Camille Martinez-Almoyna

Amélie Saillard
email: amelie.saillard@univ-grenoble-alpes.fr

Lucie Zinger

Clément Lionnet
email: clement.lionnet@univ-grenoble-alpes.fr

Cindy Arnoldi
email: cindy.arnoldi@univ-grenoble-alpes.fr

Arnaud Foulquier
email: arnaud.foulquier@univ-grenoble-alpes.fr

Ludovic Gielly
email: ludovic.gielly@univ-grenoble-alpes.fr

Gabin Piton
email: gabinpiton@gmail.com

Orchamp Consortium

Tamara Münkemüller

Wilfried Thuiller
email: wilfried.thuiller@univ-grenoble-alpes.fr

Camille Martinez
email: cmartinezalm@gmail.com

Differential effects of soil trophic networks on microbial decomposition activity in mountain ecosystems

Keywords: trophic regulation, ecosystem functioning, multitrophic communities, soil food web, structural equation models, mountain ecosystems

Soil trophic networks are key to biogeochemical cycles, in particular decomposition.

However, few studies have yet quantified how microbial decomposition activity along environmental gradients is jointly driven by bacteria, fungi, and their respective consumers.

Here, we quantified these direct and indirect effects on decomposition and contrasted them between forests and open habitats using multiple elevational gradients in the French Alps.

While environmental control on microbial decomposition activity was comparable in the two habitats, the pathways and strengths of biotic predictors strongly differed. The fungal channel composition played a moderate role in forests, while the bacterial channel composition was critical in open habitats. Importantly, we found trophic regulation by consumers to be a key modulator of the direct environmental effects on decomposition in open habitats. These results highlight the need to integrate trophic regulation when predicting future ecosystem functioning.

INTRODUCTION

Soil biodiversity underpins a range of ecosystem functions and services crucial to human well-being, including carbon storage and litter decomposition [START_REF] Bardgett | Belowground biodiversity and ecosystem functioning[END_REF][START_REF] Smith | Biogeochemical cycles and biodiversity as key drivers of[END_REF]. Predicting the consequences of environmental change on soil biodiversity and associated ecosystem functions is therefore crucial [START_REF] Jansson | Soil microbiomes and climate change[END_REF][START_REF] Wall | Biodiversity in the dark[END_REF]. To do so, we need to better understand the regulatory pathways that occur in soil trophic networks, and their changes along environmental gradients and habitats (Thakur and Geisen, 2019;Wardle et al., 1998).

Bacteria and fungi are key decomposers of organic matter (De Boer et al., 2005;[START_REF] Swift | Decomposition in terrestrial ecosystems[END_REF]. Their activity is context-dependent and varies with climatic conditions, soil physico-chemistry and vegetation structure. In forests, the litter coming from the trees is of rather low quality, which generates rather acidic soils with high organic matter content.

Recycling organic matter thus requires the ability to use complex forms of carbon immobilised in recalcitrant litter [START_REF] Begon | Ecology. From Individuals to Ecosystems[END_REF]. This favours the dominance of many fungal species but also certain bacterial clades that represent the slow cycling of organic matter [START_REF] López-Mondéjar | Metagenomics and stable isotope probing reveal the complementary contribution of fungal and bacterial communities in the recycling of dead biomass in forest soil[END_REF]Wardle et al., 2004). In contrast, open habitats like grasslands, mineral surfaces and, to a lesser extent, heathlands have a higher availability of labile nutrients and a lower organic matter content [START_REF] Fierer | TOWARD AN ECOLOGICAL CLASSIFICATION OF SOIL BACTERIA[END_REF][START_REF] Hagedorn | Above-and belowground linkages shape responses of mountain vegetation to climate change[END_REF].

They are therefore more suitable for the dominance of many bacterial species but also certain groups of fungi representing the rapid cycling of organic matter [START_REF] Ruess | Decomposition pathways and successional changes[END_REF]Wardle et al., 2004). Following this observation, many studies have used the fungi to bacteria biomass ratio to quantify the importance of primary decomposers (i.e. bacteria and fungi) in relation to the environment for decomposition [START_REF] Maassen | Response of soil microbial biomass , activities , and community structure at a pine stand in northeastern Germany 5 years after[END_REF][START_REF] Malik | Soil Fungal:Bacterial Ratios Are Linked to Altered Carbon Cycling[END_REF]. In contrast, far fewer studies have compared the relative importance of bacterial community composition, fungal community composition and the environment in forests and open habitats. Even fewer studies have integrated and quantified the role of higher trophic levels for decomposition (Thakur and Geisen, 2019). Indeed, primary decomposers are part of a complex trophic network, which draws energy from the decomposition of soil organic matter, and which can also influence microbial decomposition activity. This trophic network is structured along two main channels and flows of energy (Moore et al., 2002;[START_REF] Moore | Resource compartmentation and the stability of real ecosystems[END_REF]Wardle et al., 2004). On the one hand, the bacterial channel is based on bacterial communities and the organisms that feed on bacteria, i.e. protists, nematodes and some micro-arthropods, and on the other hand, the fungal channel is based on fungal communities and their consumers, i.e. micro-arthropods (e.g. springtails and mites) and some nematodes and protists (Moore et al., 2002;Thakur and Geisen, 2019;Wardle et al., 2004). Studies of trophic regulation on the biomass of primary decomposers have shown varied results with either important or negligible top-down regulation for both fungi and bacteria [START_REF] Goncharov | Trophic Chains in the Soil[END_REF][START_REF] Mikola | NO EVIDENCE OF TROPHIC CASCADES IN AN EXPERIMENTAL MICROBIAL-BASED SOIL FOOD WEB[END_REF]Wardle et al., 1998). Moreover, the direct effects of this trophic regulation on the composition and diversity of primary decomposer communities are poorly understood, as are the indirect effects on microbial decomposition activity and the differences between channels (Thakur and Geisen, 2019). The composition of consumer communities can be expected to influence the composition of primary decomposer communities, for example when consumers preferentially feed on certain groups of bacteria or fungi (A' Bear et al., 2014; reported for nematodes, springtails and mites feeding on bacteria and fungi, [START_REF] Ruess | Food preferences of a fungal-feeding Aphelenchoides species[END_REF][START_REF] Schneider | Feeding preferences among dark pigmented fungal taxa (" Dematiacea") indicate limited trophic niche differentiation of oribatid mites (Oribatida, Acari)[END_REF]. Preferential feeding has the effect of relieving consumer pressure on the non-preferred taxa of primary decomposers, which indirectly influences microbial decomposition activity [START_REF] Crowther | Functional and ecological consequences of saprotrophic fungus-grazer interactions[END_REF]Thakur and Geisen, 2019;[START_REF] Trap | Ecological importance of soil bacterivores for ecosystem functions[END_REF].

The effect of indirect regulation by the consumers of bacteria and fungi may even be stronger than their direct effects on ecosystem functions (Hättenschwiler et al., 2005). However, so far, we still know little about how these trophic regulations influence microbial decomposition activity, and even less about how this influence varies between habitats dominated by different channels, such as forests or open habitats.

Finally, the existence of this trophic regulation could modulate the direct effects of the environment on microbial decomposition activity. Indeed, since consumers are also structured by the environment, their response could feed through to the composition of primary producers (Xiong et al., 2021). So far, these indirect environmental effects have never been quantified along varying environmental conditions and habitat types.

Here, we quantified the direct and indirect effects of the environment, primary decomposers, and their respective consumers on decomposition along elevational gradients in both forests and open habitats. We used extracellular enzymes to estimate microbial decomposition activity and environmental DNA (eDNA) extracted from the same soil samples to describe local compositions in fungi and bacteria, and their respective consumers among protists, nematodes and arthropods. We then built structural equation models (SEMs) to answer the following questions:

1-What are the main direct biotic and abiotic drivers of microbial decomposition activity and how does their importance vary between habitats? 2-Is there a trophic regulation of microbial decomposition activity via the composition of fungal and/or bacterial channels in both habitats? 3-Does the composition of the soil trophic network modulate the direct effect of the environment on microbial decomposition activity?

MATERIAL & METHODS

Study sites: Alpine elevation gradients characterised by two habitats

We studied the composition of soil communities in the French Alps along 18 elevational gradients of the ORCHAMP observatory (www.orchamp. osug.fr, Appendix S1).

The gradients were continuous from about 900 m to 3000 m, had a homogeneous exposure and slope, and were representative of the environmental and topographic variability of the French Alps. Between 2016 and 2018, at least five sampling plots (30*30 m, see Fig. S1 for details) were placed along each gradient, with an altitude difference of 200 m, resulting in a total of 37 forest sites and 64 open habitat sites (such as meadows or heaths, see Fig. S2 for details).

Climatic data

In mountain ecosystems, the presence of snow determines the start and length of the vegetation growing season. During this growing season, heat accumulation (i.e. energy available for growth) and soil exposure to frost (i.e. physiological limit of most metabolic processes) shape the composition of plant and microbial communities [START_REF] Choler | Consistent Shifts in Alpine Plant Traits along a Mesotopographical Gradient[END_REF][START_REF] Edwards | Changes in the seasonal snow cover of alpine regions and its effect on soil processes: A review[END_REF][START_REF] Zinger | Microbial diversity in alpine tundra soils correlates with snow cover dynamics[END_REF]. In each plot, we calculated growing degree days (GDD) to characterise heat accumulation, and freezing degree days (FDD) to characterise the intensity of freezing events during the growing season (Fig. S3). They are the sum of average daily degrees above (GDD) and below (FDD) zero accumulated over the growing season each year, averaged over the period 1988-2018, and modelled in the first soil horizon (up to 10 cm depth, see [START_REF] Martinez-Almoyna | Climate, soil resources and microbial activity shape the distributions of mountain plants based on their functional traits[END_REF], Table 1 for details). As FDD is a sum of negative degrees, the lowest FDD values correspond to the plots with the most frequent and intense frost stress events. GDD and FDD were calculated from the SAFRAN-SURFEX/ISBA-Crocus-MEPRA re-analysis [START_REF] Durand | Reanalysis of 44 yr of climate in the French Alps (1958-2002): Methodology, model validation, climatology, and trends for air temperature and precipitation[END_REF], a model that deals with weather and snow conditions in mountainous regions based on large-scale topographic features.

Sampling, physicochemical properties and total potential extracellular enzyme activity of the soil

In each plot, we sampled soil in three 2×2 m subplots with homogeneous vegetation, separated by a distance of 2 to 12 m (Fig. S1), taking particular care to avoid sample contamination. In each sub-plot, we took about ten 5-cm-diameter cores of superficial soil (ca.

1-8 cm deep) and deeper soil (ca. 8-16 cm), in order to properly describe the first soil horizons. The superficial soil cores were pooled and homogenised separately from the deeper soil cores, resulting in two composite soil samples per subplot. In total, we collected 606 soil samples over the three years, 222 in forest and 384 in open habitats.

We immediately recovered 15 g per soil sample for subsequent eDNA extractions, and sieved the remaining parts to 5.6 mm. Then, we froze 2.75 g of the sieved sample for subsequent analyses of extracellular enzyme activities, 5 g to quantify water content, and sieved the remainder to 2 mm for analyses of the soil physico-chemistry (soil pH, soil organic matter (SOM) and soil C/N as described in [START_REF] Martinez-Almoyna | Climate, soil resources and microbial activity shape the distributions of mountain plants based on their functional traits[END_REF], Appendix S2).

We estimated the potential extracellular enzyme activity of six extracellular enzymes involved in the decomposition of C-rich substrates (α-Glucosidase (AG, EC 3.2.1.20), β-1,4-Glucosidase (BG, EC 3.2.1.21), β-D-Cellobiosidase (CB, EC 3.2.1.91) and β-Xylosidase (XYL, EC 3.2.1.37)) and N-rich substrates (β-1,4-N-acetylglucosaminidase (NAG, EC 3.2.1.30) and leucine aminopeptidase (LAP,EC 3.4.11.1)) using standardised fluorimetric techniques [START_REF] Bell | High-throughput Fluorometric Measurement of Potential Soil Extracellular Enzyme Activities[END_REF]; see also [START_REF] Martinez-Almoyna | Climate, soil resources and microbial activity shape the distributions of mountain plants based on their functional traits[END_REF] for more experimental details). We calculated the total potential extracellular enzyme activity (total EAA) as the sum of the raw potential activity values of the six extracellular enzymes, which had comparable levels of potential activity (Fig. S4). We then standardized total EEA by soil organic matter (SOM) to capture difference of SOM turnover between sites [START_REF] German | Soil Biology & Biochemistry Optimization of hydrolytic and oxidative enzyme methods for ecosystem studies[END_REF]. SOMstandardisation also has the advantage of limiting confounding effects due to the correlation between total EEA and microbial biomass [START_REF] Crowther | The global soil community and its influence on biogeochemistry[END_REF][START_REF] Piton | Disentangling drivers of soil microbial potential enzyme activity across rain regimes: An approach based on the functional trait framework[END_REF].

Hereafter, we refer to this variable as microbial decomposition activity, as extracellular enzymes are the functional tools by which microbes break down organic matter (Burns 2013).

Specifically, we address the enzymatic potential for depolymerization and recycling of specific C-and N-rich substrates that dominate plant and microbial necromass (e.g., cellulose, hemicellulose, proteins, chitin and peptidoglycan, [START_REF] German | Soil Biology & Biochemistry Optimization of hydrolytic and oxidative enzyme methods for ecosystem studies[END_REF]. Note that investigating the whole decomposition process would require considering the action of other drivers (e.g. mechanical action of other soil species, action of other enzymes), and is not directly the subject of this article.

Soil eDNA extraction, amplification, sequencing and curation

From the 15 g per soil sample, we conducted the extraction of soil eDNA immediately in the field following the protocol of Taberlet et al. (2012). This method was used due to its easiness of implementation for large-scale samplings, and also because it produces ecological patterns that are similar to those obtained with total DNA, although it may under-sample rare taxa [START_REF] Zinger | Extracellular DNA extraction is a fast, cheap and reliable alternative for multi-taxa surveys based on soil DNA[END_REF]. For eDNA amplification, we targeted Eukaryotes, Fungi and Bacteria using the DNA markers Euka02 (18S rRNA), Fung02 (ITS) and Bact01 (16S rRNA) described in Taberlet et al. (2018). We added unique eight-base long tags to the 5' end of each primer in order to be able to find the original sample of each sequencing read after sequencing. PCR reactions were conducted in quadruplicates for each DNA extract and control. Finally, the PCR samples were pooled and purified prior to sequencing. We built one library per marker in 2016 and 2018 and two libraries per marker in 2017, using the METAFAST protocol. Sequencing was performed with pair-end sequencing technology on the Illumina platform at Fasteris, Geneva, Switzerland (2*125, Hiseq 2000, for Euka02, and 2*250 miseq, for Bact01 and Fung02).

We curated the sequencing reads using the obitools package [START_REF] Boyer | obitools: a unix-inspired software package for DNA metabarcoding[END_REF] and the R metabaR package (Zinger et al., 2021, Fig. S5). For each sequencing library, paired-end reads were assembled, assigned to their original samples and markers, and dereplicated with the OBITools package. We then removed PCR errors and grouped the remaining sequences into molecular operational taxonomic units (MOTUs). MOTUs were assigned a taxonomy using different databases depending on the marker. Cross-samples, reagents, sequencing contaminants, dysfunctional PCRs and other artefacts were eliminated using metabaR using conservative quality criteria [START_REF] Zinger | metabaR: An r package for the evaluation and improvement of DNA metabarcoding data quality[END_REF]. Next, we excluded samples for which only two or fewer PCR replicates remained. 562 samples remained after this curation procedure, of which 212 were from forest soils and 350 from open habitats. We finally performed a final clustering step to unify the MOTUs across the libraries for each marker. For more details, see Appendix S3 (1, Fig. S5 and Tables S1,S2)

Trophic classes and groups of MOTUs

In order to assess the importance of trophic regulation, we only considered organisms directly involved in decomposition (i.e. fungi and bacteria) and their consumers. Finally, we grouped them following their known role in the soil trophic network (i.e. according to their resources and predators, [START_REF] Elton | Animal Ecology[END_REF][START_REF] Gauzens | Trophic groups and modules: Two levels of group detection in food webs[END_REF] and according to their function in the ecosystem. MOTUs were therefore aggregated at two levels: first, at the level of their trophic class (e.g. fungi), and second, at the level of a subgroup of this trophic class, which we call a trophic group for simplicity (e.g. saprophytic fungi).

At trophic class level:

We separated bacterial MOTUs and fungal MOTUs into two different trophic classes, which respectively form the basis of bacterial and fungal channels (Wardle et al., 2004). We referred to the trophic classes of bacteria and fungi together as primary decomposers.

We then separated MOTUs feeding on bacteria from those feeding on fungi in two different trophic classes, using literature, expert knowledge and databases (Fig. S6). Among the bacterivores, we found mainly protists and nematodes, while among the fungivores, we found mainly mites and springtails. We removed the MOTUs referenced as feeding on both bacteria and fungi as they were very rare (~ 15 MOTUs and less 1 % of reads in forests and open habitats). This classification represents the knowledge we have been able to extract from the current literature and databases. We are aware that it may only represent the preferred diet (and not obligate, Geisen, 2016). Details on the number of MOTUs within each trophic class are in Appendix S3 (Table S3).

At the trophic group level:

We assigned MOTUs within each trophic class to more resolved trophic groups, to facilitate interpretation of compositional variation within trophic classes. For fungi, we used FUNGuild [START_REF] Nguyen | FUNGuild: An open annotation tool for parsing fungal community datasets by ecological guild[END_REF]), FungalTraits (Põlme et al., 2020) and Tedersoo et al. (2014) to divide the MOTUs into eleven trophic groups (Table S3). For bacteria, we used FAPROTAX combined with expert knowledge to derive six trophic groups (Louca et al., 2016;Sansupa et al., 2021, Table S3). Then, for the two consumer trophic classes, bacterivores and fungivores, we used taxonomy to assign MOTUs to more resolved trophic groups, as closely related taxa exhibit similar functions (Potapov et al., 2016;[START_REF] Schaefer | Oribatid mites show that soil food web complexity and close aboveground-belowground linkages emerged in the early Paleozoic[END_REF][START_REF] Wiens | Niche conservatism as an emerging principle in ecology and conservation biology[END_REF]. Details on the number of MOTUs within each trophic group are in Appendix S3 (Table S3).

Statistical analyses

To quantify the direct and indirect effects of the environment and the compositions of the four trophic classes on microbial decomposition activity, we first constructed variables representing the composition of each trophic class, and integrated these variables into a SEM. This was done for both forest and open habitats independently to highlight differences in decomposition regulation in the two habitats.

2.6.1. First step: Composition of each trophic class

2.6.1.1.Variables construction

To summarise the main variation of composition of a trophic class between samples of a given habitat, we performed a correspondence analysis (CA) for each trophic class for each habitat, using the dudi.coa function of the ade4 package in R (R version 3.6.1., Core Development Team, 2020; [START_REF] Dray | The ade4 Package: Implementing the Duality Diagram for Ecologists[END_REF], and we kept the first axis. Two samples with similar coordinates on the first axis had similar compositions of the trophic class considered. To describe the composition of a trophic class in a sample, we therefore used its coordinate on the first axis of the corresponding CA. CA is based on the chi2 distance, which is hardly sensitive to sampling depth, allowing it to be performed directly on the numbers of reads (Appendix S4, Figs. S7, S8).

2.6.1.2.Interpretation of the compositional CA axes

For each trophic class, we then used the more resolved information of the trophic groups to facilitate the interpretation, the aim being to visualise which compositional variations structure the position of the samples on the first CA axes. First, for each sample, we calculated the relative abundance of each trophic group within each trophic class as the sum of the number of reads of all MOTUs belonging to that trophic group standardised by the total sum of the number of reads of all MOTUs in the same trophic class. We then fitted generalised linear models to visualise the relative abundance of each trophic group as a function of the sample coordinates on the first CA axis (Appendix S5, Figs. S9, S10). Second, for each trophic class, we also used the variations in trophic diversity to interpret the compositional variations represented by the first CA axes. To do this, we calculated the trophic diversity of each trophic class in each sample, via the Shannon index applied to the relative abundances of trophic groups (Figs. 1,2).

Second step: Structural Equation Models

2.6.2.1.Construction of the SEMs and evaluation of path coefficients

For each habitat, we built a SEM to explain microbial decomposition activity. In this SEM, we used the first CA axes to describe the compositions of the different trophic classes and tested the existence of a trophic regulation of decomposition, through cascading links between the compositions of the different trophic classes (i.e. representing their trophic interactions). The SEM also allowed us to consider the direct effects of the environment on the composition of the trophic classes and on decomposition (see Appendix S6 for the a priori model).

We first tested the structure of the a priori model using the sem function in R [START_REF] Lefcheck | piecewiseSEM: Piecewise structural equation modelling in r for ecology, evolution, and systematics[END_REF]. This function locally estimates the path coefficients and then assesses the fit between the a priori model and the data by testing the conditional independence of variables that are not linked by a path [START_REF] Shipley | A new inferential test for path models based on directed acyclic graphs[END_REF]. Then, where necessary, we added some of the missing paths based on their ecological relevance to our model [START_REF] Grace | On the specification of structural equation models for ecological systems[END_REF]. Finally, we optimised the model structure through a stepwise procedure using the BIC criterium [START_REF] Hertzog | How robust are Structural Equation Models to model miss-specification? A simulation study 1-24[END_REF].

2.6.2.2.Interpretation of the SEM in each habitat

For interpretations, we used the standardised coefficients extracted from the most parsimonious model [START_REF] Grace | On the specification of structural equation models for ecological systems[END_REF]. We also used the information on relative abundances of trophic groups and trophic diversity within each trophic class to better understand the direction of the effects of the composition of each trophic class on microbial decomposition activity.

RESULTS

The direct effects of abiotic and biotic predictors on microbial decomposition activity varied between habitats

The two SEMs for forest and open habitats explained a very similar percentage of the total variation in microbial decomposition activity (R 2 =0.16). Interestingly, although close in terms of explained variance, the pathway strengths differed strongly between forests and open habitats (Figs. 1 & 2). In forests, microbial decomposition activity was strongly influenced by a direct and negative effect of GDD and FDD, to a lesser extent by a direct and negative effect of soil C-N ratio, and less, but still significantly, by the composition of the fungal class (Fig. 1). In contrast, for open habitats, the composition of the trophic class of bacteria was the main factor controlling microbial decomposition activity, while climate and soil played a similar direct role as in forests, but to a lesser extent (Fig. 2).

Different roles of biotic channels in decomposition activity in the two habitats

To quantify the overall influence of fungal vs. bacterial channels, we extracted from the SEMs the standardised direct and indirect effects through both the bacterial channel (i.e., bacterivores and bacteria), and the fungal channel (i.e., fungivores and fungi, Fig. 3).

Contrasting forests and open habitats, we found that the biotic effect on microbial decomposition activity was due to different channels. In forests, it was the composition of the fungi channel that influenced microbial decomposition activity, with a weak influence of primary decomposers (i.e., fungi) and no trophic regulation by fungivores. In contrast, in open habitats, the bacterial channel strongly influenced microbial decomposition activity, with a strong effect of the compositions of the trophic classes of bacteria and bacterivores, but also through strong trophic regulation, i.e. a strong indirect effect of bacterivores (Fig. 3).

The composition of the trophic classes of primary decomposers and trophic regulation modulate the effects of the environment on microbial decomposition activity.

Based on SEMs, we calculated the standardised indirect effects of the environment on microbial decomposition activity through its direct effect on the composition of primary decomposers, but also through its direct effect on the composition of consumers, which then cascades indirectly on primary decomposers (i.e. trophic regulation, (Fig. 4)). The indirect influence of the environment on microbial decomposition activity was much lower in forests than in open habitats. This highlighted that the modulation of environmental effects by the soil trophic network depended on the habitat type (Figs. 3,4).

In forests, we found a weak indirect effect of the environment on decomposition through its effect on the composition of primary decomposers. Acidic, rather cold soils with poor organic matter slightly enhanced the presence of different groups of saprophytic fungi (i.e. wood, soil and litter saprotrophs) compared to ectomycorrhizal fungi (ECM) in the fungal composition (Fig. 1 (D)), which slightly increased the microbial decomposition activity. On the other hand, the indirect effect of the environment via its effect on the composition of consumers was minor in forests (Fig. 4), although the environment explained the composition of consumers well (R 2 bacterivores = 0.79, R 2 fungivores = 0.26). The composition of fungivores was mainly determined by the quality of soil organic matter, with a dominance of mites on low fertility soils (i.e. high C/N ratio) and a co-dominance of mites and springtails on soils where organic matter was of better quality (Figs. 1 (B), S9 (B)). However, fungivore composition was not directly related to fungi composition, which negated the indirect effect of the environment via fungivores. In addition, soil pH was the main predictor of the composition of bacterivores, with Rhizaria-dominated communities occurring in acidic soils, and communities with a co-dominance of Rhizaria and Ciliophora found in less acidic conditions In open habitats, however, we found a strong indirect effect of the environment, both through primary decomposers and their consumers (Fig. 4). Acidic soils favoured the dominance of saprophytic bacteria, which clearly increased microbial decomposition activity (Figs. 2 (C),4). In contrast, the strong effect of pH on fungal composition resulted in a dominance of soil saprophytes on acidic soils and arbuscular mycorrhizal fungi (AMF) under less acidic conditions, but did not affect decomposition (Fig. 2 (D)). The indirect effect of the environment via the consumers was explained on the one hand by the strong influence of bacterivores on microbial decomposition activity and on the other hand by trophic regulation.

As in forests, the environment explained the composition of the consumer classes fairly well (R 2 bacterivores= 0.64, R 2 fungivores= 0.14). In particular, pH was the main predictor of bacterivore composition, leading to the same compositional trends as in forests (Figs. 2 (A),S10 (A)).

This effect cascaded to microbial decomposition activity, firstly because bacterivore communities with a greater trophic diversity were themselves strongly associated with higher microbial decomposition activity. On the other hand, the indirect effect of pH was moderated by the effect of trophic regulation (Figs. 3,[START_REF]Communauté de communes de la vallée de Chamonix Mont-Blanc Table S1: Number of reads and MOTUs (or unique sequences, steps 1 & 2) for the sample at 1048 each cleaning or preparation step, for each marker and each library[END_REF]. In acidic soils, Rhizaria-dominated bacterivore communities were associated with communities of bacteria dominated by saprophytic bacteria and with increased microbial decomposition activity. In less acidic soils, where bacterivore communities were more trophically diverse, bacterial communities were also characterised by greater trophic diversity and associated with decreased decomposition activity (Figs. 2 (A,C), S10 (A, C), 4).

DISCUSSION

While the importance of soil decomposers in ecosystem functioning is increasingly acknowledged, the regulating effects of other trophic soil classes on microbial decomposition remain poorly understood, as does their dependence on habitat (Thakur and Geisen, 2019).

Here, we compared the role of four environmental variables and of the consumers of primary decomposers in structuring the composition and activity of primary decomposers in two different habitats. In respect to our three initial questions, we found that (i) the environment, Climate, much more than soil physico-chemistry, was an important direct driver of microbial decomposition activity in both habitats. The negative relationships between GDD, FDD and microbial decomposition activity showed that the investment of microbes in extracellular enzyme production was favoured in difficult climatic conditions during the vegetation season, i.e. low heat energy input (i.e. low GDD) and frequent and/or intense frost events (i.e. low FDD). Lower (or more frequently negative) average temperatures can indeed reduce (or even stop) extracellular enzyme activity in situ [START_REF] Steinweg | Modeling the effects of temperature and moisture on soil enzyme activity: Linking laboratory assays to continuous field data[END_REF], thus increasing the amount of extracellular enzymes needed to capture the same amount of resource [START_REF] Allison | Soil-carbon response to warming dependent on microbial physiology[END_REF].

This study corroborates the hypotheses of Wardle et al. (2004) on a large scale, with a complete shift from a moderate direct effect of the composition of fungi in forests to a strong direct effect of the composition of bacteria in open habitats. In forests, our results are supported by the work of [START_REF] Schneider | Who is who in litter decomposition Metaproteomics reveals major microbial players and their biogeochemical functions[END_REF][START_REF] Schneider | Proteome analysis of fungal and bacterial involvement in leaf litter decomposition[END_REF], who showed that fungi produce most extracellular enzymes in forest soils. Indeed, some trophic groups of fungi are key to forest decomposition, because they open access to organic compounds, such as cellulose, blocked in recalcitrant plant debris (e.g. wood and litter saprotrophs, De Boer et al. 2005). This could explain why the increase in the relative abundance of saprophytic fungal groups compared to ECM is associated with an increase in microbial decomposition activity in our study (Fig. 1).

In contrast, open habitats are characterised by plants with fewer recalcitrant compounds, generally associated with less acidic soils (Appendix S2, Fig. S3). This could relieve bacteria from the aforementioned constraints and a pH-related environmental stress, and make them more competitive and active in open habitats overall (Rousk et al., 2010b). The increase in relative abundance of parasitic and autotrophic bacterial groups (Figs. 1 (C), S9 (C)) is likely driven by the presence of a mineral nutrient availability gradient in the mountainous open habitats of our study [START_REF] Fierer | TOWARD AN ECOLOGICAL CLASSIFICATION OF SOIL BACTERIA[END_REF][START_REF] Guo | Diversity and distribution of autotrophic microbial community along environmental gradients in grassland soils on the Tibetan Plateau[END_REF], and could explain the association of the composition of bacteria with a decrease in microbial decomposition activity.

We also demonstrated the importance of a top-down regulation of the composition of the consumer trophic classes on microbial decomposition activity. In particular, in line with Xiong et al. (2021), the link between the composition of bacterivores and bacteria was central in our two habitats (Figs. 1,2). This can arise from the high degree of specificity of bacterivores, which had already been observed for bacterivorous protists, dominant in our study [START_REF] Adl | Protists in soil ecology and forest nutrient cycling[END_REF][START_REF] Trap | Ecological importance of soil bacterivores for ecosystem functions[END_REF]. More generally, it is known that the presence of specific groups of protists and nematodes influences the composition of primary decomposer classes [START_REF] Gao | Protists: Puppet Masters of the Rhizosphere Microbiome[END_REF][START_REF] Geisen | Soil protists: A fertile frontier in soil biology research[END_REF][START_REF] Griffiths | CHANGES IN SOIL MICROBIAL COMMUNITY STRUCTURE IN THE PRESENCE OF MICROBIAL-FEEDING NEMATODES AND PROTOZOA[END_REF]. Furthermore, we found a fairly strong link between the composition of bacterivores and fungi in forest habitats, possibly due to the quite generalist feeding behaviour of some groups of protists (Geisen, 2016). In the forests of our study, for example, Rhizaria-dominated bacterivore communities seemed to favour the presence of saprophytic fungi over ECMs (Fig. 1). In contrast, the effect of fungivore community composition on fungal community composition and function was less clear in both habitats (Figs. 1,2,3). This lack of signal could be caused by a biased representation of specific animal organisms in eDNA data, because it includes not only signal from current communities, but also to a lesser extent, from those from the recent past. For example, fungivores with stable bodies (e.g. mites) might be decomposed more slowly and thus be slightly over-represented in fungivore communities (Table S3). However, the literature also reports more contrasting effects of fungivores, suggesting that fungivores are less specific [START_REF] Crowther | Functional and ecological consequences of saprotrophic fungus-grazer interactions[END_REF][START_REF] Hanlon | The effects of collembola grazing on microbial activity in decomposing leaf litter[END_REF][START_REF] Maraun | Adding to "the enigma of soil animal diversity": Fungal feeders and saprophagous soil invertebrates prefer similar food substrates[END_REF].

The weak links we found between the compositions of fungivores and fungi could also be explained by the length and spatial extent of mycelia, which makes the effect of grazing by fungivores less drastic. Up to a certain grazing pressure, fungivores do not suppress the whole individual, which makes the trophic interaction less likely to influence the composition of the fungal community [START_REF] Crowther | Functional and ecological consequences of saprotrophic fungus-grazer interactions[END_REF][START_REF] Hanlon | The effects of collembola grazing on microbial activity in decomposing leaf litter[END_REF].

Moreover, we provided new information on environmental drivers of the composition of multi-trophic soil assemblages, and how environmentally induced compositional changes can affect microbial decomposition activity. Soil physico-chemical properties had a stronger effect on the composition of four soil trophic classes than climatic variables. In particular, soil pH was a key predictor of the composition of bacteria, fungi and bacterivores (Figs. 1,2). This result is known for bacteria and fungi [START_REF] Donhauser | Alpine soil microbial ecology in a changing world[END_REF][START_REF] Fierer | The diversity and biogeography of soil bacterial communities[END_REF][START_REF] Griffiths | The bacterial biogeography of British soils[END_REF][START_REF] Looby | Diversity and function of soil microbes on montane gradients: The state of knowledge in a changing world[END_REF]Rousk et al., 2010a;Tedersoo et al., 2014), but is more surprising for bacterivores, which are mainly protists in our study. Indeed, studies on protist biogeography suggest that protist community composition is mainly determined by climatic and topographic factors [START_REF] Bates | Global biogeography of highly diverse protistan communities in soil[END_REF][START_REF] Seppey | Soil protist diversity in the Swiss western Alps is better predicted by topo-climatic than by edaphic variables[END_REF].

However, most of the bacterivorous protists in our study belong to the Ciliophora and Rhizaria clades that have recently been shown to be strongly affected by soil pH (Oliverio et al., 2020).

Finally, we showed that the environment has a top-down influence on trophic network composition that cascades down to microbial decomposition activity. This effect was rather moderate within both habitats (Fig. 4), but our study suggests that larger long-term climate changes, i.e. causing a habitat switch, would induce a radical change in decomposition regulation. Moreover, our results imply that more abrupt compositional changes in the trophic network, especially in consumer classes that are particularly sensitive to anthropogenic effects (Xiong et al., 2021), could induce greater indirect environmental effects [START_REF] O'neill | Responses of soil biota to elevated atmospheric carbon dioxide[END_REF]Wardle et al., 1998).

CONCLUSIONS

Our comparison of the regulation of microbial decomposition activity in forests vs. open habitats demonstrates the complexity of direct and indirect environmental effects on an ecosystem function. By hierarchically integrating the environment and soil trophic network, we showed how top-down regulation through the soil trophic network can affect the outcome of decomposition, and modulate the direct effects of the environment. Experimental studies testing the trophic pathways identified here will help further understanding the biological mechanisms involved in our observations. Moreover, predictive models are needed to predict the cascading effects of future environmental and biodiversity changes on ecosystem functioning and multifunctionality. doi:10.1016/j.soilbio.2020.107881 Põlme, S., Abarenkov, K., Nilsson, R.H., Lindahl, B.D., Clemmensen, K.E., Kauserud, H., Nguyen, N., Kjøller, R., Bates, S.T., Baldrian, P., Frøslev, T.G., Adojaan, K., Vizzini, A., Suija, A., Pfister, D., Baral, H., Järv, H., Madrid, H., Nordén, J., Liu, J., Pawlowska, J., Põldmaa, K., Pärtel, K., Runnel, K., Hansen, K., Larsson, K., Hyde, K.D., Sandovaldenis, M., Smith, M.E., Toome-heller, M., Wijayawardene, N.N., Menolli, N., Reynolds, N.K., Drenkhan, R., Maharachchikumbura, S.S.N., Gibertoni, T.B., Laessøe, T., Davis, W., Tokarev, Y., Corrales, A., Soares, A.M., Agan, A., Machado, A.R., Argüellesmoyao, A., Detheridge, A., Meiras-ottoni, A. De, Verbeken, A., Dutta, A.K., Cui, B., Pradeep, C.K., Marín, C., Stanton, D., Gohar, D., Wanasinghe, D.N., Otsing, E., Aslani, F., Griffith, G.W., Lumbsch, T.H., Grossart, H., Masigol, H., Timling, I., Hiiesalu, I., Oja, J., Kupagme, J.Y., Geml, J., Alvarez-manjarrez, J., Ilves, K., Loit, K., Adamson, K., Nara, K., Küngas, K., Rojas-jimenez, K., Bitenieks, K., Irinyi, L., Nagy, L.L., Soonvald, L., Zhou, L., Wagner, L., Aime, M.C., Öpik, M., Mujica, M.I., Metsoja, M., Ryberg, M., Vasar, M., Murata, M., Nelsen, M.P., Cleary, M., Samarakoon, M.C., Doilom, M., Bahram, M., Hagh-doust, N., Dulya, O., Johnston, P., Kohout, P., Chen, Q., Tian, Q., Nandi, R., Amiri, R., Perera, R.H., Chikowski, S., Mendes-alvarenga, R.L., Garibayorijel, R., Gielen, R., Phookamsak, R., Jayawardena, R.S., Rahimlou, S., Karunarathna, S.C., Tibpromma, S., Brown, S.P., Sepp, S., Mundra, S., Luo, Z., Bose, T., Vahter, T., Netherway, T., Yang, T., May, T., Varga, T., Li, W., Rafael, V., Coimbra, M., Rodrigo, V., Oliveira, T. De, Lima, V.X. De, Mikryukov, V.S., Lu, Y., Matsuda, Y., Miyamoto, Y., Kõljalg, U., Tedersoo, L., 2020. FungalTraits : a userfriendly traits database of fungi and funguslike stramenopiles. Fungal Diversity. doi:10.1007/s13225-020-00466-2 Potapov, A.A., Semenina, E.E., Korotkevich, A.Y., Kuznetsova, N.A., Tiunov, A. V., 2016. 1, which were simplified for the sake of readability.

FIGURE CAPTIONS

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0.00 0.25 0.50 0.75 1.00 D.

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(

  Figs. 1 (A), S9 (A)). The indirect effect of the environment on microbial decomposition activity in forests was only modulated via bacterivores and their weak effect on fungal composition.

  in particular climate, have a stronger effect on microbial decomposition activity in forests than in open habitats, where the biotic effect is more important. (ii) In forests, the composition of the soil network, and primarily of fungal communities, was moderately linked to microbial decomposition activity, whereas in open habitats the effects were much stronger, driven by bacteria, but also strongly influenced by bacterivores. (iii) The environment influences the composition of the different trophic classes, which indirectly influences the microbial decomposition activity through trophic interactions.

Figure 1 :

 1 Figure 1: SEM of the effects of environment and composition of the soil trophic network on

Figure 2 :

 2 Figure 2: SEM of the effects of environment and composition of the soil trophic network on

Figure 3 :

 3 Figure 3: Comparison of direct and indirect standardized effects of soil trophic network on

Figure 4 :

 4 Figure 4: Comparison of the indirect standardized effects of the environment (soil physico-

Figure 1

 1 Figure 1: 807

  Figure 2: 809

Figure S1 :

 S1 Figure 3:

Figure S2 :

 S2 Figure S2: Overview of the 18 gradients of ORCHAMP used in this study: Repartition of the elevations and habitat per gradient (A), location of gradients (B), names and year of sampling (A, B).

Figure S4 :

 S4 Figure S4: Comparison of the original variables used to compute microbial decomposition 990 activity -six potential extracellular enzymatic activities (LAP, NAG, AG, CB, BG, XYL resulting 991 in the so called total potential extracellular enzymatic activity, Tot_EEA) and the soil organic 992 matter (SOM) -in forests and open habitats. Red points represent the mean of each variable 993 for each habitat. 994 995

Figure S5 :

 S5 Figure S5: Overview of the cleaning and preparation steps of metabarcoding data from sequences to a MOTUs-sample matrix for each trophic class. Green boxes and arrows correspond to steps done with functions belonging to the OBITools package. Blue boxes and arrows correspond to steps done with functions belonging to the metabaR package. Yellow and oranges boxes and arrows correspond respectively to assignment to a taxonomical path and a trophic class and trophic group. Arrows on the right lists the function used at each step A.

Table S2 :

 S2 Number of extraction-, negative PCR-, positive PCR-, and sequencing controls, Number of samples before and after data curation, Number of PCR replicates before and after data curation for each marker and each library.

	library/year	Nb of extraction controls	Nb of negative PCR controls	Nb of positive PCR controls	Nb of sequencing controls	Nb of samples before data curation	bact01 Nb of samples after data curation fung02 euka02	Nb of pcr replicates before data curation	bact01 Nb of pcr replicates after data fung02 euka02
	2016	40	24	0	392	174	162	166	174	696	519	544	639
	2017 A	52	40	24	96	139	136	134	136	556	511	528	499
	2017 B	88	44	24	96	129	127	128	128	516	483	509	494
	2018	88	76	36	144	202	200	202	200	808	755	794	795

Table S3 :

 S3 Characteristics of the trophic classes and trophic groups. Total number of MOTUs and reads, average number of MOTUs and reads of trophic classes, number of MOTUs and reads of trophic groups, also expressed as percentage of the corresponding trophic class.

	Appendix S5:

Compositional change Figure S9. Compositional change between samples in forests

  , for A. bacterivores, B. fungivores, C. bacteria, D. fungi. This supplementary figure corresponds to the panels A, B, C & D of the main figure

Compositional change between samples in open habitats, for

  A. bacterivores, B. fungivores, C. bacteria, D. fungi. This supplementary figure corresponds to the panels A, B, C&D of the main figure2, which were simplified for the sake of readability.

	Figure S10.
		fungV.Acari	fungV.Ciliophora	fungV.Collembola	fungV.Nematode	fungV.Rhizaria	B.	D.
		•	•	•	•	•
	FUNGIVOROUS, Comp1 FUNGIVORES						(explained 4.75% var)	FUNGI	First component in COA (explained 1.39% var)
							A.	C.
	BACTERIVORES						(explained 4.00% var)	BACTERIA	(explained 7.16% var)
	FORESTS	Consumers						Primary	Decomposers

Acknowledgements:

The research presented here belongs to the ORCHAMP monitoring program. We thank the few dozens of interns, field assistants, park managers and rangers for their help conducting the field observations and the lab measurements. In particular, we thank Cindy Arnoldi for her help with the laboratory work on the AEEM technical platform. The S2M data were provided by Météo-France -CNRS, CNRM Centre d'Etudes de la Neige, through AERIS. Funding -Part of the funding came from the French Agence Nationale de la Recherche (ANR) through the GlobNets (ANR-16-CE02-0009), from two 'Investissement d'Avenir' grants (Trajectories: ANR-15-IDEX-02; Montane: OSUG@2020: ANR-10-LAB-56) and the METRO Grenoble Alpes and the Isere departement. We also received support from the Lautaret Garden-UMS 3370 (Univ. Grenoble Alpes, CNRS, SAJF, 38000 Grenoble, France), member of AnaEE-France (ANR-11-INBS-0001AnaEE-Services, Investissements d'Avenir frame) and of the LTSER Zone Atelier Alpes, a member of the eLTER-Europe network

Author statement: CMA, TM and WT conceptualized the overall study and methodology.

WT, AS, CMA, TM and the ORCHAMP Consortium collected the soil data. LG, GP and AS carried out the lab work for eDNA extractions, PCRs and measures of extracellular enzyme activities and soil properties. CL and CMA run the bioinformatic pipelines. LZ helped with the trophic assignment and bioinformatic pipelines. CMA run all statistical analyses. CMA, TM and WT led the interpretation of the results. CMA drafted the initial draft of the paper together with WT and TM. All authors contributed to editing and finalizing the paper.

SUPPLEMENTARY MATERIALS

Appendix S1: ORCHAMP: Spatio-temporal observatory of biodiversity and ecosystem functioning of mountains' socio-ecosystems 1. Brief description ORCHAMP is a long-term observatory of mountain ecosystems aiming to observe, understand and model biodiversity and ecosystem functioning over space and time. It relies on the active involvement of local actors, managers and researchers with the objective to better safeguard the contribution of biodiversity to human society.

ORCHAMP is built around multiple elevational gradients representative of the pedo-climatic environmental space of the French Alps (Fig. S1). In this article we used 18 gradients (Figs. S1,S2). Each gradient consists of 4 to 8 permanent plots distributed regularly, each with a difference of 200 m of altitude, from down the valley to the top, with the lowest site at 280 m and the highest site at 3160 m (Fig. S2). While this work does not use temporal replicates, in OR-CHAMP the gradients are resampled on average every 5 years using a rotating sampling scheme. Measures include physical properties (soil temperature, physicochemical measures, and pedology), biodiversity estimates (botanical surveys, multi-trophic biodiversity using soil environmental DNA, tree growth, deadwood in forests), ecosystem functions (productivity, enzymatic activities, soil organic matter) and human land use. Data are open-access and synthetized following GEOBON recommendations on Essential Biodiversity Variables. Specific data used in this paper are described in the main text.

Appendix S3: Description of the soil eDNA data processing 1. Soil eDNA extraction, amplification and sequencing

We conducted the soil eDNA extraction immediately in the field. In order to liberate eDNA from clay and silica particles, each sample was shaken for 15 minutes by rotation in a 15 mL saturated phosphate buffer solution (Na2HPO4; 0.12 M; pH ≈ 8). We then sampled 2 mL of sediment/buffer mixture and centrifuged it for 10 minutes at 10,000 g. We collected a 400 µL aliquot of supernatant and used it as starting material for eDNA extraction using the NucleoSpin® Soil extraction kit (Macherey-Nagel GmbH, Düren, Germany), following manufacturer's instructions except skipping the lysis cell step (Taberlet et al. 2012). We stored the samples in a dry environment before the elution step, which we performed back at LECA, Laboratoire d'ECologie Alpine, France. After elution, soil eDNA extracts were diluted 10 times before being used as templates for amplification. A total of 268 negative extraction controls were also performed over the three years to identify potential extraction contaminants.

For eDNA amplification, we targeted Eukaryotes, Fungi and Bacteria using respectively the DNA markers Euka02 (18S rRNA), Fung02 (ITS) and Bact01 (16S rRNA) described in Taberlet et al. 2018. We added unique eight-base long tags at the 5' end of each primer so that the sample of origin of each sequencing read could be retrieved after sequencing. These tags modified from [START_REF] Binladen | The use of coded PCR primers enables high-throughput sequencing of multiple homolog amplification products by 454 parallel sequencing[END_REF][START_REF] Valentini | New perspectives in diet analysis based on DNA barcoding and parallel pyrosequencing: The trnL approach[END_REF] differed by at least five bases. DNA amplifications were carried out in a final volume of 20 μL containing i) 2 μL of eDNA sample, ii) 10 μL of AmpliTaq Gold 360 Master Mix 2X (Applied Biosystems™, Foster City, CA, USA), iii) 2 μL of primers mix at initial concentration of 5 μM each and iv) 0.16 μL of Bovine Serum Albumin. We included a total of 184 negative-and 84 positive PCR-controls over the three years to identify potential PCR-contaminants. Four PCR-replicates were amplified for each sample and control. We performed PCRs as follow: 1) samples and controls were first incubated 10 min at 95°C, 2) they underwent 40 cycles of i) 30 s at 95°C, ii) 30 s at 57°C (Bact01), 55°C (Fung02) or 45°C (Euka02) and iii) 60 s at 72°C, and 3) they went through a final elongation at 72°C for seven minutes. Then, we 4) checked the amplification success using capillary electrophoresis (QIAxcel System; Qiagen), 5) mixed PCR products in an equi-volume way (15 µl each) and 6) purified eight*100 µL-aliquots of the resulting mix using MinElute Purification kit (Qiagen GmbH, Hilden, Germany). Finally, we pooled purified products together before sequencing.

Sequencing was performed by pair-end sequencing on Illumina HiSeq 2000 platform (2*125) for Euka02, and Illumina Miseq platform (2*250) for both Bact01 and Fung02) at Fasteris, Geneva, Switzerland. Data for each marker correspond to four libraries for sequencing (2016, 2017 A&B, 2018). We also included 728 negative sequencing controls over the three years to retrieve false positives caused by tag-switching events during the sequencing. For the first question, we expected abiotic ((1), climate in purple and soil physico-chemistry in orange, Burns et al. 2013;[START_REF] Hendriksen | Soil exo-enzyme activities across Europe-The influence of climate, land-use and soil properties[END_REF]) and biotic ((2), [START_REF] Moore | Detritus, trophic dynamics and biodiversity[END_REF]Wardle et al. 2004;Hättenschwiler et al. 2005;Thakur & Geisen 2019) predictors to be important factors in microbial decomposition activity (in green). For the second question, we expected the compositions of the bacterial and fungal channel to be crucial for microbial decomposition activity in open habitats and forests, respectively ((2), in particular hypotheses of Wardle et al. 2004). We expected to find evidence of trophic regulation by consumers of primary decomposers in both habitats, having a significant topdown effect on microbial decomposition activity ((3), consumers are in blue, primary decomposers in yellow, Hättenschwiler et al. 2005;Thakur & Geisen 2019). Furthermore, this trophic regulation could occur between the channels, as they are not completely impermeable (( 4), Geisen 2016). We set a correlation between the compositions of bacteria and fungi in the two habitats to account for their usually strong interactions (( 5), e.g. facilitation, competition, De Boer et al. 2005). Finally, in considering the third question, we expected that environmental effects on trophic class compositions would have cascading effects on microbial decomposition activity and modulate direct environmental effects ((6 and 2), Wardle et al. 1998;[START_REF] Wu | Molecular study of worldwide distribution and diversity of soil animals[END_REF]Tedersoo et al. 2014;[START_REF] Delgado-Baquerizo | A global atlas of the dominant bacteria found in soil[END_REF][START_REF] Van Den Hoogen | Soil nematode abundance and functional group composition at a global scale[END_REF]Oliverio et al. 2020;[START_REF] Potapov | Towards a global synthesis of Collembola knowledge: challenges and potential solutions[END_REF]Xiong et al. 2021).

(5) (6) (6) (6) (6)

(3)


Yassine Laguel
email: yassine.laguel@univ-grenoble-alpes.fr

Krishna Pillutla

Jerôme Malick

Zaid Harchaoui

A Superquantile Approach to Federated Learning with Heterogeneous Devices

We present a federated learning framework that allows one to handle heterogeneous client devices that may not conform to the population data distribution. The proposed approach hinges upon a parameterized superquantile-based objective, where the parameter ranges over levels of conformity. We introduce a stochastic optimization algorithm compatible with secure aggregation, which interleaves device filtering steps with federated averaging steps. We conclude with numerical experiments with neural networks on computer vision and natural language processing data.

I. INTRODUCTION

In federated learning [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF][START_REF] Kairouz | Advances and Open Problems in Federated Learning[END_REF], a number of client devices with privacy-sensitive data collaboratively learn a machine learning model under the orchestration of a central server, while keeping their data decentralized. This is achieved by pushing the computation to the devices while the server coordinates with the devices for aggregation of model updates. Federated learning has found myriad applications such as smartphone apps [START_REF] Yang | Applied Federated Learning: Improving Google Keyboard Query Suggestions[END_REF][START_REF] Hard | Federated Learning for Mobile Keyboard Prediction[END_REF] and healthcare [START_REF] Huang | Patient Clustering Improves Efficiency of Federated Machine Learning to Predict Mortality and Hospital stay time using Distributed Electronic Medical Records[END_REF].

A key feature of federated learning is statistical heterogeneity, i.e., client data distributions are not identical. Each user has unique characteristics which are reflected in the data they generate. These characteristics are influenced by personal, cultural, and geographical factors. For instance, the varied use of language contributes to data heterogeneity in a next word prediction task.

Vanilla federated learning [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF], aims to minimize the prediction loss of a given model on average over a population of devices available for training. While this approach works for users with local data distribution close to the average distribution, it is liable to fail on individuals who do not conform to the population, leading to poor user experience. The goal of this work is to present a framework to improve the experience of these diversely non-conforming users without sacrificing the good experience of conforming users.

In this paper, we introduce the ∆-FL framework, summarized in Fig. 1, to handle heterogeneity of client data distributions. The framework relies on a superquantile-based objective parameterized by the conformity level, which is a scalar summary of how closely a device conforms to the population. To optimize the ∆-FL objective, we present an algorithm which interleaves device filtering with federated optimization steps. We discuss its special features compared to the standard FedAvg and to recent algorithms [START_REF] Mohri | Agnostic Federated Learning[END_REF]- [START_REF] Li | Fair Resource Allocation in Federated Learning[END_REF] about heterogeneity in federated learning. We analyze our algorithm in the convex setting and establish bounds on total communication cost.

We demonstrate the breadth of our framework with numerical experiments using convolutional and recurrent neural networks on tasks including image classification, and sentiment analysis based on public datasets. The simulations demonstrate superior performance of ∆-FL over state-of-the-art baselines on the upper quantiles of the error on test devices, while being competitive on the mean error.

Outline. Section II describes the setting and precisely defines conformity. Section III discusses the ∆-FL framework, the training objective, and the related approaches. Section IV gives an optimization algorithm for the ∆-FL objective and analyzes its convergence. Section V presents numerical experiments of the proposed method. Full proofs and additional details can be found in [START_REF] Laguel | A Superquantile Approach to Federated Learning with Heterogeneous Devices[END_REF]. The code and the scripts to reproduce numerical results are publicly available at https://github.com/krishnap25/ simplicial-fl.

II. PROBLEM SETTING

We introduce the notion of conformity in federated learning to measure how conforming are test devices from the population of train devices.

Train Devices and Trend Distribution. Consider N clients devices with respective probability distribution q k and weights α k > 0. We assume N k=1 α k = 1 and that the data on device k are distributed i.i.d. according to q k . The loss of a device with distribution q is F (w; q) := E ξ∼q [f (w; ξ)], where f (w; ξ) is the loss on the input-output pair ξ under model w ∈ R d . The loss on k th training device is then F k (w) := F (w; q k ).

The standard federated learning objective is the weighted average of the local losses of the training devices:

min w∈R d N k=1 α k F k (w) + λ 2 w 2 2 , (1)
with a regularization parameter λ. We define the trend distribution as p α = n i=1 α k q k , the intrinsic distribution on which standard federated learning models are trained. Indeed, (1) exactly minimizes F (w; p α ) + (λ/2) w 2 2 . Our approach will consider test distribution that depart from the trend distribution.

∈ ∆ N -1 is p π := N k=1 π k q k .
Here, ∆ N -1 is the probability simplex in R N . We now define the conformity of mixture with respect to the trend distribution.

Definition 1. The conformity conf(p π ) ∈ (0, 1] of a mixture p π with weight π is defined as min k∈[N ] α k /π k . The conformity of a test device refers to the conformity of its data distribution. Given a large representative set of training devices, it is reasonable to assume that each test device can be wellapproximated by a mixture p π . Then, the conformity of a device is a scalar summary of how close it is to the population. A test device with conformity θ ≈ 1 has its local distribution close to the trend distribution. Then, a model trained with the distribution p α is expected to have a high predictive power for this test device. In contrast, a test device with θ ≈ 0 would be vastly different from the distribution p α , and the predictive power of a model trained on p α could be poor.

There is a trade-off between fitting to the population and supporting non-conforming test devices. The conformity θ presents a natural way to encapsulate this tradeoff in a scalar parameter. That is, given a conformity θ ∈ (0, 1), we choose to only support test distributions p π with conf(p π ) ≥ θ.

III. THE ∆-FL FRAMEWORK

The ∆-FL framework supplies each test device with a model appropriate to its conformity. Given a discretization {θ 1 , . . . , θ r } of (0, 1], ∆-FL maintains r models, one for each conformity level θ j , as laid down in Fig. 1. The local data is not allowed to leave a device due to privacy restrictions; hence, the conformity of a test device cannot be measured. Instead, we allow each test device to tune their conformity.

Superquantiles come into play. To train a model with a given conformity, we aim to do well on all mixtures p π with conf(p π ) ≥ θ. This leads us to the minimax objective

min w∈R d F θ (w) := max π∈P θ F (w; p π ) + λ 2 w 2 2
where

P θ := π ∈ ∆ N -1 : conf(p π ) ≥ θ .
Using duality [START_REF] Rockafellar | Optimization of Conditional Valueat-Risk[END_REF] we can show that F θ can be written as a minimum:

F θ (w) = min η∈R G θ (w, η),
where

G θ (w, η) := η+ 1 θ N k=1 α k max F k (w)-η, 0 + λ 2 w 2 2 . ( 2
)
This reveals that F θ is a (1 -θ)-superquantile; see e.g. the overview [START_REF] Rockafellar | Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk[END_REF] and the drawing of Fig. 2. ∆-FL Objective. We now introduce the ∆-FL objective as the dual to the minimax objective, i.e., min

w∈R d ,η∈R G θ (w, η) .
In Algorithm 1, we propose to solve this problem by alternating updates of w and η. From standard results on superquantiles [START_REF] Rockafellar | Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk[END_REF], the optimal solution of partial minimization in η admits a closed form as the (1-θ)-quantile of the distribution of losses (F k (w)) 1≤k≤N with weights (α k ) 1≤k≤N . In contrast, the w-step is executed by a standard federated optimization algorithm such as FedAvg until a stopping criterion is met. Precisely, given a target suboptimality ε t > 0, we require that

E [G(w t+1 , η t ) | w t ] -min w G(w, η t ) ≤ ε t . (3)
Note that existing works about the use of superquantiles always consider centralized settings and often use convex optimization approaches, such as interior point algorithms [START_REF] Rockafellar | Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk[END_REF]. The alternating minimization is suitable for our setting, by using a federated optimization algorithm for the w-step.

Putting the method into perspective. Let us discuss further some aspects of the ∆-FL framework and Algorithm 1. The theoretical bound on communication rounds presented in Section IV exhibits the same dependence on target

Sθ(Z) = E[Z | Z > Qθ(Z)] area = θ E[Z] Qθ(Z)
Fig. 2. For a continuous random variable Z, drawing of (1-θ)-quantile Q θ (Z) and (1-θ)-superquantile S θ (Z), defined as an expectation.

In this work, we heavily rely on the variational expression of the superquantile as the minimum of G θ over η in [START_REF] Kairouz | Advances and Open Problems in Federated Learning[END_REF]; see [START_REF] Rockafellar | Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk[END_REF].

accuracy ε as the base federated optimization algorithm, e.g., FedAvg. We also corroborate this observation with numerical experiments in Section V. (d) Tuning Conformity. If using a single global value of conformity for all test users, one should account for all implications of this choice. This depends on the distributions of users. Committing blindly to a single conformity level could fail to balance supporting nonconforming users with fitting the population. On the other hand, measuring the conformity of users requires transfer of user data, a violation of privacy. As such, decisions impacting fairness and privacy are questions of policy rather than of mathematics. Both ∆-FL and [START_REF] Li | Fair Resource Allocation in Federated Learning[END_REF] circumvent this issue by training a family of models for different conformity levels, and allows a test user to tune their conformity.

Related Work on Heterogeneity. Past works have aimed to tackle heterogeneity in federated learning by modifying the objective. AFL [START_REF] Mohri | Agnostic Federated Learning[END_REF] uses a minimax objective that is effective only with coarse groups of devices; this requires domain knowledge. We note that ∆-FL interpolates between AFL (θ → 0) and standard federated learning (θ → 1). Like ∆-FL, the method q-FFL [8] also interpolates between these two extremes but in an different way: it raises losses to the power 1 + q, thus penalizing large losses more, while ∆-FL minimizes the average of the largest losses (see Fig. 2). q-FFL also maintains multiple global models for different parameters q. However, q does not admit a natural interpretation and its range could be unbounded. In contrast, ∆-FL's conformity parameter θ ∈ (0, 1) is the conformity level, which has a clear meaning. We provide an experimental comparison of these methods with respect to the client heterogeneity in Section V.

Other works minimize the standard federated learning objective, but aim to improve convergence in the presence of heterogeneity by correcting for potential client drift. Examples include FedProx [START_REF] Li | Federated Optimization in Heterogeneous Networks[END_REF] and Scaffold [START_REF] Karimireddy | Scaffold: Stochastic controlled averaging for federated learning[END_REF]. Such algorithms could thus be combined with our framework to reduce the cost of each w-step but that goes beyond the scope of this paper.

IV. CONVERGENCE ANALYSIS

We now analyze Algorithm 1 in the convex setting. We assume at each F k is convex (which is the case for instance for linear models and convex losses). This yields that G θ is Algorithm 1 Alternating Minimization for ∆-FL

Input: Function G : R d × R → R, w 0 ∈ R d ,
G θ,ν (w, η) := η + 1 θ N k=1 α k h ν F k (w) -η + λ 2 w 2 , F θ,ν (w) := min η∈R G θ,ν (w, η),
where h ν (ρ) = min u max{u, 0} + (u -ρ) 2 /(2ν) is the socalled Moreau envelope of max{• , 0}, It provides a smooth and tractable uniform approximation [START_REF] Beck | Smoothing and First Order Methods: A Unified Framework[END_REF]. Our working function is convex and smooth, as formalized in the next lemma.

Lemma 2. If F k is convex, B-Lipshitz and L-smooth, then G θ,ν is jointly convex in (w, η) over R d × R and we have that w → ∇ w G θ,ν (w, η) is L w -Lipschitz for all η ∈ R where L w := λ + (L + B 2 /ν)/θ, and, η → ∂ ∂η G θ,ν (w, η) is L η - Lipschitz for all w ∈ R d where L η := (νθ) -1 .
In the next proposition, we use this lemma to establish an upper bound of the outer-loop complexity of Algorithm 1. Proposition 3. Under the assumptions of Lemma 2, denote the condition number of G θ,ν (•, η) by κ = 1 + (L + B 2 /ν)/(θλ). Consider Algorithm 1 with inputs G θ,ν and stopping criterion (3) with ε t = ε 0 e -t/κ for some ε 0 > 0. Then,

E [F θ,ν (w t )] -F θ,ν ≤ t e -t/κ F θ,ν (w 0 ) -F θ,ν + 2ε 0 ,
where appears the minimum value F θ,ν = min F θ,ν .

Proof. For each iteration t, denote ψ t (w) := G θ,ν (w, η t ), and let w t = w t -∇ψ t (w t )/L w . Lemma 2 ensures that ψ t is L w -smooth and λ-strongly convex which in turn yields [START_REF] Lan | First-order and Stochastic Optimization Methods for Machine Learning[END_REF]:

ψ t (w t ) -ψ t ( w t ) ≥ 1 2L w ∇ψ t (w t ) 2 (smoothness) ψ t (w t ) -min ψ t ≤ 1 2λ ∇ψ t (w t ) 2 (strong convexity).
Let F t denote the σ-algebra generated by w t . Putting these together with the stopping criterion (3), we get

E[ψ t (w t+1 )|F t ] ≤ min ψ t + ε t ≤ ψ t ( w t ) + ε t ≤ ψ t (w t ) -κ -1 ψ t (w t ) -min ψ t + ε t .
Using now the three following facts F θ,ν (w t+1 ) ≤ ψ t (w t+1 ) , F θ,ν (w t ) = ψ t (w t ) , and min w F θ,ν (w) ≤ min ψ t , we get,

E [F θ,ν (w t+1 )|F t ] ≤ F θ,ν (w t ) -κ -1 F θ,ν (w t ) -min F θ,ν + ε t .
Rearranging this and taking an expectation over F t yields

E [F θ,ν (w t+1 )] -min F θ,ν ≤ (1 -κ -1 ) E [F θ,ν (w t )] -min F θ,ν + ε t ≤ exp(-κ -1 ) E [F θ,ν (w t )] -min F θ,ν + ε t .
Using the shorthand ∆ t := E [F θ,ν (w t )] -min F θ,ν and unrolling the above inequality gives

∆ t+1 ≤ exp - t + 1 κ ∆ 0 + t τ =0 exp - t -τ κ ε τ ≤ t exp - t κ (∆ 0 + 2ε 0 ) ,
which completes the proof.

Next, we bound the total communication rounds (where a round refers to a secure aggregation of model updates) required by Algorithm 1, taking into account the cost of solving each w-step. We state the result with the w-step solved by FedAvg with full device participation (a.k.a. local SGD). The proof uses a result on communication rounds of FedAvg from [START_REF] Koloskova | A Unified Theory of Decentralized SGD with Changing Topology and Local Updates[END_REF].

ŵ such that E[F θ,ν ( ŵ)] -F θ,ν ≤ ε is at most O σ 2 κ 2 ∆ 0 N λτ ε + σ 2 κ 3 ∆ 0 λ 2 τ ε + D 2 κ 4 ∆ 0 λε + κ 2 ,
where ∆ 0 = (F θ,ν (w 0 ) -F θ,ν )/ε 0 + 1, and big O(•) includes constants and polylog factors.

Proof. For simplicity, C denotes some universal constant which may change from one line to the next. Let ∆ 0 := ∆ 0 + 2ε 0 = F θ,ν (w 0 ) -F θ,ν + 2ε 0 . We use the fact that for ε ≤ 2∆ 0 /3, t ≥ κ log 2κ∆ 0 ε + κ log log 2κ∆ 0 ε implies ε ≥ t exp (-t/κ) ∆ 0 . Thus, using Proposition 3, the number of outer iterations t * to get

E [F θ,ν (w t * )] -F θ,ν ≤ ε is t * = κ log 2κ∆ 0 ε + κ log log 2κ∆ 0 ε .
From [15, Theorem 2], the number n t of communication rounds to obtain w t+1 satisfying the stopping criterion (3) is

n t ≤ C   α ∞ σ 2 λτ ε t + σ 2 λ 2 τ ε t + κD 2 λε t + κ log κτ ∆ t ε t   with ∆ t = F θ,ν (w t ) -F θ,ν ≥ G θ,ν (w t , η t ) -min G θ,ν (•, η t ).
The total number of communication rounds is then

T = t * -1 t=0 n t ≤ C σ 2 N λτ ε 0 t * -1 t=0 exp(t/κ) +   σ 2 λ 2 τ ε 0 + κD 2 λε 0   t * -1 t=0 exp t/(2κ) + κt * log κτ ∆ max ε 0 + t * -1 t=0 t .
For the first term, we use exp(κ -1 ) ≥ 1 + κ -1 to get,

t * -1 t=0 exp(t/κ) ≤ 2κ 2 ∆ 0 ε log 2κ∆ 0 ε .
We similarly treat the second term to get

t -1 t=0 exp(t/(2κ)) ≤ C κ 3 ε ∆ 0 log 2κ∆ 0 ε .
The last two terms can be bounded by simply plugging in t * . Putting these terms together completes the proof.

Setting ν = θε/2 we get that T is bounded by

O 1 N τ θ 2 (λε) 3 + 1 ε 2 √ θ 3 λ 5 τ + 1 θ 2 (λε) 5/2 .

V. NUMERICAL EXPERIMENTS

We now experimentally check the performance of ∆-FL. The experiments were implemented in Python using automatic differentiation provided by PyTorch, while the data was preprocessed using LEAF [START_REF] Caldas | LEAF: A benchmark for federated settings[END_REF].

A. Numerical Setting

We optimize ∆-FL with a variant of Algorithm 1 with an iteration budget. In particular, we reduce each w-step to FedAvg with a single communication round with a fixed budget of local gradient descent on selected devices The resulting algorithm is given in Algorithm 2. Datasets, Tasks and Models. We consider two learning tasks. (a) Character Recognition: We use the EMNIST dataset [START_REF] Cohen | EMNIST: an extension of MNIST to handwritten letters[END_REF],

where the input x is a 28 × 28 grayscale image of a handwritten character and the output y is its label (0-9, a-z, A-Z). Each device is a writer of the character x.

The weight α k assigned to author k is the number of characters written by this author. We use a convolutional neural network architecture (ConvNet). (b) Sentiment Analysis: We use the Sent140 dataset [START_REF] Go | Twitter Sentiment Classification using Distant Supervision[END_REF] where the input x is a tweet and the output y = ±1 is its sentiment. Each device is a distinct Twitter user. The weight α k assigned to user k is the number of tweets publised by this user. The neural network model is a LSTM [START_REF] Hochreiter | Long Short-Term Memory[END_REF] built on the GloVe embeddings. We refer to the latter as "RNN". We trained on half the devices and tested on the rest, where each training device k is weighted by its number of datapoints.

Algorithm 2 ∆-FL with a fixed budget Input: N devices {(q k , α k )} k∈[N ] , number of local updates n local , learning rate sequence (γ t ), devices per round m, initial iterate w 0 , conformity level θ ∈ (0, 1) Server executes:

1: for t = 1, 2, • • • do 2: Sample devices S t ∼ Unif([N ]) m 3: Broadcast w t to each device k ∈ S t 4:
Each k ∈ S t computes F k (w t ) and sends to server 5:

η t ← Quantile 1 -θ, F k (w t ), α k ) k∈St 6: Filter out S t = {k ∈ S t : F k (w t ) ≥ η t } 7:
for each device k ∈ S t in parallel do 8:

w k,t ← LocalUpdate(k, w t ) 9: w t+1 ← SecureAggregate {(w k,t , α k )} k∈S t 10: function LocalUpdate(k, w)
Run on device k 11:

for i = 1, . . . , n local do 12:
Update w ← w -γ t ∇f (w; ξ i ) using ξ i ∼ q k return w

We use the logistic loss for training and misclassification error for evaluation. Each experiment is repeated 5 times with a different sampling of train and test devices.

Methods. We compare ∆-FL with the standard objective (1) optimized using FedAvg, FedProx [START_REF] Li | Federated Optimization in Heterogeneous Networks[END_REF] (which shows more stable convergence than FedAvg), and FedAvg-Sub, which is FedAvg with as many devices per round as ∆-FL for the best conformity level. We also compare ∆-FL with other heterogeneity-sensitive objectives, namely q-FFL [START_REF] Li | Fair Resource Allocation in Federated Learning[END_REF], AFL [START_REF] Mohri | Agnostic Federated Learning[END_REF].

Metrics. We track the loss F k of each training device and the misclassification error on each test device. We summarize these distributions with their mean and the 90 th percentile, where the latter measures performance on devices with low conformity.

Hyperparameters. We run FedAvg until convergence and fix the corresponding number of iterations as the budget for all other algorithms. We tuned a learning rate schedule using grid search to find the best terminal loss averaged over training devices for FedAvg. The same iteration budget and learning rate schedule were used for all other methods including ∆-FL. Each method, except FedAvg-Sub, selected 100 devices per round for training. For all methods, local SGD is run on selected devices for 1 epoch. We run q-FFL for q ∈ {10 -3 , 10 -2 , . . . , 10} and report q with the smallest 90 th percentile of misclassification error on test devices. We use q-FFL with q = 10 as a proxy for AFL, as it was found to converge faster with similar performance across devices [START_REF] Li | Fair Resource Allocation in Federated Learning[END_REF].

B. Experimental Results

Performances across Iterations. Fig. 3 compares the convergence of Algorithm 1 with FedAvg, measured in terms of the number of communication rounds. We see that ∆-FL converges on par with FedAvg, while using the same hyperparameters such as learning rate in the w-step of Algorithm 1. We noticed also that tuning the communication budget for the w-step of Algorithm 1 (set to 1 for the rest of the experiments as in Algorithm 2) allows minor improvements in the terminal error.

Still, the rate of convergence is the same. Performances and comparisons. Table I records the mean and 90 th percentile of the misclassification error on the test devices. We find that ∆-FL improves the 90 th percentile test error in three out of four dataset-model pairs, and is within one standard deviation of the best error in the other last one. A 3.3% absolute (12% relative) improvement on EMNIST is most striking. Indeed, ∆-FL accounts for test users who do not conform to the population. ∆-FL is competitive with the other methods on the mean error, and, perhaps surprisingly, attains the smallest mean error in two cases. In Sent140-RNN, ∆-FL with θ = 0.1 displays unstable behavior. This could be due to the objective being unsuitable (AFL does poorly too) coupled with η-step filtering out too many devices. Nevertheless, ∆-FL is competitive for multiple values of θ.

Performance Across Devices. Fig. 4 and 5 visualize distribution of error on test devices, respectively as a histogram and a scatter plot against the number of datapoints. We find that ∆-FL exhibits thinner upper and lower tails of the error in general and a lower variance of the distribution. Regarding Fig. 5, we observe that improvement over the worst cases is achieved regardless of the local data size of the devices. Indeed, the device filtering step operates a sorting of the loss of the devices which does not prevent small devices from being selected. In contrast, FedAvg, by averaging with respect to the weights of the devices, is likely to give more importance to the updates of devices with larger local data size. Secondly, ∆-FL appears to reduce the variance of the loss on the train devices. Lastly, note that amongst test devices with a small number of data points (e.g., < 200 for EMNIST or < 100 for Sent140), ∆-FL reduces the variance of the misclassification error.

VI. CONCLUSION AND PERSPECTIVES

We presented ∆-FL, a federated learning framework that can handle heterogeneous client devices that do not conform to the population. We provided an optimization algorithm, proved its rate of convergence and demonstrated numerically that ∆-FL boosts performance on non-conforming devices. An interesting venue for future work is the exploration of settings where new training users arrive in an online fashion and the trend distribution is revealed incrementally over time. It is also interesting to consider other divergence to measure the conformity between a test distribution p π and the trend distribution p α , for instance, the χ 2 -divergence 1/(2N ) N k=1 (π k /α k -1) 2 or the 1 norm π -α 1 .

  (a) Device Filtering. Algorithm 1 can be viewed as interleaving device filtering (where devices with F k (w) < η are filtered out) with federated optimization steps. If a device is filtered out, it means that the model fits well the data on that device. (b) Computation Cost. The computation cost for each communication round of the w-step of Algorithm 1 is the same as that of the base federated optimization algorithm used. The η-step requires a pass over the local data on each device. (c) Communication Cost. The total communication cost is dominated by the communication of model parameters.

Proposition 4 .

 4 Consider the setting of Proposition 3 with α k ≡ 1/N . Consider using local SGD with τ local steps per communication round to solve w-step of Algorithm 1. Suppose that stochastic gradients (w.r.t. w) of h ν (F k (w) -η) have a bounded variance σ 2 k , with σ 2 = k α k σ 2 k . Let D bound the diversity of F k ; see [15, Assumption 3.b] for the definition. Then the total rounds T of communication to obtain a point

1 Fig. 3 .

 13 Fig. 3. Performance across iterations of loss on training devices and misclassification error of test devices for EMNIST.

Fig. 4 . 5 Fig. 5 .

 455 Fig. 4. Histogram of misclassification error on test devices.

  Fig. 1. Schematic summary of the ∆-FL framework. Left: The server maintains multiple models w θ j , one for each level of conformity θj. Middle: During training, selected devices participate in training each model w θ j . Individual updates are securely aggregated to update the server model. Right: Each test user is allowed to select their level of conformity θ, and are served the corresponding model w θ .

	Trajectories of model parameters over time	In iteration t of training	At test time
	Model param. →	w 0	Iteration →		w θ 2 w θ 3 w θ 4 w θ 1	devices selected	+ Server Sec.Agg.	Server
		θ 1	θ 2	θ 3	θ 4			test devices select their level of conformity θ
	Test Devices and Conformity. We consider "test" devices,
	unseen during training, whose distribution can be written as
	a mixture of the training distributions. A mixture p π with
	weight π

  inexactness sequence (εt), time horizon t * 1: for t = 0, 1, . . . , t * -1 do 2: ηt ∈ arg min η∈R G(wt, η) 3: w t+1 ≈ arg min w∈R d G(w, ηt) such that (3) holds with εt 4: return w t * convex, but it is still not smooth, due to the max{•, 0} term. So, we propose to use a smooth surrogate G θ,ν and the analogous F θ,ν defined for ν > 0 as

TABLE I MEAN

 I AND 90 TH PERCENTILE OF THE DISTRIBUTION OF MISCLASSIFICATION ERROR (IN %) ON THE TEST DEVICES.

														EMNIST	Sent140
													Mean		90 th Percentile	Mean	90 th Percentile
							FedAvg				16.64 ± 0.50	28.46 ± 1.07	30.16 ± 0.44	49.67 ± 3.95
							FedAvg-Sub			16.23 ± 0.23	27.57 ± 0.81	29.86 ± 0.46	46.94 ± 3.84
							FedProx				16.02 ± 0.54	27.01 ± 1.86	30.20 ± 0.48	49.86 ± 4.07
							q-FFL (best q)		16.59 ± 0.30	28.02 ± 0.80	29.96 ± 0.56	48.66 ± 4.68
							AFL					33.01 ± 0.37	45.08 ± 1.00	37.74 ± 0.65	57.78 ± 1.19
							∆-FL, θ = 0.8		16.09 ± 0.40	26.23 ± 1.15	30.31 ± 0.33	46.46 ± 4.39
							∆-FL, θ = 0.5 15.49 ± 0.30	23.69 ± 0.94	33.59 ± 2.44	50.48 ± 8.24
							∆-FL, θ = 0.1		16.37 ± 1.03	25.46 ± 2.77	51.98 ± 11.81	86.45 ± 10.95
	0.14		EMNIST	= 0.5		0.12			Sent140		= 0.8
	0.12						0.10
	0.10						0.08
	0.08						0.06
	0.04 0.06						0.04
	0.02						0.02
	0.0 0.00	0.1	0.2	0.3	0.4	0.5	0.0 0.00	0.1	0.2	0.3	0.4	0.5	0.6	0.7
		FedAvg	-FL		10 th percentile	Mean Value		90 th percentile

ACKNOWLEDGMENTS

We would like to thank the authors of [15] for fruitful discussions. We acknowledge support from NSF DMS 2023166, DMS 1839371, CCF 2019844, the CIFAR program "Learning in Machines and Brains", faculty research awards, and a JP Morgan PhD fellowship. This work has been partially supported by MIAI -Grenoble Alpes, (ANR-19-P3IA-0003).