On resource allocation problems in distributed MIMO wireless networks

E.V. Belmega I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

Elena Veronica Belmega

Advisor: Samson Lasaulce Co-advisor: M´ erouane Debbah

II. Energyefficient communications III. Learning NE in games IV. Perspectives

14 December 2010

1 / 43

Multi-user distributed wireless networks

Ph.D. defense E.V. Belmega

Motivations :

I

Limited resources → resource allocation problems.

I

Decision-wise distributed policies may be preferred to centralized ones.

I

Aim towards autonomous, flexible, self-optimizing networks.

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

I Mutual interference creates interaction among users. I Game theory [vonNeumann-book-1944] provides a set of tools for analyzing interactive decision problems. I Non-cooperative game theory [Fudenberg-book-1991] studies strategic and competitive interactions where players choose their best strategies maximizing their individual performance. 2 / 43

Non-cooperative games

Ph.D. defense E.V. Belmega

Strategic-form game G = (K, {Sk }k∈K , {uk }k∈K ) I K, set of players I Sk , set of actions/strategies that player k can take Q I uk : S → R+ , S = `∈K S` , user’s k payoff function depending on its own choice but also others choices

Definition

[Nash-nas-1950] A strategy profile s ∗ ∈ S is a Nash equilibrium (NE) for G if, for ∗ ) ≤ u (s ∗ , s ∗ ). all k ∈ K and all sk ∈ Sk : uk (sk , s−k k k −k Interpretations of NE : I [On-shot Games] Natural solutions in non-cooperative interactive situations

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

I [Myopic algorithms] Outcomes of iterative best-response type algorithms I [Evolution] May arise as equilibrium points in dynamical systems (interaction among automata)

3 / 43

Outline

Ph.D. defense E.V. Belmega

I. Shannon-rate efficient power allocation games 1. Fast fading MIMO multiple access channels 2. Static parallel interference relay channel II. Energy-efficient communications for single-user MIMO channels

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

III. Learning Nash equilibria in non-cooperative games

IV. Perspectives

4 / 43

Outline

Ph.D. defense E.V. Belmega

I. Shannon-rate efficient power allocation games 1. Fast fading MIMO multiple access channels 2. Static parallel interference relay channel II. Energy-efficient communications for single-user MIMO channels

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

III. Learning Nash equilibria in non-cooperative games

IV. Perspectives

5 / 43

Outline

Ph.D. defense E.V. Belmega

I. Shannon-rate efficient power allocation games 1. Fast fading MIMO multiple access channels 2. Static parallel interference relay channel II. Energy-efficient communications for single-user MIMO channels

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

III. Learning Nash equilibria in non-cooperative games

IV. Perspectives

6 / 43

Multiple access channels with coordinating signal

Ph.D. defense E.V. Belmega

I

Basic multi-user scenario : K transmitters and a single receiver

I

Decoding at the receiver : successive interference cancellation (SIC) or single user decoding (SUD) [Lai-it-2008]

I

MIMO system, terminals equipped of multiple antennas (nr , nt,k )

I

SIC involves a coordination signal dictating the decoding order

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

I Toy example : K = 2 and SIC decoding technique. I The coordination signal is a Bernoulli distributed random variable S ∈ {1, 2}, Pr [S = 1] = p ∈ [0, 1] where p ∈ [0, 1]. I If S = 1, user 1 is decoded in the second place and ”sees” no multiple access interference. E. V. Belmega, S. Lasaulce, and M. Debbah, “Power control in distributed multiple access channels with coordination”, IEEE/ACM WIOPT WNC3, Berlin, Germany, Apr. 2008. 7 / 43

The game

Ph.D. defense

, ukSIC k∈K G SIC = K, ASIC k k∈K

E.V. Belmega I. Shannon-rate efficient PA games

I The players are the transmitters I Strategy of user k : precoding matrices SIC Ak

=

n

(1) (2) Qk = Qk , Qk o (2) (1) (1) (2) (1) (2) Qk = (Qk , Qk ) Qk 0, Qk 0, pTr(Qk ) + pTr(Qk ) ≤ nt P k

I Assuming fast fading (the channel matrices Hk ergodic and stationary processes), perfect channel state information at the receiver and only distribution knowledge at the transmitters, the payoff is the ergodic achievable rates : SIC

uk

(1)

(2)

(1)

(1)

(1)

(2)

(2)

2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

(2)

(1)

(2)

=

E log |I + ρH1 Q1 HH 1 |,

=

H H E log |I + ρH1 Q1 HH 1 + ρH2 Q2 H2 | − E log |I + ρH1 Q1 H1 |,

=

H H E log |I + ρH1 Q1 HH 1 + ρH2 Q2 H2 | − E log |I + ρH2 Q2 H2 |,

=

E log |I + ρH2 Q2 HH 2 |,

(Q1 , Q1 , Q2 , Q2 ) = pRk (Q1 , Q2 ) + (1 − p)Rk (Q1 , Q2 )

(1) (1) (1) R1 (Q1 , Q2 ) (1) (1) (1) R2 (Q1 , Q2 ) R (2) (Q(2) , Q(2) ) 1 1 2 (2) (2) (2) R2 (Q1 , Q2 )

1. FF MIMO MAC

(1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

where ρ = 12 σ

8 / 43

Analysis of the Nash equilibria [Rosen-eco-1965]

Ph.D. defense E.V. Belmega

Theorem [Existence of an NE] Let G = (K, {Sk }k∈K , {uk }k∈K ) be a strategic-form game. If each uk is continuous in the strategy profile s ∈ S and concave in sk ∈ Sk and each Sk is a compact and convex set, then G has at least one pure-strategy NE.

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

Theorem [Uniqueness of the NE] Consider the K -player concave game in the previous Theorem. If the the payoff functions are differentiable and the following diagonally strict concavity condition is met : for all (s 0 , s 00 ) ∈ S 2 with s 0 6= s 00 : K X

II. Energyefficient communications III. Learning NE in games IV. Perspectives

0 00 (sk00 − sk0 )T ∇sk uk (sk0 , s−k ) − ∇sk uk (sk00 , s−k ) >0

k=1

then the uniqueness of the pure-strategy NE is insured.

Theorem There exists a unique pure-strategy NE in the game G SIC . E. V. Belmega, S. Lasaulce, M. Debbah, M. Jungers, and J. Dumont, “Power allocation games in wireless networks of multi-antenna terminals”, Springer Telecommunications Systems Journal, May 2010. 9 / 43

Matrix trace inequality

Ph.D. defense E.V. Belmega I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

Theorem Let A, B be two positive definite matrices and C, D, two positive semi-definite matrices. Then Tr (A − B)(B−1 − A−1 ) + (C − D)[(B + D)−1 − (A + C)−1 ] ≥ 0,

II. Energyefficient communications III. Learning NE in games IV. Perspectives

with equality iff A = B and C = D.

E. V. Belmega, S. Lasaulce, and M. Debbah, “A trace inequality for positive definite matrices”, J. of Inequalities in Pure and Applied Mathematics, Jan. 2009. E. V. Belmega, M. Jungers, and S. Lasaulce, “A generalization of a trace inequality for positive definite matrices”, The Australian J. of Mathematical Analysis and Applications, to appear, Dec. 2010.

10 / 43

Ph.D. defense

Determination of the NE

E.V. Belmega

I Optimizing the ergodic rates is difficult → approximate the payoffs I We make two additional assumptions : I

The channel gains follow the unitary-independent-unitary model ˜ k Wk , E|H ˜ k (i, j)|2 = [Tulino-it-2005] Hk = VH

I

Large systems

σk (i,j) , nt

nt,k = nt

nr → ∞, nt → ∞, nnr → β t

Optimal eigenvectors : (2)

Qk I

(2)

(s)

= Wk Pk WkH and Pk

(2)

(1)

Qk = (Qk (1) , Qk ), Qk (s)

1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications

I Objective is to use Random matrix theory results I

I. Shannon-rate efficient PA games

III. Learning NE in games = Wk Pk (1) Wk H ,

(s)

= Diag(Pk (1), . . . , Pk (nt ))

IV. Perspectives

Optimal eigenvalues : determinant approximation under the large systems assumption

11 / 43

Ph.D. defense

Optimal eigenvalues of user 1

E.V. Belmega

(2)

(1)

Optimizing the approximated utilities w.r.t. P1 (j) and P1 (j) leads to the following water-filling type equations :

(1),NE

P1

(2),NE

P1

φ(j)

ψ(j)

=

=

(j) =

(j) =

∀j ∈ {1, . . . , nt } : nr σ1 (i, j) 1 X nt X nt i=1 1 1+ σ1 (i, m)ψ(m) nt m=1 (1) (j) ρP 1 . (1) 1 + ρP (j)φ(j) 1

1 1 − λ1 ln 2 ρφ(j)

+

1. FF MIMO MAC 2. Static parallel IRC

1 1 − λ1 ln 2 2ργ1 (j)

γk (j)

δk (j)

=

II. Energyefficient communications

+ ,

∀j ∈ {1, . . . , nt }, k ∈ {1, 2} : nr σk (i, j) 1 X 2nt i=1

=

I. Shannon-rate efficient PA games

1+

1 2nt

nt 2 X X

σ` (i, m)δ` (m)

`=1 m=1 (2) 2ρP (j) k (2) 1 + 2ρP (j)γk (j) k

III. Learning NE in games IV. Perspectives

.

where λ1 ≥ 0 is the Lagrangian multiplier tuned in order to meet the power constraint.

The iterative water-filling algorithm : I Converges empirically towards a fixed point (the ”approximated” NE) [Dumont-it-2010] [Biglieri-issta-2002] I Knowledge needed at the transmitter side : global channel distribution information, the strategies of all the users at the previous iteration 12 / 43

Ph.D. defense

Braess paradox

E.V. Belmega

I

General PA n: o (1) (2) (1) (2) (1) (2) ASIC = Qk = (Qk , Qk ) Qk 0, Qk 0, pTr(Qk ) + pTr(Qk ) ≤ nt P k k

I

Space PA : o n (2) (1) (2) (1) (2) (1) SIC,SPA Ak = Qk = (Qk , Qk ) Qk 0, Qk 0, Tr(Qk ) ≤ nt P k , Tr(Qk ) ≤ nt P k

I

Temporal PA : n o (1) (2) (2) (1) (2) (1) SIC,TPA Ak = Qk = (αk P k I, αk P k I) αk ≥ 0, αk ≥ 0, pαk + pαk ≤ 1

I

Price of Anarchy R NE POA = Csum [%] vs the sum

distribution of the coordination signal p ∈ [0, 1] [Koutsoupias-stacs-1999]

I

The spatial PA outperforms the joint space-time PA at the NE.

I

The performance at the NE is close to the centralized sum-capacity.

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

E. V. Belmega, S. Lasaulce, and M. Debbah, “Power allocation games for MIMO multiple access channels with coordination”, IEEE Trans. on Wireless Communications, Jun. 2009. 13 / 43

Comparison with SUD

Ph.D. defense E.V. Belmega

I Achievable rate region for the distributed network vs the distribution of the coordination signal p ∈ [0, 1] [Tse-book-2005] I SIC (Space PA) performs much closer to the sum-capacity upper bound than SUD.

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

14 / 43

Outline

Ph.D. defense E.V. Belmega

I. Shannon-rate efficient power allocation games 1. Fast fading MIMO multiple access channels 2. Static parallel interference relay channel II. Energy-efficient communications for single-user MIMO channels

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

III. Learning Nash equilibria in non-cooperative games

IV. Perspectives

15 / 43

Ph.D. defense

Parallel interference relay channel

E.V. Belmega I. Shannon-rate efficient PA games

I I I

[Sahin-globecom-2007]

1. FF MIMO MAC 2. Static parallel IRC

[Sahin-asilomar-2007] K = 2 transmitter-receiver pairs

II. Energyefficient communications III. Learning NE in games IV. Perspectives

I Static AWGN parallel channels, i.e., Q ≥ 2 non-overlapping frequency bands I The relay node operates in the full-duplex mode I No interference cancellation at the decoding steps E. V. Belmega, B. Djeumou, and S. Lasaulce “What happens when cognitive terminals compete for a relay node ?”, IEEE ICASSP, Taipei, Taiwan, Apr. 2009. 16 / 43

Single-band interference relay channel

Ph.D. defense E.V. Belmega

Three relaying protocols : I Amplify-and-Forward I Relay node amplifies its observation I Amplification gain that saturates the relay power constraint is not necessarily optimal

I Decode-and-Forward I The relay decodes perfectly the source messages I Achievable rates in [Sahin-globecom-2007] I In a distributed scenario, optimizing the relay-transmitter cooperation degrees → strategic interaction

I Estimate-and-Forward I The relay estimates it’s observation I Modified protocol : relay may build either one or two quantized versions of its observation

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

B. Djeumou, E. V. Belmega, Samson Lasaulce, “Interference Relay Channels - Part I : Transmission Rates”, arXiv :0904.2585v1, Apr. 2009. 17 / 43

Ph.D. defense

The power allocation game

E.V. Belmega

I Players : transmitter-receiver pairs I Strategies : θk = (θk(1) , . . . , θk(Q) ) in Sk I Payoff functions : the Shannon

o n (q) P θ k ∈ [0, 1]Q Q q=1 θk ≤ 1 X achievable rates µk (θk , θ−k ) = Rk(q) (θk , θ−k ) =

q

Nash equilibrium analysis : I Non-cooperative transmissions and the relay node → non-concave transmission rates → difficult problem w.r.t. the parallel interference channel [Yu-jsac-2002][Chung-isit-2003][Scutari-it-2008] I Sufficient conditions on the channel parameters that ensure existence of the Nash equilibrium.

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

I Complete characterization of the pure-strategy NE set and convergence of iterative best-response algorithms for AF with constant amplification gain

E. V. Belmega, B. Djeumou, and S. Lasaulce, “Power allocation games in interference relay channels : Existence analysis of Nash equilibria”, EURASIP JWCN, accepted for publication, Nov. 2010. 18 / 43

Ph.D. defense

The optimal relay location I I I

E.V. Belmega Scenario : Q = 2, pathloss channel gains model, relay operates in only one band using AF protocol 2

Achievable network sum-rate at the NE as a function of (xr , yr ) ∈ [−L, L] . The optimal relay position lies on the segment between S1 and D1 .

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

19 / 43

The NE allocation policy I I

θ1NE the fraction of power allocated by user 1 in the IRC band Power allocation policy at the NE θ1NE as a function of (xR , yR ) ∈ [−L, L]2 .

Ph.D. defense E.V. Belmega I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

20 / 43

The NE allocation policy I I

θ2NE the fraction of power allocated by user 2 in the IRC band Power allocation policy at the NE θ2NE as a function of (xR , yR ) ∈ [−L, L]2 .

Ph.D. defense E.V. Belmega I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

21 / 43

The NE allocation policy I I

Power allocation policy at the NE (θ1 , θ2NE ) as a function of (xR , yR ) ∈ [−L, L]2 . The regions where the uses allocate their power to IRC are almost non overlapping.

Ph.D. defense E.V. Belmega I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

22 / 43

Contributions

Ph.D. defense E.V. Belmega

Power allocation games I Fast fading MIMO MAC I Existence and uniqueness of the NE for K ≥ 2 and SUD or SIC with coordination I Iterative algorithms based on water-filling and deterministic approximation of the ergodic rates I At the cost of the coordination signal the sum-rate at the NE is close to the centralized one

I Static parallel IRC I Three relaying protocols : AF, DF, EF I Sufficient conditions ensuring the existence of the NE I AF with constant amplification game : characterization of the set of NE and convergence of iterative best-response algorithms

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

23 / 43

Outline

Ph.D. defense E.V. Belmega

I. Shannon-rate efficient power allocation games 1. Fast fading MIMO multiple access channels 2. Static parallel interference relay channel II. Energy-efficient communications for single-user MIMO channels

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

III. Learning Nash equilibria in non-cooperative games

IV. Perspectives

24 / 43

Energy-efficiency performance measure

Ph.D. defense E.V. Belmega

I Transmit power and Shannon transmission rate are two measures of the cost and benefit of a communication I Energy-efficiency measure : ratio transmission rate (say in bit/s) to transmit power (in J/s) I Two approaches : I Pragmatic [Shah-pimrc-1998][Goodman-pcom-2000][Meshkati-spmag-2007] I Information theoretic [Verdu-it-1990] Capacity-per-unit cost : log2 1 + σP2 log2 1 + σP2 1 = lim = 2 sup P→0 P P σ ln 2 P>0 Similarly to minimum energy-per-bit approach in [ElGamal-it-2006][Verdu-it-2002].

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

25 / 43

Single-user MIMO channel

Ph.D. defense E.V. Belmega I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

I Assume that there are no zero-cost symbols (silence does not convey information). I Only the energy consumed to achieve the transmission rates is taken into account. I Extend the result in [Verdu-it-1990]. 26 / 43

Ph.D. defense

Static and fast fading links

E.V. Belmega

I The energy efficiency function : Gstatic (Q) =

log2 I+ρHQHH Tr(Q)

,

Gfast (Q) =

i h E log2 I+ρHQHH Tr(Q)

I. Shannon-rate efficient PA games 1. FF MIMO MAC

where ρ =

2. Static parallel IRC

1 σ2

I Optimal covariance matrix Q∗ → 0 ∗ Gstatic →

I I

H 1 Tr(HH ) ln 2 nt σ 2

,

∗ Gfast →

1 ln 2

h i Tr(E HHH ) nt σ 2

Message : the transmission rates will go to zero as well. Introduce minimum rate constraints.

II. Energyefficient communications III. Learning NE in games IV. Perspectives

E. V. Belmega, and S. Lasaulce, “Energy-efficient precoding for multiple-antenna terminals”, IEEE Trans. on Signal Processing, accepted for publication, Sep. 2010. 27 / 43

Ph.D. defense

Slow fading links

E.V. Belmega

I The Shannon achievable rate is zero. I A new energy-efficiency metric : Γ(Q, R) =

R[1 − Pout (Q, R)] Tr(Q)

where R(1 − Pout (Q, R)) is the expected throughput [Katz-it-2005] and Pout (Q, R) is the outage probability [Ozarow-vt-1994] : R Pout (Q, R) = Pr log2 I + ρHQHH < R0

I I

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

Problem : the outage probability Pout (Q, R) minimization is still an open issue (conjecture [Telatar-ett-1999]). Main difficulty : intractable p.d.f. and c.d.f. of the instantaneous mutual information

E. V. Belmega, S. Lasaulce, M. Debbah, and A. Hjørungnes “A new energy efficiency function for quasi-static MIMO channels”, IWCMC, Leipzig, Germany, Jun. 2009. 28 / 43

Ph.D. defense

Our conjecture for i.i.d. Rayleigh fading

E.V. Belmega

Conjecture There exists a power threshold P 0 such that : 1. if P ≤ P 0 then Q∗ ∈ arg minPout (Q, R) ⇒ Q∗ ∈ arg max Γ(Q, R) Q

Q

(maximizing Γ(Q, R) is equivalent to minimizing Pout (Q, R)) ; ∗

2. if P > P 0 then Γ(Q, R) has a unique maximum in Q∗ = pn I where p ∗ ≤ P t (uniform power allocation over all the antennas is optimal but not necessarily using all the available power).

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games

I Solved for MISO, SIMO particular cases

IV. Perspectives

E. V. Belmega, and S. Lasaulce, “Energy-efficient precoding for multiple-antenna terminals”, IEEE Trans. on Signal Processing, accepted for publication, Sep. 2010. 29 / 43

Ph.D. defense

Uniform Power Allocation

E.V. Belmega

I Energy-efficiency vs. transmit power p for the UPA Q = I R = 1 bpcu, ρ = 10 dB

p I , n n I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

I Energy-efficiency function is quasi-concave w.r.t. p and saturating all the available power is not optimal I Having several transmit antennas improves the energy-efficiency I A relatively large outage probability has to be tolerated at the optimal point 30 / 43

Contributions

Ph.D. defense E.V. Belmega

I Extended result in [Verdu-it-1990] to MIMO static and fast fading links I Introduced a new efficiency measure for slow fading channel I Conjectured the optimal covariance matrix for slow fading channel I Conjecture solved for MISO and SIMO cases

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

31 / 43

Outline

Ph.D. defense E.V. Belmega

I. Shannon-rate efficient power allocation games 1. Fast fading MIMO multiple access channels 2. Static parallel interference relay channel II. Energy-efficient communications for single-user MIMO channels

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

III. Learning Nash equilibria in non-cooperative games

IV. Perspectives

32 / 43

Interpretations of the Nash equilibrium

Ph.D. defense E.V. Belmega

I One-shot game I Rational players and rationality is common knowledge I Perfect knowledge of the game G (complete information) I Multiplicity of NE may lead the system to a non-equilibrium operating state

I Iterative algorithms I High amount of signaling among players I Perfect knowledge of the payoff functions

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

I Learning algorithms in games [Fudenberg-book-1998] I Each user knows its own action set I At each iteration the transmitters update their mixed strategies based on a feedback from nature I In the long-run, the updating rules may converge to the Nash equilibria of the mean game

33 / 43

Ph.D. defense

Toy example

E.V. Belmega I. Shannon-rate efficient PA games

I I

Two-user MIMO interference channel Decoding at the receiver : single user decoding

1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

E. V. Belmega, H. Tembine, and S. Lasaulce, “Learning to precode in outage minimization games over MIMO interference channels”, IEEE Asilomar Conf., USA, Nov. 2010. 34 / 43

Ph.D. defense

Discrete game GD Non-cooperative discrete power allocation game : I The players are the two transmitter-receiver pairs I The action set is discrete ( Dk =

) nt X Pk Diag(e ` ) ` ∈ {1, . . . , nt }, e ` ∈ {0, 1}nt , e` (i) = ` . ` i=1

I Users’ payoffs are their outage probabilities Pout,k (Qk , Q−k ) = Pr [µk (Qk , Q−k ) < Rk ] µk (Qk , Q−k )

=

log2 Inr + ρk Hk Qk HH + ρk H−k Q−k HH − k −k H log2 Inr + ρk H−k Q−k H−k

E.V. Belmega I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

I Existence of mixed strategy Nash equilibria [Nash-nas-1950]

35 / 43

Ph.D. defense

Reinforcement learning algorithm

E.V. Belmega

I At step t > 0 [Sastri-smc-1994] [n−1]

[n]

I

User k chooses a random Qk ∈ Dk following the distribution p k

I

Nature feedbacks an ACK/NACK bit sk = sk (Qk , Q−k ) n 0 , if µk (Qk , Q−k ) < Rk sk (Qk , Q−k ) = 1 , otherwise

I

[n]

[n]

[n]

1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications

Update : [n]

pk,j =

p [n−1] − γ [n] s [n] p [n−1] , k,j k k,j p [n−1] + γ [n] s [n] (1 − p [n−1] ), k,j k k,j

[n]

(j)

[n]

(j)

if

Qk 6= Dk ,

if

Qk = Dk ,

I. Shannon-rate efficient PA games

(1)

I Statistical convergence [Bena¨ım-lecturenotes-1999][Kushner-book-1997][Borkar-book-2008] to the pure-strategy Nash equilibria of the game GD (provided it exists)

III. Learning NE in games IV. Perspectives

36 / 43

Ph.D. defense

Simulation I

Scenario : nr ,k = nt,k = 2, σ12 = σ22 = 1 W, P 1 = P 2 = 1 W, R1 = 2 bpcu, R2 = 3 bpcu

I I I

Convergence is slow.

Selection of pure strategy NE of GD . Trade-off between the convergence time and convergence to the optimum point.

E.V. Belmega I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

User 2

BF

UPA

BF

(0.631 ; 0.540) (0.402 ; 0.731)

UPA

(0.801 ; 0.214) (0.535 ; 0.305)

II. Energyefficient communications

User 1

III. Learning NE in games IV. Perspectives

37 / 43

Contributions

Ph.D. defense E.V. Belmega

I Based on one bit of feedback the users converge to the NE of GD I Long convergence time I Trade-off between the convergence time and convergence to the optimum I Analysis for the single-user case I To be done : existence of the pure-strategy NE

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

38 / 43

Outline

Ph.D. defense E.V. Belmega

I. Shannon-rate efficient power allocation games 1. Fast fading MIMO multiple access channels 2. Static parallel interference relay channel II. Energy-efficient communications for single-user MIMO channels

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

III. Learning Nash equilibria in non-cooperative games

IV. Perspectives

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Conclusions and perspectives

Ph.D. defense E.V. Belmega

Open issues : I Power allocation non-cooperative games : I NE set analysis in the static MIMO MAC I NE analysis in the parallel IRC

I Energy-efficiency : I I I I

Solve conjectures (Telatar’s conjecture) Rice channel model Multi-user scenario (centralized and distributed) New energy-efficiency metrics

I Learning in games : I Reduce the convergence time

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

Long-term perspective is to develop a general framework to analyze wireless networks combining Information theory, Game theory (non-cooperative and cooperative).

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Bck : System Models I I I I

Ph.D. defense E.V. Belmega

Fast fading MIMO multiple access channel Slow fading parallel interference relay channel General single-user MIMO channel Two user interference channel

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

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Ph.D. defense

Bck : FF MIMO MAC determinant approximation of user’s 1 payoff ˜ (1) (P(1) ) R 1 1

=

˜ (2) (P(2) , P(2) ) R 1 1 2

=

nt 1 X

nr j=1 nr 1 X

(1)

log2 (1 + ρP1 (j)φ(j)) +

nr 1 X

nr i=1

log2 1 +

nt 1 X

nt j=1

E.V. Belmega I. Shannon-rate efficient PA games

σ1 (i, j)ψ(j) −

1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications

φ(j)ψ(j) log2 e nr j=1 n t i 1 Xh (2) (2) log2 (1 + 2ρP1 (j)γ1 (j)) + log2 1 + 2ρP2 (j)γ2 (j) + nr j=1 nt 1 X 1 log2 1 + (σ1 (i, j)δ1 (j) + σ2 (i, j)δ2 (j)) − nr 2nt j=1

III. Learning NE in games IV. Perspectives

nt 1 X ˜ (2) , (γ1 (j)δ1 (j) + γ2 (j)δ2 (j)) log2 e − R 2 nr j=1

with φ(j) ψ(j)

=

=

∀j ∈ {1, . . . , nt } : nr 1 X σ1 (i, j) nt X nt i=1 1 1+ σ1 (i, m)ψ(m) nt m=1 (1) ρP (j) 1 . (1) 1 + ρP (j)φ(j) 1

γk (j) δk (j)

=

∀j ∈ {1, . . . , nt }, k ∈ {1, 2} : nr 1 X σk (i, j) 2nt i=1

=

1+

1 2nt

nt 2 X X `=1 m=1

σ` (i, m)δ` (m)

.

(2) 2ρP (j) k (2) 1 + 2ρP (j)γk (j) k

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Bck : Publications

Ph.D. defense E.V. Belmega

Journal Papers (4 published, 3 accepted for publication) J1 J2 J3 J4 J5 J6 J7

E. V. Belmega, S. Lasaulce, M. Debbah, M. Jungers, and J. Dumont, “Power allocation games in wireless networks of multi-antenna terminals”, Springer Telecommunications Systems Journal, DOI : 10.1007/s11235-010-9305-3, May 2010. E. V. Belmega, S. Lasaulce, and M. Debbah, “Power allocation games for MIMO multiple access channels with coordination”, IEEE Trans. on Wireless Communications, vol. 8, no. 6, pp. 3182–3192, Jun. 2009. E. V. Belmega, M. Jungers, and S. Lasaulce, “A generalization of a trace inequality for positive definite matrices”, The Australian Journal of Mathematical Analysis and Applications (AJMAA), to appear, Dec. 2010. E. V. Belmega, S. Lasaulce, and M. Debbah, “A trace inequality for positive definite matrices”, Journal of Inequalities in Pure and Applied Mathematics (JIPAM), vol. 10, no. 1, pp. 1-4, 2009. E. V. Belmega, B. Djeumou, and S. Lasaulce, “Power allocation games in interference relay channels : Existence analysis of Nash equilibria”, EURASIP Journal on Wireless Communications and Networking (JWCN), accepted for publication, Nov. 2010. E. V. Belmega, and S. Lasaulce, “Energy-efficient precoding for multiple-antenna terminals”, IEEE Trans. on Signal Processing, accepted for publication, Sep. 2010. E. V. Belmega, B. Djeumou, and S. Lasaulce, “Gaussian broadcast channels with an orthogonal and bidirectionnal cooperation link”, EURASIP J. on Wireless Communications and Networking (JWCN), pp.1–16, doi :10.1155/2008/341726, 2008.

Book Chapters (1 published, 1 accepted for publication) BC1 BC2

E. V. Belmega, S. Lasaulce, and M. Debbah, “Shannon rate efficient power allocation games”, Game Theory for Wireless Communications and Networking, Auerbach Publications, Taylor and Francis Group, CRC Press, accepted for publication, 2009. E. V. Belmega, S. Lasaulce, and M. Debbah, “Capacity of cooperative channels : three terminals case study”, Cooperative Wireless Communication, ISBN 142006469X, Auerbach Publications, Taylor and Francis Group, CRC Press, Oct. 2008.

Conference Papers (14 published, 1 submitted) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15

E. V. Belmega, S. Lasaulce, and M. Debbah “Power Control in Distributed Multiple Access Channels with Coordination”, IEEE/ACM Proc. of the Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks and Workshops (WIOPT), Berlin, Germany, 1–8 Apr. 2008. S. Medina Perlaza, E. V. Belmega, S. Lasaulce, and M. Debbah, “On the base station selection and base station sharing in self-configuring networks”, International Conference on Performance Evaluation Methodologies and Tools (VALUETOOLS), Pisa, Italy, invited paper, Oct. 2009. E. V. Belmega, S. Lasaulce, and M. Debbah, “Decentralized handovers in cellular networks with cognitive terminals”, in the IEEE Proc. of the 3rd International Symposium on Communications, Control and Signal Processing (ISCCSP), St Julians, Malta, invited paper, 12–14 Mar. 2008. P. Mertikopoulos, E.V. Belmega, A. Moustakas and S. Lasaulce, “Power Allocation Games in Parallel Multiple Access Channels”, VALUETOOLS submitted, 2010. E. V. Belmega, B. Djeumou, and S. Lasaulce “Resource allocation games in interference relay channels”, IEEE Intl. Conference on Game Theory for Networks (Gamenets), Istanbul, Turkey, invited paper, May 2009. E. V. Belmega, B. Djeumou, and S. Lasaulce “What happens when cognitive terminals compete for a relay node ?”, IEEE Intl. Conference on Acoustics, Speech and Signal Processing (ICASSP), Taipei, Taiwan, 1–4 Apr. 2009. E. V. Belmega, B. Djeumou, and S. Lasaulce “Jeux d’allocation de puissance pour les canaux ` a interf´ erence ` a relais”, GRETSI, Dijon, France, Sep. 2009. B. Djeumou, E. V. Belmega, and S. Lasaulce, “R´ egions de taux atteignables pour le canal ` a interf´ erence ` a relais”, GRETSI, Dijon, France, Sep. 2009. E. V. Belmega, S. Lasaulce, and M. Debbah, “A survey on energy-efficient communications”, IEEE Intl. Symp. on Personal, Indoor and Mobile Radio Communications (PIMRC 2010), Istanbul, Turkey, Sep. 2010. E. V. Belmega, and S. Lasaulce, “An information-theoretic look at MIMO energy-efficient communications”, International Conference on Performance Evaluation Methodologies and Tools (VALUETOOLS), Pisa, Italy, Oct. 2009. E. V. Belmega, S. Lasaulce, M. Debbah, and A. Hjørungnes “A new energy efficiency function for quasi-static MIMO channels”, International Wireless Communications and Mobile Computing Conference (IWCMC), Leipzig, Germany, invited paper, Jun. 2009. E. V. Belmega, H. Tembine, and S. Lasaulce, “Learning to precode in outage minimization games over MIMO interference channels”, IEEE Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, USA, invited paper, Nov. 2010. E. V. Belmega, S. Lasaulce, M. Debbah, and A. Hjørungnes “A new energy efficiency function for quasi-static MIMO channels”, International Wireless Communications and Mobile Computing Conference (IWCMC), Leipzig, Germany, invited paper, Jun. 2009. B. Djeumou, E. V. Belmega, and S. Lasaulce, “Recombinaison de signaux d´ ecod´ es et transf´ er´ es pour le canal ` a relais ` a division fr´ equentielle”, Actes du GRETSI, Troyes, France, 1–4 Sep. 2007. E. V. Belmega, B. Djeumou, and S. Lasaulce, “Performance analysis for the AF-based frequency division cooperative broadcast channel”, in the IEEE Proceedings of the Signal Processing Advances in Wireless Communications conference (SPAWC), Helsinki, Finland, 1–5 Jun. 2007.

I. Shannon-rate efficient PA games 1. FF MIMO MAC 2. Static parallel IRC

II. Energyefficient communications III. Learning NE in games IV. Perspectives

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