Julianna Devillers
email: julianna.devillers@student.isae-supaero.frbrun

Olivier Brun

Balakrishna J Prabhu
email: balakrishna.prabhu@laas.fr

Data Summarization for Federated Learning ⋆

Keywords: Data summarization, FedAvg, convergence

We explore data summarization techniques as a mean to reduce the energy footprint of Federated Learning (FL). We formulate the problem of selecting a small subset of data points that best represent the gradient of each local dataset as a submodular maximization problem and provide sufficient conditions under which the FL training is guaranteed to converge to the same global model as if the whole local datasets have been used on each client. Experimental results on IID and non-IID datasets show that this approach yields a similar accuracy as training on the full local datasets, but with a significant reduction of runtimes. There is however no clear advantage of data summarization over random sampling.

Introduction

Training state-of-the-art deep learning models has an ever increasing computational cost, which double every few months, and consumes a lot of energy [START_REF] Schwartz | Green ai[END_REF][START_REF] Strubell | Energy and policy considerations for deep learning in NLP[END_REF]. For instance, it is known that training the GPT-3 language model (175 billion parameters) consumed around 1,287 MWh of energy, which represents an amount of energy equivalent to the yearly consumption of 126 Danish homes and a CO 2 emission equivalent to 552 round-trip flights between New York and Paris [START_REF] Patterson | Carbon emissions and large neural network training[END_REF]. It is now broadly recognized that the current race for improved model accuracy in ML is not sustainable both from an environmental and economic point of view.

There is an increasing interest in a new distributed ML paradigm called Federated Learning (FL) in which many clients collaboratively train a shared model under the orchestration of a central server, while keeping the training data private. At each round, the central server selects a subset of clients and sends them the parameters of the shared model. The selected clients train the model on their local data and send back their gradient information to the central server for aggregation and update of the shared model. FL has a wide range of applications, especially for sensitive data for example in the fields of healthcare or finance or considering data on peoples' localization. FL has however a surprisingly large carbon footprint, which can even be up to two orders of magnitude higher than its more traditional centralized counterparts [18]. As FL is becoming more and more prevalent, the reduction of its energy footprint therefore becomes a real issue.

A first approach aims at reducing the energy spent for the communications between the clients and the central server. Gradient sparsification and gradient quantization techniques (see, e.g., [START_REF] Jiang | A linear speedup analysis of distributed deep learning with sparse and quantized communication[END_REF][START_REF] Cui | Clustergrad: Adaptive gradient compression by clustering in federated learning[END_REF][START_REF] Haddadpour | Federated learning with compression: Unified analysis and sharp guarantees[END_REF]) have been recently proposed for that purpose. Since compression is a lossy process, the gains in terms of communication costs are usually achieved at the expense of a worse iteration complexity and there is not a good understanding of how these techniques impact the total energy consumption and whether they are worth applying in general.

Another approach, which is explored in this paper, aims at reducing the energy spent for local model updates by using data summarization techniques. The general idea is to extract a small representative set from the original local dataset on each client and to train on that smaller set. It seems reasonable to expect that training on a succinct summary rather than on the entire dataset directly can substantially reduce the energy consumption of the training phase. This raises however two main questions. The first one is: "how to choose the data summary on each client while preserving data privacy and converging to the same global model as if the whole local datasets have been used? ". The second one is: "is he extra energy required for data summarization offset by the energy saved during the training phase? ". These two questions are investigated in this paper.

Contributions

From a theoretical point of view, we propose to constitute the data summary on each client by choosing the data points in the local dataset which best represent the local gradient of the loss function. We show that the computation of the data summary can be cast as a submodular maximization problem or as a submodular cover problem. With the latter setup for data summarization, we provide sufficient conditions under which the FL training is guaranteed to converge to the same global model as if the whole local datasets have been used on each client.

From a practical point of view, we study the empirical test accuracy and the efficiency of the gradient matching method based on submodular maximization. Unfortunately, it is difficult to estimate the energy cost of our experiments and we use the number of rounds before convergence and the runtime instead. We note that runtime is a good proxy for the computational cost, although it does not capture the costs of communication in a real federated environment. We use the Flower toolkit to train Deep Neural Networks (DNN) on the MNIST and CIFAR10 datasets and evaluate these performance metrics in scenarios with IID and non-IID datasets. They are compared against that of the FL with the full datasets, and against random sampling. In contrast to [START_REF] Mirzasoleiman | Coresets for data-efficient training of machine learning models[END_REF], our experimen-tal results did not show a clear advantage of data summarization over random sampling.

Organization

In Sec. 3, a federated learning algorithm that uses data summary is first presented, and its convergence guarantee is proven. Results from numerical experiments are detailed in Sec. 4. A discussion on our observations from these experiements as well as future work is given in Sec. [START_REF] Feige | A threshold of ln n for approximating set cover[END_REF].

Before presenting the algorithm in Sec. 3, we start by presenting in the next section the related work as well as the concepts from federated learning and submodular maximization that will be useful later on.

Preliminaries

In this section, we will summarize the two main objects of this paper. FL algorithms and data summarization techniques. First, we will present various FL algorithms and their convergence rates. This will be followed by known results in data summarization. These concepts will be used in the next section to show the convergence rate of a specific FL algorithm when it is combined with data summarization.

Federated learning (FL) [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF] enables multiple actors to build a common, robust machine learning model without sharing data, thus allowing to address critical issues such as data privacy, data security, data access rights and access to heterogeneous data. It is a distributed learning paradigm with two key challenges that differentiate it from traditional distributed optimization: (1) significant variability in terms of the characteristics (hardware, data rate, etc.) of each device or client in the network (system heterogeneity), and (2) non-identically distributed data across the devices (statistical heterogeneity) which can lead to model bias during training.

In the FL setting, it is assumed that there are K clients over which the data are partitioned, with P k the set of indexes of data points on client k, with n k = |P k |. The objective of the server is the following optimization problem:

min w f (w) = K k=1 n k n F k (w) where F k (w) = 1 n k i∈P k f i (w). ( 1
)
where f i (w) = ℓ(x i ; y i , w) is the loss of the prediction on example (x i ; y i ) made with the model parameters w.

If the partition P k was formed by distributing the training examples over the clients uniformly at random, then we would have

E P k [F k (w)] = f (w)
, where the expectation is over the set of examples assigned to a fixed client k. This is the IID assumption typically made by distributed optimization algorithms. The IID assumption is not always satisfied in FL settings since the server has no control on how the data is distributed among the clients. The degree of heterogeneity of the data in this non-IID setting can be quantified by Γ = F *k p k F * k , where F * (resp. F * k ) is the optimal loss on the whole (resp. local) dataset and p k = n k /n [START_REF] Li | On the convergence of fedavg on non-iid data[END_REF].

One of the first algorithm specifically designed for FL was FedAvg [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF] which proposed to cooperatively train a global model. The algorithm works in discretetime steps or rounds. At the beginning of each round, the clients receive the parameters of the global model from the central server. They then update the parameters using some standard learning algorithm on their local dataset and send the updated parameters back to the central server at the end of the round. The server averages the received model parameters and sends them to the end devices to signal the start of a new round. The pseudo-code of FedAvg is given in Alg. 1 with w t being the parameters of current model at round t. The amount of computation is controlled by three key parameters: C the fraction of clients that perform computation on each round; E, the number of training passes each client makes over its local dataset on each round; and B, the local mini-batch size used for the client updates. The convergence of FedAvg has been analysed in several works [START_REF] Li | On the convergence of fedavg on non-iid data[END_REF][23] [START_REF] Karimireddy | Scaffold: Stochastic controlled averaging for federated learning[END_REF] and the algorithm has established itself as the algorithm of choice for FL due to its simplicity and relatively low communication cost.

Algorithm 1 FedAvg algorithm

When there is a large statistical heterogeneity in the client datasets, the local updates in FedAvg can push the local model parameters towards the local optimum which is different from the global optimum. The global parameter models obtained from averaging can thus be different from the actual global optimum. FedProx [START_REF] Li | Federated optimization in heterogeneous networks[END_REF] proposes to add a regularization term to the objective function at every client. The new objective is

h k (w, w t ) = F k (w) + µ 2 ∥w -w t ∥ 2 . ( 2
)
with the second term on the RHS being the proximal term.

Convergence in FL

The main challenge in FL being the statistical heterogeneity, the analysis techniques used for centralized learning must be adapted for the non-IID case by adding assumptions on data dissimilarities. A number of works have studied the convergence of FL algorithms, trying to relax the less realistic assumptions. For example, the authors of [START_REF] Li | On the convergence of fedavg on non-iid data[END_REF] propose a convergence analysis for FedAvg where only a subset of clients participate in the training at each round. The authors of [START_REF] Yuan | On convergence of fedprox: Local dissimilarity invariant bounds, non-smoothness and beyond[END_REF] note that the assumption taken to bound gradient dissimilarity in the work introducing FedProx [START_REF] Li | Federated optimization in heterogeneous networks[END_REF] is quite unrealistic and propose to relax this assumption and extend the analysis to loss functions that are not smooth. However, they assume that the local client functions are L-Lipschitz which is also a restrictive condition on gradient dissimilarity. In Table 3 are some examples of convergence results along with the assumptions taken for FedAvg and FedProx. The explanations of the assumptions are in Table 1 and Table 2. Recall that here T the total number of communication rounds, E the number of local rounds between each communication round, K the total number of clients, C the number of clients participating in each round.

Table 1. Non-IID assumptions.

Symbol Full name Explanation

(G, B)-BGD Bounded gradient dissimilarity E k [||∇F k (w)|| 2 ] ≤ G 2 + ||∇f (w)|| 2 • B 2 BCGV Bounded inter-client gradient variance E k [||∇F k (w) -∇f (w)|| 2 ] ≤ δ 2
There are of course other significant works and approaches covering the topic of FL due to the growing interest in FL, both in industry and research. The goal of this section was however to explain some of the main concepts in FL and show the different challenges arising in the FL setup. Some of the methods covered in this section are orthogonal to each other and can be combined, some of them are only relevant in certain cases. Use proximal gradient steps on clients.

Table 3. Convergence rates.

Method

Non-IID Other assumptions Variant Rate Yu et al. [START_REF] Yu | Parallel restarted sgd with faster convergence and less communication: Demystifying why model averaging works for deep learning[END_REF] (G, 0)-BGD SMO; BLGV; BLGN AC

O( 1 √ KT ) Khaled et al. [9] (G-B)-BGD SMO; CVX; BLGV AC; LBG O( K T ) + O( 1 √ KT ) Li et al. [12] (G, 0)-BGD SMO; SCVX; BLGV; BLGN - O( E T ) Karimireddy et al. [8] (G, B)-BGD SMO; BLGV - O( T (1-C/K) T C
) FedProx [START_REF] Li | Federated optimization in heterogeneous networks[END_REF] (0, B)-

BGD SMO; BNCVX Prox O( 1 √ T )

Data Summarization

We are given a large dataset V of size n. The goal is to extract from V a subset S ⊂ V of data points which are most representative according to some objective function f : 2 V → R + . For each S ⊆ V , f (S) quantifies the utility of S. A set function f is naturally associated a discrete derivative, also called marginal gain,

∆ f (e|S) = f (S ∪ {e}) -f (S) ,
which quantifies the increase in utility obtained when adding e ∈ V to S.

Definition 1. A set function f : 2 V → R + is submodular if ∆ f (e|A) ≥ ∆ f (e|B) for every subsets A ⊆ B ⊆ V and every data point e ∈ V \ B. Further- more, f is monotone if and only if f (A) ≤ f (B) for every subsets A ⊆ B ⊆ V .
Submodular functions naturally model notions of information, diversity, and coverage in many applications. For example, let s i,j be the similarity between two elements i, j ∈ V (e.g., s i,j = e ∥i-j∥/σ ). Then, the following function is submodular (non-monotone):

f (S) = i∈V j∈S s i,j -λ i∈S j∈S s i,j ,
where λ ∈ [0, 1]. The first term is the traditional sum-coverage metric, while the second one penalizes similarity within S. Note that the function f (S) = i∈V max j∈S s i,j is another example of submodular function (see Chapter 3 of [START_REF] Mirzasoleiman | Big Data Summarization Using Submodular Functions[END_REF] for other examples).

We can distinguish two different data summarization (DS) problems of interest:

-Submodular maximization : the goal here is to find a summary S * of size at most k that maximizes the utility, that is,

S * = argmax |S|≤k f (S).
-Submodular cover : the goal is to find a subset S * of data elements which achieves a target fraction of the utility provided by the full dataset, that is,

S * = argmin {|S| : S ⊆ V s.t. f (S) ≥ (1 -ϵ)f (V )} .
These optimization problems are NP-hard for many classes of submodular functions [START_REF] Krause | Near-optimal nonmyopic value of information in graphical models[END_REF][5] [START_REF] Weinberg | Lecture notes in advanced algorithm design -lecture 7: Submodular functions, lovász extension and minimization[END_REF]. However, a simple greedy algorithm proposed by Nemhauser in [START_REF] Nemhauser | An analysis of approximations for maximizing submodular set functions -i[END_REF] is known to be very effective (see Algorithm 2).

Algorithm 2 Greedy algorithm

1: procedure Greedy 2: S ← ∅ 3: while |S| < k do 4: v ← argmax e∈V ∆ f (e|S) 5:
S ← S ∪ {v} 6: end while 7:

return S 8: end procedure Theorem 1 ( [START_REF] Nemhauser | An analysis of approximations for maximizing submodular set functions -i[END_REF]). For the submodular maximization problem of any nonnegative and monotone submodular function f , the greedy heuristic produces a solution S g of size k that achieves at least a constant factor (1 -1/e) of the optimal solution:

f (S g ) ≥ 1 - 1 e max [S]≤k f (S).
For the submodular cover problem, the approximation ratio of the greedy heuristic is 1 + log (max e f (e)), that is

|S g | ≤ 1 + log max e f (e) |S * | , where S * is the smallest subset (in cardinality) of V such that f (S * ) ≥ (1 - ϵ)f (V ).
Interestingly, [START_REF] Mirzasoleiman | Big Data Summarization Using Submodular Functions[END_REF] proposes an accelerated version of the greedy algorithm called Stochastic-Greedy that scales to voluminous datasets.

Federated learning with data summary

To reduce computational and energy costs, we propose that each client extracts a data summary S k from its local dataset P k . Further, we also modify how updates are done by clients in each epoch. Instead of training on mini-batches and covering the whole local dataset, we propose that client k perform one fullbatch update per epoch on S k .

The pseudo-code of the proposed FedAvg algorithm with data summary, which we call FedAVgDS, is shown in Alg. 3. We point out two main differences with the FedAvg algorithm both of which are in the ClientUpdate procedure. First, a data summary is performed in each local epoch. And, second, the local training is performed on the whole set S k at once unlike the batch-based training in FedAvg. Since the size of S k is expected to be small, we think that dividing it into batches is not necessary and the gradient can be computed on S k in it entirety. return w to server 20: end procedure

Algorithm 3 FedAvgDS algorithm

Convergence of FedAvgDS

We will prove convergence of FedAvgDS in the restricted setting of full node participation by mimicking the one of [START_REF] Li | On the convergence of fedavg on non-iid data[END_REF]. In fact, to show convergence of FedAvg, [START_REF] Li | On the convergence of fedavg on non-iid data[END_REF] modify slightly the original algorithm and assume that only one update is performed per epoch on a mini-batch. That is, the ClientUpdate procedure for the modified FedAvg in [START_REF] Li | On the convergence of fedavg on non-iid data[END_REF] is the same as that of FedAvgDS except that S k is chosen randomly and not by the GreedyDataSummary procedure.

For the convergence guarantees, we will need the same four assumptions as in [START_REF] Li | On the convergence of fedavg on non-iid data[END_REF] except that the third and the fourth ones are modified to account for the DS step as we explain below.

A.1 F k and the F S k are all L-smooth. A.2 F k and the F S k are µ-strongly convex. A.3 ∥∇F k (w) -∇F S k (w)∥ ≤ σ k , ∀w. Since we do not draw samples randomly, we do not need expectations in our bounds. A.4 ∥∇F S k (w)∥ ≤ G, ∀w. In FedAvg, this inequality is on the second moment of ∇F computed on the random mini-batches. Again, since we train over all of S k , we need the bound on the norm of ∇F S K . A.5 All nodes participate in all the iterations. For a random variable X taking values

x k , k = 1, . . . K, we have E[X] = K k=1 p k x k with K k=1 p k = 1. For example, p k = n k n . Theorem 2. Assume A.1 to A.5. Choose γ ≥ max{ 8L µ , E} and η t = 2 µ(t+γ)
for all t ≥ 0. Then,

F ( wt ) -F * ≤ L 2 v t + γ .
where v = max{v + , (γ + 1)∆ 1 },

v + = β 2 B + 2β 2 √ CD µβ -1 + (2β √ D) 2 + (2β √ D) 4 + 4(β 2 B + 2β 2 √ CD)(µβ -1)(2β √ D) 2 2(µβ -1) 2 , with C = 8(E -1) 2 G 2 + 6LΓ , D = K k=1 p 2 k σ 2 k , B = C + D and βµ > 1.
The proof of the above theorem is in Appendix A of [START_REF] Devillers | Data summarization methods for energy efficient federated learning[END_REF].

Finding a good DS subset

We consider two methods for extracting S k at client k. Both are based on distances computed from the average gradient of the loss function, and motivated from the observation that, after the local training, in an ideal setting, we would like the weights of client k to be updated as:

w ← w -η∇ w F k (w), (3)
that is, by following the average gradient of the loss on the whole local dataset P k . Since we want the updates with DS to closely follow those without DS, one natural way is to compute S k such that the average gradient of the loss on S k is close to that of P k . Define

∇ w F S (w) = 1 |S| i∈S ∇ w f i (w), (4)
to be the average gradient on S. If on S k , we have that ∇ w F S k (w) is a good approximation of ∇ w F k (w), then it will lead to a fairly similar updating of weights to that of FedAvg, and we can potentially hope to obtain convergence guarantees fairly close to those of FedAvg (i.e., without DS). Finding such a S k corresponds to minimizing

∥∇F k (w) -∇ w F S k (w)∥ , (5)
under a cardinality constraint on S k . However, we are faced with the problem that ( 5) is not submodular, a property that is desirable if we want to call upon efficient DS algorithm we saw in Sec. 2.2.

A first method around this problem is to instead use the objective function as in Exemplar Based Clustering which can then be transformed into a monotone submodular function (see [START_REF] Mirzasoleiman | Big Data Summarization Using Submodular Functions[END_REF] for this method). Towards this end, define

L k (S k ) = 1 |P k | i∈P k min j∈S k ∥∇ w f i (w) -∇ w f j (w)∥ , (6)
which computes how far the points in set P k are from their closest counterparts in S k . Here, the distance between two points is defined through the norm of the gradients.

The quantity in ( 6) can be related to a slight modification of that in (5) as follows. With σ(i) = argmin j∈S k ∥∇ w f i (w) -∇ w f j (w)∥, A j = {i ∈ P k , σ(i) = j} and α j = |A j |, we have:

L k (S k ) = 1 |P k | i∈P k ∇ w f i (w) -∇ w f σ(i) (w) ≥ 1 |P k | i∈P k (∇ w f i (w) -∇ w f σ(i) (w)) ≥ 1 |P k | i∈P k ∇ w f i (w) - j∈S k α j ∇ w f j (w) . (7)
Thus, by modifying the definition of ∇ w F S k to

∇ w F S k (w) = 1 |P k | i∈Sj α j ∇ w f j (w), (8)
it can inferred that if we find a S k with L k (S k ) ≤ ϵ, its norm in ( 5) is also at most ϵ.

To transform L k into a monotone submodular function, we follow the steps in [START_REF] Mirzasoleiman | Big Data Summarization Using Submodular Functions[END_REF]. Define

g k (S) = L k ({e 0 }) -L k (S ∪ {e 0 }),
with e 0 a phantom point chosen such that max

i ′ ∈P k ∥∇ w f i (w) -∇ w f i ′ (w)∥ ≤ ∥∇ w f i (w) -∇ w f e0 (w)∥, ∀i ∈ P k .
That is, e 0 is a point that is farther than any point in P k in the gradient distance. Then, g k monotone submodular.

Finally, the data summary is obtained by applying the greedy algorithm for the submodular cover problem, i.e. for a given ϵ k ∈ ]0, 1[, we look for the smallest possible subset S k ⊆ P k such that g k (S k ) ≥ (1-ϵ k )g k (P k ). The greedy algorithm will then find a S g k such that |S g k | ≤ (1 + log(max e f (e))) |S * |, where S * is the optimal subset.

It can be shown that ∥∇F k (w) -∇F S k (w)∥ is bounded with this method as is required in A.4 for the convergence guarantee.

Lemma 1 ∥∇F k (w) -∇F S k (w)∥ ≤ ϵ k L k ({e 0 }) =: σ k .
Proof. From the definition of g k and the fact that

g k (S k ) ≥ (1 -ϵ k )g k (P k ), we have L k ({e 0 }) -L k (S k ∪ {e 0 }) ≥ (1 -ϵ k ) • [L k ({e 0 }) -L k (P k ∪ {e 0 })].
Thanks to our choice of e 0 , L k (S ∪ {e 0 }) is L k (S) for any set S, and the above inequality reduces to

L k ({e 0 }) -L k (S k ) ≥ (1 -ϵ k ) • [L k ({e 0 }) -L k (P k )] which becomes L k (S k ) ≤ ϵ k L k ({e 0 }), since L k (P k ) = 0.
The claim follows by substituting [START_REF] Jiang | A linear speedup analysis of distributed deep learning with sparse and quantized communication[END_REF] in the above inequality. ⊓ ⊔ However, this method has the drawback that it requires to find e 0 satisfying max i ′ ∈P k ∥∇ w f i (w) -∇ w f i ′ (w)∥ ≤ ∥∇ w f i (w) -∇ w f e0 (w)∥, ∀i ∈ P k and that minimize i∈P k |∇ w f i (w) -∇ w f e0 (w)∥ under the previous constraint. This is in itself another optimization problem. Other works usually take the 0 point as their auxiliary element [START_REF] Mirzasoleiman | Big Data Summarization Using Submodular Functions[END_REF], however the theoretic results obtained with the optimal e 0 can't be verified in that case.

A second method for extracting S k is to transform the problem in (5) into a monotone submodular one by putting it in the Facility Location form and to maximize:

i∈P k max j∈S k s(i, j) (9)
where s(i, j) is a similarity measure between points i and j. To do so, we define

d(i, j) = ∇ w f i (w) -∇ w f j (w) and s k (i, j) = max e,e ′ ∈P k d(e, e ′ ) -d(i, j).
This formulation is very close to the problem addressed in [START_REF] Mirzasoleiman | Coresets for data-efficient training of machine learning models[END_REF], and it becomes the same for their application to DNNs. The greedy algorithm and its accelerated versions are equally applicable to this method. In the following, we will refer to Problem (9) mainly as the CRAIG problem. In the numerical experiments, we use the last-layer approximation as in CRAIG (see [START_REF] Mirzasoleiman | Coresets for data-efficient training of machine learning models[END_REF] for details).

Numerical experiments

All the experiments were performed using the Flower [START_REF] Beutel | Flower: A friendly federated learning research framework[END_REF] Python package which provides an easy-to-use library for federated learning. Its source code is available in the GitHub repository [START_REF]Flower github repository[END_REF]. We used a weighted variant of the cross-entropy loss and the Adam optimizer with its default parameters (except for the learning rate) for all clients in all our experiments. The dataset was either MNIST or CIFAR10.

In the legends, the terms 'ds' or 'craig' refer to the data summary obtained by solving Problem [START_REF] Khaled | Better communication complexity for local SGD[END_REF]. The terms 'r' or 'rd' refer to random sampling. The term 'full' refers to using the full set. The number following the terms in the legend is the subset size.

For our tests, we will show the test accuracy as a function of the number of rounds or as a function of the runtime. Indeed, the communication cost depends on the number of rounds, while the computational cost depends on the running time.

IID datasets

Our first tests were done using on a Quad Core Intel Core i7-7700 CPU. We first used MNIST dataset, splitting the training dataset into 10 datasets and splitting each of that dataset into a training and a validation set with a ratio of 90/10%. Therefore, each of our ten clients had a training set of 4500 images and a validation set of 500 images. We gave the MNIST test set to the server and used the implemented FedAvg [START_REF] Mcmahan | Communication-Efficient Learning of Deep Networks from Decentralized Data[END_REF] strategy of Flower.

On MNIST, we compared extracting a data summary using the CRAIG method, with random sampling, or the full dataset, with the same initial model parameters. The results presented here are for the accuracy on the test set. For Figs 2 and3, the subset size is of 500 images while it is of 200 for Fig. 1. The FL training included R = 10 rounds of training where at each round the server sends the current global parameters to the ten clients, the clients locally train the model for E = 5 epochs on the required dataset (full, random or data summary) and then sends its updated parameters to the server that aggregates all the responses to update the global model and so on.

Figure 1 shows that using subsets greatly accelerates training in terms of execution time when compared to the full dataset in our setup. However, this requires more rounds. In that configuration, we were able to reach a test accuracy of 0.980 in 8 rounds taking 18 minutes with the data summaries while it took 1h22 to complete 2 rounds of training on the full dataset to achieve a test accuracy of 0.976. In the FL setup, using submodular maximization seems to outperform random sampling. In 19 minutes corresponding to 10 rounds, the model was only able to reach an accuracy of 0.933 with random sampling. This comes from the fact that data selection time is much smaller compared to the whole training process.

Given that the communication rounds are very costly in FL, we tried to compensate the increased number of rounds by increasing the learning rate η. Figure 3 shows that increasing the learning rate did not really improve the training with the full dataset. For the data summaries however, increasing the learning rate in that configuration enabled the model to reach an accuracy of 0.96 in 2 rounds and 9 minutes when it took it 6 rounds and 35 minutes with the previous learning rate. In only 1 epoch, the model trained on the whole dataset was able to reach an accuracy of 0.954 in 41 minutes. Training on subsets therefore seems to greatly decrease the local training time but we were still not able on IID data to achieve a given accuracy in the same number of rounds as the full dataset training. This could however be possible on certain tasks or by adjusting some hyperparameters such as the learning rate η, the rate of subset selection (every E epochs in that configuration) or the subset size. Data subset selection seems far more relevant here, especially as the training takes more time. This shows that data subset selection, using random sampling or CRAIG sampling, can significantly speed up training for bigger models (with long training time) or for devices with limited system resources. This could help reduce the energy consumption in the case where reducing the dataset does not induce a drop in training accuracy and therefore the need for more computation rounds. Several hyperparameters are to be tuned for that, including the subset size, the number of local epochs and even the learning rate.

Non-IID datasets

In this part we tested two configurations. In the first one, 10 Flower clients run on a single machine with an Intel 5218R 2.1G CPU (20 cores) and one GPU RTXA6000. The datasets are divided as in the previous section. In the second configuration, 4 Flower clients run on the same machine, 2 on one RTXA6000 GPU and 2 on another RTXA6000 GPU. The dataset is split so that each client has images corresponding to 5 labels : 1 to 5 for the first, 6 to 10 for the second, only the even-numbered for the third and the uneven for the last one. This second configuration is therefore less non-IID than the first one. Each local dataset contains 12500 images while they contain 5000 images in the first configuration. On both configurations, the server is run on another machine (Intel E5-2695 v3 2.3G CPU). As the computing time is reduced with the use of GPU, we also decided to use ResNet-18 as this model is the one commonly used in the other works for tests on CIFAR-10. [START_REF] Mirzasoleiman | Coresets for data-efficient training of machine learning models[END_REF][18] [START_REF] Reddi | Adaptive federated optimization[END_REF]. As shown in Figure 4, the configuration with 10 clients, using CRAIG for datasummary is far too time-consuming compared with random sampling or even using the full set, as mentioned earlier in this report. However, random sampling seems to perform better than using the full set, even if we reduce the number of training epochs for each round to E = 1. This probably comes from the chosen data distribution. As we are in a non-iid configuration, each local model will be updated towards a local optimum which is not the same as the global one. If we let each client make a large number of local updates, their local In the configuration with 4 clients as shown in Figure 5, the conclusion for the CRAIG data summarization is even clearer. As each local dataset here contains 12500 images, with 1250 in the validation set and the rest in the training set, extracting a subset containing 1250 images (a 0.11 fraction as before) is even more time-consuming. Full set strategies perform better than random and CRAIG sampling in terms of test accuracy at each round, although random and CRAIG sampling soon catch up.

The results in both Figure 4 and 5 suggest that there are cases where using the full set even for only one epoch is not necessary, making the use of subsets relevant.

Frequency of subset selection In the previous section, using CRAIG was too time-consuming. One way of limiting this is to play on the frequency with which the subset is selected. On all our previous experiments, we selected a new subset at each round but we can also decide to choose a new subset each p rounds. In this part, we use the same data partition as before with 10 clients, each having a training set of size 4500 representing 2 different labels for CIFAR-10. We distribute those 10 clients on 2 GPUs. Figure 7 compares the test accuracy obtained using the full set with the one using random or CRAIG sampling each p = 50 rounds. When looking at the accuracy as a function of the number of rounds, the three methods produce close results. Craig sampling seems slightly better in this configuration. When looking at the time, random sampling is as expected the quickest but, for the first time in our experiments using GPU, the curve for CRAIG sampling closely follows that of random sampling while the one for using the full set is clearly below.

Size of subset Of course, the size of the selected subset is a hyperparameter that can greatly impact the training. Figure 8 shows the test accuracy for different values of the subset size S for random sampling. If S is too large, the local model will move towards the local optimum which can be detrimental to the training. This is why using a subset of 2250 outperforms using the full set (S = 4500). A large value of S will also increase the runtime, especially when using submodular maximization for subset selection. If S is too small, the training will take much longer in terms of rounds which will result in an increased communication cost. This is what would happen for S = 500 where the model learns and eventually achieves the accuracy reached by bigger subsets but this takes more rounds. A balance has to be found. In addition to that, the optimal value for S depends on the chosen value of E or even of the learning rate η.

Discussion and future work

Based on the results of our experiments, we can clearly identify two cases in which the use of subsets of local datasets seems relevant. In both however it is not clear how much improvement can be made by choosing the subset using a DS method rather than randomly selecting one.

The first case concerns clients with limited computational resources. In those cases, the training can take a very long time (the straggler problem) due to clients that are holding back other faster one. In some cases, it is important to also being able to include stragglers in the training loop, especially if the data points of the stragglers don't appear sufficiently in other datasets. Using subsets instead of the full set can significantly decrease the training time, particularly if those clients have large datasets, and therefore help them participate in the FL training.

The second case concerns clients with large datasets that are not representative of the whole data distribution. Basically, this means that their local optimum for the model's parameters is different from the global optimum. To prevent them from converging too much towards their local optimum, hyperparameters have to be chosen wisely. This can be done for example by reducing the learning rate or increasing the batch size in order to reduce the number of local updates. However, this is not always the best choice for example if the training works better with the initial hyperparameters for other clients with smaller datasets and more generally if we can achieve the same performance simply by going through a smaller number of points of their datasets. This is our interpretation of what happens in Section 4.2 where using subsets leads to better results than using the full set for 1 epoch.

Selecting a subset also requires tuning some parameters including the size of the subset and the selection frequency. In our case, reducing the selection frequency did not lead to a drop in accuracy while, for the data summarization method, it helped reduce the running time as shown in Section 4.2. Considering the size of the subset, a balance has to be found as explained in part 4.2. Tuning those parameters can be done locally in order to avoid additional communication rounds. It could of course result in an additional computational cost but this could easily be offset for long and costly trainings, for example in a FL hyperparameter tuning context. However, it is important to point out that the use of subsets also influences the choice of certain hyperparameters, such as the number of epochs for local training in each round. More indirectly, the choice of other parameters, such as the learning rate, may depend on whether or not subsets are used.

Our approach focused on finding a subset approximating the full set gradient for the loss. Our study did not show clear improvements in the test accuracy or the loss. Considering the longer runtimes, this means that using submodular maximization rather than random sampling was not relevant in our case. This could be linked to the datasets we used or the way we distributed them between clients. Indeed, in our experiments, each client's local dataset would actually be quite homogeneous, with 5000 images of only 2 labels resulting in a lot of redundancy. Based on that, it is not surprising that random performed well. One might think that results would be different on more complex datasets, but this has not been studied here and is part of our future work, Another avenue of investigation is to determine the actual energy consumption of the algorithms with and without energy and determine the tradeoff between the accuracy and the energy consumption.

  (a) Nb of epochs as x-axis. (b) Runtime as x-axis.

Fig. 1 .

 1 Fig. 1. Test accuracy compared between random sampling, CRAIG sampling or the full dataset for MNIST IID data. E = 5, R = 10, η = 1e -3. Full set size is 50000 and subset size is 0.0044.

Fig. 2 .

 2 Fig. 2. Test accuracy compared between random sampling, CRAIG sampling or the full dataset for MNIST IID data. E = 5, R = 10, η = 1e -3. Full set size is 50000 and subset fraction size is 0.011. FL setup.

  (a) Nb of epochs as x-axis. (b) Runtime as x-axis.

Fig. 3 .

 3 Fig. 3. Test accuracy compared between random sampling, CRAIG sampling or the full dataset for MNIST IID data. E = 5, R = 10, η = 1e -2 for the high learning rate ( hlr) and η = 1e -3 for the other. Full set size is 50000 and subset fraction size is 0.011.

  (a) Nb of rounds as x-axis. (b) Runtime as x-axis.

Fig. 4 .

 4 Fig. 4. Test accuracy compared between random sampling, CRAIG sampling or the full dataset for CIFAR-10 non-IID data, using ResNet-18. R = 100, η = 1e -3, µ = 0.01. Subset fraction size is 0.11. FL setup with 10 clients on 1 GPU.

  (a) Nb of rounds as x-axis. (b) Runtime as x-axis.

Fig. 5 .

 5 Fig. 5. Test accuracy compared between random sampling, CRAIG sampling or the full dataset for CIFAR-10 non-IID data, using ResNet-18. R = 100, η = 1e -3, µ = 0.01. Subset fraction size is 0.11. FL setup with 4 clients on 2 GPU.

  (a) Test accuracy per round for CRAIG sampling. (b) Test accuracy as a function of time for CRAIG sampling. (c) Test accuracy per round for random sampling.

Fig. 6 .

 6 Fig. 6. Test accuracy for random sampling and CRAIG sampling for different values of the period p for extracting the subsets. CIFAR-10 non-IID data, using ResNet-18. R = 100, η = 1e -3, µ = 0.01. Subset fraction size is 0.11. FL setup with 10 clients on 2 GPU.

Figure 6

 6 Figure6shows the test accuracy obtained for different values of the period p for random and CRAIG sampling. Interestingly enough, selecting the subset less frequently does not lead to a significant drop in precision. Meanwhile, this indeed allows us to save some computing time on CRAIG sampling.

  (a) Nb of rounds as x-axis. (b) Runtime as x-axis.

Fig. 7 .

 7 Fig. 7. Test accuracy compared between full set or random sampling and CRAIG sampling with p = 50 for CIFAR-10 non-IID data, using ResNet-18. R = 100, η = 1e -3, µ = 0.01. Subset fraction size is 0.11. FL setup with 10 clients on 2 GPU.

  (a) Nb of rounds as x-axis. (b) Runtime as x-axis.

Fig. 8 .

 8 Fig. 8. Test accuracy compared for various sizes of the subset selected by random sampling for CIFAR-10 non-IID data, using ResNet-18. R = 100, η = 1e -3, µ = 0.01. Subset fraction size varies. FL setup with 10 clients on 2 GPU.

Table 2 .

 2 Other assumptions and variants. (.) is convex. SCVX Each client function F k (.) is µ-strongly convex. BNCVX Each client function has bounded nonconvexity with ∇ 2 F k (x) ⪰ -l • I. SMO Each client function F k (.) is L-smooth. BLGV The variance of stochastic gradients on local clients is bounded. BLGN The expected squared norm of any stochastic gradient is bounded. LBG Clients use the full batch of local samples to compute updates. AC All clients participate in each round. Prox

	Symbol Explanation
	CVX	Each client function F k

⋆ This work was partially financed by French Agence Nationale de la Recherche (ANR) through grant ANR-22-CE23-0024 (project DELIGHT).


Tristan Asset
email: t.asset@unistra.fr

The Runt of Ammonia Production by N 2 Reduction: Electrocatalysis in Aqueous Media

Keywords: Nitrogen Reduction Reaction, Surface Diffusion, Control Experiments, Ionic Liquids, Gas Diffusion Electrode

Highlights:

• Design of bifunctional electrocatalysts to break the scaling relationships;

• New interfaces to better control the reactant concentration in the local reaction environment;

• Updated experimental protocols to ease the emergence of new systems/electrocatalysts design.

Introduction: Ammonia is a cornerstone of our society, present and future. It is essential in the fertilizers production chain, therefore being an indispensable tool of a farming industry faced with an evergrowing human population. Furthermore, it is often envisioned as a key alternative for the hydrogen storage and, thus, the entire fuel cell industry. However, its main mean of production remains the Haber-Bosch process, which extreme conditions (200 -300 × 10 5 Pa, 300 -500°C), high energy consumption and CO2 generation [1,[START_REF] Kyriakou | An Electrochemical Haber-Bosch Process[END_REF] justify the willingness of the scientific community to investigate potential alternatives. They are inspired in this endeavour by the existence of nitrogenase, an enzymatic structure built around a FeMo complex (-co) able to produce NH3 from N2 in ambient conditions (see Figure 1a) [START_REF] Hoffman | Mechanism Of Nitrogen Fixation By Nitrogenase: The Next Stage[END_REF][START_REF] Sippel | A Bound Reaction Intermediate Sheds Light On The Mechanism Of Nitrogenase[END_REF]. The latter has been extensively investigated during the past decade, leading to the identification of activity and selectivity defining features, i.e., the ability to control (i) the local reaction environment (LRE) of the FeMo-co (e.g., concentration in protons, nitrogen and oxygen) and (ii) the rate of delivery of the two former to the FeMo-co [1, [START_REF] Burgess | Mechanism Of Molybdenum Nitrogenase[END_REF][START_REF] Durrant | Controlled Protonation Of Iron-Molybdenum Cofactor By Nitrogenase : A Structural And Theoretical[END_REF]. The goal thus became to 'mimic' nitrogenase, while adding advantageous features of other (electro)catalytic systems, e.g., increased active site density, ease of preparation, stability when exposed to O2, etc. The three main alternatives are, nowadays, well identified (see Figure 1b-d):

(i)

Homogenous catalysis [START_REF] Yandulov | Catalytic Reduction of Dinitrogen to Ammonia at a Single Molybdenum Center[END_REF];

(ii) Li-mediated electrocatalysis [START_REF] Lazouski | Understanding Continuous Lithium-Mediated Electrochemical Nitrogen Reduction[END_REF][START_REF] Westhead | Is Lithium The Key For Nitrogen Electroreduction?[END_REF][START_REF] Tsuneto | Efficient Electrochemical Reduction of N2 to NH3 Catalyzed by Lithium[END_REF][START_REF] Du | Electroreduction Of Nitrogen With Almost 100% Current-To-Ammonia Efficiency[END_REF];

(iii) Electrocatalysis in aqueous media (e.g., nitrogen reduction reaction, NRR); Among those, homogenous catalysis and Li-mediated systems are most often pursued, owing to the shortcomings of the N2 electroreduction in NH3 in aqueous environment. Indeed, (i) theoretical calculations evidenced that, in the presence of an aqueous proton source, most electrocatalyst surfaces favour the hydrogen evolution reaction (HER) vs. the NRR, resulting in extremely low faradaic efficiencies [START_REF] Skúlason | A Theoretical Evaluation Of Possible Transition Metal Electro-Catalysts For N2 Reduction[END_REF]. Furthermore, albeit the NRR has been extensively studied throughout the past decade, it has been plagued by incorrect assessment of the NH3 generation, due to (i) the ease of contamination of the reactive system by NH3 and (ii) the low amount of NH3 generated by electrochemical means [START_REF] Choi | Identification And Elimination Of False Positives In Electrochemical Nitrogen Reduction Studies[END_REF][START_REF] Andersen | A Rigorous Electrochemical Ammonia Synthesis Protocol With Quantitative Isotope Measurements[END_REF][START_REF] Greenlee | The Use of Controls for Consistent and Accurate Measurements of Electrocatalytic Ammonia Synthesis from Dinitrogen[END_REF]. This resulted in a general disbelief in the future of the generation of NH3 by NRR in aqueous environment, coupled with extremely complex experimental procedures to certify the findings. Here, we shall nevertheless discuss it, and its perspectives. As the two other approaches are not discussed here, any interested reader is encouraged to read the excellent review by Westhead et al.

[1] on these topics.

Directions for the nitrogen reduction reaction:

The NRR potential has long been seen in the prism of the theoretical calculations of the Gibbs energy of adsorption of N* (ΔGN*) vs. the Gibbs energy of adsorption of H*, ΔGH*. This started with the pivotal work of Skúlason et al. [START_REF] Skúlason | A Theoretical Evaluation Of Possible Transition Metal Electro-Catalysts For N2 Reduction[END_REF] underlining the potential gap between the NRR equilibrium potential (E = 0.057 V vs. SHE) and the highest onset potential achievable on a mono-element surface, at E = ca. -0.5 V vs. SHE. Such discrepancy is due to the existence of 'scaling relationships' [START_REF] Montoya | The Challenge Of Electrochemical Ammonia Synthesis: A New Perspective On The Role Of Nitrogen Scaling Relations[END_REF], i.e., the fact that the binding strength of the NRR intermediates are correlated (see Figure 2a). They imply that an electrocatalyst surface cannot be optimal for each intermediate step as it cannot bind all intermediates with the 'ideal' binding strength, but should instead be a compromise of 'acceptable' binding strength for all of them. In addition, Skúlason et al. also evidenced a favourable adsorption of hydrogen, and subsequent reduction in H2 vs. the NRR, for the overwhelming majority of the elements they investigated [START_REF] Skúlason | A Theoretical Evaluation Of Possible Transition Metal Electro-Catalysts For N2 Reduction[END_REF].

The limitations of theoretical findings -Promising approaches for the NRR were nevertheless underlined by theoretical calculations and modelling, e.g., metal nitrides (operating through a Mars-van Krevelen mechanism), atomically dispersed single or dual atoms, etc. [START_REF] Lv | High-Throughput Screening Of Synergistic Transition Metal Dual-Atom Catalysts For Efficient Nitrogen Fixation[END_REF][START_REF] Abghoui | Electroreduction of N2 to Ammonia at Ambient Conditions on Mononitrides of Zr, Nb, Cr, and V: A DFT Guide for Experiments[END_REF][START_REF] Abghoui | Electrochemical Synthesis Of Ammonia Via Mars-Van Krevelen Mechanism On The (111) Facets Of Group III-VII Transition Metal Mononitrides[END_REF]. Unfortunately, modelling and theoretical calculations were also almost systematically used to explain the NH3 generation of electrocatalysts published during the last decade. This is an issue, as it is nowadays understood that most of those electrocatalysts were unlikely to be active for the NRR, but that the NH3 observed came either from environmental pollution or NOx reduction [START_REF] Choi | Identification And Elimination Of False Positives In Electrochemical Nitrogen Reduction Studies[END_REF]. The main problem here arise that modelling does not often consider some reactive pathways and critical steps, e.g., the non-electrochemical N2 dissociation in 2N*. Furthermore, it is often performed in unrealistic conditions, e.g., in vacuum, on model slabs, without a time dimension, etc. Thus, it fails to see the hinderances in the LRE (e.g., local pH, cations distribution, etc.) and other parameters that contribute to the modification of the reactivity and surface state of the electrocatalyst. As such (i) theoretical calculations and modelling should be carefully used, and never as the sole justification of an electrocatalyst' performance and (ii) substantial time should be invested to develop modelling (and experimental [START_REF] Monteiro | Measuring Local Ph In Electrochemistry[END_REF]) tools that are able to adequately mimic the NRR' LRE [START_REF] Fernández-Alvarez | Molecular Dynamics Study of Reaction Conditions at Active Catalyst-Ionomer Interfaces in Polymer Electrolyte Fuel Cells[END_REF][START_REF] Sakong | The Structure Of The Electric Double Layer: Atomistic Versus Continuum Approaches[END_REF][START_REF] Soares | Real Electrolyte Solutions in the Functionalized Mean Spherical Approximation: A Density Functional Theory for Simple Electrolyte Solutions[END_REF].

Bifunctional electrocatalysts -Hence, the design of the NRR interface in aqueous environment should be built either from our understanding of the multi-steps nature of its mechanism or of the nitrogenase.

Specifically, the aim should be to (i) break the scaling relationships and/or to (ii) control the presence of H* at the reactive interface. It is expected that N* binding energy on an active site is strongly dependent on its local environment. A relevant example is that, for the oxygen reduction reaction (ORR), the OH* binding strength to a platinum atom is correlated to the platinum coordination number and its distance with its closest neighbours. Modifying those parameters by increasing the structural disorder leads to the emergence of active sites with a near-perfect local environment (i.e., coordination number and distance with its nearest neighbours), therefore being at the apex of the volcano plot [START_REF] Bacq | Effect of Atomic Vacancies on the Structure and the Electrocatalytic Activity of Pt-rich/C Nanoparticles: A Combined Experimental and Density Functional Theory Study[END_REF][START_REF] Chattot | Surface Distortion As A Unifying Concept And Descriptor In Oxygen Reduction Reaction Electrocatalysis[END_REF]. But this remains limited by the scaling relationships, and, to go 'beyond', an option that ought to be envisioned is the use of bifunctional electrocatalysts [START_REF] Montoya | The Challenge Of Electrochemical Ammonia Synthesis: A New Perspective On The Role Of Nitrogen Scaling Relations[END_REF], i.e., surfaces where two types of active sites are in close proximity while presenting widely different ΔGN* (arising from various coordination/atomic distance with nearest neighbours, different elementssee Figure 2b, etc.) and thus being each optimal for different steps of the reaction. The two main questions on those structures would then be (i) what is the minimal dimension of the domain for each type of active site and (ii) will the surface diffusion (SD) of the reactive intermediates be possible? Indeed, an alloy, resulting in an atomic-ranged alternance of each domains, might result in a common, and not two individual ΔGN*, whereas nanoparticules decorating NRR-active supports or Janus-like nanoparticles (Figure 2c) might be adequate. The idea of SD to ease the reduction of NRR intermediates arises from the field of alcohols and CO* oxidation. Lebedeva et al. [START_REF] Lebedeva | Role Of Crystalline Defects In Electrocatalysis: Mechanism And Kinetics Of CO Adlayer Oxidation On Stepped Platinum Electrodes[END_REF] observed CO* diffusion from terraces to steps, i.e., toward more favourable oxidation sites.

Dubau et al. [START_REF] Dubau | Electrooxidation Of Methanol At Platinum-Ruthenium Catalysts Prepared From Colloidal Precursors: Atomic Composition And Temperature Effects[END_REF][START_REF] Dubau | On The Structure Effects Of Bimetallic PtRu Electrocatalysts Towards Methanol Oxidation[END_REF] then observed an enhanced activity for a mixture of platinum and ruthenium nanoparticles (vs. the individual nanoparticles or the alloy) for the methanol oxidation reduction, arguing that the methanol adsorption-dehydrogenation was occurring on platinum, followed by the CO* diffusion to the ruthenium sites, where it was oxidized in CO2. Here, it is critical to underline that (i) SD is not limited by the N* energy of adsorption (ΔEN*), but by the energy required to move out of an adsorption site to a neighbouring one (i.e., ΔEdiff, with ΔEdiff << ΔEN*) and that (ii) SD is fast, even on > 3 nm Pt nanoparticles, for strongly bound CO* [START_REF] Maillard | Size Effects On Reactivity Of Pt Nanoparticles In CO Monolayer Oxidation: The Role Of Surface Mobility[END_REF] (e.g. Kobayashi et al. [START_REF] Kobayashi | An NMR Determination Of CO Diffusion On Platinum Electrocatalysts[END_REF] estimated a 36 nm 2 /s diffusion rate for CO* on Pt nanoparticles of 7 nm). Hence, it would be meaningful to investigate, through microkinetic modelling, if the same can apply to the SD of the NRR intermediates (e.g., N2H*, N*) from surfaces on which they are strongly bound to surfaces that are adequate for hydrogenation and are weakly binding them. Using heterogenous surfaces has been touched upon in the NRR field, through the study of vacancies, heteroatoms, single and dual-atoms sites [START_REF] Han | Atomically Dispersed Molybdenum Catalysts for Efficient Ambient Nitrogen Fixation[END_REF][START_REF] Liu | FeMo Sub-Nanoclusters/Single Atoms For Neutral Ammonia Electrosynthesis[END_REF], etc. [START_REF] Guo | Defect And Interface Engineering For Electrochemical Nitrogen Reduction Reaction Under Ambient Conditions[END_REF][START_REF] Huang | Recent Progress In Electrocatalytic Nitrogen Reduction To Ammonia (NRR)[END_REF] but decorrelating NH3 generation arising from the NRR and the contamination in these works is extremely difficult [START_REF] Choi | Identification And Elimination Of False Positives In Electrochemical Nitrogen Reduction Studies[END_REF]. However, albeit the performances themselves can be debated, electrocatalysts such as the gold nanoparticules supported on Fe3O4 developed by Zhang et al. [START_REF] Zhang | Adsorbing and Activating N2 on Heterogeneous Au-Fe3O4 Nanoparticles for N2 Fixation[END_REF] echoes the concept of a bifunctional electrocatalyst.

Controlling the local reaction environment -A fine tuning of the LRE itself would help in (i) increasing the N2 concentration and (ii) minimizing the concentration in aqueous proton sources at the reactive interface. Similarly to other reactions in aqueous environment, the solubility of N2 is extremely low (i.e., ca. 0.7 mmol L -1 in H2O at room temperature -RT), this being partly responsible for the almost-inexistent selectivity toward the NRR in aqueous media (see Figure 3a). Such phenomenon has been deemed an issue when comprehensively studying electrocatalysts for the ORR, with O2 as the reactant (i.e., ca.

1.2 mmol L -1 in H2O at RT), resulting in the design of 'gas diffusion electrode' (GDE)-cell [START_REF] Riasse | Benchmarking Proton Exchange Membrane Fuel Cell Cathode Catalyst At High Current Density: A Comparison Between The Rotating Disk Electrode, The Gas Diffusion Electrode And Differential Cell[END_REF][START_REF] Ehelebe | Benchmarking Fuel Cell Electrocatalysts Using Gas Diffusion Electrodes: Inter-lab Comparison and Best Practices[END_REF], in which the O2 is provided to the LRE under gaseous form, through the working electrode (see Figure 3c). This substantially decreased the O2 mass transport related limitations. In the NRR field, this approach was suggested by Kani et al. [START_REF] Kani | Competing Effects of pH, Cation Identity, H2O Saturation, and N2 Concentration on the Activity and Selectivity of Electrochemical Reduction of NH3 on Electrodeposited Cu at Ambient Conditions[END_REF], along with optimizing the pH of the solution and the nature of the cation. Lazouski et al. [START_REF] Lazouski | Non-Aqueous Gas Diffusion Electrodes For Rapid Ammonia Synthesis From Nitrogen And Water-Splitting-Derived Hydrogen[END_REF] also discussed this topic, suggesting the use of carbon or stainless steel cloth in non-aqueous environment for the NRR. Of greater challenge is the control of the concentration of the aqueous proton sources in the LRE. Removing altogether the aqueous electrolyte has been investigated, by using ionic liquids (ILs). An increased selectivity [START_REF] Suryanto | Rational Electrode-Electrolyte Design for Efficient Ammonia Electrosynthesis under Ambient Conditions[END_REF][START_REF] Zhou | Electrosynthesis Of Ammonia From Nitrogen At Ambient Temperature And Pressure In Ionic Liquids[END_REF] was observed at the cost of a low NRR activity, likely induced by a nearinexistent concentration of a proton source in the IL, required for the NRR intermediates hydrogenation. could be addressed by adding in the electrolyte an alternative, non-aqueous proton source, similarly to the Li-mediated system approach (Figure 1c) [START_REF] Lazouski | Non-Aqueous Gas Diffusion Electrodes For Rapid Ammonia Synthesis From Nitrogen And Water-Splitting-Derived Hydrogen[END_REF][START_REF] Lazouski | Proton Donors Induce a Differential Transport Effect for Selectivity toward Ammonia in Lithium-Mediated Nitrogen Reduction[END_REF][START_REF] Suryanto | Nitrogen Reduction To Ammonia At High Efficiency And Rates Based On A Phosphonium Proton Shuttle[END_REF]. But, if the objective is to maintain a less expensive, aqueous like design, an encapsulation (vs. a bulk presence) of the IL should be favoured (see

Figure 3b

). This approach was explored for the ORR by Snyder et al. [START_REF] Snyder | Oxygen Reduction Reaction Performance of [MTBD][beti]-Encapsulated Nanoporous NiPt Alloy Nanoparticles[END_REF] to ease the transport of O2 in the complex porous network of an active material, or, more recently, by Avid et al. [45] to increase the accessibility of the active sites, and would allow a fine tuning of the N2, and aqueous proton source concentration, in the LRE. Controlling the electron availability, by tuning the conductivity of the electrocatalyst support, should not be overlooked either. Indeed, Singh et al. [START_REF] Singh | Electrochemical Ammonia Synthesis -The Selectivity Challenge[END_REF] calculated that the HER would be greatly inhibited by decreasing the electron availability while the NRR activity would remain identical. Easing scientific innovation in the NRR field: The exploration of new directions for the NRR in aqueous environment requires a less tedious characterization process. Indeed, our community became extremely dismissive of new findings, ascribing the performances of any supposed NH3-generating electrocatalyst to a yet-to-be-identified source of pollution. This resulted in the development of extremely rigorous, yet imperfect, assessment procedures (e.g., including mandatory characterization with 15 N2. However, without a proper purification, the latter might contain substantial amount of easy-to-reduce 15 NOx, as evidenced by Dabundo et al. [START_REF] Dabundo | The Contamination Of Commercial 15 N2 Gas Stocks With 15 N-Labeled Nitrate And Ammonium And Consequences For Nitrogen Fixation Measurements[END_REF] in a systematic analysis of the 15 N2 from different providers, which would render the 15 N2 characterization meaningless) that did not manage to stop the propagation of false positives while putting a notable barrier of entry to the field (e.g., the purchase of the requested amount of 15 N2 and/or the development of an adequate gas recirculation and purification system is, based on the author experience, > 10,000 euros), thus greatly limiting the spread of new ideas. Therefore, a less constraining, more collaborative, protocol is here suggested: (i) characterization of new material through a rigorous, multi-reference samples approach in Ar-, N2-and NOx-saturated environment (see Figure 4), as the promising activity of a given electrocatalyst might simply arises of traces of 15 NOx or often assessed as promising NRR electrocatalysts, were instead catalysing the NOx reduction reaction -NOxRR); (ii) transfer of the promising electrocatalysts to an independent research group that possesses the equipment and the know-how to fully assess the promises of the this electrocatalyst (e.g., similarly to the recent works of Choi and Du et al. [START_REF] Choi | Identification And Elimination Of False Positives In Electrochemical Nitrogen Reduction Studies[END_REF][START_REF] Choi | Electroreduction of Nitrates, Nitrites, and Gaseous Nitrogen Oxides: A Potential Source of Ammonia in Dinitrogen Reduction Studies[END_REF][START_REF] Du | Critical Assessment of the Electrocatalytic Activity of Vanadium and Niobium Nitrides toward Dinitrogen Reduction to Ammonia[END_REF][START_REF] Du | Is Molybdenum Disulfide Modified with Molybdenum Metal Catalytically Active for the Nitrogen Reduction Reaction?[END_REF] to debunk several 'so-called' NRR electrocatalysts) and insure not only its robustness while exposed to more elaborated characterization protocols, but also its reproducibility, along to promoting interactions between different NRR-related fields (electrochemistry, systems, materials design, etc.); (iii) if the electrocatalysts validates this first set and second set of characterization, to publish it as an electrocatalyst for the NRR.

Conclusion:

The nitrogen reduction reaction in aqueous media is an uphill battleone that might eventually conclude in the foretold conclusion that this field was always hopeless. However, directions to be explored remain. First, at the scientific level: (i) the use of bifunctional electrocatalysts might allow to go beyond the scaling relationships, enhancing the intrinsic activity for the NRR, whereas (ii) tuning the LRE (e.g., N2 and aqueous proton sources concentration near the reactive interface) should greatly improve the electrocatalyst' selectivity for the NRR. Both directions might allow the community to generate sufficient amount of NH3 to be freed of the substantial contribution of contaminants, NO3RR, etc. However, since this goal is yet to be achieved, the means of characterization should also be thought upon. Echoing the message sent by Greenlee et al. [START_REF] Greenlee | The Use of Controls for Consistent and Accurate Measurements of Electrocatalytic Ammonia Synthesis from Dinitrogen[END_REF] and Choi et al. [START_REF] Choi | Identification And Elimination Of False Positives In Electrochemical Nitrogen Reduction Studies[END_REF] in 2018 and 2020, respectively, the overwhelming complexity of this field requires to rethink the way we do share our results and interact with our so-called competitors: validating new concepts should be done carefully and with the contribution of research labs with the proper expertise (i.e., the so-called 'certification laboratories' [START_REF] Choi | Identification And Elimination Of False Positives In Electrochemical Nitrogen Reduction Studies[END_REF]) for each new material; the publication of negative results should be encouraged and valorised [START_REF] Greenlee | The Use of Controls for Consistent and Accurate Measurements of Electrocatalytic Ammonia Synthesis from Dinitrogen[END_REF], to avoid the community losing itself in the wrong direction. This aims to eventually rebuild trust in this domainand not to see it, yet again, being swept by incorrect findings, and identified, yet again, as the runt of the litter in the future of ammonia generation.

Figure 1 .

 1 Figure 1. Illustration of the main alternatives to the Haber-Bosch process to reduce N2 in NH3, i.e., (a) the nitrogenase enzyme and the clusters of interest (with the FeMo-co including a R-homocitrate group, here illustrated with a dotted line); (b) catalysis for the homogeneous reduction of N2 on Fe/Mo complexes (the electrons are provided by the Fe/Mo oxidoreduction cycle), taking advantage of 'bulky ligands' to control the environment near

Figure 2 .

 2 Figure 2. Schematic representation of (a) the 'volcano-plot' relationship between the theoretical potential and the free energy of adsorption of N* (ΔGN*) (in pink, the domain in which N* is more favourable than H*) and the scaling relationship between different intermediate steps of the nitrogen reduction reaction (NRR); (b) a surfacediffusion-based alternative to the NRR, to 'break' the scaling relationships through the replacement of

Figure 3 .

 3 Figure 3. Schematic representation of the different types of interfaces to be envisioned for the nitrogen reduction reaction (NRR): (a) the 'conventional' interface, with aqueous environment and N2 dissolved in H2O, leading to a hydrogen evolution reaction (HER) dominated interface; (b) an aqueous interface with the active material encapsulated in ionic liquid (IL), thus tuning the solubility of the reactants at the interface; (c) a 'gas diffusion electrode' design, in which the N2 is provided under gaseous form directly in the LRE.

Figure 4 .

 4 Figure 4. Illustration of different cases of NH3 detection in a potential NRR system in aqueous environment, using several reference points to rigorously assess any parasitic NH3, e.g., under Ar, in absence of electrocatalyst (empty), at open-circuit potential and two potential of interest (OCP, E1, E2) and at OCP and two potentials of interest under N2 or in presence of NOx (e.g., NO3 - , NO2, NO): (yellow) a substantial increase of the NH3 generation is observed at E1 and E2 under N2, and no NH3 generation is observed in presence of NOx = promising NRR electrocatalyst; (purple) a substantial increase of the NH3 generation is observed at E1 and E2 under N2 and NOx = promising NOxRR electrocatalyst; (blue) albeit NH3 'generation' is observed at E1 and E2, it is below, or equivalent, to the highest reference point in non-NRR conditions = non-NRR active electrocatalyst.

Acknowledgments -T.A. thanks Elena Savinova for the insightful comments and discussion on the original version of this manuscript. This work was partially supported by the energy branch of the French national center for scientific research (CNRS) through a PEPS grant.