We study automated intrusion response and formulate the interaction between an attacker and a defender on an IT infrastructure as a stochastic game where attack and defense strategies evolve through reinforcement learning and self-play. Direct application of reinforcement learning to any non-trivial instantiation of this game is impractical due to the exponential growth of the state and action spaces with the number of components in the infrastructure. We propose a decompositional approach to deal with this challenge and prove that under assumptions generally met in practice the game decomposes into a) additive subgames on the workflow-level that can be optimized independently; and b) subgames on the component-level that satisfy the optimal substructure property. We further show that the optimal defender strategies on the component-level exhibit threshold structures. To solve the decomposed game we develop Decompositional Fictitious Self-Play (\dfsp), an efficient fictitious self-play algorithm that learns Nash equilibria through stochastic approximation. We show that \dfsp outperforms a state-of-the-art algorithm for our use case. To evaluate the learned strategies, we deploy them in a a virtual IT infrastructure in which we run real network intrusions and real response actions. From our experimental investigation we conclude that our approach can produce effective defender strategies for a practical IT infrastructure.

1. 1/54
Learning Optimal Intrusion Responses for
IT Infrastructures via Decomposition
Visit to MIT CSAIL ALFA group
Kim Hammar
kimham@kth.se
Division of Network and Systems Engineering
KTH Royal Institute of Technology
May 19, 2023

2. 2/54
Use Case: Intrusion Response
I A defender owns an infrastructure
I Consists of connected components
I Components run network services
I Defender defends the infrastructure
by monitoring and active defense
I Has partial observability
I An attacker seeks to intrude on the
infrastructure
I Has a partial view of the
infrastructure
I Wants to compromise specific
components
I Attacks by reconnaissance,
exploitation and pivoting
Attacker Clients
. . .
Defender
1 IPS
1
alerts
Gateway
7 8 9 10 11
6
5
4
3
2
12
13 14 15 16
17
18
19
21
23
20
22
24
25 26
27 28 29 30 31

3. 3/54
Automated Intrusion Response: Current Landscape
Levels of security automation
No automation.
Manual detection.
Manual prevention.
No alerts.
No automatic responses.
Lack of tools.
1980s 1990s 2000s-Now Research
Operator assistance.
Manual detection.
Manual prevention.
Audit logs.
Security tools.
Partial automation.
System has automated functions
for detection/prevention
but requires manual
updating and configuration.
Intrusion detection systems.
Intrusion prevention systems.
High automation.
System automatically
updates itself.
Automated attack detection.
Automated attack mitigation.

4. 4/54
Can we use decision theory and learning-based methods to
automatically find effective security strategies?1
π
Σ Σ
security
objective
feedback
control
input
target
system
security
indicators
disturbance
1
Kim Hammar and Rolf Stadler. “Finding Effective Security Strategies through Reinforcement Learning and
Self-Play”. In: International Conference on Network and Service Management (CNSM 2020). Izmir, Turkey, 2020,
Kim Hammar and Rolf Stadler. “Learning Intrusion Prevention Policies through Optimal Stopping”. In:
International Conference on Network and Service Management (CNSM 2021). Izmir, Turkey, 2021, Kim Hammar
and Rolf Stadler. “Intrusion Prevention Through Optimal Stopping”. In: IEEE Transactions on Network and
Service Management 19.3 (2022), pp. 2333–2348. doi: 10.1109/TNSM.2022.3176781, Kim Hammar and
Rolf Stadler. Learning Near-Optimal Intrusion Responses Against Dynamic Attackers. 2023. doi:
10.48550/ARXIV.2301.06085. url: https://arxiv.org/abs/2301.06085.

5. 5/54
Our Framework for Automated Network Security
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

6. 5/54
Our Framework for Automated Network Security
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

7. 5/54
Our Framework for Automated Network Security
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

8. 5/54
Our Framework for Automated Network Security
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

9. 5/54
Our Framework for Automated Network Security
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

10. 5/54
Our Framework for Automated Network Security
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

11. 5/54
Our Framework for Automated Network Security
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

12. 5/54
Our Framework for Automated Network Security
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

13. 6/54
Creating a Digital Twin of the Target Infrastructure
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

14. 6/54
Theoretical Analysis and Learning of Defender Strategies
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

15. 7/54
Creating a Digital Twin of the Target Infrastructure
I An infrastructure is
defined by its
configuration.
I Set of configurations
supported by our
framework can be
seen as a
configuration space
I The configuration
space defines the
class of
infrastructures for
which we can create
digital twins.
Σ Configuration Space
σi
*
* *
172.18.4.0/24
172.18.19.0/24
172.18.61.0/24
Digital Twins
R1 R1 R1

16. 8/54
The Target Infrastructure
I 64 nodes. 24 ovs switches, 3
gateways. 6 honeypots. 8 application
servers. 4 administration servers. 15
compute servers.
I Topology shown to the right
I 11 vulnerabilities (cve-2010-0426,
cve-2015-3306, cve-2015-5602, etc.)
I 4 zones: dmz, r&d zone, admin
zone, quarantine zone
I 9 workflows
I Management: 1 sdn controller, 1
Kafka server, 1 elastic server.
r&d zone
App servers Honeynet
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
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19
20
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22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

17. 9/54
Emulating Physical Components
I We emulate physical components with
Docker containers
I Focus on linux-based systems
I The containers include everything
needed to emulate the host: a runtime
system, code, system tools, system
libraries, and configurations.
I Examples of containers: IDPS
container, client container, attacker
container, CVE-2015-1427 container,
Open vSwitch containers etc.
Containers
Physical server
Operating system
Docker engine
CSLE

18. 10/54
Emulating Network Connectivity
Management node 1
Emulated IT infrastructure
Management node 2
Emulated IT infrastructure
Management node n
Emulated IT infrastructure
VXLAN VXLAN . . . VXLAN
IP network
I We emulate network connectivity on the same host using
network namespaces.
I Connectivity across physical hosts is achieved using VXLAN
tunnels with Docker swarm.

19. 11/54
Emulating Network Conditions
I We do traffic shaping using
NetEm in the Linux kernel
I Emulate internal
connections are full-duplex
& loss-less with bit
capacities of 1000 Mbit/s
I Emulate external
connections are full-duplex
with bit capacities of 100
Mbit/s & 0.1% packet loss
in normal operation and
random bursts of 1% packet
loss
User space
. . .
Application processes
Kernel
TCP/UDP
IP/Ethernet/802.11
OS
TCP/IP
stack
Queueing
discipline
Device driver
queue (FIFO)
NIC
Netem config:
latency,
jitter, etc.
Sockets

20. 12/54
Emulating Actors
I We emulate client arrivals
with Poisson processes
I We emulate client
interactions with load
generators
I Attackers are emulated by
automated programs that
select actions from a
pre-defined set
I Defender actions are
emulated through a custom
gRPC API.
Markov Decision Process
s1,1 s1,2 s1,3 . . . s1,4
s2,1 s2,2 s2,3 . . . s2,4
Digital Twin
. . .
Virtual
network
Virtual
devices
Emulated
services
Emulated
actors
IT Infrastructure
Configuration
& change events
System traces
Verified security policy
Optimized security policy

21. 13/54
System Identification
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Digital Twin
Target
Infrastructure
Model Creation &
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning &
Generalization
Strategy evaluation &
Model estimation
Automation &
Self-learning systems

22. 14/54
Outline
I Use Case & Digital Twin
I Use case: intrusion response
I Digital twin for data collection & evaluation
I System Model
I Discrete-time Markovian dynamical system
I Partially observed stochastic game
I System Decomposition
I Additive subgames on the workflow-level
I Optimal substructure on component-level
I Learning Near-Optimal Intrusion Responses
I Scalable learning through decomposition
I Digital twin for system identification & evaluation
I Efficient equilibrium approximation
I Conclusions & Future Work

23. 14/54
Outline
I Use Case & Digital Twin
I Use case: intrusion response
I Digital twin for data collection & evaluation
I System Model
I Discrete-time Markovian dynamical system
I Partially observed stochastic game
I System Decomposition
I Additive subgames on the workflow-level
I Optimal substructure on component-level
I Learning Near-Optimal Intrusion Responses
I Scalable learning through decomposition
I Digital twin for system identification & evaluation
I Efficient equilibrium approximation
I Conclusions & Future Work

24. 14/54
Outline
I Use Case & Digital Twin
I Use case: intrusion response
I Digital twin for data collection & evaluation
I System Model
I Discrete-time Markovian dynamical system
I Partially observed stochastic game
I System Decomposition
I Additive subgames on the workflow-level
I Optimal substructure on component-level
I Learning Near-Optimal Intrusion Responses
I Scalable learning through decomposition
I Digital twin for system identification & evaluation
I Efficient equilibrium approximation
I Conclusions & Future Work

25. 14/54
Outline
I Use Case & Digital Twin
I Use case: intrusion response
I Digital twin for data collection & evaluation
I System Model
I Discrete-time Markovian dynamical system
I Partially observed stochastic game
I System Decomposition
I Additive subgames on the workflow-level
I Optimal substructure on component-level
I Learning Near-Optimal Intrusion Responses
I Scalable learning through decomposition
I Digital twin for system identification & evaluation
I Efficient equilibrium approximation
I Conclusions & Future Work

26. 14/54
Outline
I Use Case & Digital Twin
I Use case: intrusion response
I Digital twin for data collection & evaluation
I System Model
I Discrete-time Markovian dynamical system
I Partially observed stochastic game
I System Decomposition
I Additive subgames on the workflow-level
I Optimal substructure on component-level
I Learning Near-Optimal Intrusion Responses
I Scalable learning through decomposition
I Digital twin for system identification & evaluation
I Efficient equilibrium approximation
I Conclusions & Future Work

27. 15/54
System Model
I G = h{gw} ∪ V, Ei: directed graph
representing the virtual infrastructure
I V: finite set of virtual components.
I E: finite set of component
dependencies.
I Z: finite set of zones.
r&d zone
App servers Honeynet
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

28. 15/54
System Model
I G = h{gw} ∪ V, Ei: directed graph
representing the virtual infrastructure
I V: finite set of virtual components.
I E: finite set of component
dependencies.
I Z: finite set of zones.
r&d zone
App servers Honeynet
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

29. 15/54
System Model
I G = h{gw} ∪ V, Ei: directed graph
representing the virtual infrastructure
I V: finite set of virtual components.
I E: finite set of component
dependencies.
I Z: finite set of zones.
r&d zone
App servers Honeynet
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

30. 16/54
State Model
I Each i ∈ V has a state
vt,i = (v
(Z)
t,i
|{z}
D
, v
(I)
t,i , v
(R)
t,i
| {z }
A
)
I System state st = (vt,i )i∈V ∼ St.
I Markovian time-homogeneous
dynamics:
st+1 ∼ f (· | St, At)
At = (A
(A)
t , A
(D)
t ) are the actions.
s1 s2 s3
s4 s5 s4
.
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.
.

31. 16/54
State Model
I Each i ∈ V has a state
vt,i = (v
(Z)
t,i
|{z}
D
, v
(I)
t,i , v
(R)
t,i
| {z }
A
)
I System state st = (vt,i )i∈V ∼ St.
I Markovian time-homogeneous
dynamics:
st+1 ∼ f (· | St, At)
At = (A
(A)
t , A
(D)
t ) are the actions.
s1 s2 s3
s4 s5 s4
.
.
.
.
.
.
.
.
.

32. 16/54
State Model
I Each i ∈ V has a state
vt,i = (v
(Z)
t,i
|{z}
D
, v
(I)
t,i , v
(R)
t,i
| {z }
A
)
I System state st = (vt,i )i∈V ∼ St.
I Markovian time-homogeneous
dynamics:
st+1 ∼ f (· | St, At)
At = (A
(A)
t , A
(D)
t ) are the actions.
s1 s2 s3
s4 s5 s4
.
.
.
.
.
.
.
.
.

33. 17/54
Workflow Model
I Services are connected into workflows W = {w1, . . . , w|W|}.

34. 17/54
Workflow Model
I Services are connected into workflows W = {w1, . . . , w|W|}.
gw fw idps lb
http
servers
auth
server
search
engine
db
cache
Dependency graph of an example workflow representing a web
application; gw, fw, idps, lb, and db are acronyms for gateway,
firewall, intrusion detection and prevention system, load balancer, and
database, respectively.

35. 18/54
Workflow Model
I Services are connected into
workflows
W = {w1, . . . , w|W|}.
I Each w ∈ W is realized as a
directed acyclic subgraph (dag)
Gw = h{gw} ∪ Vw, Ewi of G
I W = {w1, . . . , w|W|} induces a
partitioning
V =
[
wi ∈W
Vwi such that i 6= j =⇒ Vwi ∩ Vwj = ∅
Zone a
Zone b Zone c
gw
1 2 3
4 5 6
7
A workflow dag

36. 18/54
Workflow Model
I Services are connected into
workflows
W = {w1, . . . , w|W|}.
I Each w ∈ W is realized as a
directed acyclic subgraph (dag)
Gw = h{gw} ∪ Vw, Ewi of G
I W = {w1, . . . , w|W|} induces a
partitioning
V =
[
wi ∈W
Vwi such that i 6= j =⇒ Vwi ∩ Vwj = ∅
Zone a
Zone b Zone c
gw
1 2 3
4 5 6
7
A workflow dag

37. 19/54
Client Model
Client population
. . .
Arrival rate λ Departure
Service time µ
. . .
.
.
.
.
.
.
.
.
.
w1 w2 w|W|
Workflows (Markov processes)
I Homogeneous client population
I Clients arrive according to Po(λ), Service times Exp(1
µ)
I Workflow selection: uniform
I Workflow interaction: Markov process

38. 20/54
Observation Model
I idpss inspect network traffic and
generate alert vectors:
ot ,
ot,1, . . . , ot,|V|
∈ N
|V|
0
ot,i is the number of alerts related to
node i ∈ V at time-step t.
I ot = (ot,1, . . . , ot,|V|) is a realization
of the random vector Ot with joint
distribution Z
idps
idps
idps
idps
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
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24
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28
29
30 31
32
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41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

39. 20/54
Observation Model
I idpss inspect network traffic and
generate alert vectors:
ot ,
ot,1, . . . , ot,|V|
∈ N
|V|
0
ot,i is the number of alerts related to
node i ∈ V at time-step t.
I ot = (ot,1, . . . , ot,|V|) is a realization
of the random vector Ot with joint
distribution Z
idps
idps
idps
idps
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
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22 23
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30 31
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40. 21/54
probability
ZO1
ZO2
ZO3
ZO4
ZO5
ZO6
ZO7
ZO8
probability
ZO9
ZO10
ZO11
ZO12
ZO13
ZO14
ZO15
ZO16
probability
ZO17
ZO18
ZO19
ZO20
ZO21
ZO22
ZO23
ZO24
probability
ZO25
ZO26
ZO27
ZO28
ZO29
ZO30
ZO31
ZO32
probability
ZO33
ZO34
ZO35
ZO36
ZO37
ZO38
ZO39
ZO40
probability
ZO41
ZO42
ZO43
ZO44
ZO45
ZO46
ZO47
ZO48
probability
ZO49
ZO50
ZO51
ZO52
ZO53
ZO54
ZO55
ZO56
250 500 750
O
probability
ZO57
250 500 750
O
ZO58
250 500 750
O
ZO59
250 500 750
O
ZO60
250 500 750
O
ZO61
250 500 750
O
ZO62
250 500 750
O
ZO63
250 500 750
O
ZO64
Distributions of # alerts weighted by priority ZOi
(Oi | S
(D)
i , A
(A)
i ) per node i ∈ V
no intrusion intrusion

41. 22/54
Monitoring and Telemetry
Devices
Event bus
Security Policy
Storage Systems
Control actions
Data pipelines
Events
I Emulated devices run monitoring agents that periodically
push metrics to a Kafka event bus.
I The data in the event bus is consumed by data pipelines that
process the data and write to storage systems.
I The processed data is used by an automated security policy to
decide on control actions to execute in the digital twin.

42. 23/54
Feature Selection
I Our framework collects 100s of metrics every time-step.
I We focus on the IDPS alert metric as it provides the most
information for detecting the type of attacks we consider.
0 2500 5000 7500
Probability
Weighted idps alerts
DKL(ZO|0 k ZO|1) = 0.49
−100 0 100
New login attempts
DKL(ZO|0 k ZO|1) = 0.07
−500 0
# New processes
DKL(ZO|0 k ZO|1) = 0.01
−100 0 100
Probability
New tcp connections
DKL(ZO|0 k ZO|1) = 0.01
0 10 20 30
# Blocks written to disk
DKL(ZO|0 k ZO|1) = 0.12
0 20 40
# Blocks read from disk
DKL(ZO|0 k ZO|1) = 0.0
ZO|s=0 (no intrusion) ZO|s=1 (intrusion)

43. 24/54
Defender Model
I Defender action:
a
(D)
t ∈ {0, 1, 2, 3, 4}|V|
I 0 means do nothing. 1 − 4 correspond
to defensive actions (see fig)
I A defender strategy is a function
πD ∈ ΠD : HD → ∆(AD), where
h
(D)
t = (s
(D)
1 , a
(D)
1 , o1, . . . , a
(D)
t−1, s
(D)
t , ot) ∈ HD
I Objective: (i) maintain workflows; and
(ii) stop a possible intrusion:
J ,
T
X
t=1
γt−1
η
|W|
X
i=1
uW(wi , st)
| {z }
workflows utility
− (1 − η)
|V|
X
j=1
cI(st,j, at,j)
| {z }
intrusion and defense costs
!
dmz
rd
zone
admin
zone
Old path
New path
Honeypot App server
Defender
Revoke
certificates
Blacklist
IP
1) Server migration 2) Flow migration and blocking
3) Shut down server 4) Access control

44. 24/54
Defender Model
I Defender action:
a
(D)
t ∈ {0, 1, 2, 3, 4}|V|
I 0 means do nothing. 1 − 4 correspond
to defensive actions (see fig)
I A defender strategy is a function
πD ∈ ΠD : HD → ∆(AD), where
h
(D)
t = (s
(D)
1 , a
(D)
1 , o1, . . . , a
(D)
t−1, s
(D)
t , ot) ∈ HD
I Objective: (i) maintain workflows; and
(ii) stop a possible intrusion:
J ,
T
X
t=1
γt−1
η
|W|
X
i=1
uW(wi , st)
| {z }
workflows utility
− (1 − η)
|V|
X
j=1
cI(st,j, at,j)
| {z }
intrusion and defense costs
!
dmz
rd
zone
admin
zone
Old path
New path
Honeypot App server
Defender
Revoke
certificates
Blacklist
IP
1) Server migration 2) Flow migration and blocking
3) Shut down server 4) Access control

45. 24/54
Defender Model
I Defender action:
a
(D)
t ∈ {0, 1, 2, 3, 4}|V|
I 0 means do nothing. 1 − 4 correspond
to defensive actions (see fig)
I A defender strategy is a function
πD ∈ ΠD : HD → ∆(AD), where
h
(D)
t = (s
(D)
1 , a
(D)
1 , o1, . . . , a
(D)
t−1, s
(D)
t , ot) ∈ HD
I Objective: (i) maintain workflows; and
(ii) stop a possible intrusion:
J ,
T
X
t=1
γt−1
η
|W|
X
i=1
uW(wi , st)
| {z }
workflows utility
− (1 − η)
|V|
X
j=1
cI(st,j, at,j)
| {z }
intrusion and defense costs
!
dmz
rd
zone
admin
zone
Old path
New path
Honeypot App server
Defender
Revoke
certificates
Blacklist
IP
1) Server migration 2) Flow migration and blocking
3) Shut down server 4) Access control

46. 25/54
Attacker Model
I Attacker action: a
(A)
t ∈ {0, 1, 2, 3}|V|
I 0 means do nothing. 1 − 3 correspond
to attacks (see fig)
I An attacker strategy is a function
πA ∈ ΠA : HA → ∆(AA), where HA
is the space of all possible attacker
histories
h
(A)
t = (s
(A)
1 , a
(A)
1 , o1, . . . , a
(A)
t−1, s
(A)
t , ot) ∈ HA
I Objective: (i) disrupt workflows; and
(ii) compromise nodes:
− J
.
.
.
Attacker
login attempts
configure
Automated
system
Server
2) Brute-force
1) Reconnaissance
3) Code execution
Attacker Server
TCP SYN
TCP SYN ACK
port open
Attacker Service Server
malicious
request inject code
execution

47. 25/54
Attacker Model
I Attacker action: a
(A)
t ∈ {0, 1, 2, 3}|V|
I 0 means do nothing. 1 − 3 correspond
to attacks (see fig)
I An attacker strategy is a function
πA ∈ ΠA : HA → ∆(AA), where HA
is the space of all possible attacker
histories
h
(A)
t = (s
(A)
1 , a
(A)
1 , o1, . . . , a
(A)
t−1, s
(A)
t , ot) ∈ HA
I Objective: (i) disrupt workflows; and
(ii) compromise nodes:
− J
.
.
.
Attacker
login attempts
configure
Automated
system
Server
2) Brute-force
1) Reconnaissance
3) Code execution
Attacker Server
TCP SYN
TCP SYN ACK
port open
Attacker Service Server
malicious
request inject code
execution

48. 25/54
Attacker Model
I Attacker action: a
(A)
t ∈ {0, 1, 2, 3}|V|
I 0 means do nothing. 1 − 3 correspond
to attacks (see fig)
I An attacker strategy is a function
πA ∈ ΠA : HA → ∆(AA), where HA
is the space of all possible attacker
histories
h
(A)
t = (s
(A)
1 , a
(A)
1 , o1, . . . , a
(A)
t−1, s
(A)
t , ot) ∈ HA
I Objective: (i) disrupt workflows; and
(ii) compromise nodes:
− J
.
.
.
Attacker
login attempts
configure
Automated
system
Server
2) Brute-force
1) Reconnaissance
3) Code execution
Attacker Server
TCP SYN
TCP SYN ACK
port open
Attacker Service Server
malicious
request inject code
execution

49. 25/54
The Intrusion Response Problem
maximize
πD∈ΠD
minimize
πA∈ΠA
E(πD,πA) [J] (1a)
subject to s
(D)
t+1 ∼ fD · | A
(D)
t , A
(D)
t
∀t (1b)
s
(A)
t+1 ∼ fA · | S
(A)
t , At
∀t (1c)
ot+1 ∼ Z · | S
(D)
t+1, A
(A)
t ) ∀t (1d)
a
(A)
t ∼ πA · | H
(A)
t
, a
(A)
t ∈ AA(st) ∀t (1e)
a
(D)
t ∼ πD · | H
(D)
t
, a
(D)
t ∈ AD ∀t (1f)
where E(πD,πA) denotes the expectation of the random vectors
(St, Ot, At)t∈{1,...,T} under the strategy profile (πD, πA).
(1) can be formulated as a zero-sum Partially Observed Stochastic
Game with Public Observations (a PO-POSG):
Γ = hN, (Si )i∈N , (Ai )i∈N , (fi )i∈N , u, γ, (b
(i)
1 )i∈N , O, Zi

50. 26/54
Existence of a Solution
Theorem
Given the po-posg Γ (2), the following holds:
(A) Γ has a mixed Nash equilibrium and a value function
V ∗ : BD × BA → R that maps each possible initial pair of
belief states (b
(D)
1 , bA
1 ) to the expected utility of the defender
in the equilibrium.
(B) For each strategy pair (πA, πD) ∈ ΠA × ΠD, the best response
sets BD(πA) and BA(πD) are non-empty and correspond to
optimal strategies in two Partially Observed Markov Decision
Processes (pomdps): M (D) and M (A). Further, a pair of
pure best response strategies (π̃D, π̃A) ∈ BD(πA) × BA(πD)
and a pair of value functions (V ∗
D,πA
, V ∗
A,πD
) exist.

51. 27/54
The Curse of Dimensionality
I While (1) has a solution (i.e the game Γ has a value (Thm
1)), computing it is intractable since the state, action, and
observation spaces of the game grow exponentially with |V|.
1 2 3 4 5
104
105
2
105
|S|
|O|
|Ai |
|V|
Growth of |S|, |O|, and |Ai | in function of the number of nodes |V|

52. 28/54
The Curse of Dimensionality
I While (1) has a solution (i.e the game Γ has a value (Thm
1)), computing it is intractable since the state, action, and
observation spaces of the game grow exponentially with |V|.
1 2 3 4 5
104
105
2
105
|S|
|O|
|Ai |
|V|
Growth of |S|, |O|, and |Ai | in function of the number of nodes |V|
We tackle the scability challenge with decomposition

53. 29/54
Intuitively..
rd zone
App servers Honeynet
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 65
The optimal
action here...
Does not directly
depend on the state or
action of a node
down here

54. 30/54
Intuitively..
rd zone
App servers Honeynet
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 65
The optimal
action here...
But they are
not completely
independent either.
How can we
exploit this
structure?
Does not directly
depend on the state
or action of a node
down here

55. 31/54
System Decomposition
To avoid explicitly enumerating the very large state, observation,
and action spaces of Γ, we exploit three structural properties.
1. Additive structure across workflows.
I The game decomposes into additive subgames on the
workflow-level, which means that the strategy for each
subgame can be optimized independently
2. Optimal substructure within a workflow.
I The subgame for each workflow decomposes into subgames on
the node-level that satisfy the optimal substructure property
3. Threshold properties of local defender strategies.
I The optimal node-level strategies for the defender exhibit
threshold structures, which means that they can be estimated
efficiently

56. 31/54
System Decomposition
To avoid explicitly enumerating the very large state, observation,
and action spaces of Γ, we exploit three structural properties.
1. Additive structure across workflows.
I The game decomposes into additive subgames on the
workflow-level, which means that the strategy for each
subgame can be optimized independently
2. Optimal substructure within a workflow.
I The subgame for each workflow decomposes into subgames on
the node-level that satisfy the optimal substructure property
3. Threshold properties of local defender strategies.
I The optimal node-level strategies for the defender exhibit
threshold structures, which means that they can be estimated
efficiently

57. 32/54
Additive Structure Across Workflows (Intuition)
“=”
I If there is no path between i and j in G, then i and j are
independent in the following sense:
I Compromising i has no affect on the state of j.
I Compromising i does not make it harder or easier to
compromise j.
I Compromising i does not affect the service provided by j.
I Defending i does not affect the state of j.
I Defending i does not affect the service provided by j.

58. 33/54
Additive Structure Across Workflows
Definition (Transition independence)
A set of nodes Q are transition independent iff the transition
probabilities factorize as
f (St+1 | St, At) =
Y
i∈Q
f (St+1,i | St,i , At,i )
Definition (Utility independence)
A set of nodes Q are utility independent iff there exists functions
u1, . . . , u|Q| such that the utility function u decomposes as
u(St, At) = f (u1(St,1, At,1), . . . , u1(St,|Q|, At,Q))
and
ui ≤ u0
i ⇐⇒ f (u1, . . . , ui , . . . , u|Q|) ≤ f (u1, . . . , u0
i , . . . , u|Q|)

59. 34/54
Additive Structure Across Workflows
Theorem (Additive structure across workflows)
(A) All nodes V in the game Γ are transition independent.
(B) If there is no path between i and j in the topology graph G,
then i and j are utility independent.
Corollary
Γ decomposes into |W| additive subproblems that can be solved
independently and in parallel.
π
(w1)
k
π
(w2)
k
π
(w|W|)
k
ot,w1
ot,w2
ot,w|W|
.
.
.
⊕
a
(k)
w1
a
(k)
w2
a
(k)
w|W|
a
(k)
t

60. 35/54
Additive Structure Across Workflows: Minimal Example
Zone 1 Zone 2
1
2
3
gw
w1
w2
V = {1, 2, 3},
E = {(1, 2)},
W = {w1, w2},
Z = {1, 2}
a) IT infrastructure b) Transition dependencies
St, At St+1, Ot+1
S
(D)
t+1,1
S
(D)
t,1
A
(D)
t,1 S
(A)
t+1,1
S
(A)
t,1
Ot,1
A
(A)
t,1
S
(D)
t+1,2
S
(D)
t,2
A
(D)
t,2 S
(A)
t+1,2
S
(A)
t,2
Ot,2
A
(A)
t,2
S
(D)
t+1,3
S
(D)
t,3
A
(D)
t,3 S
(A)
t+1,3
S
(A)
t,3
Ot,1
A
(A)
t,3
c) Utility dependencies
St, At Ut
S
(D)
t,1
A
(D)
t,1
Ut,1
S
(A)
t,1
S
(D)
t,2
A
(D)
t,2
Ut,2
S
(A)
t,2
S
(D)
t,3
A
(D)
t,3
Ut,3
S
(A)
t,3

61. 35/54
System Decomposition
To avoid explicitly enumerating the very large state, observation,
and action spaces of Γ, we exploit three structural properties.
1. Additive structure across workflows.
I The game decomposes into additive subgames on the
workflow-level, which means that the strategy for each
subgame can be optimized independently
2. Optimal substructure within a workflow.
I The subgame for each workflow decomposes into subgames on
the node-level that satisfy the optimal substructure property
3. Threshold properties of local defender strategies.
I The optimal node-level strategies for the defender exhibit
threshold structures, which means that they can be estimated
efficiently

62. 36/54
Optimal Substructure Within a Workflow
I Nodes in the same workflow are utility
dependent.
I =⇒ Locally-optimal strategies for
each node can not simply be added
together to obtain an optimal strategy
for the workflow.
I However, the locally-optimal strategies
satisfy the optimal substructure
property.
I =⇒ there exists an algorithm for
constructing an optimal workflow
strategy from locally-optimal
strategies for each node.
Zone 1 Zone 2
1
2
3
gw
w1
w2
V = {1, 2, 3},
E = {(1, 2)},
W = {w1, w2},
Z = {1, 2}
IT infrastructure
Utility dependencies
St, At Ut
S
(D)
t,1
A
(D)
t,1
Ut,1
S
(A)
t,1
S
(D)
t,2
A
(D)
t,2
Ut,2
S
(A)
t,2
S
(D)
t,3
A
(D)
t,3
Ut,3
S
(A)
t,3

63. 36/54
Optimal substructure within a workflow
I Nodes in the same workflow are utility
dependent.
I =⇒ Locally-optimal strategies for
each node can not simply be added
together to obtain an optimal strategy
for the workflow.
I However, the locally-optimal strategies
satisfy the optimal substructure
property.
I =⇒ there exists an algorithm for
constructing an optimal workflow
strategy from locally-optimal
strategies for each node.
Zone 1 Zone 2
1
2
3
gw
w1
w2
V = {1, 2, 3},
E = {(1, 2)},
W = {w1, w2},
Z = {1, 2}
IT infrastructure
Utility dependencies
St, At Ut
S
(D)
t,1
A
(D)
t,1
Ut,1
S
(A)
t,1
S
(D)
t,2
A
(D)
t,2
Ut,2
S
(A)
t,2
S
(D)
t,3
A
(D)
t,3
Ut,3
S
(A)
t,3

64. 37/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
Algorithm 1: Algorithm for combining local strate-
gies
1 Input: Γ: the game,
2 πk: a vector with local strategies
3 Output: (πD, πA): global game strategies
4 Algorithm composite-strategy(Γ, πk)
5 for player k ∈ N do
6 πk ←λ (s
(k)
t , b
(k)
t )
7 a
(k)
t = ()
8 for workflow w ∈ W do
9 for node
i ∈ topological-sort(Vw) do
10 a
(k,i)
t ← π
(i)
k (s
(k)
t , b
(k)
t )
11 if gw 6→
a
(k)
t
t i then
12 a
(k,i)
t ← ⊥
13 end
14 a
(k)
t = a
(k)
t ⊕ a
(k,i)
t
15 end
16 end
17 return a
(k)
t
18 end
19 return (πD, πA)
π
(1)
k
→1
ot,1 a
(k)
t,1 a
(k),0
t,1
π
(2)
k
→2
⊕
ot,2 a
(k)
t,2 a
(k),0
t,2
π
(3)
k
→3
⊕
ot,3 a
(k)
t,3 a
(k),0
t,3
.
.
.
π
(|Vw|)
k
→|Vw|
⊕
ot,|Vw|
a
(k)
t,|Vw| a
(k),0
t,|Vw|
⊕ a
(k)
w

65. 38/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
(π
(i)
D )i∈Vw : local strategies in the same workflow w ∈ W

66. 38/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
a
(D,1)
t ∼ π
(1)
D
Step 1; select action for node 1 according to its local strategy
Workflow action:
(a
(D,1)
t )

67. 38/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
a
(D,2)
t ∼ π
(2)
D
Step 2; update the topology based on the previous local action;
select action a = 0 for unreachable nodes;
move to the next node in the topological ordering (i.e. 2);
select the action for the next node according to its local strategy.
Workflow action:
(a
(D,1)
t , a
(D,2)
t )

68. 38/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
Step 3; update the topology based on the previous local action;
select action a = 0 for unreachable nodes;
move to the next node in the topological ordering (i.e. 3);
select the action for the next node according to its local strategy.
Workflow action:
(a
(D,1)
t , a
(D,2)
t )

69. 38/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
a
(D,4)
t = 0
Step 3; update the topology based on the previous local action;
select action a = 0 for unreachable nodes;
move to the next node in the topological ordering (i.e. 3);
select the action for the next node according to its local strategy.
Workflow action:
(a
(D,1)
t , a
(D,2)
t , ·, 0)

70. 38/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
a
(D,3)
t ∼ π
(3)
D
Step 3; update the topology based on the previous local action;
select action a = 0 for unreachable nodes;
move to the next node in the topological ordering (i.e. 3);
select the action for the next node according to its local strategy.
Workflow action:
(a
(D,1)
t , a
(D,2)
t , a
(D,3)
t , 0)

71. 38/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
a
(D,5)
t ∼ π
(5)
D
Step 4; update the topology based on the previous local action;
select action a = 0 for unreachable nodes;
move to the next node in the topological ordering (i.e. 5);
select the action for the next node according to its local strategy.
Workflow action:
(a
(D,1)
t , a
(D,2)
t , a
(D,3)
t , 0, a
(D,5)
t )

72. 38/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
Step 5; update the topology based on the previous local action;
select action a = 0 for unreachable nodes;
Workflow action:
(a
(D,1)
t , a
(D,2)
t , a
(D,3)
t , 0, a
(D,5)
t )

73. 38/54
Algorithm for Combining Locally-Optimal Node Strategies
into Optimal Workflow Strategies
π
(2)
D
π
(1)
D
π
(3)
D
π
(4)
D
π
(5)
D
π
(6)
D
a
(D,6)
t = 0
Step 5; update the topology based on the previous local action;
select action a = 0 for unreachable nodes;
Workflow action:
(a
(D,1)
t , a
(D,2)
t , a
(D,3)
t , 0, a
(D,5)
t , 0)

74. 39/54
Computational Benefits of Decomposition
I ∴ we can obtain an optimal (best response) strategy for the
full game Γ by combining the solutions to V simpler
subproblems that can be solved in parallel and have
significantly smaller state, observation, and action spaces.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
101
102
103
104
105
log10 |S| log10 |O| log10 |Ak|
log10 |S(i)| log10 |O(i)| log10 |A
(i)
k |
|V|
4 oom
3 oom
2 oom
Space complexity comparison between the full game and the decomposed
game.

75. 40/54
System Decomposition
To avoid explicitly enumerating the very large state, observation,
and action spaces of Γ, we exploit three structural properties.
1. Additive structure across workflows.
I The game decomposes into additive subgames on the
workflow-level, which means that the strategy for each
subgame can be optimized independently
2. Optimal substructure within a workflow.
I The subgame for each workflow decomposes into subgames on
the node-level that satisfy the optimal substructure property
3. Threshold properties of local defender strategies.
I The optimal node-level strategies for the defender exhibit
threshold structures, which means that they can be estimated
efficiently

76. 41/54
Can we Solve the Local Problems with Dynamic
Programming?

77. 42/54
Scalable learning through decomposition (Simulation)
0.00
0.25
0.50
0.75
1.00
avg
defender
utility
|V| = 2, |W| = 1 |V| = 4, |W| = 2 |V| = 8, |W| = 2 |V| = 16, |W| = 2
0 50 100 150
running time (min)
0.00
0.25
0.50
0.75
1.00
avg
attacker
utility
0 50 100 150
running time (min)
0 50 100 150
running time (min)
0 50 100 150
running time (min)
πdecomposition
D πworkflow
D πfull
D upper bound πdecomposition
A πworkflow
A πfull
A
Learning curves obtained during training of ppo to find best response
strategies against randomized opponents; red, purple, blue and brown
curves relate to decomposed strategies; the orange and green curves
relate to the non-decomposed strategies.

78. 43/54
Scalable learning through decomposition (Simulation)
1 2 3 4 5 6 7 8 9 10
2
4
6
8
10
linear
measured
# parallel processes n
|V| = 10
Speedup
S
n
Speedup of completion time when computing best response strategies for
the decomposed game with |V| = 10 nodes and different number of
parallel processes; the subproblems in the decomposition are split evenly
across the processes; let Tn denote the completion time when using n
processes, the speedup is then calculated as Sn = T1
Tn
; the error bars
indicate standard deviations from 3 measurements.

79. 44/54
Threshold Properties of Local Defender Strategies.
I The local problem of the defender can be decomposed in the
temporal domain as
max
πD
T
X
t=1
J = max
πD
τ1
X
t=1
J1 +
τ2
X
t=1
J2 + . . . (2)
where τ1, τ2, . . . are stopping times.
I =⇒ (1) selection of defensive actions is simplified; and (2)
the optimal stopping times are given by a threshold strategy
that can be estimated efficiently:
Belief space B
(j)
D
Switching curve
Υ
Continuation set
C
Stopping set
S
(1, 0, 0)
j healthy
(0, 1, 0)
j discovered
(0, 0, 1)
j compromised

80. 45/54
Threshold Properties of Local Defender Strategies.
Belief space B
(j)
D
Switching curve
Υ
Continuation set
C
Stopping set
S
(1, 0, 0)
j healthy
(0, 1, 0)
j discovered
(0, 0, 1)
j compromised
I A node can be in three attack states s
(A)
t : Healthy,
Discovered, Compromised.
I The defender has a belief state b
(D)
t

81. 46/54
Threshold Properties of Local Defender Strategies.
We estimate the optimal switching curves using a linear
approximation
πD(b(D)
) =







Stop if
h
0 1 θT
i
b(D)
−1
#
0
Continue otherwise
(3)
subject to θ ∈ R2
, θ2 0 and θ1 ≥ 1 (4)
(1, 0, 0)
j healthy
(0, 1, 0)
j discovered
(0, 0, 1)
j compromised
(1, 0, 0)
j healthy
(0, 1, 0)
j discovered
(0, 0, 1)
j compromised
(1, 0, 0)
j healthy
(0, 1, 0)
j discovered
(0, 0, 1)
j compromised
(a) (b) (c)
Examples of learned linear switching curves.

82. 47/54
Learning Best Responses for the Target Infrastructure
(Simulation)
0 2 4 6 8
running time (min)
0.0
0.5
1.0
avg
defender
utility
0 20 40 60 80
running time (min)
0.0
0.5
1.0
avg
attacker
utility
upper bound
our method
ppo
random
ot,i 0
rd zone
App servers Honeynet
dmz
admin
zone
workflow
Gateway idps
quarantine
zone
alerts
Defender
. . .
Attacker Clients
2
1
3 12
4
5
6
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

83. 48/54
Proof Sketch (Threshold Properties)
I Let L(e1, b̂) denote the line segment
that starts at the belief state
e1 = (1, 0, 0) and ends at b̂, where b̂ is
in the sub-simplex that joins e2 and e3.
I All beliefs on L(e1, b̂) are totally
ordered according to the Monotone
Likelihood Ratio (MLR) order. =⇒ a
threshold belief state αb̂ ∈ L(e1, b̂)
exists where the optimal strategy
switches from C to S.
I Since the entire belief space can be
covered by the union of lines L(e1, b̂),
the threshold belief states αb̂1
, αb̂2
, . . .
yield a switching curve Υ.
Belief space B
(j)
D
(the 2-dimensional unit simplex)
sub-simplex B
(j)
D,e1
joining e2 and e3
b̂5
b̂4
b̂3
b̂2
b̂1
b̂6
b̂7
b̂8
b̂9
L(e1, b̂5)
Switching curve
Υ
Threshold
belief state αb̂9
e1
(1, 0, 0)
e2
(0, 1, 0)
e3
(0, 0, 1)

84. 49/54
Decompositional Fictitious Play (DFSP) to Approximate
an Equilibrium
π̃2 ∈ B2(π1)
π2
π1
π̃1 ∈ B1(π2)
π̃0
2 ∈ B2(π0
1)
π0
2
π0
1
π̃0
1 ∈ B1(π0
2)
. . .
π∗
2 ∈ B2(π∗
1)
π∗
1 ∈ B1(π∗
2)
Fictitious play: iterative averaging of best responses.
I Learn best response strategies iteratively through the parallel
solving of subgames in the decomposition
I Average best responses to approximate the equilibrium

85. 50/54
Learning Equilibrium Strategies
0 20 40 60 80
running time (h)
0
2
4
6
8
Approximate exploitability
0 20 40 60 80
running time (h)
0.0
0.2
0.4
0.6
0.8
1.0
Defender utility per episode
0 20 40 60 80
running time (h)
2.5
5.0
7.5
10.0
12.5
Episode length
dfsp simulation dfsp digital twin upper bound ot,i 0 random defense
Learning curves obtained during training of dfsp to find optimal
(equilibrium) strategies in the intrusion response game; red and blue
curves relate to dfsp; black, orange and green curves relate to baselines.

86. 51/54
Evaluation in the Digital Twin
s1,1 s1,2 s1,3 . . . s1,n
s2,1 s2,2 s2,3 . . . s2,n
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Digital Twin
Target
Infrastructure
Model Creation
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation System
Reinforcement Learning
Generalization
Strategy evaluation
Model estimation
Automation
Self-learning systems

87. 51/54
Learning Equilibrium Strategies
0 20 40 60 80
running time (h)
0
2
4
6
8
Approximate exploitability
0 20 40 60 80
running time (h)
0.0
0.2
0.4
0.6
0.8
1.0
Defender utility per episode
0 20 40 60 80
running time (h)
2.5
5.0
7.5
10.0
12.5
Episode length
dfsp simulation dfsp digital twin upper bound ot,i 0 random defense
Learning curves obtained during training of dfsp to find optimal
(equilibrium) strategies in the intrusion response game; red and blue
curves relate to dfsp; black, orange and green curves relate to baselines.

88. 52/54
Learning Equilibrium Strategies (Comparison against
NFSP)
0 10 20 30 40 50 60 70 80
running time (h)
0.0
2.5
5.0
7.5
Approximate exploitability
dfsp nfsp
Learning curves obtained during training of dfsp and nfsp to find
optimal (equilibrium) strategies in the intrusion response game; the red
curve relate to dfsp and the purple curve relate to nfsp; all curves show
simulation results.

89. 53/54
Conclusions
I We develop a framework to
automatically learn security strategies.
I We apply the method to an intrusion
response use case.
I We design a novel decompositional
approach to find near-optimal
intrusion responses for large-scale IT
infrastructures.
I We show that the decomposition
reduces both the computational
complexity of finding effective
strategies, and the sample complexity
of learning a system model by several
orders of magnitude.
s1,1 s1,2 s1,3 . . . s1,n
s2,1 s2,2 s2,3 . . . s2,n
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Emulation
Target
System
Model Creation
System Identification
Strategy Mapping
π
Selective
Replication
Strategy
Implementation π
Simulation
Learning

90. 54/54
Current and Future Work
Time
st
st+1
st+2
st+3
. . .
rt+1
rt+2
rt+3
rrT
1. Extend use case
I Heterogeneous client population
I Extensive threat model of the attacker
2. Extend solution framework
I Model-predictive control
I Rollout-based techniques
I Extend system identification algorithm
3. Extend theoretical results
I Exploit symmetries and causal structure