity of adaptors and innovators[END_REF][START_REF] Runco | Implicit theories of artistic, scientific, and everyday creativity[END_REF][START_REF] Sternberg | Implicit theories of intelligence, creativity, and wisdom[END_REF]. Regarding stimulating creativity, people tend to believe that people get more creative ideas under the influence of alcohol or marijuana (which is a creativity myth), or that when stuck on a problem, it is helpful to continue working on it after taking a break (which is a creativity fact) [START_REF] Benedek | Creativity myths: Prevalence and correlates of misconceptions on creativity[END_REF]. The strength of LIToCs varies across individuals, depending in part on individuals' age [START_REF] Delany | Adolescents' implicit theories of a creative person: A longitudinal investigation in three countries[END_REF][START_REF] Ho | Understanding the influence of epistemological beliefs on the creative learning process[END_REF], cognitive abilities (Saunders [START_REF] Wickes | Measuring gifted adolescents' implicit theories of creativity[END_REF], culture [START_REF] Delany | Adolescents' implicit theories of a creative person: A longitudinal investigation in three countries[END_REF][START_REF] Paletz | Implicit theories of creativity across cultures: Novelty and appropriateness in two product domains[END_REF][START_REF] Paletz | In the world or in the head: External and internal implicit theories of creativity[END_REF][START_REF] Seng | Implicit theories of creativity: a comparison of student-teachers in Hong Kong and Singapore[END_REF], education [START_REF] Konstantinidou | Primary physical education perspective on creativity: The nature of creativity and creativity fostering classroom environment[END_REF], or job [START_REF] Spiel | Implicit theories of creativity: The conceptions of politicians, scientists, artists and school teachers[END_REF][START_REF] Stone | Art teachers' beliefs about creativity[END_REF][START_REF] Tang | Implicit theories of creativity in computer science in the United States and China[END_REF].

Influence of LIToCs on creative performance

LIToCs about the Nature and Process of creativity can have a direct impact on people's behaviour. For instance, O'Connor, [START_REF] O'connor | Consequences of Beliefs about the Malleability of Creativity[END_REF] showed that the belief that creativity is malleable (i.e., that creativity can be trained and requires effort to manifest) produces better problem-solving outcomes and more innovative ideas, a relationship mediated by creativity motivation [START_REF] Li | From implicit theories to creative achievements: The mediating role of creativity motivation in the Do you believe Red Bull gives wings? relationship between stereotypes, growth mindset, and creative achievement[END_REF]. As an addition, Lucas and Nordgen (2020) demonstrated that people mistakenly believe that creative idea generation decreases over time and showed that this belief negatively influences actual creative performance. Thus, LIToCs can have a direct impact on creative performance. Other research has shown that different beliefs about creativity will guide individuals to socialize and organise differently [START_REF] Ho | Understanding the influence of epistemological beliefs on the creative learning process[END_REF][START_REF] Paletz | In the world or in the head: External and internal implicit theories of creativity[END_REF][START_REF] Simmons | The influence of goal orientation and risk on creativity[END_REF]. If you believe that creativity is based on internal Do you believe Red Bull gives wings? traits, you will focus on thinking, writing, designing, whereas beliefs about external traits will promote social interactions, building, sports performance, etc. LIToCs about the creative Person is related to creative performance as well: individuals believing in the stereotype of the mad genius, i.e., that creativity and mental illness are connected, perform better on a creative task [START_REF] Kaufman | Insane, poetic, lovable: Creativity and endorsement of the "Mad Genius" stereotype. Imagination[END_REF].

Managers deploy great efforts in organizations to manage creativity and stimulate creative performance [START_REF] Tan | Managing creativity in organizations: A total system approach[END_REF][START_REF] Isaksen | Managing for innovation: The two faces of tension in creative climates[END_REF]). Yet, the specific influence of LIToCs about creativity Stimulation on creative performance has been relatively ignored. For instance, while there is a general belief that alcohol produces favorable effects on creative output, and various studies have examined performance on many cognitive tasks while either sober or under the influence of alcoholic beverages (e.g., [START_REF] Benedek | Creativity myths: Prevalence and correlates of misconceptions on creativity[END_REF][START_REF] Benson | Attentional and working memory performance following alcohol and energy drink: A randomised, double-blind, placebocontrolled, factorial design laboratory study[END_REF][START_REF] Lapp | On the Enhancement of Creativity by Alcohol: Pharmacology or Expectation?[END_REF], the effect of the specific belief regarding the link between alcohol and creativity is completely overlooked. Baas and colleagues (2015) are the only ones who studied the effects of LIToCs regarding creativity drivers. They show that holding such LIToCs influences decisions about how to manage organizational creativity: for instance, people who believe that creativity is enhanced by positive activating moods tend to choose a stimulating room over a relaxing one to manage creativity. However, they do not further investigate the influence of LIToCs on creative performance per se. Given that individuals and collectives make choices and invest considerable efforts in creating conditions that foster creativity (e.g. [START_REF] Chaoji | Managers' search practices at the front end of radical manufacturing technology innovations[END_REF][START_REF] Epstein | How is creativity best managed? Some empirical and theoretical guidelines[END_REF][START_REF] Lenfle | When project management meets design theory: revisiting the Manhattan and Polaris projects to characterize 'radical innovation' and its managerial implications[END_REF], it is crucial to assess the influence of LIToCs regarding these drivers on creative performance.

Conceptual model

In many domains, laypeople's implicit theories influence perceptions, efforts, beliefs, goals, and affects [START_REF] Zedelius | The science of lay theories[END_REF]. They create the framework of meaning within which attributions occur and are important for understanding motivation and coping strategies [START_REF] Hong | Implicit theories, attributions, and coping: a meaning system approach[END_REF]. As such, laypeople's implicit theories could ultimately shape individuals' outcomes, such as performance [START_REF] Rattan | Embedding mindsets in context: Theoretical considerations and opportunities for studying fixed-growth lay theories in the workplace[END_REF].

Strictly speaking, the effect of a LIToC about what can foster creativity should be conceptualized as the effect of holding beliefs about the drivers of creativity while precisely not activating those drivers. This echoes the phenomenon of the placebo effect, which can be defined as the therapeutic effect produced by a substance or procedure that has no specific activity for the condition being treated, but an effect that would not have occurred without the administration of said placebo [START_REF] Kienle | The powerful placebo effect: fact or fiction?[END_REF][START_REF] Moerman | Deconstructing the placebo effect and finding the meaning response[END_REF]. When applied to the context of stimulating creativity, this implies to suggest that a creativity driver is activated while not activating it. In such a context, if individuals hold the belief that the driver can enhance their creative abilities, receiving a placebo (perceiving that the driver is activated) could augment their self-assurance in their ability to perform, thereby increasing creative self-efficacy beliefs [START_REF] Tierney | Creative self-efficacy: Its potential antecedents and relationship to creative performance[END_REF]Puente-Diaz, 2016;[START_REF] Shaw | Measuring creative self-efficacy: An item response theory analysis of the creative self-efficacy scale[END_REF] defined as beliefs in one's capability to successfully complete a creative task. Conversely, if individuals do not hold the belief that the driver can enhance their creative abilities, their creative self-efficacy will remain unaffected. This is because these individuals will not experience any psychological mechanisms being triggered when that specific driver is activated. In essence, the absence of belief in the efficacy of a certain creativity driver means that there is no corresponding psychological response that can impact their creative selfefficacy. This leads to the formulation of Hypothesis 1:

H1: The strength of a LIToC regarding a creativity driver moderates the influence of the perceived activation of that creativity driver on creative self-efficacy beliefs, such

Do you believe Red Bull gives wings? that the perception of the activation of that creativity driver will improve creative selfefficacy beliefs, but only for individuals displaying a strong LIToC regarding that creativity driver.

Generating ideas typically involves a challenging process that demands both time and mental exertion, and it comes with a considerable level of uncertainty and potential for failure. Consequently, it is crucial for individuals to cultivate the self-assurance required to persevere in their creative brainstorming efforts, ultimately nurturing creative self-efficacy [START_REF] Tierney | Creative self-efficacy: Its potential antecedents and relationship to creative performance[END_REF]. Creative self-efficacy thus plays a crucial role in shaping creative behaviors and outcomes [START_REF] Mathisen | Organizational antecedents of creative self-efficacy[END_REF][START_REF] Nisula | The antecedents of individual innovative behaviour in temporary group innovation[END_REF]. When individuals possess high levels of creative self-efficacy, they are more likely to take on challenging creative tasks, persist in the face of obstacles, and generate novel solutions. On the other hand, individuals scoring low on creative self-efficacy might be hesitant to engage in creative tasks and may doubt their capacity to come up with innovative ideas. This leads to the formulation of the mediated moderation Hypothesis 2: H2: Creative self-efficacy beliefs will mediate the interactive relationship of the perception of the activation of a creativity driver and the strength of a LIToC regarding that creativity driver on creative performance, such that the perception of the activation of a creativity driver will only enhance creative self-efficacy beliefs and creative performance among individuals displaying a strong LIToC regarding that creativity driver.

Thus, a moderated mediation model through the creative self-efficacy route may predict how LIToCs regarding creativity drivers could increase creative performance. But creative self-efficacy beliefs are not the only antecedent of individual creative behaviors. As a cognitive process, creativity requires certain cognitive efforts [START_REF] Doyle | The creative process: Effort and effortless cognition[END_REF]. Indeed, creativity involves intricate mental activities that encompass divergent thinking, problemsolving, pattern recognition, and the generation of novel ideas. These cognitive processes are an integral part of the creative act, as individuals use a variety of mental pathways to explore Do you believe Red Bull gives wings? alternative perspectives and original solutions. This means that creativity demands an engagement of cognitive faculties beyond routine thinking. In this perspective, the dual pathway to creativity model, initially proposed by De Dreu et al. ( 2008) and further explored by [START_REF] Baas | A meta-analysis of 25 years of research on mood and creativity: Hedonic tone, activation, or regulatory focus?[END_REF] and [START_REF] Nijstad | The dual pathway to creativity model: Creative ideation as a function of flexibility and persistence[END_REF], offers the core premise that creative outcomes, such as original ideas and problem-solving, can be achieved through two distinct cognitive pathways: the flexibility pathway and the persistence pathway. The flexibility pathway encompasses a creative cognitive style characterized by flexible, fluent, and divergent thinking [START_REF] Winkielman | The hedonic marking of processing fluency: Implications for evaluative judgment[END_REF][START_REF] De Dreu | Hedonic tone and activation in the mood-creativity link: Towards a dual pathway to creativity model[END_REF][START_REF] Roskes | Necessity is the mother of invention: avoidance motivation stimulates creativity through cognitive effort[END_REF], while the persistence pathway entails a creative cognitive style that relies on persistent and systematic thinking [START_REF] Nijstad | The dual pathway to creativity model: implications for workplace creativity[END_REF]. The latter approach involves high effort, perseverance, and a slower pace of cognitive operation. It relies more heavily on executive control processes and is constrained by working memory capacity [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF][START_REF] Dietrich | A review of EEG, ERP, and neuroimaging studies of creativity and insight[END_REF].

Besides, ideas generated during an ideation phase tend to bear a strong resemblance to solutions applied to analogous past issues, unless deliberate cognitive exertion is channeled toward broadening the exploration and encoding of information beyond the scope initially activated [START_REF] Weisberg | Problem solving and creativity[END_REF]. Consequently, the level of effort exerted in a creative task can influence the resulting creative performance. This prompts to consider how the perception of being exposed to something that is believed to enhance creativity could affect the amount of cognitive effort dedicated to creativity.

While no specific research has investigated the link between beliefs and efforts in a creativity situation, notable findings have emerged from the literature on complex problemsolving. Studies investigating the impact of one's belief in God on self-regulation reveal that individuals engaged in a word game containing several God-related terms displayed reduced effort in a subsequent anagram task [START_REF] Laurin | Divergent effects of activating thoughts of God on self-regulation[END_REF]. This decrease in effort was more pronounced if these individuals had earlier communicated to the researchers their conviction in the potential influence of external forces on their probability of achieving success in complex tasks. Similarly, research on beliefs about free will has shown that rejecting free will attenuates the activation of brain areas involved in the decision to perform a specific action, thus affecting intentional commitment to the action [START_REF] Rigoni | Inducing disbelief in free will alters brain correlates of preconscious motor preparation: The brain minds whether we believe in free will or not[END_REF].

Overall, this suggests that when people believe that external forces could help them accomplish a task, they would attribute their success not only to their own ability but also to external factors. Thus, they would regulate their engagement in the task, which could affect their performance. Applied to the perception of the activation of a creativity driver, this would mean that those holding a strong LITOC regarding that creativity driver could perceive the activation of that creativity driver as a "magic pill", something that helps the creative process, leading them to put less effort into the task, and harming creative performance, resulting in a counterproductive effect on creative performance. Hypotheses 3 and 4 are thus formulated as follows:

H3: The strength of a LIToC regarding a creativity driver moderates the influence of the perceived activation of that creativity driver on the efforts dedicated to creativity, such that the perception of the activation of that creativity driver will decrease the efforts dedicated to creativity, but only for individuals displaying a strong LIToC regarding that creativity driver. H4: The efforts dedicated to creativity will mediate the interactive relationship of the perception of the activation of a creativity driver and the strength of a LIToC regarding that creativity driver on creative performance, such the perception of the activation of a creativity driver will only decrease the efforts dedicated to creativity and creative performance among individuals displaying a strong LIToC regarding that creativity driver.

METHOD

Activating LIToCs: the case of energy drinks

In the following study, we investigated the implicit theories people may have about the stimulating effects of energy drinks on creativity to determine whether they promote or hinder creativity. We opted to employ energy drinks as a catalyst for creativity that could be introduced in a creative task due to its practicality and straightforward implementation.

Energy drinks are popular and continue to represent a rapidly growing share of the beverage market in Western countries. Students around the world consume them to improve their academic performance [START_REF] Badar | Determinants of energy drinks consumption among the students of a Saudi University[END_REF][START_REF] Ghozayel | Energy drinks consumption and perceptions among University Students in Beirut, Lebanon: A mixed methods approach[END_REF]. A recent study suggests that 50% of members of the U.S. military consume energy drinks, hoping for an energy boost and/or increased mental alertness [START_REF] Attipoe | Energy beverage use among US service members[END_REF]. Other reports suggest that healthcare providers also regularly consume energy drinks to boost performance and delay fatigue in the workplace [START_REF] Guilbeau | Health Risks of Energy Drinks: What Nurses and Consumers Need to Know[END_REF]. In sum, energy drinks have found their way in the workplace because of the implicit theory that individuals have about their cognitive effects. Indeed, several studies have described the placebo effect of drinking juice labeled as an energy drink on motivation and risk-taking behaviors (e.g., [START_REF] Cornil | Does Red Bull give wings to vodka? Placebo effects of marketing labels on perceived intoxication and risky attitudes and behaviors[END_REF]Schmidt et al., 2017). Thus, energy drinks have great potential to activate LIToCs in individuals, while being increasingly present in organizations.

Procedure and measures

The experiment took place in the offices of a professional market research institute and was advertised as a test for a new beverage. In the placebo condition, participants were greeted and offered a plastic glass containing a pre-mixed beverage, supposedly consisting of fruit juices mixed with Red Bull. However, the beverage was only composed of a mixture of various strong and unusual tasting fruit juices (e.g., guava, passion fruit, pineapple) in order to make it plausible for participants that it could contain Red Bull. The protocol was the same for participants in the control condition, except that the beverage was presentedwhich was trueas being composed solely of fruit juices. To reinforce the placebo condition, piles of Red Bull cans were displayed in the experimental room in the placebo condition.

Participants were then given 10 minutes to drink their beverage while answering questions regarding their self-perceived creativity using 4 7-point items adapted from [START_REF] Amabile | Creativity in Context: Update to the Psychology of Creativity[END_REF] and [START_REF] George | Dual tuning in a supportive context: Joint contributions of positive mood, negative mood, and supervisory behaviors to employee creativity[END_REF] (e.g., "I generally approach problems in a creative manner", "I like to use my imagination to come up with several ideas") to be controlled for, along with participants' age and gender. They also reported their drinking habits for different beverages, including energy drinks, on a 5-point frequency scale. We then measured the LIToCs by asking questions about their beliefs about the stimulating effects of energy drinks, using 3 ad hoc 7-point items (i.e., "Drinking energy drinks stimulates the brain / encourages risk-taking / has a disinhibiting effect"). Fiability analysis encouraged us to only retain 2 items (i.e., "Drinking energy drinks stimulates the brain / encourages risk-taking).

We did not ask direct questions about their belief regarding the influence of energy drinks on creativity as we did not want participants to know that this was an experiment investigating creativity per se. Then, participants were asked to spend the next 10 minutes writing down as many creative ideas as possible to complete the "egg task", which is commonly used in research on individual creativity (e.g., Agogué et al., 2014a[START_REF] Agogué | The impact of age and training on creativity: A design-theory approach to study fixation effects[END_REF][START_REF] Agogué | Nudging individuals' creativity using social labeling[END_REF][START_REF] Bissola | Organizing individual and collective creativity: Flying in the face of creativity clichés[END_REF]. The egg task is presented as follows: "Please suggest as many original solutions as possible to the following problem: Make sure that a hen's egg dropped from a height of 10 meters does not break." We then measured creative self-efficacy using 4 7-point items from Shaw, Kapnek and Morelli (2021) (e.g., "I was confident in my capacity to generate creative ideas / answer creatively to the exercise"). Finally, participants were asked to answer other survey questions about their experience with the creative task and to guess what the study was about. No participant guessed that our purpose was to study LIToCs. To measure the efforts dedicated to the creativity task, we chose not to rely on selfassessment of efforts, as such self-report may be tainted by the belief held by the participants regarding their performance and involvement in the task. We therefore chose to use fluency, i.e., the number of solutions proposed by each participant, as a proxy for the efforts deployed.

In line with the dual pathway to creativity model (see De Dreu et al, 2008), we measured individual creative performance through participants' responses on two criteria: flexibility (the ability to generate solutions in different directions, measured by the number of solution categories) and originality (the ability to provide some highly original solutions). To assess flexibility and originality, we draw on previous studies (i.e., Agogué et al., 2014a;[START_REF] Camarda | Do we need inhibitory control to be creative? Evidence from a dual-task paradigm[END_REF] that proposed a coding matrix consisting of 62 categories of solutions so that each idea (for instance "we can attach a big helium balloon to the egg") can be affected to one of the 62 categories (for instance Category "Balloon / Air Balloon"). Each category within this matrix is then linked to a specific originality score, which was calculated with a 10-point scale Do you believe Red Bull gives wings?

ranging from 1 ("not at all original") to 10 ("highly original") using [START_REF] Amabile | Social psychology of creativity: A consensual assessment technique[END_REF] Consensual Assessment Technique: two trained raters evaluated each category on this scale, and interrater agreement was excellent (97%). For each participant, another independent trained rater then used this coding matrix to calculate the number of categories explored and the Top 2 originality index proposed by [START_REF] Silvia | Assessing creativity with divergent thinking tasks: Exploring the reliability and validity of new subjective scoring methods[END_REF], which consists of the sum of the scores of each participant's two most original responses. Following this approach, participants were evaluated by the best level of performance they could achieve on a constant number of responses. In line with Pichot and colleagues' ( 2022) argument that novelty is primarily the driver of creativity, we did not assess the value or utility of ideas.

Participants

While most studies on creative cognition rely on student samples, our experiment involved 69 subjects (35 females and 34 males, mean age=43) recruited through a professional market research institute. Each participant was randomly assigned to one of the two experimental conditions, i.e., the control (n=34) vs. the placebo condition (n=35). The two groups did not differ in terms of gender (² (1) =.01, ns), age (F (1,67) =1.74, ns), selfperceived creativity (F (1,67) =.00, ns), beliefs about the stimulating effects of energy drinks (F (1,67) =.18, ns), or frequency of energy drink consumption (F (1,67) =.07, ns).

All the participants provided written consent and were tested in accordance with international norms governing the study of human research participants. The Ethic Committee of one of the authors' institutions approved this study. For ethical purposes, after completing the entire procedure, the participants in the placebo condition were informed that the beverage offered to them was only a mixture of various fruit juices.

RESULTS

To test our hypotheses, we used a floodlight analysis, i.e., a spotlight analysis repeated for all the values of the moderating variable [START_REF] Cadario | Reconsidering the practice of discretizing quantitative variables: Towards a new analysis of moderation in experimental research[END_REF]. Specifically, we used SPSS's PROCESS macro [START_REF] Hayes | Introduction to mediation, moderation, and conditional process analysis: A regression-based approach[END_REF] to perform such floodlight analysis for each of our mediators, and each of our 2 creativity measures, considering our manipulation, participants' beliefs, and their interaction as independent variables, and controlling for participants' age, gender, frequency of energy drink consumption, and self-perceived creativity.

We first acknowledge the correlations between the variables of our model, as illustrated in Table 2 : Regarding the influence on creative self-efficacy, a moderation analysis was conducted using SPSS's PROCESS macro [START_REF] Hayes | Introduction to mediation, moderation, and conditional process analysis: A regression-based approach[END_REF], Model 1. The analysis shows no direct effect of the activation of a creativity driver on creative self-efficacy (p=.730). The interaction between the activation of a creativity driver and the strength of a LIToC regarding that driver was not significant (p = .813>.05), indicating that the relationship between the activation of a Do you believe Red Bull gives wings? creativity driver and creative self-efficacy was not moderated by the strength of the LIToC. This doesn't support H1. We didn't further investigate H2.

Regarding the influence on efforts deployed in the task, a moderation analysis was conducted using SPSS's PROCESS macro [START_REF] Hayes | Introduction to mediation, moderation, and conditional process analysis: A regression-based approach[END_REF], Model 1. The interaction between the activation of a creativity driver and the strength of a LIToC regarding that driver was significant (F=9.59, p = .003), indicating that the relationship between the activation of a creativity driver and the efforts put into the creative task was moderated by the strength of the LIToC, supporting H3. The simple slope of X on Y was significant at high strength of LIToC (t = -2.57, p= .012) but also tends to be significant at low strength of LIToC (t = 1.90, p = .06). Figure 2 To investigate H4, a moderated mediation analysis was conducted using SPSS's PROCESS macro [START_REF] Hayes | Introduction to mediation, moderation, and conditional process analysis: A regression-based approach[END_REF], Model 7. Hypothesis H4 suggested that the indirect effect of Do you believe Red Bull gives wings? the activation of a creativity driver on Creative performance through the efforts put into the creative task will be moderated by the strength of LIToC regarding the creativity driver. Table 3 displays our analysis. H4 was supported as the index of moderated mediation (index = -3.41, 95% CI = [-5.8827/-1.0796] for flexibility and index = -1.41, 95% CI = [-3. 3611/-0.1269] for originality) is significant since the 95% CI does not include zero. Figure 3 displays our results.

Figure 3 -The relationship between the activation of a creativity driver and creative performance (flexibility and originality) was moderated by the strength of the LIToC.

DISCUSSION

Do you believe Red Bull gives wings?

Theoretical contributions

In seeking to sustain creativity, organizations adopt several strategies, including recruiting individuals that exhibit more creative qualifications [START_REF] Ford | Interpretive Style, Motivation, Ability and Context as Predictors of Executives' Creative Performance[END_REF], establishing corporate cultures that promote creativity [START_REF] Amabile | How to kill creativity[END_REF][START_REF] Nemeth | Managing Innovation: When Less is More[END_REF], using training and specific methods to facilitate the generation, sharing and distribution of ideas [START_REF] O'connor | Major innovation as a dynamic capability: A systems approach[END_REF][START_REF] Rampa | Developing radical innovation capabilities: Exploring the effects of training employees for creativity and innovation[END_REF][START_REF] Yekanialibeiglou | Enhancing creativity in activity-based offices: A critical incident study of knowledge workers[END_REF] or designing spaces for creative activities [START_REF] Kristensen | The physical context of creativity[END_REF][START_REF] Haner | Spaces for creativity and innovation in two established organizations[END_REF]). Yet such strategies mostly focus on alterations to the environment or creating incentives to stimulate creativity. Less dominant in the organizational literature is how an individual's beliefs, attitudes, and expectations about how creativity may be stimulated influence their creativity. We contribute to this gap by offering insights on the potential influence of LIToCs regarding the stimulation of creativity.

This research investigated how the potency of Lay Implicit Theories of Creativity (LIToCs) concerning factors that drive creativity can impact the creative performance of individuals who believed these factors were triggered. We conducted an experiment where participants consumed an identical beverage comprising a blend of different fruit juices. They were informed that it was either a standard fruit juice blend (the control group) or a mixture of Red Bull and fruit juices (the placebo group). Consequently, we focused exclusively on the psychological influence of beliefs about the invigorating effects of energy drinks while accounting for potential physiological effects. Our results showed an interaction effect with subjects' LIToC, such that, among subjects displaying strong LIToC about the stimulating effects of energy drinks, creative performance was lower in the placebo condition than in the control condition, due to a decreased level of efforts deployed in the creative task. Oddly, having strong LIToCs did not influence creative self-efficacy. Overall, this suggests an underlying cognitive mechanism: having strong beliefs in the effects of some creativity factors might trigger a complacent attitude when these drivers are activated and reduce the effort invested in generating creative ideas. However, our results also underline two surprising additional elements.

First, having weak beliefs regarding a creativity driver trigger additional efforts in the task when individuals perceived that this driver is activated, which leads to greater creativity as well. This is quite surprising to see that having little belief is actually beneficial for creativity. This resonates with the fact that just as people cope with their own beliefs or lay theories, they may be aware that others hold lay theories as well and form perceptions of others' lay theories, i.e., meta-lay theories [START_REF] Rattan | Meta-lay theories of scientific potential drive underrepresented students' sense of belonging to science, technology, engineering, and mathematics (STEM)[END_REF][START_REF] Lou | The role of meta-lay theories in ESL learners' mindsets and need satisfaction[END_REF], in line implicit theory of mind, i.e., "humans' ability to reason about the mental processes (e.g., beliefs) of others and to recognize that these may be different from their own" (Schneider et al., 2014, p. 43). Such meta-lay theories may then influence behavior. Indeed, in our experiment, just before performing the creative task, participants were asked about their beliefs about energy drinks and given a beverage to drink. This made the lay theory about creativity salient When a specific presumed driver of creativity is activated, individuals holding a weak belief about this driver may try to attribute its activation and form the metalay theory that greater creativity is expected. Therefore, when having a weak LIToC regarding a creativity driver and being confronted with this driver, individuals would anticipate high expectations of creative performance, and therefore put more effort and involvement in the task, as such enhancing creative performance.

Second, our results show an interesting effect of LIToCs regarding creativity drivers in a condition where these creativity drivers are not present: indeed, in the condition where the creativity driver was not activated, in the condition where participants were told they were drinking a mixture of different juices (and indeed they were), those with strong LIToCs regarding energy drinks performed better than those with weak LIToCs, due to more effort deployed in the creativity task. As previously mentioned, questions about LIToCs about the stimulating effects of energy drinks were asked before engaging in the creative task, highlighting the hypothesized link between energy drinks and creative performance. For individuals displaying strong LIToCs, not receiving Red Bull may have led them to believe that they were on their ownnot supported by any energy drinkand that in order to perform well, they had to compensate and double their effort, leading to better creative performance. This means that contexts that are perceived to be difficult or hostile for creativity, where resources that are supposed to be helpful are not present, are beneficial for individuals that believe in the positive influence of such context. This echoes the work on the role of constraints in the creative process [START_REF] Caniëls | Organizing Creativity: Creativity and Innovation under Constraints[END_REF], that showed that supportive aspects within the organization help counteract the adverse influence of constraining organizational factors on cultivated creativity (ibid).

Managerial implications

Individual creativity is influenced by numerous factors that may be managed, including mood [START_REF] Baas | A meta-analysis of 25 years of research on mood and creativity: Hedonic tone, activation, or regulatory focus?[END_REF], creative self-efficacy [START_REF] Tierney | Creative self-efficacy: Its potential antecedents and relationship to creative performance[END_REF]Mathisen & Bronnick, 2009), efforts invested in the task or beliefs regarding creativity. This malleability of individual creative performance is consistent with previous studies that have shown that LIToCs regarding the Nature and Process of creativity will guide individuals to socialize and organise differently (Baas, et.al., 2015;[START_REF] Paletz | Implicit theories of creativity across cultures: Novelty and appropriateness in two product domains[END_REF][START_REF] Simmons | The influence of goal orientation and risk on creativity[END_REF][START_REF] Stone | Art teachers' beliefs about creativity[END_REF]. Yet, our study offers new insights into the specific influence of LIToCs regarding the Stimulation of creativity on creative performance. We argue that when individuals believe they are supported by a favorable external factorthe combination of strongly believing that something will help them be creative and apparently receiving itthey adopt a complacent attitude. They expect the favorable external factor to help them perform well, and thus put less effort into the task, which leads to poorer creative performance. On the other end, not being providing with something that is supposed to boost creative performance ends up Do you believe Red Bull gives wings?

stimulating efforts in the creative task. This offers another perspective on direct incentives to boost creativity: it is indeed usually among the managers' tasks to shape an organization's culture likely to disseminate values and beliefs regarding creativity, thus providing psychological safety [START_REF] Hu | Leader humility and team creativity: The role of team information sharing, psychological safety, and power distance[END_REF] and sustaining individual ideation through an innovative climate (Liu et al., 2021). Direct instructions from managers to employeessuch as instructions to "be creative"have been shown to have a concrete impact on creative performance [START_REF] Harrington | Effects of explicit instructions to "be creative" on the psychological meaning of divergent thinking test scores[END_REF][START_REF] Nusbaum | Ready, set, create: What instructing people to "be creative" reveals about the meaning and mechanisms of divergent thinking[END_REF] Besides, finding ways to boost individual and collective creative behaviors [START_REF] Stieler | Innovation nudging-A novel approach to foster innovation engagement in an incumbent company[END_REF] is at the heart of many managerial arrangementsand we argue that beliefs about creativity interact with these arrangements. In their seminal study on the influence of LIToCs on individual behaviors, Baas and colleagues (2015) showed that a strong belief that creativity is facilitated by defocused and flexible thinking, positive moods, and approach motivation rather than systematic and persistent thinking, negative moods, and avoidance motivation shaped the decisions people made when asked to design a creativity room. In this study, participants chose a relaxation room nearly twice as often as the activation room and having strong LIToCs on creativity reinforced by positive activating moods was a positive predictor of choosing the activation room over the relaxing room. This led the authors to conclude that "managers may seek more creative problem solving in their employees by having expensive, but unproductive relaxation rooms built " (p. 351-352). As such, LIToCs can influence the design of a creativity space, a topic that has been of interest for partitioners and academics [START_REF] Schmidt | Open creative labs: Spatial settings at the intersection of communities and organizations[END_REF]. Baas and colleagues' (2015) study found that Do you believe Red Bull gives wings?

LIToCs lead decision makers to propose an environment that is actually irrelevant to creativity.

Our study complements these findings by examining the potential behaviors that users of these spaces may engage in based on their own LIToCs. If a person strongly believes that a certain type of physical environment has a direct positive effect on creativity, when working in a creative space that matches those beliefs, that person will feel supported, helped, and may become complacent. This will lead them to invest less creative effort, and therefore produce less creativity. On the other hand, if this person has no a priori beliefs, when placed in this type of environment while being made aware of the potential influence of this environment (e.g., the merits of this creativity room are explicitly praised), they may have the intuition that they are expected to be creative and will therefore make more effort and be more creative. In other words, managers need to be creative in managing the antecedents of creativity in organizations: if they play on levers that people already have strong beliefs about their effectiveness, it will be counterproductive. Consequently, managers in charge of designing and facilitating creative devices to support creativity need to regularly renew the way they stimulate creative activities within their organizations in order to counteract any mechanism of self-indulgence. As such, they need to be creative about means to be creative.

Limitations and future research

This research has several limitations that offer paths for future research. First, we chose to investigate a very specific LIToC about the stimulation of creativity (namely the effects of energy drinks), which was practical for our experimental design, but may not be the most disseminated LIToC about stimulation. As such, additional research is needed to assess the effects of a broader range of LIToCs regarding the stimulation of creativity on creative performance. The present study extends previous research on the influence of beliefs about the potential cognitive stimulating effect of energy drinks [START_REF] Champlin | Is the consumption of energy drinks associated with academic achievement among college students?[END_REF][START_REF] Marinoni | Psychological and socio-educational correlates of energy drink consumption in children and adolescents: a systematic review[END_REF], but could be replicated in the contexts of other creativity drivers, such as factors related to the physical environment of creativity (e.g., creative room colors, furniture style, object) that are regularly mobilized in organizations to sustain creativity (e.g. [START_REF] Yekanialibeiglou | Enhancing creativity in activity-based offices: A critical incident study of knowledge workers[END_REF][START_REF] Chen | The influence of objects on creativity[END_REF]. Second, our experimental design focused on a very specific part of the creative process, namely the generation phase of the ideation process, and at an individual level. Future studies are needed to examine the effect of LIToCs about creativity drivers using other creative tasks than the egg's task, such as convergent thinking tasks, and on other cognitive processes critical to creativity such as idea construction [START_REF] Gillier | Group creativity in the wild: When building on ideas enhances the generation and selection of creative ideas[END_REF] or idea communication [START_REF] Steils | Effective creative crowdsourcing: A multi-dimensional evaluation framework[END_REF], taking into account beliefs in one's own ability to be creative [START_REF] Puente-Díaz | Idea generation, selection, and evaluation: A metacognitive approach[END_REF]. Finally, we did not impact the LIToCs of our participants, while research has shown that lay theories are malleable and can evolve [START_REF] Benedek | Creativity myths: Prevalence and correlates of misconceptions on creativity[END_REF]. In that vein, research could investigate the influence of individual characteristics on the nature, strength and potential evolution of LIToCs regarding creativity in order to derive more relevant implications for managers wishing to manage creativity processes in their organizations.

  . Laypeople's Implicit Theories of Creativity (LIToC) describe lay beliefs and myths about creativity. Such LIToCs are not necessarily stable and homogeneous, even within the same individual: previous research indeed indicates that people's responses to both specific and open-ended questions about their beliefs regarding creativity were inclined to mirror what they thought was socially desirable.

  Do you believe Red Bull gives wings?LIToCs about the Nature of creativity influence laypeople's ability to judge what is creative

Figure 1

 1 Figure 1 synthesizes our conceptual model, with our 4 hypotheses and two (potentially

Figure 1

 1 Figure 1 -Conceptual model

  displays our result.

Figure 2 -

 2 Figure 2 -The relationship between the activation of a creativity driver and the efforts put into the creative task was moderated by the strength of the LIToC.

  , while sometimes generating psychological reactance (O'Hara & Sternberg, 2001). Influencing LIToCs of employees thus become a new way to nudge creativity (Agogué & Parguel, 2020), as long as there is a clear diagnosis of the non-alignment of the LIToCs and the organizational context provided for creativity.

Table 1

 1 presents elements regarding the psychometric scales used in our research design.

	Do you believe Red Bull gives wings?

Table 2 -

 2 Psychometric scales

	Variable	Items	Factor	Mean	Cronbach Alpha /
			loadings	(SD)	Correlation for 2 items
	Self-perceived	To what extent do you …
	creativity
		Approach problems in a creative manner	.807
		Enjoy discovering new ways to consider an	.732	5.70	.801
		issue
		Like to engage my imagination to generate	.857	(0.87)
		multiple ideas
		Enjoy working on exceptional ideas	.701
		To what extent do you think that drinking
		energy beverages …
	LIToC	Stimulates the brain	.819	4.73	.707
		Encourages risk taking	.892	(1.35)
		During the creative task, to what extent were
		you confident in your capacity
	Creative self-	To do well in the task	.878
	efficacy	To realize efficiently the task	.868	5.29	.892
		To answer creatively	.872	(1.27)
		To propose original solutions	.745

Table 2 -

 2 Correlations between the main variable of the model . The correlation is significant at the 0.05 level (two-tailed).

				Efforts
		Flexibility	Originality	dedicated to the creativity	Creative self-efficacy	Frequency of energy drink consumption	Self-perceived creativity
				task
	Flexibility	1	,496 **	,808 **	0,106	-0,159	0,065
	Originality		1	,376 **	0,059	0,098	-0,028
	Efforts
	dedicated to the creativity			1	0,15	-0,097	0,099
	task
	Creative self-efficacy				1	,306 *	,440 **
	Frequency of
	energy drink					1	,254 *
	consumption
	Self-
	perceived						1
	creativity
	**. The correlation is significant at the 0.01 level (two-tailed).

*

Table 3 -

 3 Regression analysis using SPSS's Hayes PROCESS macro

	Creative Self-efficacy (SPSS's Hayes PROCESS macro -Model 1)
	R2 = .3144
		Coeff	t	p
	Constant	1.84	1.82	.073
	Activation of a creativity driver	.14	.34	.730
	LIToC	.54	1.34	.183
	Interaction between the activation of a creativity driver and the
	strength of a LIToC	-.13	-.23	.813
	Frequency of energy drink consumption	.12	.83	.409
	Gender	-.61	-2.04	.045
	Age	.01	.84	.399
	Self-perceived creativity	.56	3.32	.001
	Efforts dedicated to creativity task (SPSS's Hayes PROCESS macro -Model 1)
	R2 = .2377
		Coeff	t	p
	Constant	5.79	1.89	.062
	Activation of a creativity driver	2.41	1.90	.061
	LIToC	4.45	3.65	.001
	Interaction between the activation of a creativity driver and the
	strength of a LIToC	-5.22	-3.09	.003
	Frequency of energy drink consumption	-.80	-1.76	.082
	Gender	1.07	1.20	.234
	Age	-.07	-1.96	.052
	Self-perceived creativity	.68	1.32	.191
	Flexibility (SPSS's Hayes PROCESS macro -Model 7)
	R2 = .6730
		Coeff	t	p
	Constant	2.59	1.53	.129
	Activation of a creativity driver	-.51	-1.13	.261
	Efforts dedicated to creativity task	.65	10.45	.001
	Frequency of energy drink consumption	-.30	-1.28	.204
	Gender	-1.17	-.36	.714
	Age	-.01	-73	.468


Angelo Rodio

Giovanni Neglia

Fabio Busacca

Stefano Mangione

Sergio Palazzo

Francesco Restuccia

Ilenia Tinnirello

Federated Learning with Packet Losses

Keywords:

This paper tackles the problem of training Federated Learning (FL) algorithms over real-world wireless networks with packet losses. Lossy communication channels between the orchestrating server and the clients affect the convergence of FL training as well as the quality of the learned model. Although many previous works investigated how to mitigate the adverse effects of packet losses, this paper demonstrates that FL algorithms over asymmetric lossy channels can still learn the optimal model, the same model that would have been trained in a lossless scenario by classic FL algorithms like FedAvg. Convergence to the optimum only requires slight changes to FedAvg: i) while FedAvg computes a new global model by averaging the received clients' models, our algorithm, UPGA-PL, updates the global model by a pseudo-gradient step; ii) UPGA-PL accounts for the potentially heterogeneous packet losses experienced by the clients to unbias the pseudo-gradient step. Still, UPGA-PL maintains the same computational and communication complexity as FedAvg. In our experiments, UPGA-PL not only outperforms existing stateof-the-art solutions for lossy channels (by more than 5 percentage points on test accuracy) but also matches FedAvg's performance in lossless scenarios after less than 150 communication rounds.

I. INTRODUCTION

Federated Learning (FL) [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF], [START_REF] Reddi | Adaptive Federated Optimization[END_REF] involves a population of devices (typically referred to as clients) iteratively training a Machine Learning (ML) model over a network under the orchestration of a central server. At each global training round, the server sends the current global FL model to the clients, who individually train on their local datasets and send their locally-trained models back to the server. In many FL applications, such as training Google keyboard next-word prediction model, the clients are mobile devices such as smartphones or Internet of Things (IoT) devices, and the models are exchanged on wireless networks incurring potential transmission losses.

Lossy channels necessarily degrade the performance of FL training on wireless networks. Theoretical and experimental work [START_REF] Eris | Unveiling the Wireless Network Limitations in Federated Learning[END_REF]- [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF] has shown that packet losses affect the quality of the final model towards which the FL training algorithms converge as well as their convergence rate. Under medium/high network background traffic, the authors in [START_REF] Eris | Unveiling the Wireless Network Limitations in Federated Learning[END_REF] measured a twofold training duration and a halved accuracy in the early stages of the training. Prior work [START_REF] Yang | Scheduling Policies for Federated Learning in Wireless Networks[END_REF]- [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF] analyzed the convergence of state-of-the-art FL algorithms under different channel assumptions. Specifically, the authors of [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF] proved the existence of a non-vanishing error due to the lossy channels, which prevents the convergence of their direct model aggregation scheme to the optimal model, and proposed to reduce this error by opportunely allocating resources (e.g., transmission power, radio blocks) to control packet losses. Similar approaches to the one proposed in [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF] have been considered to mitigate the effect of packet loss on wireless networks, relying on automatic repeat request (ARQ) and forward error correction (FEC) techniques [START_REF] Wen | An Overview of Data-Importance Aware Radio Resource Management for Edge Machine Learning[END_REF], [START_REF] Su | On Model Transmission Strategies in Federated Learning With Lossy Communications[END_REF].

Despite these efforts, packet losses are typically caused by external factors beyond the control of the orchestrating server and can therefore be unavoidable. Communication protocols may define a maximum number of retransmissions, but these retransmissions can still fail. More importantly, we point out that targeting high transmission reliability in FL-oriented applications may be sub-optimal, as it usually comes at the detriment of training time and/or resource usage, e.g., in terms of wider sub-channel bandwidth, higher energy consumption, or both. These issues are even more exacerbated in resourceconstrained scenarios, such as the IoT, where increasing communication reliability may result in a reduced device lifetime or may not be feasible. Moreover, the iterative nature of the gradient methods used for ML model training makes them robust against limited errors at intermediate calculations [START_REF] Xing | Strategies and Principles of Distributed Machine Learning on Big Data[END_REF].

For the aforementioned reasons, this paper diverges from prior work [START_REF] Chandrasekaran | FHDnn: Communication efficient and robust federated learning for AIoT networks[END_REF]- [START_REF] Su | On Model Transmission Strategies in Federated Learning With Lossy Communications[END_REF], which primarily focused on loss mitigation. Instead, we address the fundamental question of whether FL algorithms can achieve optimal model convergence despite packet losses. Our response is affirmative, necessitating only slight adjustments to the classic FedAvg [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF] algorithm.

More in detail, we consider a FL framework where losses can occur in the downlink, uplink, or both, and loss probabilities can differ among clients. Indeed, the channel quality in wireless networks can vary according to per-user characteristics, such as the relative positioning of the transmitter and the receiver, the device transmission power, the selected frequency channel. As a result, the clients will not participate evenly in the training process, potentially leading to learning a biased ML model [START_REF] Wang | Tackling the Objective Inconsistency Problem in Heterogeneous Federated Optimization[END_REF], [START_REF] Rodio | Federated Learning under Heterogeneous and Correlated Client Availability[END_REF]. Thus, the design of an aggregation strategy becomes critical to ensure the convergence of the FL model in the presence of packet losses. Previous works [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF], [START_REF] Su | On Model Transmission Strategies in Federated Learning With Lossy Communications[END_REF] proposed direct model aggregation schemes, denoted in the following as DMA-PL and UDMA-PL. We will discuss and compare our approach to them in the rest of the paper.

This work makes the following novel contributions:

• We propose UPGA-PL, a novel algorithm that aggregates pseudo-gradients, instead of models, and considers the client loss probabilities. Its complexity is comparable to FedAvg;

• We analytically prove UPGA-PL's convergence to the optimal model, the same model which would have been learned over ideal, lossless channels. This result proves UPGA-PL's ability to filter out the noise due to packet losses. • We validate our analysis through numerical experiments.

While losses largely affect the performance of the state-ofthe-art algorithms [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF], [START_REF] Su | On Model Transmission Strategies in Federated Learning With Lossy Communications[END_REF], UPGA-PL is robust to them and gains 5-10 percentage points on the test accuracy. Most importantly, even under severe losses, UPGA-PL achieves the same model's accuracy as FedAvg under lossless channels in less than 150 communication rounds.

II. PROBLEM DESCRIPTION AND BACKGROUND

A central server and a set of clients K = {1, . . . , K} collaborate to train a global ML model w ∈ R n over a wireless network. For example, the model w ∈ R n can be the vector of parameters of a neural network architecture. Each client k ∈ K holds a local dataset D k and has access to a loss function ℓ(w, d k ) → R + that evaluates the performance of the model w on a data sample

d k ∈ D k . We define F k (w) := 1 |D k | d k ∈D k ℓ(w, d k )
as the average loss computed after evaluating the performance of the model w on the k-th client's local dataset. The clients solve, under the coordination of the central server, the minimization of the global objective F (w) via the following optimization problem:

min w∈R n F (w) := k∈K α k F k (w) , (1)
where {α k } k∈K are positive coefficients, chosen by the server, such that k∈K α k = 1. They represent the weight assigned to each client's objective function F k . Typical choices are: 1) α k = 1/K ∀k ∈ K, the server giving equal weight to all clients, 2)

α k = |D k |/
w k t,j+1 = w k t,j -η t ∇F k (w k t,j , B k t,j ) j = 0, . . . , E -1, (2)
where η t > 0 is an appropriately chosen learning rate, B k t,j is a random batch sampled from client-k's local dataset at round t and local iteration j, and ∇F k (•, B) := = k∈K α k w k t,E : this scheme was introduced in the algorithm FedAvg [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF], and we refer to it as Direct Model Aggregation (DMA). Alternatively, the server can consider the model updates ∆ k t received by the clients as pseudo-gradients, aggregate them as ∆ t = k∈K α k ∆ k t , and finally apply a global pseudo-SGD step as w PGA t+1 = w t + ∆ t . This aggregation scheme was proposed in [START_REF] Reddi | Adaptive Federated Optimization[END_REF]: it corresponds to FedOpt with SGD used both as server and client optimizer. We denote it as Pseudo-Gradients Aggregation (PGA).

The DMA and PGA aggregation schemes are equivalent under lossless channels. However, in typical FL applications the information is transmitted over lossy channels, which ultimately affect the workflow of the considered FL algorithm.

We consider the same scenario as in [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF], [START_REF] Su | On Model Transmission Strategies in Federated Learning With Lossy Communications[END_REF]: due to downlink losses, only a subset of clients P D t ⊆ K correctly receives the model w t sent by the server and computes the local models {w k t,E } k∈P D t . On the other hand, due to losses in the upstream, the server gathers the updates (either the models w k t,E or the pseudo-gradients ∆ k t ) only from a subset of clients P t = P U t ⊆ P D t . Note that if transmissions span multiple packets and only some packets are affected by losses, the recipient could still leverage the partial information correctly received. Previous literature [START_REF] Yang | Scheduling Policies for Federated Learning in Wireless Networks[END_REF], [START_REF] Ye | Decentralized Federated Learning With Unreliable Communications[END_REF], [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF]- [START_REF] Su | On Model Transmission Strategies in Federated Learning With Lossy Communications[END_REF] has ignored this possibility, which we plan to investigate in the future.

Since losses are random and can potentially differ among clients, the aggregation scheme plays an important role in the quality of the global ML model w t+1 learned by the FL training algorithm. Previous works [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF], [START_REF] Su | On Model Transmission Strategies in Federated Learning With Lossy Communications[END_REF] considered the problem of FL training under lossy channels and proposed to generalize FedAvg's DMA aggregation strategy by letting the server aggregate all received models, i.e., the models of all clients k ∈ P t :

w DMA-PL t+1 = k∈Pt α k w k t,E k∈Pt α k . (3)
We refer to this strategy as Direct Model Aggregation with Packet Loss (DMA-PL). The authors of [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF] also analyzed the convergence of the DMA-PL aggregation scheme under the effect of packet losses. They showed the existence of a generally non-vanishing error between the model trained under a non-zero loss rate and the optimal model towards which the training converges in the absence of losses:

E[F (w t+1 )] -F * ≤ A t (F (w 1 ) -F * )
vanishing term for small statistical heterogeneity

+ 2ζ 1 L k∈K α k p k 1 -A t 1 -A ,
non-vanishing error due to statistical heterog. and packet loss [START_REF] Chandrasekaran | FHDnn: Communication efficient and robust federated learning for AIoT networks[END_REF] where p k denotes the probability that the server does not receive client-k's local model, A = 1 -µ L + 4µζ2 L k∈K α k p k , L and µ are the L-smooth and µ-strongly convex constants (they will be introduced in Assumptions 1, 2), and ζ 1 , ζ 2 are parameters that quantify the statistical heterogeneity of the local datasets (the larger ζ 1 and ζ 2 , the more heterogeneous the clients' data). We observe that, for non-zero loss probabilities and high statistical heterogeneity (large ζ 2 ), it is possible that the bound does not guarantee convergence (when A ≥ 1). On the contrary, for sufficiently small ζ 2 , (4) predicts linear convergence to a neighborhood of the optimal solution, whose size is proportional to the loss probabilities {p k } k∈K . Motivated by these results, reference [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF] focuses on resource allocation to reduce loss probabilities and minimize the non-vanishing term.

Moreover, due to losses, only a subset of the clients contributes to updating the new model at each round. Previous works [START_REF] Wang | Tackling the Objective Inconsistency Problem in Heterogeneous Federated Optimization[END_REF], [START_REF] Li | On the Convergence of FedAvg on Non-IID Data[END_REF] have studied partial client participation due to client sampling, i.e., when the server samples at each round a subset of clients S t ⊆ K. Convergence results in [START_REF] Li | On the Convergence of FedAvg on Non-IID Data[END_REF] require unbiased sampling for DMA to converge to the optimal model, i.e., the sampling scheme should satisfy Lemma 4], so that in expectation the k-th client contributes proportionally to its weight in the global objective [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF]. This observation suggests to unbias the DMA-PL scheme in (3) as follows:

E St [w t+1 ] = k∈K α k w k t,E [14,
w UDMA-PL t+1 = k∈Pt α k 1 -p k w k t,E , (5)
so that the server counterbalances the more severe losses experienced by some clients with larger aggregation weights. We refer to this aggregation as Unbiased DMA-PL (UDMA-PL). However, by directly aggregating models, the UDMA-PL scheme suffers a possibly large variance due to the randomness in the set P t . Our analysis in Lemma 1 confirms that this variance leads to a non-vanishing term, which prevents UDMA-PL from converging to the optimal model. Moreover, our experimental results in Section IV confirm that UDMA-PL is not a practical solution.

In the next section, we present UPGA-PL, an unbiased aggregation scheme like UDMA-PL that filters out the noise due to losses and then succeeds in converging to the optimal model. To the best of our knowledge, only reference [START_REF] Ye | Decentralized Federated Learning With Unreliable Communications[END_REF] showed a similar result for a decentralized FL algorithm, but it required uplink and downlink channels to have the same loss probabilities, which is uncommon in wireless networks.

III. PROPOSED ALGORITHM AND ITS ANALYSIS

To solve the issues which characterized the DMA-PL and UDMA-PL aggregation schemes, we propose the Unbiased Pseudo-Gradient Aggregation strategy (UPGA-PL):

w UPGA-PL t+1 = w t + k∈Pt α k 1 -p k ∆ k t . (6)
Note that both UDMA-PL and UPGA-PL rely on the knowledge of the loss probabilities {p k } k∈K . In practical scenarios, these probabilities can be estimated through channel measurements [START_REF] Benko | A Passive Method for Estimating End-to-End TCP Packet Loss[END_REF], [START_REF] Yajnik | Measurement and Modelling of the Temporal Dependence in Packet Loss[END_REF].

As UDMA-PL, UPGA-PL is unbiased because it aggregates the pseudo-gradients with weights that compensate for clients' different loss probabilities. At the same time, by aggregating the pseudo-gradients {∆ k t } k∈Pt rather than the local models {w k t,E } k∈Pt (as UDMA-PL does), UPGA-PL can be seen as a stochastic approximation algorithm [START_REF] Borkar | Stochastic Approximation: A Dynamical Systems Viewpoint[END_REF]

← w t + k∈Pt α k 1-p k ∆ k t ; Output: Final model w T +1 .
the optimal model is guaranteed if η t decreases fast enough to filter out the noise due to the randomness in the set P t (i.e., t η 2 t < +∞), but also slow enough for the algorithm to be able to move from the initial tentative model (w 1 ) to the optimal one (i.e., t η t = +∞). Our theoretical analysis below confirms these qualitative considerations: the UPGA-PL aggregation strategy enables the convergence of FL training algorithms to the optimal model even in the presence of lossy channels.

With abuse of language, we refer to the FL algorithm defined by the local update rule in (2) and the UPGA-PL aggregation scheme in (6) simply as UPGA-PL. The complete procedure is summarized in Algorithm 1. Similarly, we denote by DMA-PL and UDMA-PL the FL algorithms obtained replacing line 8 in Algorithm 1 with (3) and (5), respectively.

In the following, we analyze the convergence of UPGA-PL.

A. Convergence Analysis

For the analysis of the UPGA-PL algorithm, we make the following hypotheses. Assumptions 1 and 2 are standard in the literature on convex optimization [18, Sections 4.1, 4.2]. Assumptions 3 and 4 are standard hypothesis in the analysis of federated optimization algorithms [START_REF] Li | On the Convergence of FedAvg on Non-IID Data[END_REF], [19, Section 6.1]. Assumption 1. {F k } k∈K are L-smooth: for all v and w,

F k (v) ≤ F k (w) + ⟨∇F k (w), v -w⟩ + L 2 ∥v -w∥ 2 2 . Assumption 2. {F k } k∈K are µ-strongly convex: for all v and w, F k (v) ≥ F k (w) + ⟨∇F k (w), v -w⟩ + µ 2 ∥v -w∥ 2
2 . Assumption 3. Let B k t,j be a random batch sampled from the k-th device's local data uniformly at random. The variance of stochastic gradients in each device is bounded:

E ∇F k (w k t,j , B k t,j ) -∇F k (w k t,j ) 2 ≤ σ 2 k for k ∈ K. Assumption 4. The expected squared norm of stochastic gra- dients is uniformly bounded, i.e., E ∇F k (w k t,j , B k t,j ) 2 ≤ G 2
for k ∈ K and t ∈ T , j = 0, . . . , E -1.

We use the indicator variable ξ k t to denote the outcome of the t-th communication round between the server and the client k: ξ k t equals one if and only if the server correctly receives client-k's local model at round t. Assumption 5. At each round t ∈ T , the communication outcomes {ξ k t } k∈K are independent among clients. For each client k ∈ K, the outcomes {ξ k t } t∈T are independent and identically distributed (iid) over time with mean E[ξ k t ] = 1 -p k . In Assumption 5, p k denotes the probability that the overall communication between the server and client k fails either because client k does not receive the global model w t or because later the server does not receive client-k's update ∆ k t . If these events are independent, and p sk and p ks denote the downstream and upstream loss probabilities, respectively, then p k = 1 -(1 -p sk )(1 -p ks ). If ARQ or FEC techniques are employed, then p k can be interpreted as the residual loss probability experienced by the k-th client after potential retransmissions and/or error corrections, therefore our analysis remains agnostic to these methods.

Assumption 5 provides the flexibility for different loss probabilities across clients in the uplink and downlink transmissions, but does not consider spatial or temporal correlations, such as those arising from inter-channel interference or fading effects. We believe that results for Markov Chain gradient descent methods (where random samples are taken on the trajectory of a Markov chain) could be used to study the convergence of UPGA-PL under correlated channels [START_REF] Rodio | Federated Learning under Heterogeneous and Correlated Client Availability[END_REF], [START_REF] Sun | On Markov Chain Gradient Descent[END_REF]. However, we defer this analysis to future work.

Convergence results for FL algorithms require to bound statistical heterogeneity in terms of some metric (e.g., [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF], [START_REF] Wang | Tackling the Objective Inconsistency Problem in Heterogeneous Federated Optimization[END_REF], [START_REF] Li | On the Convergence of FedAvg on Non-IID Data[END_REF]). We adopt the same metric introduced in [14]: If the local datasets are identical, then the functions {F k } k∈K coincide and Γ = 0. In general, Γ is larger the more heterogeneous the local data distributions are.

Theorem 1 (proof in Appendix A) establishes UPGA-PL convergence under lossy channels. It builds upon [START_REF] Li | On the Convergence of FedAvg on Non-IID Data[END_REF], which considers the ideal lossless scenario. Our primary technical contribution is captured in Lemma 1 (Appendix A), with the additional term in (8) that accounts for the lossy channels.

Theorem 1 (Convergence under lossy channels). Let Assumptions 1-5 hold and L, µ, σ k , G, p k defined therein. Choose diminishing learning rates as η t+1 = 2/µ 8κ+t , with κ := L/µ. Then, for each t ∈ T , UPGA-PL satisfies:

E F (w UPGA-PL t+1 ) -F * ≤ κ 8κ + t 2EC µ + 4L ∥w 1 -w * ∥ 2 asymptotically vanishing term , (7)
where:

C = k∈K α 2 k σ 2 k + 2(E-1) 2 G 2 + 6LΓ + EG 2 k∈K α 2 k p k 1 -p k effect of lossy channels . (8)
B. Discussion a) UPGA-PL enables convergence under lossy channels: Theorem 1 proves that the objective F (w), evaluated on the sequence of models {w t } t>0 computed by UPGA-PL, converges in expectation to its minimum value F * . Moreover, as the function F is strongly convex and then has a unique minimizer, the trained model converges also to the optimal one, i.e., lim t→∞ E[w UPGA-PL t ] = w * , where w * ∈ R n ∈ arg min w F (w). The UPGA-PL aggregation strategy (with decreasing learning rates) does not suffer then from residual convergence errors as DMA-PL and UDMA-PL do.

b) The effect of packet losses on the convergence: The constant C (see [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF]) quantifies the impact of lossy channels on the convergence in terms of the clients' loss probabilities {p k } k∈K . As expected, the larger the loss probabilities, the larger is C and the slower the convergence predicted by the bound in [START_REF] Baccarelli | AFAFed-Asynchronous Fair Adaptive Federated learning for IoT stream applications[END_REF]. Moreover, the convergence rate in Theorem 1 (O(1/t)) is comparable to the convergence rate of FedAvg in absence of losses under the same assumptions (O(1/(Et))) [START_REF] Li | On the Convergence of FedAvg on Non-IID Data[END_REF].

c) Convergence speed vs. residual error: The bound in (4) suffers from a non-zero residual error but may achieve linear convergence (A t decreases exponentially fast); our bound in [START_REF] Baccarelli | AFAFed-Asynchronous Fair Adaptive Federated learning for IoT stream applications[END_REF] removes such error at the cost of a sublinear convergence rate, i.e., of slower convergence. One may then think that for a short duration of the training period, DMA-PL is preferable to UPGA-PL. In reality, the bound (4) achieves such a rate requiring the use of full gradients at each client (i.e., σ 2 k = 0) and a single local gradient update at each communication round (i.e., E = 1) [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF]; however, these assumptions do not correspond to FL practice [START_REF] Kairouz | Advances and Open Problems in Federated Learning[END_REF].

IV. EXPERIMENTAL RESULTS

A. Experimental setup

In the experiments, we consider a population with K = 10 clients. We split the population into two groups G i , i = 1, 2, to which we associate different packet loss probabilities p i , i = 1, 2. After evaluating different loss configurations, we present a challenging setting with p 1 = 0.1 and p 2 = 0.9.

We perform experiments on two datasets: the LEAF Synthetic Dataset for multinomial classification [START_REF] Caldas | LEAF: A Benchmark for Federated Settings[END_REF] and the realworld MNIST dataset for handwritten digit recognition [START_REF] Deng | The MNIST Database of Handwritten Digit Images for Machine Learning Research[END_REF]. To introduce statistical heterogeneity in the clients' datasets, we distribute the data among clients in a non-IID fashion. The LEAF Synthetic Dataset allows direct control of statistical heterogeneity through the parameters α and β: in our experiments, we set α = β = 1. For MNIST, we generate a non-IID data distribution by splitting the labels among clients, with each client containing samples from only two classes [START_REF] Achituve | Personalized Federated Learning With Gaussian Processes[END_REF]. We define Problem [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF]

with α k = |D k |/|D| ∀k ∈ K.
For the Synthetic LEAF dataset, we train a linear classifier with a ridge penalization of parameter 5 × 10 -4 , which defines a strongly convex objective function that well aligns with our theoretical assumptions. As for MNIST, we use a CNN architecture with two convolutional and two fully connected layers, resulting in a non-convex objective function that introduces additional complexity to the learning process.

We compare UPGA-PL, in Algorithm 1, with DMA-PL (aggregation strategy in (3)), UDMA-PL (aggregation strategy in ( 5)), and an ideal lossless FedAvg (when p k = 0 ∀k ∈ K).

In the experiments, UDMA-PL and UPGA-PL rely on the knowledge of {p k } k∈K . For all algorithms, we tuned the learning rate η = {η t } t∈T via grid search with values η = {10 -3 , 10 -2.5 , 10 -2 , 10 -1.5 , 10 -1 }. The reported results are averaged over 10 random seeds.

B. Experimental results

Figures 1-2 compare the train loss and test accuracy of DMA-PL, UDMA-PL, and UPGA-PL on the Synthetic LEAF and MNIST datasets. For both datasets, we include the reference performance of the ideal lossless FedAvg.

a) UPGA-PL outperforms the baselines and converges to the optimal model: The experimental results unanimously confirm the advantages of the UPGA-PL aggregation strategy in terms of train loss and test accuracy on the two datasets. Indeed, UPGA-PL improves the state-of-the-art solutions by 12 percentage points on the Synthetic LEAF dataset (Fig. 1b) and by 6 percentage points on the MNIST dataset (Fig. 2b). Moreover, UPGA-PL nearly attains the same performance as the FedAvg algorithm in lossless scenarios after around 150 communication rounds for both the Synthetic LEAF dataset (Fig. 1a) and the MNIST dataset (Fig. 2a).

In line with our theoretical analysis, the numerical experiments also confirm the effects of lossy channels on the convergence captured by Theorem 1 and discussed in Section III-B.

b) The effects of packet losses on the convergence: FL algorithms perform best under the ideal scenario with lossless channels (p k = 0 ∀k ∈ K). Nevertheless, our experiments show that a severe amount of packet losses (p 1 = 0.1, p 2 = 0.9) slows down but does not prevent convergence to the optimal model, provided that UPGA-PL is used (UPGA-PL curve overlaps with FedAvg curve in the absence of losses). c) Residual errors: DMA-PL and UDMA-PL suffer from non-vanishing errors. The residual error of DMA-PL is evident in Figure 2a, where its loss curve reaches a plateau around the value 0.3, while UPGA-PL converges to the value 0.0, as the ideal FedAvg. On the other hand, the UDMA-PL aggregation strategy, which should, in theory, unbias the DMA-PL scheme, does not filter the variance introduced by the lossy channels and suffers a noisy convergence: the UDMA-PL performance dramatically oscillates in the experiments, and its mean accuracy lies around 50-70%.

V. CONCLUSIONS

This paper studied the problem of training FL algorithms over real-world wireless networks with lossy channels. We considered the presence of independent and identically distributed packet losses in the communication channels between the orchestrating server and the clients and we showed that the quality of the learned model is highly sensitive to the choice of the aggregation strategy. To mitigate the negative effects of packet losses, we proposed UPGA-PL, an algorithm that aggregates pseudo-gradients rather than models and that effectively converges to the optimal model under asymmetric lossy channels. While its complexity is comparable to FedAvg, under severe lossy settings UPGA-PL significantly outperformed the state-of-the-art solutions [START_REF] Chen | A Joint Learning and Communications Framework for Federated Learning Over Wireless Networks[END_REF], [START_REF] Su | On Model Transmission Strategies in Federated Learning With Lossy Communications[END_REF] and attained very close performance to the optimal scenario with ideal, lossless channels at the cost of a slower convergence. Our work enabled optimal FL training under lossy channels, and-we believe-opened interesting research questions. For example, if losses affect only a part of the transmitted model, would it be possible for the clients or the server to take advantage of the partial information received instead of ignoring it (as DMA-PL, UDMA-PL, and UPGA-PL do)? What happens if the losses are correlated (e.g., due to inter-channel interference and/or fading)? What if they change over time? APPENDIX A. Proof of Theorem 1

For the proof, we define the sequence wt,j = k∈K α k w k t,j . We denote: B t = {B t,0 , B t,1 , . . . , B t,E-1 };

(9) H t = {B 1 , P 1 , B 2 , P 2 , . . . , B t-1 , P t-1 }, [START_REF] Su | On Model Transmission Strategies in Federated Learning With Lossy Communications[END_REF] where B t,j = {B k t,j } k∈K is the set of random batches sampled at time (t, j) and H t includes all history up to the t-th round.

Finally:

E Pt,Bt|Ht ∥w t+1 -wt,E ∥ 2 = k∈K α 2 k p k 1 -p k E Bt|Ht w k t,E -w t 2 ≤ k∈K α 2 k p k 1 -p k η 2 t E 2 G 2 . ( 14
)
Conversely, for w t = w UDMA-PL t , the same proof technique leads to the bound in [START_REF] Wang | Tackling the Objective Inconsistency Problem in Heterogeneous Federated Optimization[END_REF], but the steps in ( 14) do not hold.

From [14, Lemma 1], [14, Lemma 2], and [14, Lemma 3], recursively:

E Bt|Ht ∥ wt,E -w * ∥ 2 ≤ ≤ (1 -η t µ) E ∥ wt -w * ∥ 2 + η 2 t B E-1 j=0
(1 -η t µ) j ( 16)

≤ (1 -η t µ) ∥ wt -w * ∥ 2 + η 2 t EB, (17)
where B = k∈K α 2 k σ 2 k + 6LΓ + 2(E -1) 2 G 2 . Combining ( 15) and ( 17), and applying Lemma 1, we have:

E Pt,Bt|Ht ∥w t+1 -w * ∥ 2 ≤ (1 -η t µ) ∥ wt -w * ∥ 2 + η 2 t EC. ( 18
)
The conclusion of the proof follows similar steps as [14, Theorem 1]. We require a learning rate η t ≤ ( 1 µ , 1 4L ) = 1 4L . Set η t+1 ≤ 2/µ 8κ+t , with κ := L/µ, such that η 1 = 1 4L . Then:

E [F (w t+1 )] -F * ≤ κ 8κ + t 2EC µ + 4L ∥w 1 -w * ∥ 2 . (19)

Definition 1 .

 1 Let F * and F * k be the minimum values of F and F k , respectively. The parameter Γ := F * -k∈K α k F * k quantifies the degree of data heterogeneity.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Train loss/test accuracy on the Synthetic LEAF dataset.

  ∥w t+1 -wt,E ∥ 2 + ∥ wt,Ew * ∥ 2 .

  with stepsize η t (it is easy to verify that each ∆ k t is proportional to η t ). Stochastic approximation theory suggests that convergence to Algorithm 1: UPGA-PL Input : Initial model w 1 ; Weights α = {α k } k∈K ; Client loss probabilities p = {p k } k∈K ; Learning rates {η t } t∈T ; Local steps E. 1 for global round t ∈ T do 2 for client k ∈ K, in parallel do

	3	w k t,0 = w t ;
	4	for j = 0, . . . , E -1 do
	5	w k t,j+1 = w k t,j -η t ∇F k (w k t,j , B k t,j );
	6	∆ k t ← w k t,E -w t ;
	7	Receive {∆ k t } from a subset P t ⊆ K of clients;
	8	w UPGA-PL t+1

  Lemma 1. Let Assumptions 4-5 hold, and w t = w UPGA-PL

	Conversely, for w t = w UDMA-PL t	, we have:
	E Pt|Ht,Bt	∥w t+1 -wt,E ∥ 2 =	k∈K	α 2 k	p k 1 -p k	w k t,E	2 . (12)
	Proof of Lemma 1.
								2
	E Pt|Ht,Bt	w t +	k∈K	α k 1 -p k	w k t,E -w t ξ k t -wt,E
			= Var	k∈K	α k 1 -p k	w k t,E -w t ξ k t	=
			=	k∈K	Var		α k 1 -p k	w k t,E -w t ξ k t	=
			=	k∈K	α k (1 -p k ) 2 w k 2 t,E -w t	2 Var ξ k t =
			=	k∈K	α 2 k	p k 1 -p k	w k t,E -w t	2 .
								t	.
								Then:
								E Pt,Bt|Ht	∥w t+1 -wt,E ∥ 2 ≤ η 2 t E 2 G 2	k∈K	α 2 k	p k 1 -p k	. (11)

This research was funded by the French National Research Agency (ANR) through the 3IA Côte d'Azur Investments in the Future project with reference number ANR-19-P3IA-0002; by Groupe La Poste, sponsor of Inria Foundation, in the framework of FedMalin Inria Challenge; by the U.S. Air Force Office of Scientific Research under contract number FA9550-23-1-0261; and by the U.S. Office of Naval Research under award number N00014-23-1-2221.