# FedCLIP: Fast Generalization and Personalization for CLIP in Federated
Learning

Wang Lu1 Xixu Hu2 Jindong Wang3 Xing Xie3
1 Chinese Academy of Sciences, Beijing, China
2 City University of Hong Kong, Hong Kong
3 Microsoft Research Asia, Beijing, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>{jindong.wang,
<EMAIL_ADDRESS>
Corresponding author: Jindong Wang<EMAIL_ADDRESS>
###### Abstract

Federated learning (FL) has emerged as a new paradigm for privacy-preserving
computation in recent years. Unfortunately, FL faces two critical challenges
that hinder its actual performance: data distribution heterogeneity and high
resource costs brought by large foundation models. Specifically, the non-IID
data in different clients make existing FL algorithms hard to converge while
the high resource costs, including computational and communication costs that
increase the deployment difficulty in real-world scenarios. In this paper, we
propose an effective yet simple method, named FedCLIP, to achieve fast
generalization and personalization for CLIP in federated learning. Concretely,
we design an attention-based adapter for the large model, CLIP, and the rest
operations merely depend on adapters. Lightweight adapters can make the most
use of pretrained model information and ensure models be adaptive for clients
in specific tasks. Simultaneously, small-scale operations can mitigate the
computational burden and communication burden caused by large models.
Extensive experiments are conducted on three datasets with distribution
shifts. Qualitative and quantitative results demonstrate that FedCLIP
significantly outperforms other baselines ($\mathbf{9}\%$ overall improvements
on PACS) and effectively reduces computational and communication costs (283x
faster than FedAVG). Our code will be available at:
https://github.com/microsoft/PersonalizedFL.

##  Sec. 1 Introduction

The success of machine learning, especially deep learning, is inseparable from
a large amount of data. However, data, as an important resource, usually
scatter across different individuals or organizations. In recent years, people
pay more attention to data privacy and security and some organizations even
enact relevant regulations and laws, e.g. The EU general data protection
regulation (GDPR) [48] and China’s cyber power [20]. Under this circumstance,
direct raw data communication can be impossible in reality, making traditional
data-centric machine learning paradigms unlikely to work. To cope with this
challenge, federated learning (FL) [51] emerges as a new distributed machine
learning paradigm and has been widely adopted in various applications.

Federated learning makes it possible to perform model aggregation without
directly accessing the raw user data from different clients. One of the
earliest works in FL is called FedAVG [33] which aggregates distributed
information using a simple and powerful averaging algorithm. FedAVG mainly
contains four steps, including training local models with local data,
uploading local models to the server, aggregating models in the server, and
distributing the aggregated model to each individual or organization. These
four steps are executed for multiple rounds for better information
aggregation. FedAVG can ensure that raw data does not leave the local client
and thus protect data privacy and security. Due to its simplicity and great
performance, FedAVG quickly became popular in many areas [24, 39, 3].

(a) Data distribution shifts

(b) Huge resource demands

Figure 1: Existing issues in federated learning. In Figure 1(a), circles
denote participated clients while squares denote unseen targets.

In this paper, we are specially interested in federated learning under the
_large foundation models_ era [4]. Foundation models, as suggested by the
name, have become increasingly popular in different machine learning tasks,
such as Vision Transformer in computer vision [54] and the GPT series in
natural language processing [37]. Since these models are extremely large,
e.g., GPT-3 [5] has 175 billion parameters, our key question is: _how to
perform effective and efficient federated learning using these large models?_

Specifically, two critical research challenges arise in this situation: _data
distribution shifts_ and _huge resource demands_. On the one hand, data
distribution shifts widely exist in the real world, e.g. figures shown in
Figure 1(a). When meeting heterogeneous data, common federated learning
methods can suffer from slow convergence and low accuracy due to inconsistent
optimization directions, local optima, or some other factors [12]. A qualified
FL model can cope with both various clients and unseen targets, i.e.
personalization and generalization. On the other hand, huge resource demands
of increasingly popular large models lead to conflicts with realistically
constrained resources, as shown in Figure 1(b). In addition to high
computational costs, communication cost is also a critical metric in federated
learning. For instance, the CLIP [36] model based on VIT-B/32 contains more
than $10^{8}$ trainable parameters and most existing networks cannot afford to
transmit it quickly. Achieving fast generalization and personalization with
minimal resource costs is an urgent issue to be addressed.

Some existing work tried to address the issues mentioned above [32, 53, 14].
FedAP [32] attempted to learn the similarity among clients and then leveraged
the learned similarity matrix to guide aggregation. FedAP could achieve
acceptable personalization results but it ignored generalization. Another
paper [53] discussed two gaps, including the out-of-sample gap and the
participation gap. These two gaps correspond to goals of generalization and
personalization respectively. This paper performed extensive empirical studies
to analyze these issues but it did not offer a possible solution for large
models. PromptFL [14] only updated the prompts instead of the whole model to
accelerate the whole process. However, clients still require large amounts of
computation and PromptFL is not designed for personalization and
generalization.

In this paper, we propose FedCLIP to achieve fast generalization and
personalization for CLIP in federated learning. Since larger pretrained
models, e.g. CLIP, have contained enough prior information, our goal is to
find where we should focus in specific tasks. The core part of FedCLIP is
AttAI, an attention-based adapter for the image encoder in CLIP. Instead of
finetuning whole networks, AttAI directly utilizes fixed features extracted by
pretrained models and explores where FedCLIP should pay attention to for
specific tasks. Simply training AttAI can ensure FedCLIP preserving prior
information as much as possible while it allows models adapted for specific
tasks. Through AttAI, FedCLIP does not rely on pretrained models anymore once
obtaining diversified and robust features and thus FedCLIP can save large
amounts of computational costs and communication costs. Therefore, FedCLIP is
extensible and can be deployed to many applications.

Our contributions are as follows.

  1. 1.

We propose FedCLIP, a fast generalization and personalization learning method
for CLIP in federated learning. It can achieve personalization for
participating clients and its remarkable generalization ability can attract
new clients.

  2. 2.

Extensive experiments on three public image benchmarks demonstrate that
FedCLIP can have achieved personalization and generalization performance at
the same time ($\textbf{9}\%$ overall improvements on PACS). More importantly,
FedCLIP reduces the number of trainable parameters thus saving communication
costs and computational costs (283x faster than FedAVG).

  3. 3.

FedCLIP is extensible and can be applied in many real applications, which
means it can work well in many circumstances. We can even embed it in some
other architectures, e.g. BERT [45] and ViT [16]. Our code will be available
at: https://github.com/microsoft/PersonalizedFL.

The remainder of this paper is organized as follows. In Sec. 2, we introduce
related work. And then we elaborate on the proposed method in Sec. 3.
Extensive experiments are reported and analyzed in Sec. 4. Finally, we
conclude the paper and provide possible future work in Sec. 5.

##  Sec. 2 Related Work

###  2.1 Challenges in Machine Learning

Machine learning has achieved great success and gradually entered people’s
daily lives [41, 34, 49]. It has been applied to many fields, e.g. human
activity recognition [29], face recognition [28], and healthcare [8].
Successful machine learning applications, especially deep learning based
applications, often require a large amount of data and lots of computational
resources. In most cases, a deluge of data and computing resources can lead to
easy success, such as ChatGPT [46]. However, data and computation also mean
money and resources. In reality, it is impossible to aggregate all data
together in some situations. There seems to be a contradiction between the
massive resource requirements of traditional methods and the limited real
environment.

Generalization is another challenging problem caused by data distribution
shifts. Its goal is to learn a generalized model with limited data and it
expects that the learned model can work well on unseen targets with unknown
distributions. [50] gives a survey on domain generalization and first groups
existing methods into three categories, including data manipulation [31],
representation learning [30], and learning strategy [19].

###  2.2 Federated Learning

Data is often scatted everywhere and cannot be aggregated together due to some
factors, such as laws and regulations [48] and the awakening of people’s
awareness of data security and privacy protection. In such an environment,
federated learning came into being [51, 40]. According to [51], federated
learning can be grouped into three categories, including horizontal federated
learning, vertical federated learning, and federated transfer learning. Most
deep learning based methods belong to horizontal federated learning and so is
this paper. For a more detailed introduction, please refer to the survey [27].

FedAVG is a traditional horizontal federated learning method [33]. Although it
is simple, it was applied in many applications. When meeting data distribution
heterogeneity, FedAVG appeared powerless [42]. And many researchers proposed
various methods to solve the above problems. FedProx [25] added a proximal
regularized term to FedAVG and it allowed slight model gaps between clients
and the server. In FedBN [26], the authors thought that parameters in batch
normalization layers can represent data distribution, and keeping specific
batch normalization layers for each client could make local models
personalized. Another latest method, FedAP [32], learned the similarity
between clients based on the statistics of the batch normalization layers
while preserving the specificity of each client with different local batch
normalization. The above methods all achieve satisfactory results in their
corresponding scenarios. However, most of them focused on personalization and
ignored generalization issues [7].

Generalization in federated learning is a novel problem. In recent two years,
some papers tried to solve this problem. [53] first discussed the
generalization in federated learning and it proposed a framework to
disentangle performance gaps, including out-of-sample gaps and participation
gaps. FED-DRO [7] proposed a novel federated learning framework to explicitly
decouple a model’s dual duties with two prediction tasks and it mainly focused
on label shifts. Some other work tried to adapt existing domain generalization
methods to generalization in federated learning [15, 44, 35, 6]. FL Games [15]
utilized Nash equilibrium to learn causal features that were invariant across
clients which is similar to Invariant Risk Minimization (IRM) [2]. FedSAM [35]
proposed a general effective algorithm based on Sharpness Aware Minimization
(SAM) local optimizer [11]. Although these methods can bring generalization,
they were not designed for large models and could not make full use of
knowledge brought by pretrained models.

###  2.3 CLIP and Large Models

From perceptron [13] to AlexNet [1] to ResNet [17] to Vision transformer [54]
to CLIP [36], pretrained models have become larger and larger. The importance
of pretrained models has been increasing and pretrained models contain a
growing amount of knowledge. For specific applications, researchers usually
choose suitable backbone models and then adopt some techniques, e.g. finetune
[43], to slightly adapt pretrained models. Since pretrained models are trained
via a large amount of data, features extracted from them are often generalized
and insightful. Few works pay attention to large models in federated learning
and high demands of computational costs and communication costs hinder the
development of this field. In this paper, we focus on CLIP in federated
learning.

CLIP [36] learned SOTA image representations from scratch on a dataset of 400
million(image, text) pairs collected from the internet. The natural language
was used to reference learned visual concepts. It has been applied in many
fields and demonstrated its superiority [38, 22]. However, in federated
learning, CLIP is still in its infancy. PromptFL [14] replaced the federated
model training with the federated prompt training to simultaneously achieve
efficient global aggregation and local training by exploiting the power of
foundation models in a distributed way. However, it still requires certain
computational costs and it is not designed for data distribution heterogeneity
problems. Moreover, it is hard to tune the hyperparameters for the prompt
techniques in Transformer.In this paper, we focus on fast personalization and
generalization for CLIP.

##  Sec. 3 Method

###  3.1 Problem Formulation

In a generalization and personalization federated learning setting, $N$
different clients, denote as $\\{C_{1},C_{2},\cdots,C_{N}\\}$, participate in
exchanging information and they have data, denoted as
$\\{\mathcal{D}_{1},\mathcal{D}_{2},\cdots,\mathcal{D}_{N}\\}$ with different
distributions, which means $P(\mathcal{D}_{i})\neq P(\mathcal{D}_{j})$. In
this paper, we only focus on homogeneous data with the same input space and
output space, i.e.
$\mathcal{X}_{i}=\mathcal{X}_{j},\mathcal{Y}_{i}=\mathcal{Y}_{j},\forall i\neq
j$. Each dataset,
$\mathcal{D}_{i}=\\{(\mathbf{x}_{i,j},y_{i,j})\\}_{j=1}^{n_{i}}$, consists of
three parts, a training dataset
$\mathcal{D}_{i}^{train}=\\{(\mathbf{x}_{i,j}^{train},y_{i,j}^{train})\\}_{j=1}^{n_{i}^{train}}$,
a validation dataset
$\mathcal{D}_{i}^{valid}=\\{(\mathbf{x}_{i,j}^{valid},y_{i,j}^{valid})\\}_{j=1}^{n_{i}^{valid}}$
and a test dataset
$\mathcal{D}_{i}^{test}=\\{(\mathbf{x}_{i,j}^{test},y_{i,j}^{test})\\}_{j=1}^{n_{i}^{test}}$.
Three sub-datasets in each client have no overlap and
$n_{i}=n_{i}^{train}+n_{i}^{valid}+n_{i}^{test},\mathcal{D}_{i}=\mathcal{D}_{i}^{train}\cup\mathcal{D}_{i}^{valid}\cup\mathcal{D}_{i}^{test}$.
Our goal is to aggregate all clients’ information with preserving data privacy
and security and learn a good model $f$ for each client $\mathcal{D}_{i}$:

$\min_{f}\frac{1}{N}\sum_{i=1}^{N}\frac{1}{n_{i}^{test}}\sum_{j=1}^{n_{i}^{test}}\ell(f(\mathbf{x}_{i,j}^{test}),y_{i,j}^{test}),$
(1)

where $\ell$ is a loss function. Moreover, for generalization, we assume that
there exist $M$ different clients, denote as $\\{F_{1},F_{2},\cdots,F_{M}\\}$,
with data
$\\{\mathcal{D}_{1}^{F}=\\{(\mathbf{x}_{i,j},y_{i,j})\\}_{j=1}^{m_{1}},\mathcal{D}_{2}^{F}=\\{(\mathbf{x}_{i,j},y_{i,j})\\}_{j=1}^{m_{2}},\cdots,\mathcal{D}_{N}^{F}=\\{(\mathbf{x}_{i,j},y_{i,j})\\}_{j=1}^{m_{M}}\\}$.
These $M$ clients do not participate in training, and we hope $f$ can also be
able to perform well on these clients.

$\min_{f}\frac{1}{M}\sum_{i=1}^{M}\frac{1}{m_{i}}\sum_{j=1}^{m_{i}}\ell(f(\mathbf{x}_{i,j}),y_{i,j}),$
(2)

###  3.2 Preliminaries

#### CLIP

CLIP, Contrastive Language Image Pre-training, is an efficient and scalable
method of learning [36]. To compensate for the problems caused by the amount
of data and model parameters, it trained a large model with over $4\times
10^{8}$ pairs of data. With help of natural language supervision, CLIP can
better understand concepts of visual images and better learn the semantic
connections behind images. Usually, CLIP models contain more information and
they might be more robust.

A simple CLIP model regularly contains two parts, an image encoder $f^{I}$ and
a text encoder $f^{T}$. In common models, labels are frequently represented as
numbers or one-hot vectors. For CLIP, to better utilize semantic information,
these labels are often transformed into sentences, e.g. ’A photo of dogs’. And
then text feature vectors, $\mathbf{T}$ are extracted from these sentences via
$f^{T}$. Concurrently, images are encoded into visual feature vectors,
$\mathbf{I}$, via $f^{I}$. Cosine similarities between $\mathbf{T}$ and
$\mathbf{I}$ are used to training and predicting.

#### FedAVG

In FedAVG [33], each client trains $f$ with local clients’ data, and then
parameters of updated models, $w_{i}$, are transmitted to the server. The
server typically aggregates the parameters according to Eq. 3,

$w^{*}=\sum_{i=1}^{N}\frac{n_{i}}{\sum_{j=1}^{N}n_{j}}w_{i}$ (3)

After aggregation, $w^{*}$ is distributed. When $|w|$ is larger, the server
cannot afford communication costs.

###  3.3 FedCLIP

To reduce computational costs and communications and make the most use of
existing pretrained model information, we propose FedCLIP. Pretrained models
already have abilities to extract robust and diversified features. Tuning
whole networks with limited data can compromise the original ability of
pretrained models. What we need to do is to try our best to preserve useful
prior knowledge and let it be used to a suitable extent for our task. Besides,
tuning large networks is impractical in federated learning due to limited
resources in reality. Therefore, instead of operating on the whole model,
FedCLIP concentrates on a simple attention-based adapter for the image
encoder, AttAI.

Figure 2: The framework of FedCLIP.

Figure 2 gives the framework of FedCLIP. As shown in Figure 2, our method
mainly contains four steps.

  1. 1.

For Client $i$, we utilize a pretrained CLIP model to extract features of
data, denoted as $T_{i}$ and $I_{i}$.

  2. 2.

In each client, we utilize $\mathcal{D}^{train}_{i}$ to train the
corresponding adapter, $g_{i}$. And then we upload $\\{g_{i}\\}_{i=1}^{N}$ to
the server.

  3. 3.

In the server, the parameters of all $g_{i}$ are weighted averaged and we can
obtain $g^{*}$. The server then distributes $g^{*}$ to each client and updates
the parameters of each $g_{i}$.

  4. 4.

Repeat Step 2 and Step 3 until convergence or reaching maximum rounds.

In step 1, we utilize the pretrained CLIP model to extract features. We
consider the pretrained model is so powerful that we do not need to explore
some other features. For $(\mathbf{x},y)$, we can obtain corresponding
features,

$\mathbf{I}=f^{I}(\mathbf{x}),\mathbf{T}=f^{T}(y)$ (4)

What we need to do next is to identify which parts of features are suitable
for our specific tasks. Therefore, we introduce an attention-based adapter,
$g$, to locate where we should concentrate on. Particularly, we utilize one
linear layer, Tahn activation function, one linear layer, and Softmax
activation function to construct $g$. The Softmax function is used to ensure
our final outputs ranging from $0$ to $1$. Once we obtain the attention vector
$att=g(\mathbf{I})$, we utilize it to update the visual feature via a dot
multiply operation,

$\mathbf{I}^{*}=g(\mathbf{I})\cdot\mathbf{I}.$ (5)

Then, similar to [36], we normalize $\mathbf{I}^{*}$ and $\mathbf{T}$ to
compute the final logits.

$\displaystyle{\mathbf{I}}=\frac{\mathbf{I}^{*}}{|I^{*}|},$
$\displaystyle{\mathbf{T}}=\frac{\mathbf{T}}{|\mathbf{I}|},$ (6)
$\displaystyle\hat{\mathbf{I}}=s\times{\mathbf{I}}\times{\mathbf{T}}^{T},$
$\displaystyle\hat{\mathbf{T}}=\hat{\mathbf{I}}^{T}.$ (7)

where $s$ is a scale parameter.

Now, we can utilize the ground truth, a vector
$\tilde{\mathbf{y}}=[0,1,2,3,\cdots,B]$

$\begin{split}\ell_{cls}^{I}=\ell(\hat{\mathbf{I}},\tilde{\mathbf{y}}),\\\
\ell_{cls}^{T}=\ell(\hat{\mathbf{T}},\tilde{\mathbf{y}}),\end{split}$ (8)

where $\ell$ is CrossEntropy loss [55] while $B$ is the number of images in a
batch.

We only exchange parameters of adapters, $w^{g}$, and therefore in the server,
we replace Eq. 3 with Eq. 9.

$w^{g,*}=\sum_{i=1}^{N}\frac{n_{i}}{\sum_{j=1}^{N}n_{j}}w_{i}^{g}.$ (9)

Since $w^{g}$ contains substantially less amount of trainable parameters than
$w$, FedCLIP saves computational costs and communication costs.

###  3.4 Summary

For clarity, we give a detailed description of FedCLIP in Algorithm 1. In Line
1, directly obtaining generalized and diversified features with fixed CLIP
make it possible to utilize more prior knowledge of pretrained models. In Line
2, with adapters, we can concentrate on valuable information and eliminate the
influence of redundant information in specific tasks. Rich prior knowledge and
targeted attention make the ultimately extracted features more robust,
effective, and adaptable, resulting in our method having good generalization
and personalization capabilities. From Line 2 to Line 5, performing
computation and transmission merely with adapters can save a lot of resources
and ensure the efficiency of our method.

Algorithm 1 FedCLIP

Input: $N$ clients’ datasets $\\{\mathcal{D}_{i}\\}_{i=1}^{N}$, a pretained
CLIP model consist of an image encoder, $f^{I}$, and a text encoder, $f^{T}$
Output: An adapter $g$

1:For client $i$, computer the corresponding features
$I_{i}=f^{I}(\mathbf{X}_{i}),T_{i}=f^{T}(\mathbf{Y}_{i})$

2:For client $i$, train the local adapter, $g_{i}$, according to Eq.5 to Eq.9

3:Send the current adapter $g_{i}$ to the server

4:Aggregate adapters’ parameters via Eq. 9 and obtain $w^{g*}$

5:Transmit $w^{g*}$ to each client

6:Repeat steps $2\sim 5$ until convergence

###  3.5 Discussion

Adapter is a common technique in transfer learning [18]. It is at a small
scale and has a plug-and-play implementation. In this paper, we mainly focus
on adaptations to image encoders. Actually, we also can add adapters to text
encoders. We can even change the inputs of text encoders to incorporate more
semantic information.

##  Sec. 4 Experiments

In this section, we extensively evaluate FedCLIP in three common visual image
classification benchmarks.

###  4.1 Datasets

#### PACS

PACS [23] is a popular object classification benchmark. It is composed of four
sub-datasets, including photo, art-painting, cartoon, and sketch. There exist
$9,991$ images in total and the dataset contains $7$ classes, including dog,
elephant, giraffe, guitar, horse, house, and person. Large discrepancies in
image styles widely exist among different sub-datasets. In this paper, we view
each sub-dataset as a client. We choose three sub-datasets as participated
clients while the rest served as the target client to evaluate generalization
ability. For each participated client, we split the corresponding sub-dataset
into three parts, $60\%$ for training, $20\%$ for validation, and the rest
$20\%$ for testing. Validation parts of data are used for model selection.

#### VLCS

VLCS [10] is another widely accepted public image classification benchmark. It
also consists of four sub-datasets (VOC2007, LabelMe, Caltech10, and SUN09).
It contains $10,729$ instances with 5 classes. Feature shifts exist generally
among different sub-datasets. Similar to PACS, four sub-datasets correspond to
four clients. Three sub-datasets play the roles of participants while the rest
one act as an upcoming client.

#### Office-Home

Office-Home [47] is a larger image classification benchmark, which contains
$65$ classes. Office-Home comprises four sub-datasets (Art, Clipart, Product,
and Real_World) with about $15,500$ images. The feature shifts from Office-
Home mainly come from image styles and viewpoints, but they are much smaller
than PACS. We assess methods on Office-Home in a similar manner to PACS.

###  4.2 Implementation Details and Comparison Methods

For these three common image classification benchmarks, we use the CLIP pre-
trained model with ViT-B/32 [9] as the image encoder. For model training, we
utilize cross-entropy loss and Adam optimizer. The learning rate is tuned from
$5\times 10^{-5}$ to $5\times 10^{-3}$. We set local update epochs as $E=1$
where $E$ means the number of training epochs in one round while we set the
total communication round number as $R=200$. Since, at each time, we set one
sub-dataset as the target, i.e. upcoming client, there exist four tasks for
each benchmark. We run three trials to record the average results. To better
illustrate the function and necessity of using larger pretrained models, we
also utilize a related small architecture, AlexNet [21], to perform some base
federated learning methods.

We compare our method with two methods including a common federated learning
method, FedAVG, and a method designed for non-iid data, FedProx.

  1. 1.

FedAVG [33]. The server aggregates all client models’ parameters. FedAVG will
aggregate networks with several layers for AlexNet while FedAVG will aggregate
both image encoders and text encoders for CLIP.

  2. 2.

FedProx [25]. It adds a proximal term to FedAVG and allows the existence of
slight differences between clients and the server.

###  4.3 Results

#### Generalization Ability

Table 1: Generalization accuracy. Bold means the best.

Dataset | PACS | Office-Home
---|---|---
Backbone | Method | A | C | P | S | AVG | Backbone | Method | A | C | P | R | AVG
AlexNet | FedAVG | 31.54 | 43.69 | 44.55 | 36.29 | 39.02 | AlexNet | FedAVG | 15.70 | 17.00 | 31.56 | 28.99 | 23.31
FedProx | 29.79 | 46.80 | 44.67 | 35.12 | 39.09 | FedProx | 16.48 | 17.66 | 29.83 | 27.98 | 22.99
CLIP | FedAVG | 53.08 | 80.08 | 90.00 | 76.99 | 75.04 | CLIP | FedAVG | 65.60 | 57.64 | 71.64 | 75.42 | 67.57
FedProx | 66.06 | 87.33 | 91.68 | 78.42 | 80.87 | FedProx | 65.60 | 57.64 | 71.64 | 75.42 | 67.57
Ours | 96.34 | 97.91 | 99.76 | 85.59 | 94.90 | Ours | 78.00 | 63.69 | 87.52 | 87.79 | 79.25

We first evaluate the generalization ability of each method via accuracy on
clients that do not participate in training. Table 1 shows the generalization
results for each task on PACS and Office-Home. We have the following
observations from these results. 1) Our method achieves the best
generalization ability on average with remarkable improvements (about $14\%$
for PACS and about $12\%$ for Office-Home). Moreover, our method achieves the
best generalization ability in each task, which demonstrates the excellent
generalization ability of our method. 2) Compared to methods with AlexNet as
the backbone, methods with CLIP as the backbone can obtain better performance.
It demonstrates that large well-trained models can be able to bring better
generalization. 3) Compared to methods with CLIP as the backbone, our method
has a further improvement, which demonstrates that our method leverages prior
knowledge better.

#### Personalization Ability

Table 2: Personalization accuracy. Bold means the best.

Dataset | PACS | Office-Home
---|---|---
Target | BackBone | Method | C | P | S | AVG | Target | BackBone | Method | C | P | R | AVG
A | AlexNet | FedAVG | 72.86 | 61.08 | 78.22 | 70.72 | A | AlexNet | FedAVG | 50.74 | 63.47 | 38.81 | 51.01
FedProx | 71.37 | 56.89 | 81.53 | 69.93 | FedProx | 51.78 | 66.74 | 40.07 | 52.86
CLIP | FedAVG | 76.28 | 86.83 | 42.42 | 68.51 | CLIP | FedAVG | 64.38 | 79.14 | 78.76 | 74.09
FedProx | 90.81 | 90.42 | 63.95 | 81.73 | FedProx | 64.38 | 79.14 | 78.76 | 74.09
Ours | 97.65 | 99.40 | 86.75 | 94.60 | Ours | 68.61 | 87.37 | 88.06 | 81.35
C |  |  | A | P | S | AVG | C |  |  | A | P | R | AVG
AlexNet | FedAVG | 46.45 | 66.17 | 75.67 | 62.76 | AlexNet | FedAVG | 23.51 | 61.78 | 41.56 | 42.28
FedProx | 47.19 | 64.07 | 77.45 | 62.90 | FedProx | 24.54 | 64.04 | 40.18 | 42.92
CLIP | FedAVG | 84.11 | 92.81 | 81.02 | 85.98 | CLIP | FedAVG | 73.81 | 80.38 | 80.48 | 78.23
FedProx | 86.06 | 92.81 | 85.61 | 88.16 | FedProx | 73.81 | 80.38 | 80.48 | 78.23
Ours | 96.33 | 99.10 | 86.88 | 94.10 | Ours | 78.97 | 87.60 | 87.60 | 84.72
P |  |  | A | C | S | AVG | R |  |  | A | C | R | AVG
AlexNet | FedAVG | 37.65 | 75.00 | 81.53 | 64.73 | AlexNet | FedAVG | 23.30 | 49.94 | 40.87 | 38.04
FedProx | 35.45 | 73.93 | 83.57 | 64.32 | FedProx | 21.03 | 48.91 | 39.84 | 36.59
CLIP | FedAVG | 83.13 | 93.38 | 84.97 | 87.16 | CLIP | FedAVG | 70.93 | 68.73 | 77.73 | 72.46
FedProx | 83.86 | 93.59 | 88.54 | 88.66 | FedProx | 70.93 | 68.73 | 77.73 | 72.46
Ours | 97.56 | 97.65 | 86.75 | 93.99 | Ours | 78.35 | 68.38 | 87.94 | 78.23
S |  |  | A | C | P | AVG | P |  |  | A | C | P | AVG
AlexNet | FedAVG | 53.30 | 68.80 | 66.17 | 62.76 | AlexNet | FedAVG | 22.27 | 49.14 | 58.51 | 43.31
FedProx | 52.32 | 69.66 | 66.47 | 62.82 | FedProx | 20.21 | 50.06 | 58.29 | 42.85
CLIP | FedAVG | 90.71 | 94.02 | 94.91 | 93.21 | CLIP | FedAVG | 69.07 | 66.21 | 77.79 | 71.02
FedProx | 91.44 | 94.66 | 95.81 | 93.97 | FedProx | 69.07 | 66.21 | 77.79 | 71.02
Ours | 97.31 | 97.65 | 99.40 | 98.12 | Ours | 78.56 | 68.50 | 87.37 | 78.14

Then, we evaluate the personalization ability of each method via the accuracy
on test data of each participating client. Table 2 shows the personalization
results for each task on PACS and Office-Home. We also have some insightful
observations. 1)Although all clients share the same adapter in our method, our
method still achieves the best average accuracy. Moreover, FedCLIP almost
achieves the best performance on each client for every task. 2) Compared to
methods with AlexNet, corresponding methods with CLIP perform better overall.
For CLIP-based methods, results are quite sensitive to hyperparameters, e.g.
learning rate. And FedAVG has disappointing results on some specific clients.
3) Our method has the most use of prior knowledge since it achieves the
stablest results.

#### Comprehensive Ability

Table 3: Comprehensive average accuracy. Bold means the best

Datasets | PACS | Office-Home
---|---|---
Backbone | AlexNet | CLIP | Backbone | AlexNet | CLIP
Methods | FedAVG | FedProx | FedAVG | FedProx | Ours | Methods | FedAVG | FedProx | FedAVG | FedProx | Ours
A | 60.93 | 59.89 | 64.65 | 77.81 | 95.04 | A | 42.18 | 43.77 | 71.97 | 71.97 | 80.51
C | 57.99 | 58.88 | 84.50 | 87.95 | 95.06 | C | 35.96 | 36.60 | 73.08 | 73.08 | 79.46
P | 59.68 | 59.41 | 87.87 | 89.42 | 95.43 | P | 36.42 | 34.90 | 72.26 | 72.26 | 80.55
S | 56.14 | 55.89 | 89.16 | 90.08 | 94.99 | R | 39.73 | 39.13 | 72.12 | 72.12 | 80.55
AVG | 58.69 | 58.52 | 81.55 | 86.32 | 95.13 | AVG | 38.57 | 38.60 | 72.36 | 72.36 | 80.27

Finally, taking into account the performance of both personalization and
generalization, we provide an overall performance in Table 3. Without a doubt,
our method achieves the best overall performance with significant improvements
(about $9\%$ for PACS and $8\%$ for Office-Home). Compared to methods based on
AlexNet, corresponding methods based on CLIP perform better.

#### More results on VLCS

Table 4: Comprehensive average accuracy on VLCS. Bold means the best

Backbone | AlexNet | CLIP
---|---|---
Methods | FedAVG | FedProx | FedAVG | FedProx | Ours
C | 62.13 | 61.37 | 72.48 | 68.57 | 83.68
L | 63.01 | 63.77 | 75.04 | 76.50 | 82.62
S | 63.15 | 63.59 | 68.13 | 75.50 | 82.82
V | 62.32 | 62.04 | 69.55 | 70.09 | 83.30
AVG | 62.65 | 62.69 | 71.30 | 72.67 | 83.11

Due to space limitations, we only report comprehensive ability on VLCS. As
shown in Table 4, our method still achieves the best performance with
improvements of over $10\%$. Moreover, our method achieves the best in each
task. The results prove the superiority of our method again.

###  4.4 Analysis

#### Can more adapters bring better performance?

In our method, we only add one adapter to the image encoder. We can add
another adapter to the text encoder. As shown in Figure 3(a), adding more
adapters brings slight improvements. However, the improvements are so small
that we need to assess whether it is necessary to do so since more adapters
regularly mean more computational costs and more communication costs.

#### Can more trainable parameters bring better performance?

If we train both adapters and the backbones, the results could be worse. Since
CLIP models have a wealth of good information, it is not suitable to change
parameters with only a few data for a specific task. Changes in CLIP with few
data can destroy the feature extraction capabilities. As shown in Figure 3(b),
we train more parameters but achieve worse performance.

#### Will finetuning bring better personalization?

According to [52], finetuning can be a useful technique for better
personalization. We also add experiments with finetune. As shown in Figure
3(c), finetune has no advance in personalization, which demonstrates that our
method can be remarkable and robust when meeting non-iid.

#### Resource Cost Comparison

The number of trainable parameters represents how many resources we need to
cost in federated learning. As shown in Figure 3(d), our method merely has
$5.3E5$ parameters while FedAVG with CLIP requires $1.5E8$ trainable
parameters. Common methods via training whole networks have $283$ times as
many parameters as ours, which illustrates that our method is fast and
resource-efficient.

(a) Adapter influence.

(b) Training backbone.

(c) Finetune influence.

(d) Parameter counts.

Figure 3: Analysis on PACS.

##  Sec. 5 Conclusion and Future Work

In this article, we propose FedCLIP, a fast generalization and personalization
learning method for CLIP in federated learning. FedCLIP designs an attention
based adapter to replace updating the whole model. Therefore, FedCLIP makes
the most use of prior knowledge and saves computational costs and
communication costs. Comprehensive experiments have demonstrated the
superiority of FedCLIP. In the future, we plan to embed FedCLIP into more
architectures and design more flexible adapters for different tasks. We also
plan to apply FedCLIP for heterogeneous architectures and more realistic
applications.

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$Q_{0}^{1/n}=q_{0}$ is fixed. To prove the second condition, we start by
observing that, for all $n\in\mathbb{N}^{*}$,

$\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|s-t|\leqslant\delta\end{subarray}}\left|Q^{1/n}_{s}-Q^{1/n}_{t}\right|\leqslant
3\max_{0\leqslant k\leqslant\lfloor
T/\delta\rfloor}\sup_{t\in[0,\delta]}\left|Q^{1/n}_{k\delta+t}-Q^{1/n}_{k\delta}\right|.$

By Lemma 33 in Section E.3, for any $k\leqslant\lfloor T/\delta\rfloor$,

$\displaystyle\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|s-t|\leqslant\delta\end{subarray}}\left|Q^{1/n}_{s}-Q^{1/n}_{t}\right|$
$\displaystyle\quad\leqslant 3\max\left(\max_{0\leqslant k\leqslant\lfloor
T/\delta\rfloor}\max_{\lfloor nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|,\max_{0\leqslant k\leqslant\lfloor
T/\delta\rfloor}\max_{\lceil nk\delta\rceil\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lceil
nk\delta\rceil}\right|\right).$

Thus,

$\displaystyle\mathbb{P}\left(\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|s-t|\leqslant\delta\end{subarray}}\left|Q^{1/n}_{s}-Q^{1/n}_{t}\right|\geqslant\varepsilon\right)$
$\displaystyle\leqslant\underbrace{\mathbb{P}\left(\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|\geqslant\frac{\varepsilon}{3}\right)}_{(a)}$ (133)
$\displaystyle\ \ +\underbrace{\mathbb{P}\left(\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{\lceil
nk\delta\rceil\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lceil
nk\delta\rceil}\right|\geqslant\frac{\varepsilon}{3}\right)}_{(b)}.$

The two terms $(a)$ and $(b)$ are controlled using similar arguments; for
simplicity we make precise only the control of $(a)$. We start by writing, for
$\ell\geqslant\lfloor nk\delta\rfloor$,

$q_{1/n}^{\ell}-q_{1/n}^{\lfloor nk\delta\rfloor}=\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}\Delta_{n}(q^{i}_{1/n},G^{i+1},U^{i+1}),$

where (in view of (80) and recalling the notation $R_{\Delta t}$ from (79))

$\Delta_{n}(q,G,U)=\sqrt{\frac{2}{n}}\mathcal{D}(q)^{1/2}G\mathbbm{1}_{\\{U\leqslant
R_{1/n}(q,G)\\}}.$ (134)

By Lemma 35 in Section E.3, the following decomposition holds:

$q_{1/n}^{\ell}-q_{1/n}^{\lfloor nk\delta\rfloor}=\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}M_{n}(q^{i}_{1/n},G^{i+1},U^{i+1})+\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}T_{n}(q^{i}_{1/n}),$ (135)

with $(M_{n}(q^{i}_{1/n},G^{i+1},U^{i+1}))_{i\geqslant 0}$ a sequence of
martingale increments satisfying $M_{n}(q,G,U)=\mathrm{O}(n^{-1/2})$ and
$T_{n}(q)=\mathrm{O}(n^{-1})$. Explicit expressions for
$M_{n}(q^{i}_{1/n},G^{i+1},U^{i+1})$ and $T_{n}(q^{i}_{1/n})$ are given in
Lemma 35. From (135), we obtain

$\displaystyle\max_{0\leqslant k\leqslant\lfloor T/\delta\rfloor}\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|$ $\displaystyle\qquad\qquad\leqslant\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}M_{n}(q^{i}_{1/n},G^{i+1},U^{i+1})\right|$
$\displaystyle\qquad\qquad\ \ +\max_{0\leqslant k\leqslant\lfloor
T/\delta\rfloor}\max_{\lfloor nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}T_{n}(q^{i}_{1/n})\right|,$

so that

$\displaystyle\mathbb{P}\left(\max_{0\leqslant k\leqslant\lfloor
T/\delta\rfloor}\max_{\lfloor nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|\geqslant\frac{\varepsilon}{3}\right)$
$\displaystyle\qquad\qquad\leqslant\underbrace{\mathbb{P}\left(\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}M_{n}(q^{i}_{1/n},G^{i+1},U^{i+1})\right|\geqslant\frac{\varepsilon}{6}\right)}_{I}$
$\displaystyle\qquad\qquad\ \ +\underbrace{\mathbb{P}\left(\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}T_{n}(q^{i}_{1/n})\right|\geqslant\frac{\varepsilon}{6}\right)}_{II}.$

We next bound each of the terms on the right-hand side of the above
inequality. We use $C$ to denote a constant which can change from line to
line.

##### Control of $I$.

The term $I$ is a sum of martingale increments, and can therefore be bounded
using martingale inequalities. Consider to this end the sequence
$(S_{k}^{j})_{\ell\geqslant 0}$ for $0\leqslant k\leqslant\left\lfloor
T/\delta\right\rfloor$, defined as $S_{k}^{0}=0$ and

$\forall 1\leqslant j\leqslant\lceil n(k+1)\delta\rceil-\lfloor
nk\delta\rfloor,\qquad S_{k}^{j}=\sum_{i=\lfloor nk\delta\rfloor}^{\lfloor
nk\delta\rfloor+j-1}M_{n}(q^{i}_{1/n},G^{i+1},U^{i+1}).$

For $j>\lceil n(k+1)\delta\rceil-\lfloor nk\delta\rfloor$, we set the
increments to $0$, which yields $S_{k}^{j}=S_{k}^{\lceil
n(k+1)\delta\rceil-\lfloor nk\delta\rfloor}$. The sequence
$(S_{k}^{j})_{j\geqslant 1}$ is a martingale. We start by applying the union
bound and Markov’s inequality, which yields

$\displaystyle\mathbb{P}\left(\max_{0\leqslant k\leqslant\lfloor
T/\delta\rfloor}\max_{j\geqslant
1}\left|S_{k}^{j}\right|\geqslant\frac{\varepsilon}{6}\right)=\mathbb{P}\left(\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{j\geqslant
1}\left|S_{k}^{j}\right|^{4}\geqslant\frac{\varepsilon^{4}}{6^{4}}\right)$
(136) $\displaystyle\qquad\qquad\qquad\qquad\leqslant\sum_{k=0}^{\lfloor
T/\delta\rfloor}\mathbb{P}\left(\max_{j\geqslant
1}|S_{k}^{j}|^{4}\geqslant\frac{\varepsilon^{4}}{6^{4}}\right)\leqslant\frac{6^{4}}{\varepsilon^{4}}\sum_{k=0}^{\lfloor
T/\delta\rfloor}\mathbb{E}\left(\max_{j\geqslant 1}|S_{k}^{j}|^{4}\right).$

We are then in position to resort to the martingale inequality from [17,
Theorem 1.1], recalled here for convenience.

######  Theorem 32 (Theorem 1.1 in [17]).

Suppose that $X=(X_{1},X_{2},\ldots)$ is a martingale, and introduce (with the
convention $X_{0}=0$)

$X^{*}=\sup_{n\geqslant 1}|X_{n}|,\qquad
S(X)=\left(\sum_{k=1}^{+\infty}(X_{k}-X_{k-1})^{2}\right)^{1/2}.$

Consider a convex function $\Phi:[0,\infty)\to[0,\infty)$ such that
$\Phi(0)=0$, satisfying the following growth condition: There exists
$c\in\mathbb{R}_{+}$ such that

$\forall u>0,\qquad\Phi(2u)\leqslant c\Phi(u).$

Set $\Phi(\infty)=\lim_{u\to\infty}\Phi(u)$. Then there exists
$C_{-},C_{+}\in\mathbb{R}_{+}$ such that

$C_{-}\mathbb{E}\left[\Phi(S(X))\right]\leqslant\mathbb{E}\left[\Phi(X^{*})\right]\leqslant
C_{+}\mathbb{E}\left[\Phi(S(X))\right].$

Applying Theorem 32 to the sequence $(S_{k}^{j})_{j\geqslant 0}$ with
$\Phi(u)=u^{4}$, and using a discrete Cauchy–Schwarz inequality, it follows
that $\forall 0\leqslant k\leqslant\lfloor T/\delta\rfloor$,

$\displaystyle\mathbb{E}\left(\max_{j\geqslant 1}|S_{k}^{j}|^{4}\right)$
$\displaystyle\leqslant C_{+}\mathbb{E}\left[\left(\sum_{i=\lfloor
nk\delta\rfloor}^{\lceil
n(k+1)\delta\rceil-1}\left|M_{n}\left(q_{1/n}^{i},G^{i+1},U^{i+1}\right)\right|^{2}\right)^{2}\right]$
$\displaystyle\leqslant C_{+}(n\delta+1)\sum_{i=\lfloor
nk\delta\rfloor}^{\lceil
n(k+1)\delta\rceil-1}\mathbb{E}\left[\left|M_{n}\left(q_{1/n}^{i},G^{i+1},U^{i+1}\right)\right|^{4}\right].$

Since $M_{n}\left(q_{1/n}^{i},G^{i+1},U^{i+1}\right)=\mathrm{O}(n^{-1/2})$
with bounds uniform in $i$, it follows that

$\forall 0\leqslant k\leqslant\lfloor
T/\delta\rfloor,\qquad\mathbb{E}\left(\max_{1\leqslant j\leqslant\lceil
n\delta\rceil}|S_{k}^{j}|^{4}\right)\leqslant C\frac{(n\delta+1)^{2}}{n^{2}}.$

Combining the latter inequality with (136), we finally obtain that

$\mathbb{P}\left(\max_{0\leqslant k\leqslant\lfloor
T/\delta\rfloor}\max_{1\leqslant j\leqslant\lceil
n\delta\rceil}\left|S_{\ell}\right|\geqslant\frac{\varepsilon}{6}\right)\leqslant
C\left(\frac{T}{\delta}+1\right)\frac{(n\delta+1)^{2}}{n^{2}\varepsilon^{4}}.$

The control of $I$ is concluded by noticing that $\displaystyle\lim_{\delta\to
0}\limsup_{n\to+\infty}\left(T/\delta+1\right)(n\delta+1)^{2}n^{-2}=0$.

##### Control of $II$.

Using Markov’s inequality, then a Cauchy–Schwarz inequality,

$\displaystyle\mathbb{P}\left(\max_{0\leqslant k\leqslant\lfloor
T/\delta\rfloor}\max_{\lfloor nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}T_{n}(q^{i}_{1/n})\right|\geqslant\frac{\varepsilon}{6}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\leqslant\frac{36}{\varepsilon^{2}}\mathbb{E}\left[\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}T_{n}(q^{i}_{1/n})\right|^{2}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\leqslant\frac{36(n\delta+1)}{\varepsilon^{2}}\mathbb{E}\left[\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil n(k+1)\delta\rceil}\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}\left|T_{n}(q^{i}_{1/n})\right|^{2}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\leqslant\frac{36(n\delta+1)}{\varepsilon^{2}}\mathbb{E}\left[\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\sum_{i=\lfloor nk\delta\rfloor}^{\lceil
n(k+1)\delta\rceil-1}\left|T_{n}(q^{i}_{1/n})\right|^{2}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\leqslant\frac{36(n\delta+1)}{\varepsilon^{2}}\sum_{k=0}^{\lfloor
T/\delta\rfloor}\sum_{i=\lfloor nk\delta\rfloor}^{\lceil
n(k+1)\delta\rceil-1}\mathbb{E}\left[\left|T_{n}(q^{i}_{1/n})\right|^{2}\right].$
(137)

Yet, by Lemma 35, $T_{n}(q^{i}_{1/n})=\mathrm{O}(n^{-1})$ uniformly in $i$, so
that

$\sum_{i=\lfloor nk\delta\rfloor}^{\lceil
n(k+1)\delta\rceil-1}\mathbb{E}\left[\left|T_{n}(q^{i}_{1/n})\right|^{2}\right]\leqslant\frac{C(n\delta+1)}{n^{2}}.$

Plugging the last inequality in (137), we obtain that

$\mathbb{P}\left(\max_{0\leqslant k\leqslant\lfloor
T/\delta\rfloor}\max_{\lfloor nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|\sum_{i=\lfloor
nk\delta\rfloor}^{\ell-1}T_{n}(q^{i}_{1/n})\right|\geqslant\frac{\varepsilon}{6}\right)\leqslant
C\left(\frac{T}{\delta}+1\right)\frac{(n\delta+1)^{2}}{n^{2}\varepsilon^{2}}.$

The control of $II$ is concluded by noticing that
$\displaystyle\lim_{\delta\to
0}\limsup_{n\to\infty}C(T/\delta+1)(n\delta+1)^{2}n^{-2}=0$.

##### Conclusion of the proof.

Combining the control of the two terms $I$ and $II$, we conclude that

$\lim_{\delta\to 0}\limsup_{n\to+\infty}\mathbb{P}\left(\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|\geqslant\frac{\varepsilon}{3}\right)=0.$

It can similarly be proved that

$\lim_{\delta\to 0}\limsup_{n\to+\infty}\mathbb{P}\left(\max_{0\leqslant
k\leqslant\lfloor T/\delta\rfloor}\max_{\lceil
nk\delta\rceil\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lceil
nk\delta\rceil}\right|\geqslant\frac{\varepsilon}{3}\right)=0.$

In view of (133), we finally obtain that

$\lim_{\delta\to
0}\limsup_{n\to\infty}\mathbb{P}\left(\sup_{\begin{subarray}{c}s,t\in[0,T]\\\
|s-t|\leqslant\delta\end{subarray}}\left|Q^{1/n}_{s}-Q^{1/n}_{t}\right|\right)=0,$

which concludes the proof of Proposition 30.

###  E.2 Proof of Proposition 31

For $n\in\mathbb{N}^{*}$, let us denote by $\mathcal{L}^{n}$ the operator
defined by: for any bounded measurable function
$\phi:\mathbb{T}^{d}\to\mathbb{R}$,

$(\mathcal{L}^{n}\phi)(q)=\mathbb{E}\left[\phi(q_{1/n}^{1})\,\middle|\,q_{1/n}^{0}=q\right]-\phi(q).$
(138)

For a given function $\varphi\in\mathcal{C}^{3}(\mathbb{T}^{d})$, we start by
showing in Section E.2.1 that the sequence of functions
$(n\mathcal{L}^{n}\varphi)_{n\geqslant 1}$ converges uniformly towards
$\mathcal{L}_{\mathcal{D}}\varphi$ as $n\to\infty$, in the sense of uniform
convergence of continuous functions over the compact set $\mathbb{T}^{d}$. We
then write the martingale problem associated to $\mathcal{L}^{n}$ in Section
E.2.2, and prove that the limit $P_{\infty}$ solves the martingale problem for
the limit generator $\mathcal{L}_{\mathcal{D}}$.

####  E.2.1 Convergence of the sequence of functions
$(n\mathcal{L}^{n}\varphi)_{n\geqslant 1}$

In order to determine the limit of the sequence of functions
$(n\mathcal{L}^{n}\varphi)_{n\geqslant 1}$, we work out a Taylor expansion of
the function (138). We expand to this end $\varphi(q_{1/n}^{1})$ at second
order around $q$, seeing $q^{1}_{1/n}-q$ as a perturbation along the direction
$\mathcal{D}(q)^{1/2}G^{1}$ with step size
$\sqrt{2/n}\mathbbm{1}_{\\{U^{1}\leqslant
R_{1/n}(q,G^{1})\\}}=\mathrm{O}(n^{-1/2})$. Writing $G,U$ instead of
$G^{1},U^{1}$ to alleviate the notation,

$\displaystyle\varphi\left(q+\sqrt{\frac{2}{n}}\mathbbm{1}_{\\{U\leqslant
R_{1/n}(q,G)\\}}\mathcal{D}(q)^{1/2}G\right)=\varphi(q)+\sqrt{\frac{2}{n}}\mathbbm{1}_{\\{U\leqslant
R_{1/n}(q,G)\\}}\nabla\varphi(q)^{\top}\mathcal{D}(q)^{1/2}G$
$\displaystyle+\frac{1}{n}\mathbbm{1}_{\\{U\leqslant
R_{1/n}(q,G)\\}}G^{\top}\mathcal{D}(q)^{1/2}\nabla^{2}\varphi(q)\mathcal{D}(q)^{1/2}G+\mathrm{O}(n^{-3/2}),$

where the remainder (obtained for instance with a Taylor expansion with exact
remainder) is uniformly controlled using derivatives of $\varphi$ of order at
most 3, by which we mean that the remainder satisfies an inequality such as
(132), with a bound $K_{b,f}\leqslant
c_{b,f}\|\varphi\|_{\mathcal{C}^{3}(\mathbb{T}^{d})}$. Subtracting
$\varphi(q)$ and taking the expectation with respect to $U$,

$\displaystyle\mathbb{E}\left[\varphi\left(q+\sqrt{\frac{2}{n}}\mathbbm{1}_{\\{U\leqslant
R_{1/n}(q,G)\\}}\mathcal{D}(q)^{1/2}G\right)\middle|G\right]-\varphi(q)=\sqrt{\frac{2}{n}}R_{1/n}(q,G)\nabla\varphi(q)^{\top}\mathcal{D}(q)^{1/2}G$
$\displaystyle+\frac{1}{n}R_{1/n}(q,G)G^{\top}\mathcal{D}(q)^{1/2}\nabla^{2}\varphi(q)\mathcal{D}(q)^{1/2}G+\mathrm{O}(n^{-3/2}).$

In view of the expansion of the acceptance rate provided by Lemma 34 (with the
analytical expression (140) for $\zeta(q,G)$), and further taking the
expectation with respect to $G$, we obtain

$(\mathcal{L}^{n}\varphi)(q)=-\frac{2}{n}\nabla\varphi(q)^{\top}\mathcal{D}(q)^{1/2}\mathbb{E}[G\max\left(0,\zeta(q,G)\right)]+\frac{1}{n}\mathcal{D}(q):\nabla^{2}\varphi(q)+\mathrm{O}(n^{-3/2}),$

with a remainder term which is uniformly bounded in $q\in\mathbb{T}^{d}$. A
simple expression for the expectation

$\mathbb{E}[G\max\left(0,\zeta(q,G)\right)]=\int_{\mathbb{R}^{d}}g\max\left(0,\zeta(q,g)\right)\frac{\mathrm{e}^{-g^{2}/2}}{(2\pi)^{d/2}}\,dg$

can be derived as in [29] using the anti-symmetry of the function $\zeta$.
Indeed, for all $q\in\mathbb{T}^{d}$ and for all $g\in\mathbb{R}^{d}$, a
direct inspection of (140) reveals that $\zeta(q,g)=-\zeta(q,-g)$, so that
$g\zeta(q,g)=-g\zeta(q,-g)$. Therefore, using [29, Lemma 10] to obtain the
second line,

$\displaystyle\mathbb{E}[G\max\left(0,\zeta(q,G)\right)]$
$\displaystyle=\int_{\\{g\in\mathbb{R}^{d}\,|\,\zeta(q,g)\geqslant
0\\}}g\zeta(q,g)\frac{\mathrm{e}^{-|g|^{2}/2}}{(2\pi)^{d/2}}\,dg=\frac{1}{2}\int_{\mathbb{R}^{d}}g\zeta(q,g)\frac{\mathrm{e}^{-|g|^{2}/2}}{(2\pi)^{d/2}}\,dg$
$\displaystyle=\frac{1}{2}\left[\mathcal{D}(q)^{1/2}\nabla
V(q)-\mathcal{D}(q)^{-1/2}\operatorname{div}(\mathcal{D})(q)\right].$

We conclude that

$2\nabla\varphi(q)^{\top}\mathcal{D}(q)^{1/2}\mathbb{E}[G\min\left(0,\zeta(q,G)\right)]=\left(\mathcal{D}(q)\nabla
V(q)-\operatorname{div}(\mathcal{D})(q)\right)^{\top}\nabla\varphi(q).$

This ensures finally that
$n(\mathcal{L}^{n}\varphi)(q)\to(\mathcal{L}_{\mathcal{D}}\varphi)(q)$
uniformly in $q\in\mathbb{T}^{d}$ as $n\to+\infty$.

####  E.2.2 Limit martingale problem

The argument is standard, and we refer for example to [67, Section 11.2] and
[22, Theorem 5.8] for details. Let us however outline the reasoning for the
sake of completeness.

For any $n\geqslant 1$ and any test function
$\varphi\in\mathcal{C}^{3}(\mathbb{T}^{d})$, the definition of
$\mathcal{L}^{n}$ ensures that

$\varphi(q^{k}_{1/n})-\varphi(q^{0}_{1/n})-\frac{1}{n}\sum_{i=1}^{k-1}(n\mathcal{L}^{n}\varphi)(q^{i}_{1/n})$

is a martingale with respect to $P_{1/n}$. Now, using the definition (81),

$\displaystyle\varphi(q^{k}_{1/n})$
$\displaystyle-\varphi(q^{0}_{1/n})-\frac{1}{n}\sum_{i=1}^{k-1}(n\mathcal{L}^{n}\varphi)(q^{i}_{1/n})$
$\displaystyle=\varphi\left(Q^{1/n}_{k/n}\right)-\varphi\left(Q^{1/n}_{0}\right)-\frac{1}{n}\sum_{i=1}^{k-1}(n\mathcal{L}^{n}\varphi)\left(Q^{1/n}_{i/n}\right)$
$\displaystyle=\varphi\left(Q^{1/n}_{k/n}\right)-\varphi\left(Q^{1/n}_{0}\right)-\frac{1}{n}\sum_{i=1}^{k-1}(\mathcal{L}_{\mathcal{D}}\varphi)\left(Q^{1/n}_{i/n}\right)+\frac{1}{n}\sum_{i=1}^{k-1}(n\mathcal{L}^{n}\varphi-\mathcal{L}_{\mathcal{D}}\varphi)\left(Q^{1/n}_{i/n}\right).$

The last term converges to 0 as $n\to+\infty$ in view of the uniform
convergence
$n(\mathcal{L}^{n}\varphi)(q)\to(\mathcal{L}_{\mathcal{D}}\varphi)(q)$. The
first sum can be rewritten as

$\displaystyle\frac{1}{n}\sum_{i=1}^{k-1}(\mathcal{L}_{\mathcal{D}}\varphi)\left(Q^{1/n}_{i/n}\right)$
$\displaystyle=\sum_{i=1}^{k-1}\int_{i/n}^{(i+1)/n}(\mathcal{L}_{\mathcal{D}}\varphi)\left(Q^{1/n}_{i/n}\right)dt$
$\displaystyle=\int_{1/n}^{k/n}(\mathcal{L}_{\mathcal{D}}\varphi)\left(Q^{1/n}_{t}\right)dt+\mathrm{O}\left(\frac{1}{n}\right).$

For $t>0$ fixed, we conclude by letting $n\to\infty$ with $k=\lfloor
nt\rfloor$ that

$\varphi(Q_{t}^{\star})-\varphi(Q_{0}^{\star})-\int_{0}^{t}(\mathcal{L}_{\mathcal{D}}\varphi)(Q_{s}^{\star})$

is a martingale process for any weak limit $P_{\ast}$ of
$\\{P_{1/n}\\}_{n\geqslant 1}$, which concludes the proof.

###  E.3 Technical results

We gather in this section some technical results used in Sections E.1 and E.2.

######  Lemma 33.

For any $\delta>0$, $n\in\mathbb{N}$ and $k\in\mathbb{N}$, it holds

$\displaystyle\sup_{t\in[0,\delta]}\left|Q^{1/n}_{k\delta+t}-Q^{1/n}_{k\delta}\right|$
$\displaystyle\qquad\leqslant\max\left\\{\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|,\max_{\lceil
nk\delta\rceil\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil}\left|q_{1/n}^{\ell}-q_{1/n}^{\lceil
nk\delta\rceil}\right|\right\\}.$

###### Proof.

Note first that, since $Q^{1/n}_{k\delta}$ linearly interpolates between
$q_{1/n}^{\lfloor nk\delta\rfloor}$ and $q_{1/n}^{\lceil nk\delta\rceil}$,

$\sup_{t\in[0,\delta]}\left|Q^{1/n}_{k\delta+t}-Q^{1/n}_{k\delta}\right|\leqslant\max\left\\{\sup_{s\in[k\delta,(k+1)\delta]}\left|Q^{1/n}_{s}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|,\sup_{s\in[k\delta,(k+1)\delta]}\left|Q^{1/n}_{s}-q_{1/n}^{\lceil
nk\delta\rceil}\right|\right\\}.$ (139)

We next decompose the supremums over $s\in[k\delta,(k+1)\delta]$ into
supremums over intervals of length $1/n$, by noting that

$[k\delta,(k+1)\delta]\subset\bigcup_{\ell=\lfloor nk\delta\rfloor}^{\lceil
n(k+1)\delta\rceil-1}\left[\frac{\ell}{n},\frac{\ell+1}{n}\right].$

For instance,

$\sup_{s\in[k\delta,(k+1)\delta]}\left|Q^{1/n}_{s}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|\leqslant\max_{\lfloor
nk\delta\rfloor\leqslant\ell\leqslant\lceil
n(k+1)\delta\rceil-1}\sup_{\tau\in[0,1/n]}\left|Q^{1/n}_{\ell/n+\tau}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|.$

Since $Q^{1/n}_{\ell/n+\tau}$ linearly interpolates between $q_{1/n}^{\ell}$
and $q_{1/n}^{\ell+1}$, it holds

$\sup_{\tau\in[0,1/n]}\left|Q^{1/n}_{\ell/n+\tau}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|\leqslant\max\left\\{\left|q_{1/n}^{\ell}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|,\left|q_{1/n}^{\ell+1}-q_{1/n}^{\lfloor
nk\delta\rfloor}\right|\right\\}.$

The claimed conclusion then follows from a similar bound for the second
maximum over $s$ on the right-hand side of (139). ∎

The following lemma provides an expansion of the acceptance rate in inverse
powers of $n$.

######  Lemma 34 (Expansion of $R_{1/n}(q,G)$).

For any $q\in\mathbb{T}^{d}$ and $G\in\mathbb{R}^{d}$, it holds

$R_{1/n}(q,G)=1-\sqrt{\frac{2}{n}}\max(0,\zeta(q,G))+\mathrm{O}(n^{-1}),$

with

$\displaystyle\zeta(q,G)=$ $\displaystyle\nabla
V(q)^{\top}\mathcal{D}(q)^{1/2}G+\frac{1}{2}\Tr\left(\mathcal{D}(q)^{-1}\left[\textsf{D}\mathcal{D}(q)\cdot\mathcal{D}(q)^{1/2}G\right]\right)$
(140)
$\displaystyle-\frac{1}{2}G^{\top}\mathcal{D}(q)^{-1/2}\left[\textsf{D}\mathcal{D}(q)\cdot\mathcal{D}(q)^{1/2}G\right]\mathcal{D}(q)^{-1/2}G,$

where $\textsf{D}\mathcal{D}(q)$ is the differential matrix of
$q\mapsto\mathcal{D}(q)$, _i.e._ , for any $h\in\mathbb{R}^{d}$,
$\textsf{D}\mathcal{D}(q)\cdot h$ is the matrix with entries
$\nabla\mathcal{D}_{ij}(q)^{\top}h$.

###### Proof.

The computations are similar to the ones in [29, Section 4.7]. To control
remainder terms, we use the fact that the functions at hand are sufficiently
smooth, the diffusion matrix $\mathcal{D}$ is positive and the domain
$\mathbb{T}^{d}$ is compact. The magnitude of the acceptance probability in
the Metropolis algorithm reads

$R_{1/n}(q,G)=\min\left\\{1,\exp\left[-\alpha\left(q,\Phi_{1/n}(q,G)\right)\right]\right\\},\qquad\Phi_{1/n}(q,G)=q+\sqrt{\frac{2}{n}}\mathcal{D}(q)^{1/2}G,$

with

$\displaystyle\alpha(q,\widetilde{q})$
$\displaystyle=\frac{1}{2}\left(\ln\det[\mathcal{D}(\widetilde{q})]-\ln\det[\mathcal{D}(q)]\right)+\left[V(\widetilde{q})-V(q)\right]$
$\displaystyle\qquad\qquad+\frac{n}{4}\left[\left|\mathcal{D}(\widetilde{q})^{-1/2}(q-\widetilde{q})\right|^{2}-\left|\mathcal{D}(q)^{-1/2}(\widetilde{q}-q)\right|^{2}\right].$

First,

$V(\Phi_{1/n}(q,G))-V(q)=\sqrt{\frac{2}{n}}\nabla
V(q)^{\top}\mathcal{D}(q)^{1/2}G+\mathrm{O}(n^{-1}).$ (141)

We next consider the terms involving $\det(\mathcal{D})$. Since $\mathcal{D}$
is $\mathcal{C}^{2}$ on $\mathbb{T}^{d}$,

$\mathcal{D}(\Phi_{1/n}(q,G))=\mathcal{D}(q)+\sqrt{\frac{2}{n}}\textsf{D}\mathcal{D}(q)\cdot\mathcal{D}(q)^{1/2}G+\mathrm{O}(n^{-1}).$

It then holds, using the fact that $\mathcal{D}$ is uniformly bounded from
below by a positive constant,

$\displaystyle\det[\mathcal{D}(\Phi_{1/n}(q,G))]$
$\displaystyle=\det[\mathcal{D}(q)]\det(\mathrm{Id}+\sqrt{\frac{2}{n}}\mathcal{D}(q)^{-1}\textsf{D}\mathcal{D}(q)\cdot\mathcal{D}(q)^{1/2}G+\mathrm{O}(n^{-1}))$
$\displaystyle=\det[\mathcal{D}(q)]\left(1+\sqrt{\frac{2}{n}}\Tr\left(\mathcal{D}(q)^{-1}[\textsf{D}\mathcal{D}(q)\cdot\mathcal{D}(q)^{1/2}G]\right)+\mathrm{O}(n^{-1})\right),$

so that

$\ln\det[\mathcal{D}(\Phi_{1/n}(q,G))]-\ln\det[\mathcal{D}(q)]=\sqrt{\frac{2}{n}}\Tr\left(\mathcal{D}(q)^{-1}\left[\textsf{D}\mathcal{D}(q)\cdot\mathcal{D}(q)^{1/2}G\right]\right)+\mathrm{O}(n^{-1}).$
(142)

We finally turn to the term
$|\mathcal{D}(\Phi_{1/n}(q,G))^{-1/2}(q-\Phi_{1/n}(q,G))|^{2}-|\mathcal{D}(q)^{-1/2}(\Phi_{1/n}(q,G)-q)|^{2}$.
We use the expansion
$\mathcal{D}(q+x)^{-1}=\mathcal{D}(q)^{-1}-\mathcal{D}(q)^{-1}\left[\textsf{D}\mathcal{D}(q)\cdot
x\right]\mathcal{D}(q)^{-1}+\mathrm{O}(|x|^{2})$ (using again the uniform
positive lower bound on $\mathcal{D}$) to write

$\displaystyle\left|\mathcal{D}(\Phi_{1/n}(q,G))^{-1/2}(q-\Phi_{1/n}(q,G))\right|^{2}-\left|\mathcal{D}(q)^{-1/2}(\Phi_{1/n}(q,G)-q)\right|^{2}$
$\displaystyle\qquad\qquad\qquad\qquad=(\Phi_{1/n}(q,G)-q)^{\top}\left[\mathcal{D}(\Phi_{1/n}(q,G))^{-1}-\mathcal{D}(q)^{-1}\right](\Phi_{1/n}(q,G)-q)$
$\displaystyle\qquad\qquad\qquad\qquad=-\left(\frac{2}{n}\right)^{3/2}G^{\top}\mathcal{D}(q)^{-1/2}\left[\textsf{D}\mathcal{D}(q)\cdot\mathcal{D}(q)^{1/2}G\right]\mathcal{D}(q)^{-1/2}G+\mathrm{O}(n^{-2}).$

Combining (141) and (142) with the latter equality leads to
$\alpha(q,\Phi_{1/n}(q,G))=(2/n)^{1/2}\zeta(q,G)+\mathrm{O}(n^{-1})$, with
$\zeta(q,G)$ given by (140). The desired conclusion then follows from the
inequality $\max(x,0)-\max(x,0)^{2}/2\leqslant
1-\min(1,\mathrm{e}^{-x})\leqslant\max(x,0)$, obtained by distinguishing the
cases $x\leqslant 0$ and $x\geqslant 0$. ∎

The last result makes precise the martingale decomposition of the increments
$\Delta_{n}(q,G,U)$ defined in (134).

######  Lemma 35.

Fix $q\in\mathbb{T}^{d}$. The increment $\Delta_{n}(q,G,U)$ writes

$\Delta_{n}(q,G,U)=M_{n}(q,G,U)+T_{n}(q),$

where
$M_{n}(q,G,U)=\Delta_{n}(q,G,U)-\mathbb{E}\left[\Delta_{n}(q,G,U)\right]$ and
$T_{n}(q)=\mathbb{E}\left[\Delta_{n}(q,G,U)\right]$, the latter expectations
being with respect to realizations of the independent random variables $G$ and
$U$, with $G$ a $d$-dimensional standard Gaussian random variable and $U$ a
uniform random variable on $[0,1]$. Moreover,

$\mathbb{E}[M_{n}(q,G,U)]=0,\qquad M_{n}(q,G,U)=\mathrm{O}(n^{-1/2}),\qquad
T_{n}(q)=\mathrm{O}(n^{-1}),$ (143)

the last two estimates holding uniformly in $q\in\mathbb{T}^{d}$.

###### Proof.

The first property of (143) holds by construction, so it suffices to prove the
two other estimates. For the last one, we note that, by definition,

$T_{n}(q)=\sqrt{\frac{2}{n}}\mathcal{D}(q)^{1/2}\mathbb{E}\left[GR_{1/n}(q,G)\right]=\sqrt{\frac{2}{n}}\mathcal{D}(q)^{1/2}\mathbb{E}\left[G\left(R_{1/n}(q,G)-1\right)\right],$

so that the conclusion follows from Lemma 34 and the fact that $\mathcal{D}$,
its derivatives and its inverse, are uniformly bounded in $q$. The other
estimate is obtained by noting that

$M_{n}(q,G,U)=\sqrt{\frac{2}{n}}\mathcal{D}(q)^{1/2}\left(G\mathbbm{1}_{\\{U\leqslant
R_{1/n}(q,G)\\}}-\mathbb{E}\left[GR_{1/n}(q,G)\right]\right),$

so that, in view of Lemma 34,

$\forall
u\in[0,1],\qquad|M_{n}(q,G,u)|\leqslant\frac{C}{\sqrt{n}}\left|\mathcal{D}(q)\right|_{\mathrm{F}}^{1/2}|G|+\mathrm{O}(n^{-1}).$

This concludes the proof. ∎

##### Acknowledgements.

We thank Grégoire Allaire, Benjamin Bogosel, Andrew Duncan, Federico Ghimenti,
Frédéric Legoll, Mathieu Lewin, Danny Perez, Samuel Power, Frédéric van
Wijland and Jonathan Weare for stimulating discussions. The works of T.L.,
R.S. and G.S. benefit from fundings from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(project EMC2, grant agreement No 810367), and from the Agence Nationale de la
Recherche through the grants ANR-19-CE40-0010-01 (QuAMProcs) and
ANR-21-CE40-0006 (SINEQ). G.P. is partially supported by an ERC-EPSRC Frontier
Research Guarantee through Grant No. EP/X038645, ERC through Advanced Grant
No. 247031. This project was initiated as T.L. was a visiting professor at
Imperial College of London (ICL), with a visiting professorship grant from the
Leverhulme Trust. The Department of Mathematics at ICL and the Leverhulme
Trust are warmly thanked for their support.

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