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Federated Learning Over Wireless Edge Networks Wei Yang Bryan Lim
Wireless Networks
WeiYang Bryan Lim · Jer Shyuan Ng ·
Zehui Xiong · Dusit Niyato ·
Chunyan Miao
Federated
Learning Over
Wireless Edge
Networks
Wireless Networks
Series Editor
Xuemin Sherman Shen, University of Waterloo, Waterloo, ON, Canada
The purpose of Springer’s Wireless Networks book series is to establish the state
of the art and set the course for future research and development in wireless
communication networks. The scope of this series includes not only all aspects
of wireless networks (including cellular networks, WiFi, sensor networks, and
vehicular networks), but related areas such as cloud computing and big data.
The series serves as a central source of references for wireless networks research
and development. It aims to publish thorough and cohesive overviews on specific
topics in wireless networks, as well as works that are larger in scope than survey
articles and that contain more detailed background information. The series also
provides coverage of advanced and timely topics worthy of monographs, contributed
volumes, textbooks and handbooks.
** Indexing: Wireless Networks is indexed in EBSCO databases and DPLB **
Wei Yang Bryan Lim • Jer Shyuan Ng •
Zehui Xiong • Dusit Niyato • Chunyan Miao
Federated Learning Over
Wireless Edge Networks
Wei Yang Bryan Lim
Alibaba-NTU Joint Research Institute
Singapore, Singapore
Jer Shyuan Ng
Alibaba-NTU Joint Research Institute
Singapore, Singapore
Zehui Xiong
Singapore University of Technology and
Design
Singapore, Singapore
Dusit Niyato
School of Computer Science and
Engineering
Nanyang Technological University
Singapore, Singapore
Chunyan Miao
School of Computer Science and
Engineering
Nanyang Technological University
Singapore, Singapore
ISSN 2366-1186 ISSN 2366-1445 (electronic)
Wireless Networks
ISBN 978-3-031-07837-8 ISBN 978-3-031-07838-5 (eBook)
https://doi.org/10.1007/978-3-031-07838-5
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland
AG 2022
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether
the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse
of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and
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The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
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The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, expressed or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
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This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
The confluence of edge computing and artificial intelligence (AI) has driven the rise
of edge intelligence, which leverages the storage, communication, and computation
capabilities of end devices and edge servers to empower AI implementation at scale
closer to where data is generated. An enabling technology of edge intelligence is the
privacy-preserving machine learning paradigm known as federated learning (FL).
Amid the increasingly stringent privacy regulations, FL will enable the development
of applications that have to be built using sensitive user data and will continue to
revolutionize service delivery in finance, Internet of Things (IoT), healthcare, and
transport industries, among others. However, the implementation of FL is envisioned
to involve thousands of heterogeneous distributed end devices that differ in terms of
communication and computation resources, as well as the levels of willingness to
participate in the collaborative model training process. The potential node failures,
device dropouts, and stragglers effect are key bottlenecks that impede the effective,
sustainable, and scalable implementation of FL.
In Chap. 1, we will first present a tutorial and survey on FL and highlight
its role in enabling edge intelligence. This tutorial and survey provide readers
with a comprehensive introduction to the forefront challenges and state-of-the-art
approaches towards implementing FL at the edge. In consideration of resource
heterogeneity at the edge networks, we then provide multifaceted solutions for-
mulated via the interdisciplinary interplay of concepts derived from network
economics, optimization, game theory, and machine learning towards improving
the efficiency of resource allocation for implementing FL at scale amid information
asymmetry. In Chap. 2, we devise a multi-dimensional contract-matching approach
for optimized resource allocation for federated sensing and learning amid multi-
dimensional sources of heterogeneities. In Chap. 3, we propose a joint-auction
coalition formation framework towards facilitating communication-efficient FL
networks aided by unmanned aerial vehicles (UAVs). In Chap. 4, we propose a
two-level evolutionary game theoretic and auction approach to allocate and price
resources to facilitate efficient edge intelligence. In Chap. 5, we recap the key points
and discuss the promising research directions for future works.
v
vi Preface
We sincerely thank our collaborators for their contributions to the presented
research works. Special thanks also go to the staff at Springer Nature for their help
throughout the publication preparation process. Finally, we would like to take the
chance to dedicate this book to celebrate the birth of Lim Chen Huan Theodore, son
of Dr. Lim Wei Yang Bryan and Foo Feng Lin.
Singapore, Singapore Wei Yang Bryan Lim
Singapore, Singapore Jer Shyuan Ng
Singapore, Singapore Zehui Xiong
Singapore, Singapore Dusit Niyato
Singapore, Singapore Chunyan Miao
Contents
1 Federated Learning at Mobile Edge Networks: A Tutorial ............. 1
1.1 Introduction ............................................................. 1
1.2 Background and Fundamentals of Federated Learning ............... 6
1.2.1 Federated Learning ............................................. 6
1.2.2 Statistical Challenges of FL .................................... 8
1.2.3 FL Protocols and Frameworks ................................. 11
1.2.4 Unique Characteristics and Issues of FL ...................... 12
1.3 Communication Cost ................................................... 12
1.3.1 Edge and End Computation .................................... 13
1.3.2 Model Compression ............................................ 16
1.3.3 Importance-Based Updating.................................... 17
1.4 Resource Allocation .................................................... 18
1.4.1 Worker Selection................................................ 21
1.4.2 Joint Radio and Computation Resource Management ........ 25
1.4.3 Adaptive Aggregation .......................................... 27
1.4.4 Incentive Mechanism ........................................... 28
1.5 Privacy and Security Issues ............................................. 32
1.5.1 Privacy Issues ................................................... 33
1.5.2 Security Issues .................................................. 37
1.6 Applications of Federated Learning for Mobile Edge Computing.... 39
1.6.1 Cyberattack Detection .......................................... 42
1.6.2 Edge Caching and Computation Offloading................... 44
1.6.3 Base Station Association ....................................... 47
1.6.4 Vehicular Networks ............................................. 48
1.7 Conclusion and Chapter Discussion.................................... 50
2 Multi-dimensional Contract Matching Design for Federated
Learning in UAV Networks ................................................. 53
2.1 Introduction ............................................................. 53
2.2 System Model and Problem Formulation .............................. 56
2.2.1 UAV Sensing Model ............................................ 58
2.2.2 UAV Computation Model ...................................... 59
vii
viii Contents
2.2.3 UAV Transmission Model ...................................... 60
2.2.4 UAV and Model Owner Utility Modeling ..................... 61
2.3 Multi-dimensional Contract Design.................................... 61
2.3.1 Contract Condition Analysis ................................... 62
2.3.2 Conversion into a Single-Dimensional Contract .............. 63
2.3.3 Conditions for Contract Feasibility ............................ 64
2.3.4 Contract Optimality............................................. 68
2.4 UAV-Subregion Assignment............................................ 70
2.4.1 Matching Rules ................................................. 71
2.4.2 Matching Implementation and Algorithm ..................... 72
2.5 Performance Evaluation ................................................ 73
2.5.1 Contract Optimality............................................. 73
2.5.2 UAV-Subregion Preference Analysis .......................... 76
2.5.3 Matching-Based UAV-Subregion Assignment ................ 78
2.6 Conclusion and Chapter Discussion.................................... 80
3 Joint Auction–Coalition Formation Framework for
UAV-Assisted Communication-Efficient Federated Learning........... 83
3.1 Introduction ............................................................. 83
3.2 System Model ........................................................... 86
3.2.1 Worker Selection................................................ 88
3.2.2 UAV Energy Model............................................. 90
3.3 Coalitions of UAVs ..................................................... 93
3.3.1 Coalition Game Formulation ................................... 94
3.3.2 Coalition Formation Algorithm ................................ 97
3.4 Auction Design.......................................................... 98
3.4.1 Buyers’ Bids .................................................... 99
3.4.2 Sellers’ Problem ................................................ 100
3.4.3 Analysis of the Auction ........................................ 102
3.4.4 Complexity of the Joint Auction–Coalition Algorithm ....... 104
3.5 Simulation Results and Analysis ....................................... 105
3.5.1 Communication Efficiency in FL Network .................... 106
3.5.2 Preference of Cells of Workers................................. 107
3.5.3 Profit-Maximizing Behavior of UAVs ......................... 109
3.5.4 Allocation of UAVs to Cells of Workers....................... 111
3.5.5 Comparison with Existing Schemes ........................... 114
3.6 Conclusion and Chapter Discussion.................................... 115
4 Evolutionary Edge Association and Auction in Hierarchical
Federated Learning .......................................................... 117
4.1 Introduction ............................................................. 117
4.2 System Model and Problem Formulation .............................. 120
4.2.1 System Model................................................... 120
4.2.2 Lower-Level Evolutionary Game .............................. 121
4.2.3 Upper-Level Deep Learning Based Auction................... 121
4.3 Lower-Level Evolutionary Game ...................................... 122
4.3.1 Evolutionary Game Formulation............................... 122
Contents ix
4.3.2 Worker Utility and Replicator Dynamics ...................... 123
4.3.3 Existence, Uniqueness, and Stability of the
Evolutionary Equilibrium ...................................... 125
4.4 Deep Learning Based Auction for Valuation of Cluster Head ........ 128
4.4.1 Auction Formulation............................................ 128
4.4.2 Deep Learning Based Auction for Valuation of
Cluster Heads ................................................... 130
4.4.3 Monotone Transform Functions................................ 132
4.4.4 Allocation Rule ................................................. 133
4.4.5 Conditional Payment Rule ..................................... 134
4.4.6 Neural Network Training ....................................... 134
4.5 Performance Evaluation ................................................ 135
4.5.1 Lower-Level Evolutionary Game .............................. 136
4.5.2 Upper-Level Deep Learning Based Auction................... 140
4.6 Conclusion and Chapter Discussion.................................... 143
5 Conclusion and Future Works.............................................. 147
References......................................................................... 151
Index............................................................................... 165
List of Figures
Fig. 1.1 Edge AI approach brings AI processing closer to where
data are produced. In particular, FL allows training on
devices where the data are produced ................................ 3
Fig. 1.2 General FL training process involving N workers .................. 7
Fig. 1.3 Approaches to increase computation at edge and end
devices include (a) increased computation at end devices,
e.g., more passes over dataset before communication, (b)
two-stream training with global model as a reference, and
(c) intermediate edge server aggregation ........................... 14
Fig. 1.4 Worker selection under the FedCS and Hybrid-FL
protocol ............................................................... 22
Fig. 1.5 A comparison between (a) BAA by over-the-air
computation that reuses bandwidth (above) and (b)
OFDMA (below) that uses only the allocated bandwidth .......... 26
Fig. 1.6 A comparison between (a) synchronous and (b)
asynchronous FL ..................................................... 27
Fig. 1.7 Workers with unknown resource constraints maximize
their utility only if they choose the bundle that best
reflects their constraints .............................................. 30
Fig. 1.8 Selective parameter sharing model .................................. 35
Fig. 1.9 GAN attack on collaborative deep learning ......................... 36
Fig. 1.10 An illustration of (a) conventional FL and (b) the
proposed BlockFL architectures ..................................... 40
Fig. 1.11 FL based attack detection architecture for IoT edge
networks .............................................................. 43
Fig. 1.12 FL based (a) caching and (b) computation offloading ............. 45
Fig. 2.1 System model involving UAV-subregion contract matching........ 55
Fig. 2.2 UAV node coverage vs. auxiliary types ............................. 75
Fig. 2.3 Contract rewards vs. auxiliary types ................................. 75
Fig. 2.4 Contract items vs. UAV utilities...................................... 75
xi
xii List of Figures
Fig. 2.5 The model owner profits vs. UAV auxiliary types .................. 76
Fig. 2.6 The UAV utility for each subregion vs. types ....................... 77
Fig. 2.7 UAV matching for homogeneous subregions........................ 78
Fig. 2.8 UAV matching for subregions with different data
quantities and coverage area ......................................... 79
Fig. 2.9 UAV matching where J > N ........................................ 79
Fig. 3.1 System model consists of the cloud server (FL model
owner), the vehicles and RSUs (selected FL workers), and
the UAVs ............................................................. 85
Fig. 3.2 Illustration of the joint auction–coalition formation
framework ............................................................ 87
Fig. 3.3 Distributed FL Network with 3 cells and 6 UAVs .................. 105
Fig. 3.4 Communication time needed by UAVs and IoV vehicles .......... 107
Fig. 3.5 Maximum number of iterations under different energy
capacities ............................................................. 110
Fig. 3.6 Illustration of merge-and-split mechanism and allocation
of UAVs to cells of workers ......................................... 111
Fig. 3.7 Total profit and number of coalitions vs cooperation cost ......... 113
Fig. 3.8 Total profit and size of coalitions vs number of iterations ......... 113
Fig. 3.9 Comparison with existing schemes .................................. 114
Fig. 4.1 An illustration of the hierarchical system model ................... 118
Fig. 4.2 Neural network architecture for the optimal auction ................ 130
Fig. 4.3 Monotone transform functions ....................................... 133
Fig. 4.4 Phase plane of the replicator dynamics .............................. 137
Fig. 4.5 Evolutionary equilibrium of population states for cluster 1 ........ 137
Fig. 4.6 Evolution of population utilities ..................................... 138
Fig. 4.7 Evolution of population states for population 1 .................... 138
Fig. 4.8 Evolution of population states for population 2 .................... 138
Fig. 4.9 Evolution of population states for population 3 .................... 138
Fig. 4.10 Evolutionary dynamics under different learning rates .............. 139
Fig. 4.11 Data coverage vs. varying fixed rewards in cluster 1 ............... 139
Fig. 4.12 Data coverage vs. varying congestion coefficient in cluster 1 ...... 140
Fig. 4.13 Population states in cluster 3 vs. varying population data
for population 1 ....................................................... 140
Fig. 4.14 Revenue of cluster head 1 under different distribution of
model owners ......................................................... 141
Fig. 4.15 Revenue of cluster head 2 under different distribution of
model owners ......................................................... 141
Fig. 4.16 Revenue of cluster head 3 under different distribution of
model owners ......................................................... 141
Fig. 4.17 Revenue vs data coverage of cluster heads .......................... 142
Fig. 4.18 Revenue of cluster head 1 under different approximation
qualities ............................................................... 143
Fig. 4.19 Revenue of cluster head 2 under different approximation
qualities ............................................................... 143
List of Figures xiii
Fig. 4.20 Revenue of cluster head 3 under different approximation
qualities ............................................................... 143
List of Tables
Table 1.1 An overview of selected surveys in FL and MEC .................. 5
Table 1.2 Loss functions of common ML models ............................. 8
Table 1.3 Approaches to communication cost reduction in FL ............... 19
Table 1.4 Approaches to resource allocation in FL ........................... 31
Table 1.5 The accuracy and attack success rates for no-attack
scenario and attacks with 1 and 2 sybils in an FL system
with MNIST dataset [1] .............................................. 37
Table 1.6 A summary of attacks and countermeasures in FL ................. 41
Table 1.7 FL based approaches for mobile edge network
optimization .......................................................... 42
Table 2.1 Table of commonly used notations .................................. 56
Table 2.2 Table of key simulation parameters.................................. 74
Table 2.3 UAV types for preference analysis................................... 77
Table 2.4 UAV type and preference for subregions ............................ 78
Table 3.1 Table of commonly used notations .................................. 89
Table 3.2 Simulation parameters ............................................... 106
Table 3.3 Preference of cells for different coalitions .......................... 107
Table 3.4 Preference of UAVs for different cells............................... 108
Table 3.5 Preference of cells for individual UAVs (bolded row
implies top preference) ............................................... 108
Table 3.6 Revenue, cost of profit for different coalitions in each cell......... 110
Table 4.1 Simulation parameters ............................................... 135
xv
Chapter 1
Federated Learning at Mobile Edge
Networks: A Tutorial
1.1 Introduction
Currently, there are nearly 7 billion connected Internet of Things (IoT) devices
and 3 billion smartphones around the world [2]. These devices are equipped with
increasingly advanced sensors, computing, and communication capabilities. As
such, they can potentially be deployed for various crowdsensing tasks, e.g., for
medical purposes [3] and air quality monitoring [4]. Coupled with the rise of Deep
Learning (DL) [5], the wealth of data collected by end devices opens up countless
possibilities for meaningful research and applications.
In the traditional cloud-centric approach, data collected by mobile devices
are uploaded and processed centrally in a cloud-based server or data center. In
particular, data collected by IoT devices and smartphones such as measurements [6],
photos [7], videos [8], and location information [9] are aggregated at the data center
[10]. Thereafter, the data are used to provide insights or produce effective inference
models. However, this approach is no longer sustainable for the following reasons.
Firstly, data owners are increasingly privacy sensitive. Following privacy concerns
among consumers due to high profile cases of data leaks and data misuse, policy
makers have responded with the implementation of data privacy legislation such
as the European Commission’s General Data Protection Regulation (GDPR) [11]
and Consumer Privacy Bill of Rights in the USA [12]. In particular, the consent
(GDPR Article 6) and data minimalization principle (GDPR Article 5) limit data
collection and storage only to what is consumer-consented and absolutely necessary
for processing. Secondly, a cloud-centric approach involves long propagation delays
and incurs unacceptable latency [13] for applications in which real-time decisions
have to be made, e.g., in self-driving car systems [14]. Thirdly, the transfer of
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
W. Y. B. Lim et al., Federated Learning Over Wireless Edge Networks, Wireless
Networks, https://doi.org/10.1007/978-3-031-07838-5_1
1
2 1 FL at Mobile Edge Networks: A Tutorial
raw data to the cloud for processing burdens the backbone networks.1 This is
especially so in tasks involving unstructured data, e.g., in video analytics [15]. This
is exacerbated by the fact that cloud-centric training is relatively reliant on wireless
communications [16]. As a result, this can potentially impede the development of
new technologies.
With data sources mainly located outside the cloud today, Mobile Edge Com-
puting (MEC) has naturally been proposed as a solution. In MEC, the computing
and storage capabilities [13] of end devices and edge servers are leveraged to bring
model training closer to where data are produced [17]. As defined in [16], an end–
edge–cloud computing network comprises (a) end devices, (b) edge nodes, and (c)
cloud server. For model training in conventional MEC approaches, a collaborative
paradigm has been proposed in which training data are first sent to the edge servers
for model training up to lower-level DNN layers, before more computation intensive
tasks are offloaded to the cloud [18], [19] (Fig. 1.1). However, this arrangement
incurs significant communication costs and is unsuitable for applications that require
persistent training [16]. In addition, computation offloading and data processing
at edge servers still involve the transmission of potentially sensitive personal data.
This can discourage privacy-sensitive consumers from taking part in model training
or even violate increasingly stringent privacy laws [11]. Although various privacy
preservation methods, e.g., Differential Privacy (DP) [20], have been proposed, a
number of users are still not willing to expose their private data for fear that their
data may be inspected by external servers. In the long run, this discourages the
development of technologies as well as new applications.
To guarantee that training data remain on personal devices and to facilitate
collaborative machine learning of complex models among distributed devices, a
decentralized ML approach called Federated Learning (FL) is introduced in [21]. In
FL, mobile devices2 use their local data to cooperatively train an ML model required
by an FL server. They then send the model updates, i.e., the model’s weights, to the
FL server for aggregation. The steps are repeated in multiple rounds until a desirable
accuracy is achieved. This implies that FL can be an enabling technology for ML
model training at mobile edge networks. As compared to conventional cloud-centric
ML model training approaches, the implementation of FL for model training at
mobile edge networks features the following advantages:
1. Minimizing requirement of network bandwidth: Less information is required to
be transmitted to the cloud. For example, instead of sending the raw data over
for processing, participating devices only send the updated model parameters for
1 Note that in some cases, the communication cost of FL is not insignificant due to complexity
of large models, whereas the relatively fewer data samples a worker possesses are less costly to
transmit. As such, we discuss communication cost reduction as well in this chapter.
2 We use the term mobile devices and mobile edge in this chapter given that many of the works
reviewed have focused on how to implement FL on resource-constrained edge devices such as IoT.
However, note that the insights from this chapter can be similarly applied on edge networks in
general.
1.1 Introduction 3
Fig. 1.1 Edge AI approach brings AI processing closer to where data are produced. In particular,
FL allows training on devices where the data are produced
aggregation. As a result, this significantly reduces costs of data communication
and relieves the burden on backbone networks.
2. Privacy: Following the above point, the raw data of users need not be sent to the
cloud. Under the assumption that FL workers and servers are non-malicious, this
4 1 FL at Mobile Edge Networks: A Tutorial
enhances user privacy and reduces the probability of eavesdropping to a certain
extent. In fact, with enhanced privacy, more users will be willing to take part in
collaborative model training, and so, better inference models can be built.
3. Low latency: With FL, ML models can be consistently trained and updated.
Meanwhile, in the MEC paradigm, real-time decisions, e.g., event detection [22],
can be made locally at the edge nodes or end devices. Therefore, the latency is
much lower than when decisions are made in the cloud before transmitting them
to the end devices. This is vital for time critical applications such as self-driving
car systems in which the slightest delays can potentially be life threatening [14].
Given the aforementioned advantages, FL has seen recent successes in several
applications. For example, the Federated Averaging algorithm (FedAvg) proposed
in [21] has been applied to Google’s Gboard [23] to improve next-word prediction
models. In addition, several studies have also explored the use of FL in a number
of scenarios in which data are sensitive in nature, e.g., to develop predictive models
for diagnosis in health AI [24] and to foster collaboration across multiple hospitals
[25] and government agencies [26].
Besides being an enabling technology for ML model training at mobile edge
networks, FL has also been increasingly applied as an enabling technology for
mobile edge network optimization. Given the computation and storage constraints
of increasingly complex mobile edge networks, conventional network optimization
approaches that are built on static models fare relatively poorly in modeling dynamic
networks [16]. As such, a data-driven Deep Learning (DL) based approach [27]
for optimizing resource allocation is increasingly popular. For example, DL can
be used for representation learning of network conditions [28], whereas Deep
Reinforcement Learning (DRL) can optimize decision making through interactions
with the dynamic environment [29]. However, the aforementioned approaches
require user data as an input and these data may be sensitive or inaccessible in
nature due to regulatory constraints. As such, in this chapter, we also briefly discuss
FL’s potential to function as an enabling technology for optimizing wireless edge
networks, e.g., in cell association [30], computation offloading [2], and vehicular
networks [31].
However, there are several challenges to be solved before FL can be implemented
at scale. Firstly, even though raw data no longer need to be sent to the cloud servers,
communication costs remain an issue due to the high dimensionality of model
updates and limited communication bandwidth of participating mobile devices. In
particular, state-of-the-art DNN model training can involve the communication of
millions of parameters for aggregation. Secondly, in a large and complex mobile
edge network, the heterogeneity of participating devices in terms of data quality,
computation power, and willingness to participate has to be well managed from
the resource allocation perspective. Thirdly, FL does not guarantee privacy in the
presence of malicious workers or aggregating servers. In particular, recent research
works have clearly shown that a malicious worker may exist in FL and can infer
the information of other workers just from the shared parameters alone. As such,
privacy and security issues in FL still need to be considered.
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brazier, and the bowl ceased to smoke perfumes. Gorlois saw the
man gather his black robe with his glittering fingers, and move like a
wraith round the room, to stand beckoning by the door. In another
minute Gorlois was under the stars, with the house and its yews a
black mound against the sky. Like a sleeper half wakened he took
full breath of the night air, and stretched his arms up above his
head. But it was not to sleep that he passed back through the void
streets to the house of the knight Accolon.
To return to Igraine housed for the night in the little hermitage. At
the first creep of dawn, when daffodils were thrown up against the
eastern sky, she left her pallet bed in the cell and went out into the
hermit’s garden. The recluse was down at the brook drawing water,
whither the dog and the doves had followed him. Igraine passed
through the garden—spun over as it was with webs of dew. To her
comfort she found her ankle scarcely troubling her, for she had
feared pain or stiffness after the walk of yesterday. Going down the
dale, she patted the old dog’s head, and picked up the pitcher as the
recluse gave her good-morning.
“You are an early soul, sister. My dog and I come down to the brook
each morning as the sun peeps over the hill.”
“You are not lonely,” said Igraine.
The old man tightened his girdle, looked over the solemn piers of
the woods, sniffed the air, and hailed an autumn savour.
“Not I,” he said. “I have my dog and my doves, and folk often lodge
here, and I have word of the world and how the Saxons vex us. The
good people near bring me alms and pittances, or come to ask
prayers for their souls, and”—with a twinkle—“for their bodies, too.”
Igraine remembered the peasant’s little son.
“Was it you,” she said, “who gave a peasant fellow near here a
saint’s dust to scatter over a sick child?”
The old man shook his head and smiled enigmatically.
“I have no dealings in such marvels,” he said.
“The boy died.”
“Of course.”
“They will sell your dust some day.”
A keen look, cynical with beaming scorn, spread over the man’s
gaunt face.
“Much good may it do them,” he said; “death is monstrous flatterer
of mere clay. I may feed a rose bush with my bones; a better fate
than the cheating of superstitious women.”
He made a sign with his hand, and the birds went wheeling in circles
above him. The dog crept up and thrust his snout into the old man’s
palm. The garden lay above them, ripe with an autumn mellowness;
yet there was no regret though winter would soon be piping, and the
man’s hair was grey.
“What think you of life?” said Igraine.
“You should know, sister, as well as I.”
“But you see, father, I am not a nun,—only a novice.”
He stared at her a moment with a slight smile.
“Remain a novice,” he said.
“You advise me so!”
“Why subordinate your soul to chains forged of men.”
“These seem strange words.”
He patted his dog’s head, and, half stooping, looked at her with keen
grey eyes.
“Have you ever loved a man?”
“Yes,” she said, with a clear laugh and a slight colour.
“Is he worthy?”
“I believe him a noble soul.”
“Naturally.”
“He ran away and left me because he thought I was a nun.”
The hermit applauded.
“That sounds like honour,” he said critically.
“I am seeking him to tell him the truth.”
“And I will pray that you may soon meet,” said the old man, "for
there is nothing like the love of a good man for a clean maid. If I
had married a true woman, I should never have taken to the
scourge or the stone bed. Marry wisely and you are halfway to
Heaven."
They broke fast that morning in the garden, it being the man’s
custom to make his meals on the granite slab that served him as a
bed. The little dale looked very green and gracious in the tranquil
light, with its curling brook and dark barriers of trees. Igraine, as she
sat on the great stone and ate the hermit’s bread, followed the
brook with her thoughts, wondering whether it became the stream
that ran through Eudol’s meadows. She was for Sarum that day,
where she would throw off her grey habit and take some dress more
likely to baffle Gorlois. She had enough money in her purse.
Worldling again, she could give herself to winning sight of this Uther,
and to learning whether he was the Pelleas she sought or no.
As she sat and fingered her bread, something she saw down the
dale made her rigid and still as a priestess smitten with the vision of
a god in some heathen oratory. Her eyes were very wide, her lips
open and very white, her whole air as of one watching in a sudden
stupor of awe. Another moment and she had broken from the mood
like a torrent from a cavern. With eyes suddenly amber bright, she
touched the hermit’s hand and pointed down the dale, gave him a
word or so, then left him and ran down the hill.
A man on a black horse had ridden out from the trees, and was
pushing his horse over the brook at a shallow spot not far away. His
red armour glowed in the sun with a metallic lustre. Even at that
distance Igraine had seen the red dragon rampant on a shield of
green. As she ran down the grass slope she called the man by name,
thinking to see him turn and come to her. Pushing on sullenly as
though he had not heard the cry that went after him like winged
love, he drew up the further slope without wavering, and sank like a
red streak into the dense green of the trees.
X
Igraine forded the brook and followed the man by the winding path
that curled away into the wood.
She was ever a sanguine soul, and the mere sinister influences that
might have discouraged her in her purpose that morning were
impotent before the level convictions of her heart. She had seen
Pelleas ride in amid the trees; she was sure as death as to his
cognizance and his armour. Now Pelleas, she could vow, had not
heard her call to him, and if he had heard he had not understood; if
he had seen he had not recognised. Doubts could have no place in
the argument before such a justification by faith.
It was not long before she caught sight of the red glint of armour
going through the trees. It came and went, grew and disappeared,
as the path folded it in its curves or thrust out a heavy screen of
green to hide it like a heavy curtain. The man was going as he
pleased, now a walk, now a casual jog, now a short burst of a canter
over an open patch. One moment Igraine would see him clearly,
then not at all. Sometimes she gained, sometimes lost ground, yet
the knight of the red harness never seemed to come within lure of
her voice.
In due course she reached the place where the path ended bluntly
on the Winchester high-road, and where the way ran straight as a
spear-shaft, so that she could see Pelleas riding for Winchester with
a lead of a quarter of a mile. The distant ringing tramp of hoofs
came up to her like a mocking chuckle. Putting her hands to her
mouth, she hallooed with all the breath left her by her run through
the wood; yet, as far as she might see, the man never so much as
turned in the saddle, while the smite of hoofs died down and down
into a well of silence.
Another halloo and no echo.
“He’s asleep, or deaf in his helmet.”
She forgot the distance and the din of hoofs that might well have
drowned the thin cry that could have reached the rider. Maugre her
heat and her flushed face Igraine had no more thought of giving in
than she had of marrying Gorlois. With Pelleas so near she had
made her vow to follow him, and follow him she would like a comet’s
tail. If needs be she would wear her sandals to the flesh, but catch
the man she must in the end.
A mile more on the high-road, with her feet and the hem of her
gown dust-drenched, and she was still little nearer the man in the
red harness for all her hurrying. She could have vowed more than
once that he turned in his saddle and looked back at her as though
to see how near she had come to him on the road. A mile from the
hermitage path he turned his horse southwards from the track into a
grass valley headed by a ruined tower and hedged densely on either
hand by pine woods. Igraine, seeing from a slight rise in the road
this change of course, cut away crosswise with the notion of getting
near the man or of intercepting him before he should win clear law
again. After all, the effort added only more vexation. She saw the
black horse pressed to a canter and cross the point where she might
have cut him off, while a great stretch of furze that rolled away to
the black palisading of the pines came down and threw a
promontory in her path. Pelleas was a mile to the good when she
had skirted the furze and the bend of the wood, and taken a straight
course southwards down the valley between the pines.
All that morning the sport of hunter and hunted went on between
the novice in grey and the man on the black horse. For all her
trouble Igraine won little upon him, lost little as the hours went by;
while the rider in turn seemed in no wise desirous of being rid of her
for good. They passed the pine woods with their midnight aisles,
forded a stream, climbed up a heath, went over it amid the heather.
From the last ridge of the heath Igraine saw the country sloping
away into undulating grasslands, piled here and there with domes of
thicketed trees. Far to the south a dense black mass rose like a
rounded hill against the sky. The man in red was still about a mile in
front of her, riding slowly, a red speck in a waste of green. Igraine,
having him in view from her vantage point, lay down full length to
rest and take some food. She was tired enough, but dogged at heart
as ever. She vowed that if the man was playing with her she would
tell him her mind, love or no love, when she came up with him in the
end.
As the sun swam into the noontide arc she went on again downhill,
and found in turn that the man had halted, for he had been hidden
by trees, and getting view of him suddenly she saw him sitting on a
stone with his horse tethered near. As soon as Igraine was within
measurable distance she took advantage of a hollow, dropped on
her hands and knees, and began to crawl like a cat after a bird.
Edging round a thicket she came quite near the man, but could not
see his face. His spear stood in the ground by his horse, and he had
his shield slung about his neck, and a bare poniard in his hand. It
was clear that he was watching for Igraine, for despite her craft he
caught sight of her face peering white under the hem of a bush, and
climbed quickly into the saddle. Igraine started up, made a dash
across the open, calling to him as she ran. Perverse as hate his
horse broke into a canter and left her far in the rear. The girl shook
her fist at him with a sudden burst of temper. She was standing near
the stone where the man had been sitting. Looking at its flat face
she saw the reason of the naked poniard in his hand, for he had
been carving out thin straggling letters in the stone.
“Sancta Igraine,” she read—
“Ora pro nobis.”
The screed dispelled the doubts in Igraine’s mind on the instant.
Palpably the man knew well enough who was following him, and was
avoiding her of set purpose; but for what reason Igraine racked her
wit to discover. She ran through many things in her heart, the
possible testing of her devotion, a vacillating weakness on Pelleas’s
part that would not let him leave her altogether, a freakish wish to
give her penance. Then, she knew that he was superstitious, and the
thought flashed to her that he might think her a wraith, or some evil
spirit that had taken her shape to have him in temptation. Maugre
her vexation and her pride she held again on the trail, eating as she
went some dried plums that she had in her wallet. The man had
slackened down again and was less than half a mile away, now
limned against the sky, now folded into a hollow or shut out by
trees. Like a marsh-fire he tantalised her with a mystery of distance,
holding steadily south at a level tramp, while Igraine plodded after
him, her hair down and blowing out to the casual wind, her eyes at
gaze on the red lure in the van.
So the mellower half of the day passed, and towards evening they
neared the mount of trees Igraine had seen from the last ridge of
the heath at noon. The black horse was heading straight for the
cloudy mass in a way that set Igraine thinking and casting about for
Pelleas’s motive. Perhaps he had some quest in the solitary place
that needed his single hand. Would he take to the wood and let her
follow as before, or had he any purpose in leading her thither?
Drowned in conjecture she gave up prophecy with a vicious sense of
mystification, and accepted inevitable ignorance for the time being
as to the man’s moods and motives. She was no less obstinate to
follow him to the death. If she only had a horse she would come
near the man, pride or no pride, and tell him the truth.
Pressing on, with her strained ankle beginning to limp, she topped
the round back of a grass rise and came full in view of the wood she
had long seen in the distance. It looked very solemn in the declining
light. The great trunks of giant beeches were packed pillar upon
pillar into an impenetrable gloom. The foliage above, densely green,
billowy, touched with red and gold, rolled upwards cloud on cloud as
the ground ascended to the south and east. A great bronze carpet of
dead leaves swept away into the night of the trees. There was an
eternal hush, a gross silence, over the glooming aisles that seemed
to beckon to the soul, to draw the heart into the night of foliage as
into a cavern. Over all was the glowing ægis of the setting sun.
Igraine saw the man on the forest’s edge where an arch of gloom
struck into the inner shadows. He was facing the west, motionless as
stone on his black horse, with the slanting light plucking a dull red
gleam from his harness. There was a mystery about him that
seemed to harmonise with the stillness of the trees and the black
yawn of the forest galleries. Igraine imagined that he might be in a
mood at last to speak with her if he believed her human. At all
events, if he took to the trees, and she did not lose him, she would
have the vantage of him and his horse in such a barricaded place.
It began to grow dark very quickly as she passed down the gradual
slope towards the forest. The trees towered above her, a black mass
rising again towards the east. Keen to see the man’s mood, she
hurried on and found him still steadfast in the great arch, that
seemed like the gate of the wilderness, ready to abide her. A
hundred paces more and her heart began to beat the faster, and the
moil of the day’s march dwindled before the influx of a rosier idyl.
Every step towards Pelleas seemed to take her higher up the turret
stair of love till her lips should meet those that bent at last from the
gloom to hers. Pride and vexation lay fallen far below, dropped
incontinently like a ragged cloak; a more generous passion shone
out like cloth of gold; she was no longer weary. Her eyes were very
bright, her face full of a splendid wistfulness, as she neared the man
under the trees, looking up to see his face.
Twilight lay deep violet under the wooelshawe, while horse and man
were dim and impalpable, great shadows of themselves. Igraine
could not see the man’s face for the mask over the mezail of his
helmet, and he was silent as death. She was quite close to him now
and ready to speak his name, when he wheeled suddenly, looked
back at her, and pointed into the wood with his long spear. She ran
forward and would have taken hold of his bridle, but he waved her
back and slanted his spear at her in mute warning. Igraine, heart-
hungry, could hold herself no longer.
“Man—man, are you stone?”
He rode straight ahead into the night of the trees and said never a
word. Igraine drew her breath.
“Pelleas.”
“Ah, Igraine.”
The voice that came to her was muffled like the voice of a mourner,
yet the girl thought she caught the old deep tone of it like the low
cry of the wind.
“Why do you vex me?”
“Follow!”
“Pelleas, Pelleas, I am no nun!”
“Follow!”
“I kept this truth from you too long.”
“Follow!”
“Pelleas, would you hurt my heart more?”
“Follow; God shall make all plain and good.”
She gave in with a half-sob, and bent quietly to the man’s mood,
though she had no notion what he purposed in his heart, or what his
desires were in mystifying her thus. No doubt it would be well in the
end if Pelleas bade her follow like a penitent and promised ultimate
peace. At least he had not turned her away, and she trusted him to
the death. He was a strong, deep-sensed soul, she knew, and her
deceiving may have made him bitter in measure, and not easily
appeased. In this queer trial of endurance, this tempting of her
temper, she thought she read a penance laid upon her by the man
for the way she had used his love.
They were soon far into the wood, with the western sky dwindling
between the innumerable pillars of the trees. It began to be dark
and utterly silent save for the rustle of the dead leaves as they went,
and the shrilling chafe of bridle or scabbard, or the snort of the great
horse. Wherever the eye turned the forest piers stood straight and
solemn as the columns in a hypostyle hall in some Egyptian temple.
The fretwork of boughs roofed them in with hardly a glimmering
through of the darkening sky above. There was a pungent autumn
scent on the air that seemed to rise like the incense of years that
had fallen to decay on the brown flooring of the place, and there
was no breath or vestige of a wind.
Presently as the day died the wood went black as the winter night,
and Igraine kept close by the man, with his armour giving a dull
gleam now and again to guide her. They were passing up what
seemed to be a great arcade cut through the very heart of the
wood, as though leading to some shrine or altar, relic of Druid days,
or times yet more antique. The tunnel ran a curved course, bending
deeper and deeper as it went into the dense horde of trees. So dark
was the wood that it was possible to see but a few paces in
advance, and Igraine wondered how the man kept the track. She
was close at his stirrup now, with the dark mass of him and his
horse rising above her like a statue in black basalt. Though he never
spoke to her, and though she touched no part of him, his horse, or
his harness, she felt content with the queer sense of trust and
proximity that pervaded her. There was magic in the mere
companionship. As she had humbled her will to Pelleas’s the night
when he had taken her from the beech tree in Andredswold, so now
in like fashion she surrendered pride and liberty, and became a child.
Suddenly the trackway straightened out into a great colonnade that
ran due south between trees of yet vaster girth. Igraine felt the man
rein in and abide motionless beside her as she held to the stirrup
and waited for what next should chance. Silence seemed like depths
of black water over them, and they could hear each other take
breath like the faint flux and reflux of a sea. Igraine saw the man lift
his spear, a dim streak less black than the vault above, and hold it as
a sign for her to listen. Her blood began to tingle a very little. There
was something far away on the dead, stagnant air, a sort of swirl of
sound, shrill and harmonious, like a wind playing through the strings
of a harp. It was very gradual, very impalpable. As the volume of it
grew it seemed to rush nearer like a wind, to swell into a swaying
plaintive song smitten through with the wounded cry of flutes. It
gave a notion of wood-fays dancing, of whirling wings and flitting
gossamer moonbright in the shadows. Igraine’s blood seemed to
spin the faster, and her hand left the stirrup and touched the man’s
thigh. He gave never a word or sign in the dark. She spoke to him
very softly, very meekly.
“What place is this, Pelleas?”
She saw him bend slightly in the saddle.
“It is called the Ghost Forest,” he said.
“What are the sounds we hear?”
“Who can tell!”
Igraine had hardly heard him, when a streak of phosphor light
flickered among the trees, coming and going incessantly as the great
trunks intervened. It neared them in gradual fashion, and then
blazed out sudden into the open aisle, a man in armour riding on a
great white horse, his harness white as the moon, his face pale and
wide-eyed, his hair like a mass of twisted silver wire. A misty glow
haloed him round, and though he rode close there seemed no sound
at all to mark his passing. As he had come, so he went, with streaks
of flickering light that waxed less and less frequent till they died in
the dark, and left the place empty as before. Igraine thought the air
cold when he had gone.
She felt the black horse move beside her, and they went on as
before into the night of the trees. The noise of flute and harp that
had ceased awhile bubbled up again quite near, so that it was no
longer the ghost of a sound, but noise more definite, more discrete.
It had a queer way of dying to a sighing breath, and then gathering
gradually into an ascending burst of windy melody. Igraine could
almost fancy that she heard the sweep of wings, the soft thrill of
silks trailing through the trees, yet the man on the horse said never
a word as they went on like a pair of mutes to a grave.
The colonnade opened out abruptly on a great circular clearing in
the wood shut in by crowded trunks, its open vault above cut by a
dense ring of foliage. A grey light came down from the sky, showing
great stones piled one upon another, others fallen and sunk deep in
rank grass and brambles. The man halted his horse in the very
centre of the clearing, with Igraine beside him, watchful for what
should happen, and for the moment when Pelleas should unbend.
Hardly had she looked over the great cromlechs, black and sinister in
that solitary wilderness, than the whole wood about them seemed
dusted suddenly with points of fire. North, south, east, and west
torches and cressets came jerking redly out of the night, flitting
behind the trees in a wide circle, gathering nearer and nearer
without a sound. They might have been great fireflies playing
through the aisles and ways, or goblin lamps carried by fairy folk.
Igraine drew very close to the man’s horse for comfort, and looked
up to see his face, but found it dark and hidden. Her hand crept up
past the horse’s neck, rested on the mane a moment, and ventured
yet further to meet the man’s hand, where it gripped the bridle. For
a minute they abode thus without a sound, watching the weird
torch-dance in the wood.
With a sudden gibber of laughter and a swirl of pipes the throng of
lights seemed to seethe to the very margin of the clearing. Queer
phantastic shapes showed amid the trees, and the great circle grew
wide with light, and the grey cromlechs surprised in sleep by the
glare and piping. At that very moment Igraine had a thought of
some one looking deep into her eyes, of a will, a power, streaming in
upon her like sunlight into a sleepy pool. Her desire went from the
man on the black horse into the square shadow of the great central
cromlech, where an indefinite influence seemed to lurk. Looking long
under the roofing stone, she grew aware of a tall something
standing there, of a pair of eyes like the eyes of a panther, of a lean
white hand moving in the shadows.
The eyes under the cromlech seemed to follow Igraine like fire, and
to burn in upon her a foreign influence. Rebellious and wondering,
she stiffened herself against a spiritual combat that seemed moving
upon her out of the dark. She could have smitten the eyes that
stared her down, and yet the magnetic stupor of them kindled up
things in her heart that were strange and newly sensuous. She felt
her strength sway as though her soul were being lifted from her, and
she was warmed from top to toe like one who has taken wine, and
whose being swims into an idyllic glorification of the senses. Again
her desire seemed turned to the man in red harness, yet when she
looked the saddle was empty, and the horse held by an armed
servant, who wore a wolfs head for covering. Still mute with fear,
desire, and wonder, she saw a tall figure move into the full glare of
the torches, a figure in red harness with a shield of green, and a red
dragon thereon, and with head unhelmed. The armour was like the
armour of Pelleas, but the face was the face of the man Gorlois.
And now the eyes under the shadow of the cromlech were full and
strong upon Igraine. Breathing fast with a hand at her throat she
stepped back from Gorlois—hesitated—stood still. She was very
white, and her eyes were big and sightless like the eyes of one
walking in a dream. For all her strength, her scorn, and the tricking
of her heart, she was being swept like a cloud into the embraces of
the sun. Reason, power, love, sank away and became as nothing. A
shudder passed over her. Presently her hands dropped limp as
broken wings, and her body began to sway like a tall lily in a breeze.
A gradual stupor saw her cataleptic; she stood impotent, played
upon by the promptings of another soul.
Gorlois went near to her with hands outstretched, stooping to look
into her face. A sudden light kindled in her eyes, her lips parted, and
new life flooded red into her cheeks as at the beck of love. She bent
to Gorlois full of a gracious eagerness, a wistful desire that made her
face golden as dawn. Her hand sought his, while the shadowy shape
under the cromlech watched them with never-wavering eyes.
Gorlois’s arms were round her now all wreathed in her hair; her face
was turned to his; her hands were clasped upon his neck. Another
moment and he had touched her lips with his.
A sound of flutes, the tinkling of a bell, and a solemn company came
threading from the trees, guests, acolytes, torch-bearers, in
glittering cloth of gold, with a great crucifix to lead them. Gorlois
and Igraine were hand in hand near the stone that hid the frame of
Merlin. A priest in a gorgeous cape drew near, and began his patter.
The vows were taken, the pact sealed, with the noise of a chant and
music. Thus under the benedictions of the great trees, and the spell
of Merlin, Gorlois and Igraine were made man and wife.
BOOK III
THE WAR IN WALES
I
Aurelius Ambrosius the king was dead, taken off in Winchester by
the hand of a poisoner. He had been found stark and cold in his
great carved bed, with an empty wine-cup beside him, and a tress of
black hair and a tress of yellow laid twined together upon his lips.
The signet-ring had gone from his finger, and by the bed had been
discovered a woman’s embroidered shoe dropped under the folds of
the purple quilt. The truth, sinister enough in its bare suggestions,
was glossed over by the court folk out of honour to Aurelius, and of
love to Uther the king’s brother. It was told to the country how an
Irish monk sent by Pascentius, dead Vortigern’s son, had gained
audience of the king, and treacherously poisoned him as he drank
wine at supper. The tale went out to the world, and was believed of
many with a sincere and honest faith. Yet a certain child-eyed
woman, wandering on the shores of Wales for sight of Irish ships,
could have spoken more of the truth had she so dared.
Uther Pendragon had been hailed king at York before the bristling
spears of a victorious host. But a week before he had marched
against the heathen on the Humber, and overthrown them with such
slaughter as had not been seen in Britain since the days when
Boadicea smote the Romans. At the head of his men he had
marched south in a snowstorm to be thundered into Winchester as
king and conqueror. Twelve maidens of noble blood, clad in ermine
and minever, had run before him with boughs of mistletoe and bay.
Five hundred knights had walked bareheaded, with swords drawn,
behind his horse. The city had glistened in a white web of frosted
samite, sparkled over by the clear visage of a winter sun.
There were many great labours ready to the king’s hand. Britain lay
bruised by the onslaughts of the barbarians; her monks had been
slain, her churches desecrated. The pirate ships swept the seas, and
poured torch and sword along the sunny shores of the south.
Andredswold, dark, saturnine, mysterious, alone waved them back
with the sepulchral threatening of its trees. Yet, for all the burden of
the kingdom upon his broad shoulders, Uther gave his first care to
the honouring of the dead. Aurelius Ambrosius was buried with great
pomp of churchmen and nobles at Stonehenge, and a royal mound
raised above the tomb. At Christmastide, with snow upon the
ground, a great gathering was made at Sarum of all the petty kings,
princes, and nobles of the land. Hither came Meliograunt, king of
Cornwall, and Urience of the land of Gore. Fealty was sworn with
solemn ordinance to Uther Pendragon the king, and common league
bonded against the heathen and the whelps of the north.
There were other perils brewing for Britain over the sea. Pascentius,
dead Vortigern’s son, had been an outcast and a wanderer since the
days when the sons of Constantine had sailed from Armorica to save
the land from the blind lust and treason of his father. He had been a
drifting fire beyond the seas, an intriguer, a sower of sedition, a man
dangerous alike to friend and foe. Beaten like a vulture from the
coasts of Britain, he had turned with treasonable hope to Ireland
and its king, Gilomannius the Black, a strenuous potentate, boasting
little love for Ambrosius the king. Here, in Ireland, a kennel of
sedition had arisen. Pascentius, keen, hungry plotter, had toiled at
the task of piling enmity against those who had destroyed his father
amid the flames of Genorium. A great league arose, a banding of the
barbarians with the Irish princes, a union of the Saxons who ravaged
Kent with the wild tribesmen over the northern border. Month by
month a great host gathered on the Irish coast. Many ships came
from the east and from the south. Mid-winter was past before
Gilomannius embarked, and, setting sail with a fair wind, turned the
beaks of his galleys for the shores of Wales.
Noise of the gathering storm had been brought to Uther as he
journed through the southern coasts, rebuilding the churches,
recovering abbey and hermitage from their desolate ashes. His zeal
was great for God, and his love of Britain well-nigh as noble. Warned
thus in due season, he marched for the west, calling the land to
arms, assigning for the gathering of the host Caerleon upon Usk,
that fair city bosomed in the fulness of its woods and pastures. Many
a knight had answered to his call; many a city had sent out her
companies; the high-roads rang with the cry of steel in the crisp
winter weather.
Duke Gorlois had come from Cornwall from his castle of Tintagel,
bringing many knights and men-at-arms by sea, and the Lady
Igraine his wife, in a great galley whose bulwarks glistened with
shields. In Caerleon Gorlois had a house built of white stone, set
upon a little hill in the centre of the city. To Caerleon he brought this
golden falcon of a woman, this untamable thing that he had kept
prisoned in the high towers of Tintagel. He mewed her up like a nun
in his house of white stone, so that no man should see the fairness
of her face. She was wild as an eyas from the woods, fierce and
unapproachable, and sharp of claw. Robbed of her liberty, had she
not sought to take her own life with a sword, and to throw herself
from the battlements of Tintagel? Gorlois had won little love by
Merlin’s subtlety, and he feared the woman’s beauty and the spell of
her large eyes.
It was the month of February and clear crisp weather. The white
bellies of the Irish sails had shown up against the grey blue stretch
of the sea, a white multitude of canvas that had sent the herdsmen
hurrying their flocks to the mountains. Horsemen had galloped for
Caerleon, and the cry of war went up over wood and water. Flames
licked the night sky. From Caerleon to St. Davids, from St. Davids to
Eryri, the red blaze of beacon-fires told of the ships at sea.
The cry of the storm arose in Caerleon, and the tramp of armed men
sounded all day in her streets. The great host lodged about the city
broke camp and streamed westwards along the high-road into
Wales. Bugles blew, banners flapped, masses of sullen steel rolled
away into purple of the winter woods. Bristling spears and lines of
skin-clad shields vanished into the west like the waves of a solemn
sea. On the walls of Caerleon stood many women and children
watching the host march for the west, watching Uther the king ride
out with his great company of knights and nobles.
At the casement of an upper room in Gorlois’s house stood a woman
looking out over Caerleon towards the sea. She was clad in a mantle
of furs, and in a tunic of purple linked up with cord of gold. A tippet
of white fur clasped with a brooch of amethysts circled her throat.
Her hair was bound up in a net of fine silk, and there was a girdle of
blue silk about her loins, and an enamelled cross upon her bosom.
She stood with her elbows resting on the stone sill, and her peevish
face clasped between her hands. Her eyes looked very large and
lustrous as she stared out wistfully over the city.
In the great court below horses champed the bit and struck fire from
the ringing flags. Men in armour clanged to and fro; rough voices
cried questions and counter-questions; bridles jingled; spear-shafts
clattered on the stones. Now a clarion blared as a troop of horse
thundered by up the street, their armour gleaming dully past the
courtyard gate. The growl of war hung heavy over Caerleon, a grim
sullen sound that seemed in keeping with the restless chiding of the
wind.
Igraine’s face was hard as stone as she watched the men moving in
the courtyard below. She looked older than of yore, whiter, thinner in
cheek and neck, her great eyes staunch though sad under her
netted hair. Her face showed melancholy mingled with a constant
scorn that had rarely found expression with her in the old days, save
within the walls of Avangel. She looked like one who had endured
much, suffered much, yet lost no whit of pride in the trial. Though
she may have been blemished like a Greek vase smitten by some
barbaric sword, she was her self still, brave, headstrong, resolute as
ever. The shame of the things she had suffered had perhaps wiped
out the gentler outlines of her character and left her more stern,
more wary, less honest, more deep in her endeavours. There was no
passive humility or patience about her soul, and she was the falcon
still, though caged and guarded beyond her liberty.
As she stood at the casement with the prophetic murmur of war in
her ears, it seemed to her as though life surged to her feet and
mocked her bondage like laughing water. The desire of liberty abode
ever with her even to the welcoming of stagnant death. She thirsted
for her freedom, plotted for it, dreamt of it with a zeal that was
almost ferocious. Her life seemed a speculation, a perpetual
aspiration after a state that still eluded her. In the Avangel days she
had been wild and petulant. Then Pelleas had come through the
green gloom of early summer to soften her soul and inspire all the
best breath of the woman in her. Again, thanks to Gorlois, she had
fallen with the usual reaction of circumstance upon evil times; the
change had discovered the peevish discontent of the girl hardened
into the strong wilfulness of the woman.
She hated Gorlois with a fanatical immensity of soul. When the man
was near her she felt full of the creeping nausea of a great loathing,
and she waxed faint with hate at the veriest touch of his hands. His
breath seemed to her more unsavoury than the miasma of a gutter,
and it needed but the sound of his voice to bring all her baser
passions braying and yelping against him. He had driven the
religious instinct out of her heart, and she was in revolt against
heaven and the marriage pact forged by the authority of the Church.
She had often vowed in her heart that she could do no sin against
Gorlois, her husband. He had no claim upon her conscience. The
bondage had been of his making; let God judge her if she scorned
his honour.
Standing by the window watching the knights saddling for their
lord’s sally, she heard heavy footsteps mounting up the stairs, and
the ring of steel-tipped shoes along the gallery. The footsteps were
deliberate, and none too fast, as though the man walked under a
burden of thought. A shadow seemed to pass over Igraine’s face.
She slipped from the window, ran across the room, shot the bolt of
the door, and stood listening. A hand tried the latch. She knew well
enough whose fist it was that rattled on the oaken panels. Her face
hardened to a kind of cold malevolence, and she laughed noiselessly
in her sleeve.
A terse summons came to her from the gallery.
“Wife, we ride at once.”
The man could not have made a worse beginning. There was a
suggestion of tyranny in a particular word that was hardly
temperate. Igraine leant against the door; she was still smiling to
herself, and her hands fingered the embroidered tassel of the latch.
“We are late on the road; I can make no tarrying.”
The door quivered a moment as though shaken by a gusty wind.
Everything was quiet again, and Igraine could hear the man
breathing. Putting her mouth to the crack between post and hinge-
board she laughed stridently as though in scorn.
“Igraine!”
The voice was half-imperative, half-appealing.
“My very dear lord!”
“Are you abed?”
“No, dear lord.”
“Open to me; I would kiss your lips before I sally.”
“You have never kissed me these many days.”
“True, wife; is it fault of mine?”
“Nor shall again, dear lord, if I have strength.”
She heard the man muttering to himself a moment, but this time
there was no smiting of the door, no fume and tempest. His mood
seemed more temperate, less masterful, as though he were half
heavy at heart.
“Igraine—”
“Why do you whimper like a dog?” she said; “go, get you to war.
What are you to me?”
“When will you learn reason?”
“When you are dead, sire.”
“Perhaps I deserve all this.”
“Are you so much a penitent?”
Her mockery seemed to lift Gorlois to a higher range of passion, and
there was great bitterness in his voice as he tossed back words to
her with a quick kindling of desire.
“Woman, I have been hard in the winning of you, but, God knows,
you are something to me.”
“God knows, Gorlois, I hate you.”
His hand shook the door.
“Let me in, Igraine.”
“Break down the door; you shall come at me no other way.”
“Woman, woman, I am a fool; my heart smarts at leaving you.”
“You sound almost saintly.”
“I have left Brastias in charge of you.”
“Thanks, lord, for a jailer.”
Igraine drew back from the door and stood at her full height with
her hands crossed upon her bosom. She quivered as she stood with
the intense effort of her hate. Gorlois still waited without the door,
though she could not hear him moving. The silence seemed like the
deep hush that falls between the blustering stanzas of a storm.
“Igraine!”
It was a hoarse cry, quick and querulous. Igraine had both her fists
to her chin in an attitude of inward effort, as though she racked
herself to give utterance to the implacable temper of her scorn. Her
face had a queer parched look. When she spoke, her voice was shrill
like a piping wind.
“Gorlois.”
“Wife.”
“Would you have my blessing?”
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Federated Learning Over Wireless Edge Networks Wei Yang Bryan Lim

  • 1. Federated Learning Over Wireless Edge Networks Wei Yang Bryan Lim download https://ebookbell.com/product/federated-learning-over-wireless- edge-networks-wei-yang-bryan-lim-46364756 Explore and download more ebooks at ebookbell.com
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  • 5. Wireless Networks WeiYang Bryan Lim · Jer Shyuan Ng · Zehui Xiong · Dusit Niyato · Chunyan Miao Federated Learning Over Wireless Edge Networks
  • 6. Wireless Networks Series Editor Xuemin Sherman Shen, University of Waterloo, Waterloo, ON, Canada
  • 7. The purpose of Springer’s Wireless Networks book series is to establish the state of the art and set the course for future research and development in wireless communication networks. The scope of this series includes not only all aspects of wireless networks (including cellular networks, WiFi, sensor networks, and vehicular networks), but related areas such as cloud computing and big data. The series serves as a central source of references for wireless networks research and development. It aims to publish thorough and cohesive overviews on specific topics in wireless networks, as well as works that are larger in scope than survey articles and that contain more detailed background information. The series also provides coverage of advanced and timely topics worthy of monographs, contributed volumes, textbooks and handbooks. ** Indexing: Wireless Networks is indexed in EBSCO databases and DPLB **
  • 8. Wei Yang Bryan Lim • Jer Shyuan Ng • Zehui Xiong • Dusit Niyato • Chunyan Miao Federated Learning Over Wireless Edge Networks
  • 9. Wei Yang Bryan Lim Alibaba-NTU Joint Research Institute Singapore, Singapore Jer Shyuan Ng Alibaba-NTU Joint Research Institute Singapore, Singapore Zehui Xiong Singapore University of Technology and Design Singapore, Singapore Dusit Niyato School of Computer Science and Engineering Nanyang Technological University Singapore, Singapore Chunyan Miao School of Computer Science and Engineering Nanyang Technological University Singapore, Singapore ISSN 2366-1186 ISSN 2366-1445 (electronic) Wireless Networks ISBN 978-3-031-07837-8 ISBN 978-3-031-07838-5 (eBook) https://doi.org/10.1007/978-3-031-07838-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
  • 10. Preface The confluence of edge computing and artificial intelligence (AI) has driven the rise of edge intelligence, which leverages the storage, communication, and computation capabilities of end devices and edge servers to empower AI implementation at scale closer to where data is generated. An enabling technology of edge intelligence is the privacy-preserving machine learning paradigm known as federated learning (FL). Amid the increasingly stringent privacy regulations, FL will enable the development of applications that have to be built using sensitive user data and will continue to revolutionize service delivery in finance, Internet of Things (IoT), healthcare, and transport industries, among others. However, the implementation of FL is envisioned to involve thousands of heterogeneous distributed end devices that differ in terms of communication and computation resources, as well as the levels of willingness to participate in the collaborative model training process. The potential node failures, device dropouts, and stragglers effect are key bottlenecks that impede the effective, sustainable, and scalable implementation of FL. In Chap. 1, we will first present a tutorial and survey on FL and highlight its role in enabling edge intelligence. This tutorial and survey provide readers with a comprehensive introduction to the forefront challenges and state-of-the-art approaches towards implementing FL at the edge. In consideration of resource heterogeneity at the edge networks, we then provide multifaceted solutions for- mulated via the interdisciplinary interplay of concepts derived from network economics, optimization, game theory, and machine learning towards improving the efficiency of resource allocation for implementing FL at scale amid information asymmetry. In Chap. 2, we devise a multi-dimensional contract-matching approach for optimized resource allocation for federated sensing and learning amid multi- dimensional sources of heterogeneities. In Chap. 3, we propose a joint-auction coalition formation framework towards facilitating communication-efficient FL networks aided by unmanned aerial vehicles (UAVs). In Chap. 4, we propose a two-level evolutionary game theoretic and auction approach to allocate and price resources to facilitate efficient edge intelligence. In Chap. 5, we recap the key points and discuss the promising research directions for future works. v
  • 11. vi Preface We sincerely thank our collaborators for their contributions to the presented research works. Special thanks also go to the staff at Springer Nature for their help throughout the publication preparation process. Finally, we would like to take the chance to dedicate this book to celebrate the birth of Lim Chen Huan Theodore, son of Dr. Lim Wei Yang Bryan and Foo Feng Lin. Singapore, Singapore Wei Yang Bryan Lim Singapore, Singapore Jer Shyuan Ng Singapore, Singapore Zehui Xiong Singapore, Singapore Dusit Niyato Singapore, Singapore Chunyan Miao
  • 12. Contents 1 Federated Learning at Mobile Edge Networks: A Tutorial ............. 1 1.1 Introduction ............................................................. 1 1.2 Background and Fundamentals of Federated Learning ............... 6 1.2.1 Federated Learning ............................................. 6 1.2.2 Statistical Challenges of FL .................................... 8 1.2.3 FL Protocols and Frameworks ................................. 11 1.2.4 Unique Characteristics and Issues of FL ...................... 12 1.3 Communication Cost ................................................... 12 1.3.1 Edge and End Computation .................................... 13 1.3.2 Model Compression ............................................ 16 1.3.3 Importance-Based Updating.................................... 17 1.4 Resource Allocation .................................................... 18 1.4.1 Worker Selection................................................ 21 1.4.2 Joint Radio and Computation Resource Management ........ 25 1.4.3 Adaptive Aggregation .......................................... 27 1.4.4 Incentive Mechanism ........................................... 28 1.5 Privacy and Security Issues ............................................. 32 1.5.1 Privacy Issues ................................................... 33 1.5.2 Security Issues .................................................. 37 1.6 Applications of Federated Learning for Mobile Edge Computing.... 39 1.6.1 Cyberattack Detection .......................................... 42 1.6.2 Edge Caching and Computation Offloading................... 44 1.6.3 Base Station Association ....................................... 47 1.6.4 Vehicular Networks ............................................. 48 1.7 Conclusion and Chapter Discussion.................................... 50 2 Multi-dimensional Contract Matching Design for Federated Learning in UAV Networks ................................................. 53 2.1 Introduction ............................................................. 53 2.2 System Model and Problem Formulation .............................. 56 2.2.1 UAV Sensing Model ............................................ 58 2.2.2 UAV Computation Model ...................................... 59 vii
  • 13. viii Contents 2.2.3 UAV Transmission Model ...................................... 60 2.2.4 UAV and Model Owner Utility Modeling ..................... 61 2.3 Multi-dimensional Contract Design.................................... 61 2.3.1 Contract Condition Analysis ................................... 62 2.3.2 Conversion into a Single-Dimensional Contract .............. 63 2.3.3 Conditions for Contract Feasibility ............................ 64 2.3.4 Contract Optimality............................................. 68 2.4 UAV-Subregion Assignment............................................ 70 2.4.1 Matching Rules ................................................. 71 2.4.2 Matching Implementation and Algorithm ..................... 72 2.5 Performance Evaluation ................................................ 73 2.5.1 Contract Optimality............................................. 73 2.5.2 UAV-Subregion Preference Analysis .......................... 76 2.5.3 Matching-Based UAV-Subregion Assignment ................ 78 2.6 Conclusion and Chapter Discussion.................................... 80 3 Joint Auction–Coalition Formation Framework for UAV-Assisted Communication-Efficient Federated Learning........... 83 3.1 Introduction ............................................................. 83 3.2 System Model ........................................................... 86 3.2.1 Worker Selection................................................ 88 3.2.2 UAV Energy Model............................................. 90 3.3 Coalitions of UAVs ..................................................... 93 3.3.1 Coalition Game Formulation ................................... 94 3.3.2 Coalition Formation Algorithm ................................ 97 3.4 Auction Design.......................................................... 98 3.4.1 Buyers’ Bids .................................................... 99 3.4.2 Sellers’ Problem ................................................ 100 3.4.3 Analysis of the Auction ........................................ 102 3.4.4 Complexity of the Joint Auction–Coalition Algorithm ....... 104 3.5 Simulation Results and Analysis ....................................... 105 3.5.1 Communication Efficiency in FL Network .................... 106 3.5.2 Preference of Cells of Workers................................. 107 3.5.3 Profit-Maximizing Behavior of UAVs ......................... 109 3.5.4 Allocation of UAVs to Cells of Workers....................... 111 3.5.5 Comparison with Existing Schemes ........................... 114 3.6 Conclusion and Chapter Discussion.................................... 115 4 Evolutionary Edge Association and Auction in Hierarchical Federated Learning .......................................................... 117 4.1 Introduction ............................................................. 117 4.2 System Model and Problem Formulation .............................. 120 4.2.1 System Model................................................... 120 4.2.2 Lower-Level Evolutionary Game .............................. 121 4.2.3 Upper-Level Deep Learning Based Auction................... 121 4.3 Lower-Level Evolutionary Game ...................................... 122 4.3.1 Evolutionary Game Formulation............................... 122
  • 14. Contents ix 4.3.2 Worker Utility and Replicator Dynamics ...................... 123 4.3.3 Existence, Uniqueness, and Stability of the Evolutionary Equilibrium ...................................... 125 4.4 Deep Learning Based Auction for Valuation of Cluster Head ........ 128 4.4.1 Auction Formulation............................................ 128 4.4.2 Deep Learning Based Auction for Valuation of Cluster Heads ................................................... 130 4.4.3 Monotone Transform Functions................................ 132 4.4.4 Allocation Rule ................................................. 133 4.4.5 Conditional Payment Rule ..................................... 134 4.4.6 Neural Network Training ....................................... 134 4.5 Performance Evaluation ................................................ 135 4.5.1 Lower-Level Evolutionary Game .............................. 136 4.5.2 Upper-Level Deep Learning Based Auction................... 140 4.6 Conclusion and Chapter Discussion.................................... 143 5 Conclusion and Future Works.............................................. 147 References......................................................................... 151 Index............................................................................... 165
  • 15. List of Figures Fig. 1.1 Edge AI approach brings AI processing closer to where data are produced. In particular, FL allows training on devices where the data are produced ................................ 3 Fig. 1.2 General FL training process involving N workers .................. 7 Fig. 1.3 Approaches to increase computation at edge and end devices include (a) increased computation at end devices, e.g., more passes over dataset before communication, (b) two-stream training with global model as a reference, and (c) intermediate edge server aggregation ........................... 14 Fig. 1.4 Worker selection under the FedCS and Hybrid-FL protocol ............................................................... 22 Fig. 1.5 A comparison between (a) BAA by over-the-air computation that reuses bandwidth (above) and (b) OFDMA (below) that uses only the allocated bandwidth .......... 26 Fig. 1.6 A comparison between (a) synchronous and (b) asynchronous FL ..................................................... 27 Fig. 1.7 Workers with unknown resource constraints maximize their utility only if they choose the bundle that best reflects their constraints .............................................. 30 Fig. 1.8 Selective parameter sharing model .................................. 35 Fig. 1.9 GAN attack on collaborative deep learning ......................... 36 Fig. 1.10 An illustration of (a) conventional FL and (b) the proposed BlockFL architectures ..................................... 40 Fig. 1.11 FL based attack detection architecture for IoT edge networks .............................................................. 43 Fig. 1.12 FL based (a) caching and (b) computation offloading ............. 45 Fig. 2.1 System model involving UAV-subregion contract matching........ 55 Fig. 2.2 UAV node coverage vs. auxiliary types ............................. 75 Fig. 2.3 Contract rewards vs. auxiliary types ................................. 75 Fig. 2.4 Contract items vs. UAV utilities...................................... 75 xi
  • 16. xii List of Figures Fig. 2.5 The model owner profits vs. UAV auxiliary types .................. 76 Fig. 2.6 The UAV utility for each subregion vs. types ....................... 77 Fig. 2.7 UAV matching for homogeneous subregions........................ 78 Fig. 2.8 UAV matching for subregions with different data quantities and coverage area ......................................... 79 Fig. 2.9 UAV matching where J > N ........................................ 79 Fig. 3.1 System model consists of the cloud server (FL model owner), the vehicles and RSUs (selected FL workers), and the UAVs ............................................................. 85 Fig. 3.2 Illustration of the joint auction–coalition formation framework ............................................................ 87 Fig. 3.3 Distributed FL Network with 3 cells and 6 UAVs .................. 105 Fig. 3.4 Communication time needed by UAVs and IoV vehicles .......... 107 Fig. 3.5 Maximum number of iterations under different energy capacities ............................................................. 110 Fig. 3.6 Illustration of merge-and-split mechanism and allocation of UAVs to cells of workers ......................................... 111 Fig. 3.7 Total profit and number of coalitions vs cooperation cost ......... 113 Fig. 3.8 Total profit and size of coalitions vs number of iterations ......... 113 Fig. 3.9 Comparison with existing schemes .................................. 114 Fig. 4.1 An illustration of the hierarchical system model ................... 118 Fig. 4.2 Neural network architecture for the optimal auction ................ 130 Fig. 4.3 Monotone transform functions ....................................... 133 Fig. 4.4 Phase plane of the replicator dynamics .............................. 137 Fig. 4.5 Evolutionary equilibrium of population states for cluster 1 ........ 137 Fig. 4.6 Evolution of population utilities ..................................... 138 Fig. 4.7 Evolution of population states for population 1 .................... 138 Fig. 4.8 Evolution of population states for population 2 .................... 138 Fig. 4.9 Evolution of population states for population 3 .................... 138 Fig. 4.10 Evolutionary dynamics under different learning rates .............. 139 Fig. 4.11 Data coverage vs. varying fixed rewards in cluster 1 ............... 139 Fig. 4.12 Data coverage vs. varying congestion coefficient in cluster 1 ...... 140 Fig. 4.13 Population states in cluster 3 vs. varying population data for population 1 ....................................................... 140 Fig. 4.14 Revenue of cluster head 1 under different distribution of model owners ......................................................... 141 Fig. 4.15 Revenue of cluster head 2 under different distribution of model owners ......................................................... 141 Fig. 4.16 Revenue of cluster head 3 under different distribution of model owners ......................................................... 141 Fig. 4.17 Revenue vs data coverage of cluster heads .......................... 142 Fig. 4.18 Revenue of cluster head 1 under different approximation qualities ............................................................... 143 Fig. 4.19 Revenue of cluster head 2 under different approximation qualities ............................................................... 143
  • 17. List of Figures xiii Fig. 4.20 Revenue of cluster head 3 under different approximation qualities ............................................................... 143
  • 18. List of Tables Table 1.1 An overview of selected surveys in FL and MEC .................. 5 Table 1.2 Loss functions of common ML models ............................. 8 Table 1.3 Approaches to communication cost reduction in FL ............... 19 Table 1.4 Approaches to resource allocation in FL ........................... 31 Table 1.5 The accuracy and attack success rates for no-attack scenario and attacks with 1 and 2 sybils in an FL system with MNIST dataset [1] .............................................. 37 Table 1.6 A summary of attacks and countermeasures in FL ................. 41 Table 1.7 FL based approaches for mobile edge network optimization .......................................................... 42 Table 2.1 Table of commonly used notations .................................. 56 Table 2.2 Table of key simulation parameters.................................. 74 Table 2.3 UAV types for preference analysis................................... 77 Table 2.4 UAV type and preference for subregions ............................ 78 Table 3.1 Table of commonly used notations .................................. 89 Table 3.2 Simulation parameters ............................................... 106 Table 3.3 Preference of cells for different coalitions .......................... 107 Table 3.4 Preference of UAVs for different cells............................... 108 Table 3.5 Preference of cells for individual UAVs (bolded row implies top preference) ............................................... 108 Table 3.6 Revenue, cost of profit for different coalitions in each cell......... 110 Table 4.1 Simulation parameters ............................................... 135 xv
  • 19. Chapter 1 Federated Learning at Mobile Edge Networks: A Tutorial 1.1 Introduction Currently, there are nearly 7 billion connected Internet of Things (IoT) devices and 3 billion smartphones around the world [2]. These devices are equipped with increasingly advanced sensors, computing, and communication capabilities. As such, they can potentially be deployed for various crowdsensing tasks, e.g., for medical purposes [3] and air quality monitoring [4]. Coupled with the rise of Deep Learning (DL) [5], the wealth of data collected by end devices opens up countless possibilities for meaningful research and applications. In the traditional cloud-centric approach, data collected by mobile devices are uploaded and processed centrally in a cloud-based server or data center. In particular, data collected by IoT devices and smartphones such as measurements [6], photos [7], videos [8], and location information [9] are aggregated at the data center [10]. Thereafter, the data are used to provide insights or produce effective inference models. However, this approach is no longer sustainable for the following reasons. Firstly, data owners are increasingly privacy sensitive. Following privacy concerns among consumers due to high profile cases of data leaks and data misuse, policy makers have responded with the implementation of data privacy legislation such as the European Commission’s General Data Protection Regulation (GDPR) [11] and Consumer Privacy Bill of Rights in the USA [12]. In particular, the consent (GDPR Article 6) and data minimalization principle (GDPR Article 5) limit data collection and storage only to what is consumer-consented and absolutely necessary for processing. Secondly, a cloud-centric approach involves long propagation delays and incurs unacceptable latency [13] for applications in which real-time decisions have to be made, e.g., in self-driving car systems [14]. Thirdly, the transfer of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 W. Y. B. Lim et al., Federated Learning Over Wireless Edge Networks, Wireless Networks, https://doi.org/10.1007/978-3-031-07838-5_1 1
  • 20. 2 1 FL at Mobile Edge Networks: A Tutorial raw data to the cloud for processing burdens the backbone networks.1 This is especially so in tasks involving unstructured data, e.g., in video analytics [15]. This is exacerbated by the fact that cloud-centric training is relatively reliant on wireless communications [16]. As a result, this can potentially impede the development of new technologies. With data sources mainly located outside the cloud today, Mobile Edge Com- puting (MEC) has naturally been proposed as a solution. In MEC, the computing and storage capabilities [13] of end devices and edge servers are leveraged to bring model training closer to where data are produced [17]. As defined in [16], an end– edge–cloud computing network comprises (a) end devices, (b) edge nodes, and (c) cloud server. For model training in conventional MEC approaches, a collaborative paradigm has been proposed in which training data are first sent to the edge servers for model training up to lower-level DNN layers, before more computation intensive tasks are offloaded to the cloud [18], [19] (Fig. 1.1). However, this arrangement incurs significant communication costs and is unsuitable for applications that require persistent training [16]. In addition, computation offloading and data processing at edge servers still involve the transmission of potentially sensitive personal data. This can discourage privacy-sensitive consumers from taking part in model training or even violate increasingly stringent privacy laws [11]. Although various privacy preservation methods, e.g., Differential Privacy (DP) [20], have been proposed, a number of users are still not willing to expose their private data for fear that their data may be inspected by external servers. In the long run, this discourages the development of technologies as well as new applications. To guarantee that training data remain on personal devices and to facilitate collaborative machine learning of complex models among distributed devices, a decentralized ML approach called Federated Learning (FL) is introduced in [21]. In FL, mobile devices2 use their local data to cooperatively train an ML model required by an FL server. They then send the model updates, i.e., the model’s weights, to the FL server for aggregation. The steps are repeated in multiple rounds until a desirable accuracy is achieved. This implies that FL can be an enabling technology for ML model training at mobile edge networks. As compared to conventional cloud-centric ML model training approaches, the implementation of FL for model training at mobile edge networks features the following advantages: 1. Minimizing requirement of network bandwidth: Less information is required to be transmitted to the cloud. For example, instead of sending the raw data over for processing, participating devices only send the updated model parameters for 1 Note that in some cases, the communication cost of FL is not insignificant due to complexity of large models, whereas the relatively fewer data samples a worker possesses are less costly to transmit. As such, we discuss communication cost reduction as well in this chapter. 2 We use the term mobile devices and mobile edge in this chapter given that many of the works reviewed have focused on how to implement FL on resource-constrained edge devices such as IoT. However, note that the insights from this chapter can be similarly applied on edge networks in general.
  • 21. 1.1 Introduction 3 Fig. 1.1 Edge AI approach brings AI processing closer to where data are produced. In particular, FL allows training on devices where the data are produced aggregation. As a result, this significantly reduces costs of data communication and relieves the burden on backbone networks. 2. Privacy: Following the above point, the raw data of users need not be sent to the cloud. Under the assumption that FL workers and servers are non-malicious, this
  • 22. 4 1 FL at Mobile Edge Networks: A Tutorial enhances user privacy and reduces the probability of eavesdropping to a certain extent. In fact, with enhanced privacy, more users will be willing to take part in collaborative model training, and so, better inference models can be built. 3. Low latency: With FL, ML models can be consistently trained and updated. Meanwhile, in the MEC paradigm, real-time decisions, e.g., event detection [22], can be made locally at the edge nodes or end devices. Therefore, the latency is much lower than when decisions are made in the cloud before transmitting them to the end devices. This is vital for time critical applications such as self-driving car systems in which the slightest delays can potentially be life threatening [14]. Given the aforementioned advantages, FL has seen recent successes in several applications. For example, the Federated Averaging algorithm (FedAvg) proposed in [21] has been applied to Google’s Gboard [23] to improve next-word prediction models. In addition, several studies have also explored the use of FL in a number of scenarios in which data are sensitive in nature, e.g., to develop predictive models for diagnosis in health AI [24] and to foster collaboration across multiple hospitals [25] and government agencies [26]. Besides being an enabling technology for ML model training at mobile edge networks, FL has also been increasingly applied as an enabling technology for mobile edge network optimization. Given the computation and storage constraints of increasingly complex mobile edge networks, conventional network optimization approaches that are built on static models fare relatively poorly in modeling dynamic networks [16]. As such, a data-driven Deep Learning (DL) based approach [27] for optimizing resource allocation is increasingly popular. For example, DL can be used for representation learning of network conditions [28], whereas Deep Reinforcement Learning (DRL) can optimize decision making through interactions with the dynamic environment [29]. However, the aforementioned approaches require user data as an input and these data may be sensitive or inaccessible in nature due to regulatory constraints. As such, in this chapter, we also briefly discuss FL’s potential to function as an enabling technology for optimizing wireless edge networks, e.g., in cell association [30], computation offloading [2], and vehicular networks [31]. However, there are several challenges to be solved before FL can be implemented at scale. Firstly, even though raw data no longer need to be sent to the cloud servers, communication costs remain an issue due to the high dimensionality of model updates and limited communication bandwidth of participating mobile devices. In particular, state-of-the-art DNN model training can involve the communication of millions of parameters for aggregation. Secondly, in a large and complex mobile edge network, the heterogeneity of participating devices in terms of data quality, computation power, and willingness to participate has to be well managed from the resource allocation perspective. Thirdly, FL does not guarantee privacy in the presence of malicious workers or aggregating servers. In particular, recent research works have clearly shown that a malicious worker may exist in FL and can infer the information of other workers just from the shared parameters alone. As such, privacy and security issues in FL still need to be considered.
  • 23. Random documents with unrelated content Scribd suggests to you:
  • 24. gown was of black velvet, twined all about with serpent scrolls of white cloth. On his breast was brooched a great diamond that blazed and wavered back the glow from the fire. Gorlois sat in his carved chair stiff as any image. His strenuous soul seemed mewed up by the psychic influence of the man before him. He spoke seldom, and then only at the other’s motion—at a curious gesture of one of those long, lean hands. The room was as silent as the burial hall of a pyramid, and it had the air of being massed above by stupendous depths of stone. Presently the man in the black robe began to speak with deliberate intent, holding his voice deep in his throat so that it sounded much like the voice of an oracle declaring itself in the noise of a wind. “The woman is beautiful beyond other women.” “Like a golden May.” “And true.” “As a sapphire.” “Yet will not have you.” “Not a shred of me.” The man with the rings smiled out of his impenetrable eyes, and fingered the brooch on his breast. “The woman has great destiny before her.” “Ah!” “I have seen her star in the night. You dare take her fate on you?” “Like ivy holds a tree.” “As a wife?” Gorlois laughed. “How else?” “As a wife—by the church.”
  • 25. “Ah!” “Or no help of my hand.” Again there was silence. A coal fell in the brazier, and seemed like a rock down a precipice. The black eyes that stared down Gorlois were full of light, and strangely scintillant. Gorlois listened, with his limbs asleep and his brain in thrall, while the man spoke like a very Michael out of a cloud. The clear glittering plot given out of Merlin’s lips came like a dream vivid to the thought of the dreamer. If Gorlois obeyed he should have his desire, and catch Igraine to a white marriage-bed by law and her own willing. The fire died down in the brazier, and the bowl ceased to smoke perfumes. Gorlois saw the man gather his black robe with his glittering fingers, and move like a wraith round the room, to stand beckoning by the door. In another minute Gorlois was under the stars, with the house and its yews a black mound against the sky. Like a sleeper half wakened he took full breath of the night air, and stretched his arms up above his head. But it was not to sleep that he passed back through the void streets to the house of the knight Accolon. To return to Igraine housed for the night in the little hermitage. At the first creep of dawn, when daffodils were thrown up against the eastern sky, she left her pallet bed in the cell and went out into the hermit’s garden. The recluse was down at the brook drawing water, whither the dog and the doves had followed him. Igraine passed through the garden—spun over as it was with webs of dew. To her comfort she found her ankle scarcely troubling her, for she had feared pain or stiffness after the walk of yesterday. Going down the dale, she patted the old dog’s head, and picked up the pitcher as the recluse gave her good-morning. “You are an early soul, sister. My dog and I come down to the brook each morning as the sun peeps over the hill.” “You are not lonely,” said Igraine. The old man tightened his girdle, looked over the solemn piers of the woods, sniffed the air, and hailed an autumn savour.
  • 26. “Not I,” he said. “I have my dog and my doves, and folk often lodge here, and I have word of the world and how the Saxons vex us. The good people near bring me alms and pittances, or come to ask prayers for their souls, and”—with a twinkle—“for their bodies, too.” Igraine remembered the peasant’s little son. “Was it you,” she said, “who gave a peasant fellow near here a saint’s dust to scatter over a sick child?” The old man shook his head and smiled enigmatically. “I have no dealings in such marvels,” he said. “The boy died.” “Of course.” “They will sell your dust some day.” A keen look, cynical with beaming scorn, spread over the man’s gaunt face. “Much good may it do them,” he said; “death is monstrous flatterer of mere clay. I may feed a rose bush with my bones; a better fate than the cheating of superstitious women.” He made a sign with his hand, and the birds went wheeling in circles above him. The dog crept up and thrust his snout into the old man’s palm. The garden lay above them, ripe with an autumn mellowness; yet there was no regret though winter would soon be piping, and the man’s hair was grey. “What think you of life?” said Igraine. “You should know, sister, as well as I.” “But you see, father, I am not a nun,—only a novice.” He stared at her a moment with a slight smile. “Remain a novice,” he said. “You advise me so!”
  • 27. “Why subordinate your soul to chains forged of men.” “These seem strange words.” He patted his dog’s head, and, half stooping, looked at her with keen grey eyes. “Have you ever loved a man?” “Yes,” she said, with a clear laugh and a slight colour. “Is he worthy?” “I believe him a noble soul.” “Naturally.” “He ran away and left me because he thought I was a nun.” The hermit applauded. “That sounds like honour,” he said critically. “I am seeking him to tell him the truth.” “And I will pray that you may soon meet,” said the old man, "for there is nothing like the love of a good man for a clean maid. If I had married a true woman, I should never have taken to the scourge or the stone bed. Marry wisely and you are halfway to Heaven." They broke fast that morning in the garden, it being the man’s custom to make his meals on the granite slab that served him as a bed. The little dale looked very green and gracious in the tranquil light, with its curling brook and dark barriers of trees. Igraine, as she sat on the great stone and ate the hermit’s bread, followed the brook with her thoughts, wondering whether it became the stream that ran through Eudol’s meadows. She was for Sarum that day, where she would throw off her grey habit and take some dress more likely to baffle Gorlois. She had enough money in her purse. Worldling again, she could give herself to winning sight of this Uther, and to learning whether he was the Pelleas she sought or no.
  • 28. As she sat and fingered her bread, something she saw down the dale made her rigid and still as a priestess smitten with the vision of a god in some heathen oratory. Her eyes were very wide, her lips open and very white, her whole air as of one watching in a sudden stupor of awe. Another moment and she had broken from the mood like a torrent from a cavern. With eyes suddenly amber bright, she touched the hermit’s hand and pointed down the dale, gave him a word or so, then left him and ran down the hill. A man on a black horse had ridden out from the trees, and was pushing his horse over the brook at a shallow spot not far away. His red armour glowed in the sun with a metallic lustre. Even at that distance Igraine had seen the red dragon rampant on a shield of green. As she ran down the grass slope she called the man by name, thinking to see him turn and come to her. Pushing on sullenly as though he had not heard the cry that went after him like winged love, he drew up the further slope without wavering, and sank like a red streak into the dense green of the trees.
  • 29. X Igraine forded the brook and followed the man by the winding path that curled away into the wood. She was ever a sanguine soul, and the mere sinister influences that might have discouraged her in her purpose that morning were impotent before the level convictions of her heart. She had seen Pelleas ride in amid the trees; she was sure as death as to his cognizance and his armour. Now Pelleas, she could vow, had not heard her call to him, and if he had heard he had not understood; if he had seen he had not recognised. Doubts could have no place in the argument before such a justification by faith. It was not long before she caught sight of the red glint of armour going through the trees. It came and went, grew and disappeared, as the path folded it in its curves or thrust out a heavy screen of green to hide it like a heavy curtain. The man was going as he pleased, now a walk, now a casual jog, now a short burst of a canter over an open patch. One moment Igraine would see him clearly, then not at all. Sometimes she gained, sometimes lost ground, yet the knight of the red harness never seemed to come within lure of her voice. In due course she reached the place where the path ended bluntly on the Winchester high-road, and where the way ran straight as a spear-shaft, so that she could see Pelleas riding for Winchester with a lead of a quarter of a mile. The distant ringing tramp of hoofs came up to her like a mocking chuckle. Putting her hands to her mouth, she hallooed with all the breath left her by her run through the wood; yet, as far as she might see, the man never so much as turned in the saddle, while the smite of hoofs died down and down into a well of silence. Another halloo and no echo. “He’s asleep, or deaf in his helmet.”
  • 30. She forgot the distance and the din of hoofs that might well have drowned the thin cry that could have reached the rider. Maugre her heat and her flushed face Igraine had no more thought of giving in than she had of marrying Gorlois. With Pelleas so near she had made her vow to follow him, and follow him she would like a comet’s tail. If needs be she would wear her sandals to the flesh, but catch the man she must in the end. A mile more on the high-road, with her feet and the hem of her gown dust-drenched, and she was still little nearer the man in the red harness for all her hurrying. She could have vowed more than once that he turned in his saddle and looked back at her as though to see how near she had come to him on the road. A mile from the hermitage path he turned his horse southwards from the track into a grass valley headed by a ruined tower and hedged densely on either hand by pine woods. Igraine, seeing from a slight rise in the road this change of course, cut away crosswise with the notion of getting near the man or of intercepting him before he should win clear law again. After all, the effort added only more vexation. She saw the black horse pressed to a canter and cross the point where she might have cut him off, while a great stretch of furze that rolled away to the black palisading of the pines came down and threw a promontory in her path. Pelleas was a mile to the good when she had skirted the furze and the bend of the wood, and taken a straight course southwards down the valley between the pines. All that morning the sport of hunter and hunted went on between the novice in grey and the man on the black horse. For all her trouble Igraine won little upon him, lost little as the hours went by; while the rider in turn seemed in no wise desirous of being rid of her for good. They passed the pine woods with their midnight aisles, forded a stream, climbed up a heath, went over it amid the heather. From the last ridge of the heath Igraine saw the country sloping away into undulating grasslands, piled here and there with domes of thicketed trees. Far to the south a dense black mass rose like a rounded hill against the sky. The man in red was still about a mile in front of her, riding slowly, a red speck in a waste of green. Igraine,
  • 31. having him in view from her vantage point, lay down full length to rest and take some food. She was tired enough, but dogged at heart as ever. She vowed that if the man was playing with her she would tell him her mind, love or no love, when she came up with him in the end. As the sun swam into the noontide arc she went on again downhill, and found in turn that the man had halted, for he had been hidden by trees, and getting view of him suddenly she saw him sitting on a stone with his horse tethered near. As soon as Igraine was within measurable distance she took advantage of a hollow, dropped on her hands and knees, and began to crawl like a cat after a bird. Edging round a thicket she came quite near the man, but could not see his face. His spear stood in the ground by his horse, and he had his shield slung about his neck, and a bare poniard in his hand. It was clear that he was watching for Igraine, for despite her craft he caught sight of her face peering white under the hem of a bush, and climbed quickly into the saddle. Igraine started up, made a dash across the open, calling to him as she ran. Perverse as hate his horse broke into a canter and left her far in the rear. The girl shook her fist at him with a sudden burst of temper. She was standing near the stone where the man had been sitting. Looking at its flat face she saw the reason of the naked poniard in his hand, for he had been carving out thin straggling letters in the stone. “Sancta Igraine,” she read— “Ora pro nobis.” The screed dispelled the doubts in Igraine’s mind on the instant. Palpably the man knew well enough who was following him, and was avoiding her of set purpose; but for what reason Igraine racked her wit to discover. She ran through many things in her heart, the possible testing of her devotion, a vacillating weakness on Pelleas’s part that would not let him leave her altogether, a freakish wish to give her penance. Then, she knew that he was superstitious, and the thought flashed to her that he might think her a wraith, or some evil spirit that had taken her shape to have him in temptation. Maugre
  • 32. her vexation and her pride she held again on the trail, eating as she went some dried plums that she had in her wallet. The man had slackened down again and was less than half a mile away, now limned against the sky, now folded into a hollow or shut out by trees. Like a marsh-fire he tantalised her with a mystery of distance, holding steadily south at a level tramp, while Igraine plodded after him, her hair down and blowing out to the casual wind, her eyes at gaze on the red lure in the van. So the mellower half of the day passed, and towards evening they neared the mount of trees Igraine had seen from the last ridge of the heath at noon. The black horse was heading straight for the cloudy mass in a way that set Igraine thinking and casting about for Pelleas’s motive. Perhaps he had some quest in the solitary place that needed his single hand. Would he take to the wood and let her follow as before, or had he any purpose in leading her thither? Drowned in conjecture she gave up prophecy with a vicious sense of mystification, and accepted inevitable ignorance for the time being as to the man’s moods and motives. She was no less obstinate to follow him to the death. If she only had a horse she would come near the man, pride or no pride, and tell him the truth. Pressing on, with her strained ankle beginning to limp, she topped the round back of a grass rise and came full in view of the wood she had long seen in the distance. It looked very solemn in the declining light. The great trunks of giant beeches were packed pillar upon pillar into an impenetrable gloom. The foliage above, densely green, billowy, touched with red and gold, rolled upwards cloud on cloud as the ground ascended to the south and east. A great bronze carpet of dead leaves swept away into the night of the trees. There was an eternal hush, a gross silence, over the glooming aisles that seemed to beckon to the soul, to draw the heart into the night of foliage as into a cavern. Over all was the glowing ægis of the setting sun. Igraine saw the man on the forest’s edge where an arch of gloom struck into the inner shadows. He was facing the west, motionless as stone on his black horse, with the slanting light plucking a dull red
  • 33. gleam from his harness. There was a mystery about him that seemed to harmonise with the stillness of the trees and the black yawn of the forest galleries. Igraine imagined that he might be in a mood at last to speak with her if he believed her human. At all events, if he took to the trees, and she did not lose him, she would have the vantage of him and his horse in such a barricaded place. It began to grow dark very quickly as she passed down the gradual slope towards the forest. The trees towered above her, a black mass rising again towards the east. Keen to see the man’s mood, she hurried on and found him still steadfast in the great arch, that seemed like the gate of the wilderness, ready to abide her. A hundred paces more and her heart began to beat the faster, and the moil of the day’s march dwindled before the influx of a rosier idyl. Every step towards Pelleas seemed to take her higher up the turret stair of love till her lips should meet those that bent at last from the gloom to hers. Pride and vexation lay fallen far below, dropped incontinently like a ragged cloak; a more generous passion shone out like cloth of gold; she was no longer weary. Her eyes were very bright, her face full of a splendid wistfulness, as she neared the man under the trees, looking up to see his face. Twilight lay deep violet under the wooelshawe, while horse and man were dim and impalpable, great shadows of themselves. Igraine could not see the man’s face for the mask over the mezail of his helmet, and he was silent as death. She was quite close to him now and ready to speak his name, when he wheeled suddenly, looked back at her, and pointed into the wood with his long spear. She ran forward and would have taken hold of his bridle, but he waved her back and slanted his spear at her in mute warning. Igraine, heart- hungry, could hold herself no longer. “Man—man, are you stone?” He rode straight ahead into the night of the trees and said never a word. Igraine drew her breath. “Pelleas.”
  • 34. “Ah, Igraine.” The voice that came to her was muffled like the voice of a mourner, yet the girl thought she caught the old deep tone of it like the low cry of the wind. “Why do you vex me?” “Follow!” “Pelleas, Pelleas, I am no nun!” “Follow!” “I kept this truth from you too long.” “Follow!” “Pelleas, would you hurt my heart more?” “Follow; God shall make all plain and good.” She gave in with a half-sob, and bent quietly to the man’s mood, though she had no notion what he purposed in his heart, or what his desires were in mystifying her thus. No doubt it would be well in the end if Pelleas bade her follow like a penitent and promised ultimate peace. At least he had not turned her away, and she trusted him to the death. He was a strong, deep-sensed soul, she knew, and her deceiving may have made him bitter in measure, and not easily appeased. In this queer trial of endurance, this tempting of her temper, she thought she read a penance laid upon her by the man for the way she had used his love. They were soon far into the wood, with the western sky dwindling between the innumerable pillars of the trees. It began to be dark and utterly silent save for the rustle of the dead leaves as they went, and the shrilling chafe of bridle or scabbard, or the snort of the great horse. Wherever the eye turned the forest piers stood straight and solemn as the columns in a hypostyle hall in some Egyptian temple. The fretwork of boughs roofed them in with hardly a glimmering through of the darkening sky above. There was a pungent autumn scent on the air that seemed to rise like the incense of years that
  • 35. had fallen to decay on the brown flooring of the place, and there was no breath or vestige of a wind. Presently as the day died the wood went black as the winter night, and Igraine kept close by the man, with his armour giving a dull gleam now and again to guide her. They were passing up what seemed to be a great arcade cut through the very heart of the wood, as though leading to some shrine or altar, relic of Druid days, or times yet more antique. The tunnel ran a curved course, bending deeper and deeper as it went into the dense horde of trees. So dark was the wood that it was possible to see but a few paces in advance, and Igraine wondered how the man kept the track. She was close at his stirrup now, with the dark mass of him and his horse rising above her like a statue in black basalt. Though he never spoke to her, and though she touched no part of him, his horse, or his harness, she felt content with the queer sense of trust and proximity that pervaded her. There was magic in the mere companionship. As she had humbled her will to Pelleas’s the night when he had taken her from the beech tree in Andredswold, so now in like fashion she surrendered pride and liberty, and became a child. Suddenly the trackway straightened out into a great colonnade that ran due south between trees of yet vaster girth. Igraine felt the man rein in and abide motionless beside her as she held to the stirrup and waited for what next should chance. Silence seemed like depths of black water over them, and they could hear each other take breath like the faint flux and reflux of a sea. Igraine saw the man lift his spear, a dim streak less black than the vault above, and hold it as a sign for her to listen. Her blood began to tingle a very little. There was something far away on the dead, stagnant air, a sort of swirl of sound, shrill and harmonious, like a wind playing through the strings of a harp. It was very gradual, very impalpable. As the volume of it grew it seemed to rush nearer like a wind, to swell into a swaying plaintive song smitten through with the wounded cry of flutes. It gave a notion of wood-fays dancing, of whirling wings and flitting gossamer moonbright in the shadows. Igraine’s blood seemed to spin the faster, and her hand left the stirrup and touched the man’s
  • 36. thigh. He gave never a word or sign in the dark. She spoke to him very softly, very meekly. “What place is this, Pelleas?” She saw him bend slightly in the saddle. “It is called the Ghost Forest,” he said. “What are the sounds we hear?” “Who can tell!” Igraine had hardly heard him, when a streak of phosphor light flickered among the trees, coming and going incessantly as the great trunks intervened. It neared them in gradual fashion, and then blazed out sudden into the open aisle, a man in armour riding on a great white horse, his harness white as the moon, his face pale and wide-eyed, his hair like a mass of twisted silver wire. A misty glow haloed him round, and though he rode close there seemed no sound at all to mark his passing. As he had come, so he went, with streaks of flickering light that waxed less and less frequent till they died in the dark, and left the place empty as before. Igraine thought the air cold when he had gone. She felt the black horse move beside her, and they went on as before into the night of the trees. The noise of flute and harp that had ceased awhile bubbled up again quite near, so that it was no longer the ghost of a sound, but noise more definite, more discrete. It had a queer way of dying to a sighing breath, and then gathering gradually into an ascending burst of windy melody. Igraine could almost fancy that she heard the sweep of wings, the soft thrill of silks trailing through the trees, yet the man on the horse said never a word as they went on like a pair of mutes to a grave. The colonnade opened out abruptly on a great circular clearing in the wood shut in by crowded trunks, its open vault above cut by a dense ring of foliage. A grey light came down from the sky, showing great stones piled one upon another, others fallen and sunk deep in rank grass and brambles. The man halted his horse in the very
  • 37. centre of the clearing, with Igraine beside him, watchful for what should happen, and for the moment when Pelleas should unbend. Hardly had she looked over the great cromlechs, black and sinister in that solitary wilderness, than the whole wood about them seemed dusted suddenly with points of fire. North, south, east, and west torches and cressets came jerking redly out of the night, flitting behind the trees in a wide circle, gathering nearer and nearer without a sound. They might have been great fireflies playing through the aisles and ways, or goblin lamps carried by fairy folk. Igraine drew very close to the man’s horse for comfort, and looked up to see his face, but found it dark and hidden. Her hand crept up past the horse’s neck, rested on the mane a moment, and ventured yet further to meet the man’s hand, where it gripped the bridle. For a minute they abode thus without a sound, watching the weird torch-dance in the wood. With a sudden gibber of laughter and a swirl of pipes the throng of lights seemed to seethe to the very margin of the clearing. Queer phantastic shapes showed amid the trees, and the great circle grew wide with light, and the grey cromlechs surprised in sleep by the glare and piping. At that very moment Igraine had a thought of some one looking deep into her eyes, of a will, a power, streaming in upon her like sunlight into a sleepy pool. Her desire went from the man on the black horse into the square shadow of the great central cromlech, where an indefinite influence seemed to lurk. Looking long under the roofing stone, she grew aware of a tall something standing there, of a pair of eyes like the eyes of a panther, of a lean white hand moving in the shadows. The eyes under the cromlech seemed to follow Igraine like fire, and to burn in upon her a foreign influence. Rebellious and wondering, she stiffened herself against a spiritual combat that seemed moving upon her out of the dark. She could have smitten the eyes that stared her down, and yet the magnetic stupor of them kindled up things in her heart that were strange and newly sensuous. She felt her strength sway as though her soul were being lifted from her, and
  • 38. she was warmed from top to toe like one who has taken wine, and whose being swims into an idyllic glorification of the senses. Again her desire seemed turned to the man in red harness, yet when she looked the saddle was empty, and the horse held by an armed servant, who wore a wolfs head for covering. Still mute with fear, desire, and wonder, she saw a tall figure move into the full glare of the torches, a figure in red harness with a shield of green, and a red dragon thereon, and with head unhelmed. The armour was like the armour of Pelleas, but the face was the face of the man Gorlois. And now the eyes under the shadow of the cromlech were full and strong upon Igraine. Breathing fast with a hand at her throat she stepped back from Gorlois—hesitated—stood still. She was very white, and her eyes were big and sightless like the eyes of one walking in a dream. For all her strength, her scorn, and the tricking of her heart, she was being swept like a cloud into the embraces of the sun. Reason, power, love, sank away and became as nothing. A shudder passed over her. Presently her hands dropped limp as broken wings, and her body began to sway like a tall lily in a breeze. A gradual stupor saw her cataleptic; she stood impotent, played upon by the promptings of another soul. Gorlois went near to her with hands outstretched, stooping to look into her face. A sudden light kindled in her eyes, her lips parted, and new life flooded red into her cheeks as at the beck of love. She bent to Gorlois full of a gracious eagerness, a wistful desire that made her face golden as dawn. Her hand sought his, while the shadowy shape under the cromlech watched them with never-wavering eyes. Gorlois’s arms were round her now all wreathed in her hair; her face was turned to his; her hands were clasped upon his neck. Another moment and he had touched her lips with his. A sound of flutes, the tinkling of a bell, and a solemn company came threading from the trees, guests, acolytes, torch-bearers, in glittering cloth of gold, with a great crucifix to lead them. Gorlois and Igraine were hand in hand near the stone that hid the frame of Merlin. A priest in a gorgeous cape drew near, and began his patter.
  • 39. The vows were taken, the pact sealed, with the noise of a chant and music. Thus under the benedictions of the great trees, and the spell of Merlin, Gorlois and Igraine were made man and wife.
  • 40. BOOK III THE WAR IN WALES
  • 41. I Aurelius Ambrosius the king was dead, taken off in Winchester by the hand of a poisoner. He had been found stark and cold in his great carved bed, with an empty wine-cup beside him, and a tress of black hair and a tress of yellow laid twined together upon his lips. The signet-ring had gone from his finger, and by the bed had been discovered a woman’s embroidered shoe dropped under the folds of the purple quilt. The truth, sinister enough in its bare suggestions, was glossed over by the court folk out of honour to Aurelius, and of love to Uther the king’s brother. It was told to the country how an Irish monk sent by Pascentius, dead Vortigern’s son, had gained audience of the king, and treacherously poisoned him as he drank wine at supper. The tale went out to the world, and was believed of many with a sincere and honest faith. Yet a certain child-eyed woman, wandering on the shores of Wales for sight of Irish ships, could have spoken more of the truth had she so dared. Uther Pendragon had been hailed king at York before the bristling spears of a victorious host. But a week before he had marched against the heathen on the Humber, and overthrown them with such slaughter as had not been seen in Britain since the days when Boadicea smote the Romans. At the head of his men he had marched south in a snowstorm to be thundered into Winchester as king and conqueror. Twelve maidens of noble blood, clad in ermine and minever, had run before him with boughs of mistletoe and bay. Five hundred knights had walked bareheaded, with swords drawn, behind his horse. The city had glistened in a white web of frosted samite, sparkled over by the clear visage of a winter sun. There were many great labours ready to the king’s hand. Britain lay bruised by the onslaughts of the barbarians; her monks had been slain, her churches desecrated. The pirate ships swept the seas, and poured torch and sword along the sunny shores of the south. Andredswold, dark, saturnine, mysterious, alone waved them back
  • 42. with the sepulchral threatening of its trees. Yet, for all the burden of the kingdom upon his broad shoulders, Uther gave his first care to the honouring of the dead. Aurelius Ambrosius was buried with great pomp of churchmen and nobles at Stonehenge, and a royal mound raised above the tomb. At Christmastide, with snow upon the ground, a great gathering was made at Sarum of all the petty kings, princes, and nobles of the land. Hither came Meliograunt, king of Cornwall, and Urience of the land of Gore. Fealty was sworn with solemn ordinance to Uther Pendragon the king, and common league bonded against the heathen and the whelps of the north. There were other perils brewing for Britain over the sea. Pascentius, dead Vortigern’s son, had been an outcast and a wanderer since the days when the sons of Constantine had sailed from Armorica to save the land from the blind lust and treason of his father. He had been a drifting fire beyond the seas, an intriguer, a sower of sedition, a man dangerous alike to friend and foe. Beaten like a vulture from the coasts of Britain, he had turned with treasonable hope to Ireland and its king, Gilomannius the Black, a strenuous potentate, boasting little love for Ambrosius the king. Here, in Ireland, a kennel of sedition had arisen. Pascentius, keen, hungry plotter, had toiled at the task of piling enmity against those who had destroyed his father amid the flames of Genorium. A great league arose, a banding of the barbarians with the Irish princes, a union of the Saxons who ravaged Kent with the wild tribesmen over the northern border. Month by month a great host gathered on the Irish coast. Many ships came from the east and from the south. Mid-winter was past before Gilomannius embarked, and, setting sail with a fair wind, turned the beaks of his galleys for the shores of Wales. Noise of the gathering storm had been brought to Uther as he journed through the southern coasts, rebuilding the churches, recovering abbey and hermitage from their desolate ashes. His zeal was great for God, and his love of Britain well-nigh as noble. Warned thus in due season, he marched for the west, calling the land to arms, assigning for the gathering of the host Caerleon upon Usk, that fair city bosomed in the fulness of its woods and pastures. Many
  • 43. a knight had answered to his call; many a city had sent out her companies; the high-roads rang with the cry of steel in the crisp winter weather. Duke Gorlois had come from Cornwall from his castle of Tintagel, bringing many knights and men-at-arms by sea, and the Lady Igraine his wife, in a great galley whose bulwarks glistened with shields. In Caerleon Gorlois had a house built of white stone, set upon a little hill in the centre of the city. To Caerleon he brought this golden falcon of a woman, this untamable thing that he had kept prisoned in the high towers of Tintagel. He mewed her up like a nun in his house of white stone, so that no man should see the fairness of her face. She was wild as an eyas from the woods, fierce and unapproachable, and sharp of claw. Robbed of her liberty, had she not sought to take her own life with a sword, and to throw herself from the battlements of Tintagel? Gorlois had won little love by Merlin’s subtlety, and he feared the woman’s beauty and the spell of her large eyes. It was the month of February and clear crisp weather. The white bellies of the Irish sails had shown up against the grey blue stretch of the sea, a white multitude of canvas that had sent the herdsmen hurrying their flocks to the mountains. Horsemen had galloped for Caerleon, and the cry of war went up over wood and water. Flames licked the night sky. From Caerleon to St. Davids, from St. Davids to Eryri, the red blaze of beacon-fires told of the ships at sea. The cry of the storm arose in Caerleon, and the tramp of armed men sounded all day in her streets. The great host lodged about the city broke camp and streamed westwards along the high-road into Wales. Bugles blew, banners flapped, masses of sullen steel rolled away into purple of the winter woods. Bristling spears and lines of skin-clad shields vanished into the west like the waves of a solemn sea. On the walls of Caerleon stood many women and children watching the host march for the west, watching Uther the king ride out with his great company of knights and nobles.
  • 44. At the casement of an upper room in Gorlois’s house stood a woman looking out over Caerleon towards the sea. She was clad in a mantle of furs, and in a tunic of purple linked up with cord of gold. A tippet of white fur clasped with a brooch of amethysts circled her throat. Her hair was bound up in a net of fine silk, and there was a girdle of blue silk about her loins, and an enamelled cross upon her bosom. She stood with her elbows resting on the stone sill, and her peevish face clasped between her hands. Her eyes looked very large and lustrous as she stared out wistfully over the city. In the great court below horses champed the bit and struck fire from the ringing flags. Men in armour clanged to and fro; rough voices cried questions and counter-questions; bridles jingled; spear-shafts clattered on the stones. Now a clarion blared as a troop of horse thundered by up the street, their armour gleaming dully past the courtyard gate. The growl of war hung heavy over Caerleon, a grim sullen sound that seemed in keeping with the restless chiding of the wind. Igraine’s face was hard as stone as she watched the men moving in the courtyard below. She looked older than of yore, whiter, thinner in cheek and neck, her great eyes staunch though sad under her netted hair. Her face showed melancholy mingled with a constant scorn that had rarely found expression with her in the old days, save within the walls of Avangel. She looked like one who had endured much, suffered much, yet lost no whit of pride in the trial. Though she may have been blemished like a Greek vase smitten by some barbaric sword, she was her self still, brave, headstrong, resolute as ever. The shame of the things she had suffered had perhaps wiped out the gentler outlines of her character and left her more stern, more wary, less honest, more deep in her endeavours. There was no passive humility or patience about her soul, and she was the falcon still, though caged and guarded beyond her liberty. As she stood at the casement with the prophetic murmur of war in her ears, it seemed to her as though life surged to her feet and mocked her bondage like laughing water. The desire of liberty abode
  • 45. ever with her even to the welcoming of stagnant death. She thirsted for her freedom, plotted for it, dreamt of it with a zeal that was almost ferocious. Her life seemed a speculation, a perpetual aspiration after a state that still eluded her. In the Avangel days she had been wild and petulant. Then Pelleas had come through the green gloom of early summer to soften her soul and inspire all the best breath of the woman in her. Again, thanks to Gorlois, she had fallen with the usual reaction of circumstance upon evil times; the change had discovered the peevish discontent of the girl hardened into the strong wilfulness of the woman. She hated Gorlois with a fanatical immensity of soul. When the man was near her she felt full of the creeping nausea of a great loathing, and she waxed faint with hate at the veriest touch of his hands. His breath seemed to her more unsavoury than the miasma of a gutter, and it needed but the sound of his voice to bring all her baser passions braying and yelping against him. He had driven the religious instinct out of her heart, and she was in revolt against heaven and the marriage pact forged by the authority of the Church. She had often vowed in her heart that she could do no sin against Gorlois, her husband. He had no claim upon her conscience. The bondage had been of his making; let God judge her if she scorned his honour. Standing by the window watching the knights saddling for their lord’s sally, she heard heavy footsteps mounting up the stairs, and the ring of steel-tipped shoes along the gallery. The footsteps were deliberate, and none too fast, as though the man walked under a burden of thought. A shadow seemed to pass over Igraine’s face. She slipped from the window, ran across the room, shot the bolt of the door, and stood listening. A hand tried the latch. She knew well enough whose fist it was that rattled on the oaken panels. Her face hardened to a kind of cold malevolence, and she laughed noiselessly in her sleeve. A terse summons came to her from the gallery. “Wife, we ride at once.”
  • 46. The man could not have made a worse beginning. There was a suggestion of tyranny in a particular word that was hardly temperate. Igraine leant against the door; she was still smiling to herself, and her hands fingered the embroidered tassel of the latch. “We are late on the road; I can make no tarrying.” The door quivered a moment as though shaken by a gusty wind. Everything was quiet again, and Igraine could hear the man breathing. Putting her mouth to the crack between post and hinge- board she laughed stridently as though in scorn. “Igraine!” The voice was half-imperative, half-appealing. “My very dear lord!” “Are you abed?” “No, dear lord.” “Open to me; I would kiss your lips before I sally.” “You have never kissed me these many days.” “True, wife; is it fault of mine?” “Nor shall again, dear lord, if I have strength.” She heard the man muttering to himself a moment, but this time there was no smiting of the door, no fume and tempest. His mood seemed more temperate, less masterful, as though he were half heavy at heart. “Igraine—” “Why do you whimper like a dog?” she said; “go, get you to war. What are you to me?” “When will you learn reason?” “When you are dead, sire.” “Perhaps I deserve all this.”
  • 47. “Are you so much a penitent?” Her mockery seemed to lift Gorlois to a higher range of passion, and there was great bitterness in his voice as he tossed back words to her with a quick kindling of desire. “Woman, I have been hard in the winning of you, but, God knows, you are something to me.” “God knows, Gorlois, I hate you.” His hand shook the door. “Let me in, Igraine.” “Break down the door; you shall come at me no other way.” “Woman, woman, I am a fool; my heart smarts at leaving you.” “You sound almost saintly.” “I have left Brastias in charge of you.” “Thanks, lord, for a jailer.” Igraine drew back from the door and stood at her full height with her hands crossed upon her bosom. She quivered as she stood with the intense effort of her hate. Gorlois still waited without the door, though she could not hear him moving. The silence seemed like the deep hush that falls between the blustering stanzas of a storm. “Igraine!” It was a hoarse cry, quick and querulous. Igraine had both her fists to her chin in an attitude of inward effort, as though she racked herself to give utterance to the implacable temper of her scorn. Her face had a queer parched look. When she spoke, her voice was shrill like a piping wind. “Gorlois.” “Wife.” “Would you have my blessing?”
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