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Structural results for two-user interactive communication
Jhelum Chakravorty, Aditya Mahajan
McGill University
IEEE International Symposium on Information Theory
July 11, 2016
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 1 / 16
Motivating example: self-driven cars
Operating in a common environment - evolving, Markovian
Access to di�erent information
Goal: avoid collision
Control strategy: involving zero-delay, sequential communication.
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 2 / 16
Motivating example: self-driven cars
Operating in a common environment - evolving, Markovian
Access to di�erent information
Goal: avoid collision
Control strategy: involving zero-delay, sequential communication.
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 2 / 16
The model
Two users; sequentially observe two correlated sources.
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 3 / 16
The model
Two users; sequentially observe two correlated sources.
User 1 User 2
u�2u�
u�1u�u�1
u�
u�1u�
u�2u�
u�2u�
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 3 / 16
The model
User 1 User 2
u�2u�
u�1u�u�1
u�
u�1u�
u�2u�
u�2u�
Source: X it = hit(Zt ,W
it ); W i
t : i.i.d and independent of Zt ,
Zt Markovian.
Action (encoder):
U1t = f 1t (X 1
1:t ,U11:t−1,U
21:t−1), U2
t = f 2t (X 21:t ,U
11:t ,U
21:t−1)
Communication cost: c i : U i → R≥0
Action (decoder):
Z 1t = g1
t (X 11:t ,U
11:t ,U
21:t−1), Z 2
t = g2t (X 2
1:t ,U11:t ,U
21:t)
Distortion function: d it : Z × Z → R≥0
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 3 / 16
The model
User 1 User 2
u�2u�
u�1u�u�1
u�
u�1u�
u�2u�
u�2u�
Source: X it = hit(Zt ,W
it ); W i
t : i.i.d and independent of Zt ,
Zt Markovian.
Action (encoder):
U1t = f 1t (X 1
1:t ,U11:t−1,U
21:t−1), U2
t = f 2t (X 21:t ,U
11:t ,U
21:t−1)
Communication cost: c i : U i → R≥0
Action (decoder):
Z 1t = g1
t (X 11:t ,U
11:t ,U
21:t−1), Z 2
t = g2t (X 2
1:t ,U11:t ,U
21:t)
Distortion function: d it : Z × Z → R≥0
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 3 / 16
The model
User 1 User 2
u�2u�
u�1u�u�1
u�
u�1u�
u�2u�
u�2u�
Source: X it = hit(Zt ,W
it ); W i
t : i.i.d and independent of Zt ,
Zt Markovian.
Action (encoder):
U1t = f 1t (X 1
1:t ,U11:t−1,U
21:t−1), U2
t = f 2t (X 21:t ,U
11:t ,U
21:t−1)
Communication cost: c i : U i → R≥0
Action (decoder):
Z 1t = g1
t (X 11:t ,U
11:t ,U
21:t−1), Z 2
t = g2t (X 2
1:t ,U11:t ,U
21:t)
Distortion function: d it : Z × Z → R≥0
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 3 / 16
The model
User 1 User 2
u�2u�
u�1u�u�1
u�
u�1u�
u�2u�
u�2u�
Source: X it = hit(Zt ,W
it ); W i
t : i.i.d and independent of Zt ,
Zt Markovian.
Action (encoder):
U1t = f 1t (X 1
1:t ,U11:t−1,U
21:t−1), U2
t = f 2t (X 21:t ,U
11:t ,U
21:t−1)
Communication cost: c i : U i → R≥0
Action (decoder):
Z 1t = g1
t (X 11:t ,U
11:t ,U
21:t−1), Z 2
t = g2t (X 2
1:t ,U11:t ,U
21:t)
Distortion function: d it : Z × Z → R≥0
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 3 / 16
The model
User 1 User 2
u�2u�
u�1u�u�1
u�
u�1u�
u�2u�
u�2u�
Source: X it = hit(Zt ,W
it ); W i
t : i.i.d and independent of Zt ,
Zt Markovian.
Action (encoder):
U1t = f 1t (X 1
1:t ,U11:t−1,U
21:t−1), U2
t = f 2t (X 21:t ,U
11:t ,U
21:t−1)
Communication cost: c i : U i → R≥0
Action (decoder):
Z 1t = g1
t (X 11:t ,U
11:t ,U
21:t−1), Z 2
t = g2t (X 2
1:t ,U11:t ,U
21:t)
Distortion function: d it : Z × Z → R≥0
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 3 / 16
The model
User 1 User 2
u�2u�
u�1u�u�1
u�
u�1u�
u�2u�
u�2u�
time
t t+1
Zt X1t U1
t Z1t X2
t U2t Z2
t
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 3 / 16
The optimization problem
Encoding strategy: f i := (f i1 , · · · , f iT ), i ∈ {1, 2}Decoding strategy: gi := (g i
1, · · · , g iT ), i ∈ {1, 2}
Communication strategy: (f1, f2, g1, g2)
A key feature: The estimate Z it is generated at each step
Di�culty: Information available (and hence the domain of the
strategies) growing with time!
Example: binary alphabets; at time t, a minimum of 23t−2
possibilities for encoding-decosing strategies at each user!
T = 3; ≈ 1019 possible communication strategies!
This paper: Constant Z .
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 4 / 16
The optimization problem
Encoding strategy: f i := (f i1 , · · · , f iT ), i ∈ {1, 2}Decoding strategy: gi := (g i
1, · · · , g iT ), i ∈ {1, 2}
Communication strategy: (f1, f2, g1, g2)
Finite horizon optimization problem
Choose a communication strategy (f1, f2, g1, g2) that minimizes
J(f1, f2, g1, g2) = E[ T∑t=1
2∑i=1
[c i (U i
t) + d it (Zt , Z
it )]].
A key feature: The estimate Z it is generated at each step
Di�culty: Information available (and hence the domain of the
strategies) growing with time!
Example: binary alphabets; at time t, a minimum of 23t−2
possibilities for encoding-decosing strategies at each user!
T = 3; ≈ 1019 possible communication strategies!
This paper: Constant Z .
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 4 / 16
The optimization problem
A key feature: The estimate Z it is generated at each step
Di�culty: Information available (and hence the domain of the
strategies) growing with time!
Example: binary alphabets; at time t, a minimum of 23t−2
possibilities for encoding-decosing strategies at each user!
T = 3; ≈ 1019 possible communication strategies!
This paper: Constant Z .
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 4 / 16
The optimization problem
A key feature: The estimate Z it is generated at each step
Di�culty: Information available (and hence the domain of the
strategies) growing with time!
Example: binary alphabets; at time t, a minimum of 23t−2
possibilities for encoding-decosing strategies at each user!
T = 3; ≈ 1019 possible communication strategies!
This paper: Constant Z .
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 4 / 16
The optimization problem
A key feature: The estimate Z it is generated at each step
Di�culty: Information available (and hence the domain of the
strategies) growing with time!
Example: binary alphabets; at time t, a minimum of 23t−2
possibilities for encoding-decosing strategies at each user!
T = 3; ≈ 1019 possible communication strategies!
This paper: Constant Z .
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 4 / 16
Literature overview: real-time communication
Point-to-point case
Witsenhausen, 1979; source-coding, structure of optimal strategies
Walrand and Varaiya, 1983; source-channel coding, noisy channel
with noiseless feedback, structural results, DP decomposition
Teneketzis, 2006; source-channel coding, noisy channel, no feedback
Mahajan and Teneketzis, 2009; DP decomposition
Kaspi and Merhav, 2012; source-coding with variable-rate
quantization
Asnani and Weissman, 2013; source-coding with �nite lookahead
Multi-terminal case
Nayyar and Teneketzis, 2011; source-coding without feedback
Yuksel, 2013; source-coding with feedback
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 5 / 16
Literature overview: real-time communication
Point-to-point case
Witsenhausen, 1979; source-coding, structure of optimal strategies
Walrand and Varaiya, 1983; source-channel coding, noisy channel
with noiseless feedback, structural results, DP decomposition
Teneketzis, 2006; source-channel coding, noisy channel, no feedback
Mahajan and Teneketzis, 2009; DP decomposition
Kaspi and Merhav, 2012; source-coding with variable-rate
quantization
Asnani and Weissman, 2013; source-coding with �nite lookahead
Multi-terminal case
Nayyar and Teneketzis, 2011; source-coding without feedback
Yuksel, 2013; source-coding with feedback
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 5 / 16
Literature overview: real-time communication
Point-to-point case
Witsenhausen, 1979; source-coding, structure of optimal strategies
Walrand and Varaiya, 1983; source-channel coding, noisy channel
with noiseless feedback, structural results, DP decomposition
Teneketzis, 2006; source-channel coding, noisy channel, no feedback
Mahajan and Teneketzis, 2009; DP decomposition
Kaspi and Merhav, 2012; source-coding with variable-rate
quantization
Asnani and Weissman, 2013; source-coding with �nite lookahead
Multi-terminal case
Nayyar and Teneketzis, 2011; source-coding without feedback
Yuksel, 2013; source-coding with feedback
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 5 / 16
Literature overview: real-time communication
Point-to-point case
Witsenhausen, 1979; source-coding, structure of optimal strategies
Walrand and Varaiya, 1983; source-channel coding, noisy channel
with noiseless feedback, structural results, DP decomposition
Teneketzis, 2006; source-channel coding, noisy channel, no feedback
Mahajan and Teneketzis, 2009; DP decomposition
Kaspi and Merhav, 2012; source-coding with variable-rate
quantization
Asnani and Weissman, 2013; source-coding with �nite lookahead
Multi-terminal case
Nayyar and Teneketzis, 2011; source-coding without feedback
Yuksel, 2013; source-coding with feedback
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 5 / 16
Literature overview: real-time communication
Point-to-point case
Witsenhausen, 1979; source-coding, structure of optimal strategies
Walrand and Varaiya, 1983; source-channel coding, noisy channel
with noiseless feedback, structural results, DP decomposition
Teneketzis, 2006; source-channel coding, noisy channel, no feedback
Mahajan and Teneketzis, 2009; DP decomposition
Kaspi and Merhav, 2012; source-coding with variable-rate
quantization
Asnani and Weissman, 2013; source-coding with �nite lookahead
Multi-terminal case
Nayyar and Teneketzis, 2011; source-coding without feedback
Yuksel, 2013; source-coding with feedback
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 5 / 16
Literature overview: real-time communication
Point-to-point case
Witsenhausen, 1979; source-coding, structure of optimal strategies
Walrand and Varaiya, 1983; source-channel coding, noisy channel
with noiseless feedback, structural results, DP decomposition
Teneketzis, 2006; source-channel coding, noisy channel, no feedback
Mahajan and Teneketzis, 2009; DP decomposition
Kaspi and Merhav, 2012; source-coding with variable-rate
quantization
Asnani and Weissman, 2013; source-coding with �nite lookahead
Multi-terminal case
Nayyar and Teneketzis, 2011; source-coding without feedback
Yuksel, 2013; source-coding with feedback
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 5 / 16
Literature overview: real-time communication
So, what is new?
Our paper:
Real-time communication in tandem with interactive communication
Noiseless channel
Finite-horizon optimization problem
Structure of optimal strategies
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 6 / 16
Literature overview: real-time communication
So, what is new?
Our paper:
Real-time communication in tandem with interactive communication
Noiseless channel
Finite-horizon optimization problem
Structure of optimal strategies
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 6 / 16
Literature overview: real-time communication
So, what is new?
Our paper:
Real-time communication in tandem with interactive communication
Noiseless channel
Finite-horizon optimization problem
Structure of optimal strategies
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 6 / 16
Team-theoretic approach
We use Team Theory to characterize the qualitative properties of the
solution
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 7 / 16
Team-theoretic approach
Team ?
Multiple players,
Access to di�erent information,
Decentralized setup
Common objective
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 7 / 16
Team-theoretic approach
Team ?
Multiple players,
Access to di�erent information,
Decentralized setup
Common objective
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 7 / 16
Team-theoretic approach
Team ?
Multiple players,
Access to di�erent information,
Decentralized setup
Common objective
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 7 / 16
Team-theoretic approach
Team ?
Multiple players,
Access to di�erent information,
Decentralized setup
Common objective
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 7 / 16
Team-theoretic approach
Team ?
Multiple players,
Access to di�erent information,
Decentralized setup
Common objective
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 7 / 16
Team-theoretic approach
Team ?
Multiple players,
Access to di�erent information,
Decentralized setup
Common objective
Our model is a team.
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 7 / 16
Solution methodology for team problems
Mahajan, 2013 - Control sharing
Nayyar, Mahajan and Teneketzis, 2013 - Partial history sharing
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 8 / 16
Solution methodology for team problems
Mahajan, 2013 - Control sharing
Nayyar, Mahajan and Teneketzis, 2013 - Partial history sharing
Step 1: Person-by-person approach: su�cient statistic for Xi1:t
Step 2: Common information approach: su�cient statistic for
(U11:t−1,U
21:t−1) for user 1 and for (U1
1:t,U21:t−1) for user 2 and a
suitable dynamic program
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 8 / 16
Solution methodology for team problems
Mahajan, 2013 - Control sharing
Nayyar, Mahajan and Teneketzis, 2013 - Partial history sharing
Step 1: Person-by-person approach: su�cient statistic for Xi1:t
Step 2: Common information approach: su�cient statistic for
(U11:t−1,U
21:t−1) for user 1 and for (U1
1:t,U21:t−1) for user 2 and a
suitable dynamic program
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 8 / 16
Solution methodology for team problems
Person-by-person approach
Arbitrarily �x the strategy of one user and search for the best response
strategy of the other.
Identify a su�cient statistic ξit|t−1 of x i1:t .
No loss of optimality:
U1t = f 1t (Ξ1
t|t−1,U11:t−1,U
21:t−1),U2
t = f 2t (Ξ2t|t−1,U
11:t ,U
21:t−1).
Similar structure for the decoder.
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 8 / 16
Solution methodology for team problems
Common information approach
Based on common information available to both users, identify a
su�cient statistic: π1t of (u11:t−1, u21:t−1) at user 1 and a su�cient
statistic π2t of (u11:t , u21:t−1) at user 2
No loss of optimality: U1t = f 1t (Ξ1
t|t−1,Π1t ), U2
t = f 2t (Ξ2t|t−1,Π
2t )
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 8 / 16
Step 1: person-by person approach
Key Lemma: conditional independence
P(x11:t , x21:t | z , u11:t , u
21:t) =
∏i∈{1,2}
P(x i1:t | z , u11:t , u21:t)
P(x11:t , x21:t | z , u11:t−1, u
21:t−1) =
∏i∈{1,2}
P(x i1:t | z , u11:t−1, u21:t−1)
Similar results:
CEO problem - V. Prabhakaran, Ramchandran and Tse; Allerton, 2004
Control sharing - Mahajan; IEEE TAC, 2013
Secret key agreement - Tyagi and Watanabe; IEEE TIT, 2015
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 9 / 16
Step 1: person-by person approach
The belief states
ξit|t−1(z) = P(Z = z |X i1:t = x i1:t ,U1:t−1 = u1:t−1),
ξit|t(z) = P(Z = z |X i1:t = x i1:t ,U1:t = u1:t),
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 9 / 16
Step 1: person-by person approach
The belief states
ξit|t−1(z) = P(Z = z |X i1:t = x i1:t ,U1:t−1 = u1:t−1),
ξit|t(z) = P(Z = z |X i1:t = x i1:t ,U1:t = u1:t),
Lemma: update of ξit|t−1 and ξit|t
There exist functions F it|t , F
it+1|t , i ∈ {1, 2}, such that
ξit|t = F it|t(ξit|t−1, u1:t , f
−i), ξit+1|t = F it+1|t
(ξit|t , u1:t , x
it+1
).
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 9 / 16
Step 1: P-by-P approach - structure of optimal strategies
Decoder is solving a �ltering problem.
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 10 / 16
Step 1: P-by-P approach - structure of optimal strategies
Decoder is solving a �ltering problem.
Optimal decoding strategy
Z it = g i (Ξi
t|t), i ∈ {1, 2}, (1)
g i (ξi ) = arg minz i∈Z
∑z∈Z
d i (z , z i )ξi (z).
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 10 / 16
Step 1: P-by-P approach - structure of optimal strategies
Decoder is solving a �ltering problem.
Optimal decoding strategy
Z it = g i (Ξi
t|t), i ∈ {1, 2}, (1)
g i (ξi ) = arg minz i∈Z
∑z∈Z
d i (z , z i )ξi (z).
To �nd best performing encoder at user 1:
Fix g1, g2 as (1), f2 arbitrarily
Find su�cient statistic for the encoder at user 1
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 10 / 16
Step 1: P-by-P approach - structure of optimal strategies
Decoder is solving a �ltering problem.
Optimal decoding strategy
Z it = g i (Ξi
t|t), i ∈ {1, 2}, (1)
g i (ξi ) = arg minz i∈Z
∑z∈Z
d i (z , z i )ξi (z).
To �nd best performing encoder at user 1:
Fix g1, g2 as (1), f2 arbitrarily
Find su�cient statistic for the encoder at user 1
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 10 / 16
Step 1: P-by-P approach - structure of optimal strategies
Lemma: su�cient statistic for encoder at user 1
R1t = (Ξi
t|t−1,U1:t−1) is an information state for the encoder at user 1.
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 10 / 16
Step 1: P-by-P approach - structure of optimal strategies
Lemma: su�cient statistic for encoder at user 1
R1t = (Ξi
t|t−1,U1:t−1) is an information state for the encoder at user 1.
R1t is a function of (X 1
1:t ,U1:t−1), available at user 1.
P(R1t+1 |X 1
1:t ,U1:t−1,U1t ) = P(R1
t+1 |R1t ,U
1t )
E[ ∑i∈{1,2}
(c i (U it) + d i (Z , Z i
t )) |X 11:t ,U1:t−1,U
1t
]= E
[ ∑i∈{1,2}
(c i (U it) + d i (Z , Z i
t )) |R1t ,U
1t
]
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 10 / 16
Step 1: P-by-P approach - structure of optimal strategies
R1t is a controlled Markov process! So,
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 10 / 16
Step 1: P-by-P approach - structure of optimal strategies
R1t is a controlled Markov process! So,
No loss of optimality for Markov strategies
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 10 / 16
Step 1: P-by-P approach - structure of optimal strategies
R1t is a controlled Markov process! So,
No loss of optimality for Markov strategies
Optimal enoding strategy
U1t = f 1t (Ξ1
t|t−1,U1:t−1), U2t = f 2t (Ξ2
t|t−1,U1:t−1,U1t )
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 10 / 16
Step 2: Common information approach
Common information?
Data that is observed by all future decision makers
C 1t = U1:t−1, C 2
t = (U1:t−1,U1t ).
Remaining information - Local information: Lit = ξit|t−1
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 11 / 16
Step 2: Common information approach
Common information?
Data that is observed by all future decision makers
C 1t = U1:t−1, C 2
t = (U1:t−1,U1t ).
Remaining information - Local information: Lit = ξit|t−1
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 11 / 16
Step 2: Common information approach
Common information?
Data that is observed by all future decision makers
C 1t = U1:t−1, C 2
t = (U1:t−1,U1t ).
Remaining information - Local information: Lit = ξit|t−1
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 11 / 16
Step 2: Common information approach
Common information?
Data that is observed by all future decision makers
C 1t = U1:t−1, C 2
t = (U1:t−1,U1t ).
Remaining information - Local information: Lit = ξit|t−1
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 11 / 16
Step 2: Common information approach
Common information?
Data that is observed by all future decision makers
C 1t = U1:t−1, C 2
t = (U1:t−1,U1t ).
Remaining information - Local information: Lit = ξit|t−1
U it = f it (Lit ,C
it ).
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 11 / 16
Step 2: Common information approach
Common information?
Data that is observed by all future decision makers
C 1t = U1:t−1, C 2
t = (U1:t−1,U1t ).
Remaining information - Local information: Lit = ξit|t−1
The coordinated system
From the point of view of a virtual decision maker
Observes C it
Chooses prescriptions φit : Lit 7→ U it
Encoder i uses φit and its local information to generate U it .
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 11 / 16
Step 2: Common information approach
Common information?
Data that is observed by all future decision makers
C 1t = U1:t−1, C 2
t = (U1:t−1,U1t ).
Remaining information - Local information: Lit = ξit|t−1
The coordinated system
From the point of view of a virtual decision maker
Observes C it
Chooses prescriptions φit : Lit 7→ U it
Encoder i uses φit and its local information to generate U it .
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 11 / 16
Step 2: Common information approach
Common information?
Data that is observed by all future decision makers
C 1t = U1:t−1, C 2
t = (U1:t−1,U1t ).
Remaining information - Local information: Lit = ξit|t−1
The coordinated system
From the point of view of a virtual decision maker
Observes C it
Chooses prescriptions φit : Lit 7→ U it
Encoder i uses φit and its local information to generate U it .
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 11 / 16
Step 2: Common information approach
Common information?
Data that is observed by all future decision makers
C 1t = U1:t−1, C 2
t = (U1:t−1,U1t ).
Remaining information - Local information: Lit = ξit|t−1
The coordinated system
From the point of view of a virtual decision maker
Observes C it
Chooses prescriptions φit : Lit 7→ U it
Encoder i uses φit and its local information to generate U it .
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 11 / 16
The coordinated system
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 12 / 16
The coordinated system
Centralized system
Tools from Markov Decision Theory
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 12 / 16
The coordinated system
Centralized system
Tools from Markov Decision Theory
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 12 / 16
The coordinated system
Centralized system
Tools from Markov Decision Theory
Lemma: update of ξit|t−1ξit|t = F i
t|t(ξit|t−1, u
−it , φ−it
).
Recall: ξit|t = F it|t(ξit|t−1, u1:t , f
−i)
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 12 / 16
The coordinated system
Centralized system
Tools from Markov Decision Theory
The belief states
π1t (ξ1, ξ2) = P(Ξ1t|t−1 = ξ1,Ξ2
t|t−1 = ξ2 |U1:t−1 = u1:t−1),
π2t (ξ1, ξ2) = P(Ξ1t|t−1 = ξ1,Ξ2
t|t−1 = ξ2 |U1:t−1 = u1:t−1,U1t = u1t ),
Update similar to ξit|t−1
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 12 / 16
The coordinated system
Theorem: Su�cient statistic
No loss of optimality
U1t = f 1t (ξ1t|t−1,Π
1t ), U2
t = f 2t (ξ2t|t−1,Π2t ).
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 12 / 16
The coordinated system
Theorem: Dynamic program
D it(ξ
it|t) :=
∑z∈Z d i
t (z , g i (ξit|t))ξit|t(z), i ∈ {1, 2}
V 2T+1(π2) = 0. For t = T , T − 1, . . . , 1
V 2t (π2) = min
φ2t : ∆(Z)→U2E[c2(U2
t ) + D2t (Ξ2
t|t) + V 1t+1(Π1
t+1) |
Π2t = π2,U2
t = φ2t (Ξ2t|t−1)],
V 1t (π1) = min
φ1t : ∆(Z)→U1E[c1(U1
t ) + D1t (Ξ1
t|t) + V 2t (Π2
t ) |
Π1t = π1,U1
t = φ1t (Ξ1t|t−1)].
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 12 / 16
Discussion
Contributions? Limitations?
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 13 / 16
Discussion
Contributions
Identify a su�cient statistic at the encoder and the decoder; the
domain of which does not depend on time
The DP: identify optimal strategies
The search complexity increases linearly with time horizon (rather than
double exponentially, as for brute force search)
Natural extension to In�nite horizon setup; time-homogeneous optimal
strategies
Extension to multi-terminal setup: n users; sequentially broadcasts
their own action to all users and generates its own estimate and so on.
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 13 / 16
Discussion
Computational issues
Computationally formidable !
Special case: Finite Z (say, cardinality n). Then, ∆(Z) = Rn−1;∆(∆(Z)×∆(Z)) = R2n−2.
DP similar to POMDP. Minimization over functional space.
Point-based algorithms for continuous-state POMDPs
Discretization based algorithms developed for real-time communication
- Wood, Linder and Yuksel, ISIT 2015
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 13 / 16
Discussion
Computational issues
Computationally formidable !
Special case: Finite Z (say, cardinality n). Then, ∆(Z) = Rn−1;∆(∆(Z)×∆(Z)) = R2n−2.
DP similar to POMDP.
Minimization over functional space.
Point-based algorithms for continuous-state POMDPs
Discretization based algorithms developed for real-time communication
- Wood, Linder and Yuksel, ISIT 2015
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 13 / 16
Discussion
Computational issues
Computationally formidable !
Special case: Finite Z (say, cardinality n). Then, ∆(Z) = Rn−1;∆(∆(Z)×∆(Z)) = R2n−2.
DP similar to POMDP. Minimization over functional space.
Point-based algorithms for continuous-state POMDPs
Discretization based algorithms developed for real-time communication
- Wood, Linder and Yuksel, ISIT 2015
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 13 / 16
Discussion
Computational issues
Computationally formidable !
Special case: Finite Z (say, cardinality n). Then, ∆(Z) = Rn−1;∆(∆(Z)×∆(Z)) = R2n−2.
DP similar to POMDP. Minimization over functional space.
Point-based algorithms for continuous-state POMDPs
Discretization based algorithms developed for real-time communication
- Wood, Linder and Yuksel, ISIT 2015
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 13 / 16
Why is Markovian Zt di�cult?
Key steps:
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 14 / 16
Why is Markovian Zt di�cult?
Key steps:
Conditional independence
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 14 / 16
Why is Markovian Zt di�cult?
Key steps:
Conditional independence
Belief that is not growing with time
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 14 / 16
Why is Markovian Zt di�cult?
Key steps:
Conditional independence
P(x11:t , x21:t | z, u11:t , u
21:t) =∏
i∈{1,2}P(x i1:t | z, u11:t , u21:t)
P(x11:t , x21:t | z1:t, u
11:t , u
21:t) =∏
i∈{1,2}P(x i1:t | z1:t, u11:t , u
21:t)
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 14 / 16
Why is Markovian Zt di�cult?
Key steps:
Conditional independence
P(x11:t , x21:t | z, u11:t , u
21:t) =∏
i∈{1,2}P(x i1:t | z, u11:t , u21:t)
P(x11:t , x21:t | z1:t, u
11:t , u
21:t) =∏
i∈{1,2}P(x i1:t | z1:t, u11:t , u
21:t)
3 3
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 14 / 16
Why is Markovian Zt di�cult?
Key steps:
Belief that is not growing with time
f 1t (X 11:t ,U
11:t−1,U
21:t−1) −→
f 1t (P(Z |X 11:t ,U
11:t−1,U
21:t−1),
U11:t−1,U
21:t−1)
f 1t (X 11:t ,U
11:t−1,U
21:t−1) −→
f 1t (P(Z1:t |X 11:t ,U
11:t−1,U
21:t−1),
U11:t−1,U
21:t−1)
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 14 / 16
Why is Markovian Zt di�cult?
Key steps:
Belief that is not growing with time
f 1t (X 11:t ,U
11:t−1,U
21:t−1) −→
f 1t (P(Z |X 11:t ,U
11:t−1,U
21:t−1),
U11:t−1,U
21:t−1)
f 1t (X 11:t ,U
11:t−1,U
21:t−1) −→
f 1t (P(Z1:t |X 11:t ,U
11:t−1,U
21:t−1),
U11:t−1,U
21:t−1)
3 7
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 14 / 16
Future scope
Study of some special structure of Markovian Zt so that the belief
states do not grow with time
Finite state machines - DP decomposition
Lipsa and Martins, IEEE TAC, 2011: Optimality of threshold-based
strategies for Gaussian setup and the transmitter may transmit or not
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 15 / 16
Thank you !
Jhelum Chakravorty (McGill) Interactive communication, ISIT 2016 July 11, 2016 16 / 16
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