jhelum chakravo,rty aditya mahajanamahaj1/projects/real-time/slides/... · 2019. 1. 19. · jhelum...

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.

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.

The model

Two users; sequentially observe two correlated sources.

The model

Two users; sequentially observe two correlated sources.

User 1 User 2

u�2u�

u�1u�u�1

u�1u�

u�2u�

The model

User 1 User 2

u�2u�

u�1u�u�1

u�1u�

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

Distortion function: d it : Z × Z → R≥0

The model

User 1 User 2

u�2u�

u�1u�u�1

u�1u�

u�2u�

it ); W i

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)

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

The model

User 1 User 2

u�2u�

u�1u�u�1

u�1u�

u�2u�

it ); W i

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)

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

The model

User 1 User 2

u�2u�

u�1u�u�1

u�1u�

u�2u�

it ); W i

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)

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

The model

User 1 User 2

u�2u�

u�1u�u�1

u�1u�

u�2u�

it ); W i

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)

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

The model

User 1 User 2

u�2u�

u�1u�u�1

u�1u�

u�2u�

Zt X1t U1

t Z1t X2

t U2t Z2

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 .

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 )]].

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

Point-to-point case

quantization

Multi-terminal case

Point-to-point case

quantization

Multi-terminal case

Point-to-point case

quantization

Multi-terminal case

Point-to-point case

quantization

Multi-terminal case

Point-to-point case

quantization

Multi-terminal case

So, what is new?

Our paper:

Real-time communication in tandem with interactive communication

Noiseless channel

Finite-horizon optimization problem

Structure of optimal strategies

So, what is new?

Our paper:

Noiseless channel

So, what is new?

Our paper:

Noiseless channel

Team-theoretic approach

We use Team Theory to characterize the qualitative properties of the

solution

Team ?

Multiple players,

Access to di�erent information,

Decentralized setup

Common objective

Team ?

Multiple players,

Decentralized setup

Common objective

Team ?

Multiple players,

Decentralized setup

Common objective

Team ?

Multiple players,

Decentralized setup

Common objective

Team ?

Multiple players,

Decentralized setup

Common objective

Team ?

Multiple players,

Decentralized setup

Common objective

Our model is a team.

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

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

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.

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

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)

jhelum chakravo,rty aditya mahajanamahaj1/projects/real-time/slides/... · 2019. 1. 19. · jhelum...

Documents