Algorithms for Stochastic Games on Interference Channels - ECE@IISc

Algorithms for Stochastic Games on Interference Channels Krishna Chaitanya A, Utpal Mukherji, Vinod Sharma Department of ECE, Indian Institute of Science, Bangalore-560012 Email: {akc, utpal, vinod} @ece.iisc.ernet.in Abstract—We consider a wireless channel shared by multiple transmitter-receiver pairs. Their transmissions interfere with each other. Each transmitter-receiver pair aims to maximize its long-term average transmission rate subject to an average power constraint. This scenario is modeled as a stochastic game. We then formulate the problem of finding a Nash equilibrium (NE) as an affine variational inequality (VI) problem, and present an algorithm to solve the VI. We provide sufficient conditions for uniqueness of the NE and convergence of the algorithm which are much weaker than the sufficient conditions available in literature. We also provide a distributed algorithm to compute Pareto optimal solutions for the proposed game. Index Terms—Interference channel, stochastic game, Nash equilibrium, distributed algorithm, variational inequality, Pareto point.

I. I NTRODUCTION In the present wireless communication scenario, interference is unavoidable. In an interference channel, the transmission of each user causes interference to other receivers. In such a competitive communication setup, all the transmitters aim to maximize their communication rates. This system has been modeled in game theoretic framework and its equilibrium points have been computed ([1]-[6]). Parallel Gaussian interference channels have been considered in [1]. Each user is assumed to be selfish and intelligent and aims to maximize its own rate under total power constraint. Existence and uniqueness of its Nash equilibrium (NE) is established. It is shown that best response mapping for this problem is water-filling. The conditions for uniqueness of NE and convergence of iterative water-filling in [1] are very strong. The same authors extend these results to a multiantenna system in [7] and consider an asynchronous version of iterative water-filling in [8]. For parallel Gaussian interference channels, a stochastic game was formulated based on the channel estimates and an online algorithm was proposed in [2], that converges to its NE. Uniqueness and convergence of the online algorithm is guaranteed under very strong assumptions. In general, existence of multiple NE for this system has been established in [5]. Using variational inequalities, an algorithm was presented that converges to a NE which minimizes the overall weighted interference. In [9] for a 2-user parallel Gaussian interference channel, NE for a power allocation game was shown to be better than the NE for a channel selection game. c 2015 IEEE 978-1-4799-6619-6/15/$31.00

In [10], the minimization of transmit power was considered in parallel Gaussian interference channels subject to rate constraints for each user. Sufficient conditions were provided for uniqueness of the NE and convergence of the algorithm developed. Even though NE is a celebrated solution of a game, Pareto optimality can provide better solutions. In [3], a decentralized iterative algorithm was proposed to find Pareto optimal points for minimizing power consumption subject to QoS constraints. Only discrete power levels were considered. The problem of interference channels was formulated as a Stackelberg game in [6] and its equilibrium was studied. In [11], we considered power allocation in a non-gametheoretic framework (see also other references in [11] for such a setup). We proposed a centralized algorithm for finding the local Pareto points that maximize sum rate. In this paper, receivers have knowledge of all the channel gains and decode the messages from strong and very strong interferers instead of treating them as noise. In the above literature, authors assumed complete channel knowledge at all transmitters and receivers. In this paper, we also assume the same but in [12], we consider power allocation games under partial information about the channel state at all transmitters and receivers also. We have provided an algorithm in [12] to find NE for the games considered. We consider a stochastic game over additive Gaussian interference channels, where the users want to maximize their long term average rate and have long term average power constraints (for potential advantages of this over one shot optimization, see [13], [14]). For this system we obtain existence of NE and also develop an algorithm to obtain NE via affine variational inequality and using regularization. The convergence of the algorithm is proved under weaker conditions than would be obtained via the methods of [1]. Under the same conditions, we also obtain the uniqueness of the NE. Finally, we provide a distributed algorithm to obtain local Pareto points, which converges under complete generality. We can use this algorithm to also obtain the Pareto point in the setup of [11]. We will show it via an example. The paper is organized as follows. In Section II, we present the system model and formulate it as a stochastic game. In Section III, we study this stochastic game and define the basic terminology. In Section IV, we provide existing conditions on convergence of iterative water-filling. In Section V, we formulate the NE problem as a variational inequality problem

and present an algorithm to solve the variational inequality. In Section VI, we provide weak conditions for convergence of the algorithm. In Section VII, we discuss the Pareto optimal solutions to the proposed game. In Section VIII, we present numerical examples and Section IX concludes the paper. II. S YSTEM MODEL AND N OTATION We consider a fading additive Gaussian wireless channel being shared by N transmitter-receiver pairs. The time axis is slotted and all users’ slots are synchronized. The channel gains of each transmit-receive pair are constant during a slot and change independently from slot to slot. Although not addressed in this paper, our results extend to positive recurrent ergodic Markovian channel state processes. Let Hij (k) be the channel gain from transmitter j to receiver i (for transmitter i, receiver i is the intended receiver). We assume that, {Hij (k), k ≥ 0} is an i.i.d sequence with distribution πij . We also assume that these sequences are independent of each other. The direct channel power gains |Hii (k)|2 ∈ Hd = {h1 , h2 , . . . , hn1 } and the cross channel power gains |Hij (k)|2 ∈ Hc = {g1 , g2 , . . . , gn2 }. We denote (Hij (k), i, j = 1, . . . , N ) by H(k) and its realization vector by h(k) which takes values in H, the set of all possible channel states. The distribution of H(k) is denoted by π. If user i uses power Pi (H) then it gets rate log (1 + Γi (P (H))), where Γi (P (H)) =

α |H |2 P (H) Pi ii i 2 , 1 + j6=i |Hij | Pj (H)

(1)

H is the channel state vector, P (H) = (P1 (H), . . . , PN (H)) and αi is a constant that depends on the modulation and coding used by transmitter i and we assume αi = 1 for all i. The aim of each user i is to choose a power policy to maximize its long term average rate n

1X E[log (1 + Γi (P (H(k))))], ri (Pi , P−i ) , lim sup n→∞ n k=1

subject to average power constraint n

lim sup n→∞

1X E[Pi (H(k))] ≤ P i , for each i, n k=1

where P−i denotes the power policies of all users except user i. We address this problem as a stochastic game problem with the set of feasible power policies of user i denoted by Ai and its utility by ri . Each transmitter is assumed to know all the channel states H(k) at the beginning of slot k. Let A = ΠN i=1 Ai . We limit ourselves to stationary policies, i.e., the power policy for every user in slot k depends only on the channel state H(k) and not on k. In the current setup, it does not entail any loss in optimality. In fact now we can rewrite this optimization problem to find policy P (H) such that ri = EH [log (1 + Γi (P (H)))] is maximized subject to EH [Pi (H)] ≤ P i for all i. We express the power policy of player i by Pi = (Pi (h), h ∈ H), where transmitter i

transmits in channel state h with power Pi (h). We denote the power profile of all players by P = (P1 , . . . , PN ). Theory of variational inequalities offers various learning techniques to find NE of a given game. The equivalence of finding a NE and solving a V I is noted in [18]. A variational inequality problem denoted by V I(K, F ) is defined as follows. Definition 1. Consider a closed and convex set K ⊂ Rn , and a function F : K → K. The variational inequality problem V I(K, F ) is defined as the problem of finding x ∈ K such that F (x)T (y − x) ≥ 0 for all y ∈ K. Definition 2. We say that V I(K, F ) is T • Monotone if (F (x)−F (y)) (x−y) ≥ 0 for all x, y ∈ K. T • Strictly monotone if (F (x) − F (y)) (x − y) > 0 for all x, y ∈ K, x 6= y. • Strongly monotone if there exists an ǫ > 0 such that (F (x) − F (y))T (x − y) ≥ ǫkx − yk2 for all x, y ∈ K. III. G AME T HEORETIC R EFORMULATION In this section, we reformulate the Nash equilibrium problem at hand to an affine variational inequality
problem. N We denote our game by G = (Ai )N i=1 , (ri )i=1 , where ri (Pi , P−i ) = EH [log (1 + Γi (P (H)))] and Ai = {Pi ∈ RN : EH [Pi (H)] ≤ P i , Pi (h) ≥ 0, h ∈ H}.

∗ Definition 3. A point P
is a Nash Equilibrium (NE) of game N N G = (Ai )i=1 , (ri )i=1 if for each player i

ri (P∗i , P∗−i ) ≥ ri (Pi , P∗−i ) for all Pi ∈ Ai .

Existence of a pure NE for the strategic game G follows from the Debreu-Glicksberg-Fan Theorem ([15], page no. 69), since in our game ri (Pi , P−i ) is a continuous function in the profile of strategies P = (Pi , P−i ) ∈ A and concave in Pi . Definition 4. The best-response of player i is a function BRi : A−i → Ai such that BRi (P−i ) is a solution of the optimization problem of maximizing ri (Pi , P−i ), subject to Pi ∈ Ai . We see that the Nash equilibrium is a fixed point of the best-response function. Given other players’ power profile P−i , we use Lagrange method to evaluate the best response of player i. The Lagrangian function is defined by Li (Pi , P−i ) = ri (Pi , P−i ) + λi (P i − EH [Pi (H)]). ∂Li To maximize Li (Pi , P−i ), we solve for Pi such that ∂P = i (h) 0 for each h ∈ H. Thus, the component of the best response of player i, BRi (P−i ) corresponding to channel state h is given by ) ( P (1 + j6=i |hij |2 Pj (h)) 1 , − BRi (P−i ; h) = max 0, λi (P−i ) |hii |2 (2) where λi (P−i ) is chosen such that the average power constraint is satisfied.

It is easy to observe that the best-response of player i to a given strategy of other players is water-filling on fi (P−i ) = (fi (P−i ; h), h ∈ H) where P (1 + j6=i |hij |2 Pj (h)) . (3) fi (P−i ; h) = |hii |2 For this reason, we represent the best-response of player i by WFi (P−i ). The notation used for the overall best-response WF(P) = (WF(P (h)), h ∈ H), where WF(P (h)) = (W F1 (P−1 ; h), . . . , W FN (P−N ; h)) and W Fi (P−i ; h) is as defined in (2). We use WFi (P−i ) = (W Fi (P−i ; h), h ∈ H). It is observed in [1] that the best-response WFi (P−i ) is also the solution of the optimization problem 2

minimize kPi + fi (P−i )k , subject to Pi ∈ Ai .

(4)

As a result we can interpret the best-response as projection of (−fi,1 (P−i ), . . . , −fi,N (P−i )) on to Ai . We denote the projection of x on to Ai by ΠAi (x). We consider (4), as a game in which every player minimizes its cost function 2 kPi + fi (P−i )k with strategy set of player i being Ai . We denote this game by G ′ . This game has the same set of NEs as G because the best responses of these two games are equal. We now formulate the variational inequality problem corresponding to the game G ′ . For this, we rewrite the optimization problem (4) as : !2 P X 1 + j6=i |hij |2 Pj (h) , minimize Pi (h) + |hii |2 h∈H

subject to

Pi ∈ Ai .

(5)

We note that this is a convex optimization problem. Given P−i , necessary and sufficient condition for P∗i to be a solution of the convex optimization problem of player i ([16], page 210) is given by ! P X 1 + j6=i |hij |2 Pj (h) ∗ Pi (h) + |hii |2 h∈H

(xi (h) − Pi∗ (h)) ≥ 0,

for all xi ∈ Ai . Further, P∗ is a NE of the game G ′ if ! P X 1 + j6=i |hij |2 Pj∗ (h) ∗ Pi (h) + |hii |2 h∈H

(xi (h) − Pi∗ (h)) ≥ 0,

(6)

for each player i. We can rewrite the N inequalities in (6) in compact form as T  ˆ + HP ˆ ∗ (x − P∗ ) ≥ 0 for all x ∈ A, (7) P∗ + h

ˆ is a N1 -length block vector with N1 = |H|, and each where h ˆ ˆ block h(h), h ∈ H,is of length N and is defined by h(h) =  1 1 ˆ ˆ |h11n |2 , . . . , |hN N |2 o and H is the block diagonal matrix H = ˆ ˆ diag H(h), h ∈ H with each block H(h) defined by ( 0 if i = j, ˆ [H(h)] ij = |hij |2 else. |hii |2 ,

The characterization of Nash equilibrium in (7) corresponds to an affine variational inequality problem of solving for P such that F (P)T (x − P) ≥ 0 for all x ∈ A, ˆ We denote this variaˆ where F (P) = (I + H)P + h. tional inequality problem by V I(A, F ). We defer solving the V I(A, F ) until section V to study some existing results in the next section. IV. N OTE ON EXISTING RESULTS A condition for uniqueness of the NE, and for convergence of iterative water-filling for parallel Gaussian interference channels to the NE, was presented in [1]. We adapt it to our system in this section. In the next section, using VI, we will substantially weaken this condition. The condition in [1] in the current setup is given by ρ(S max ) < 1, where ρ(S max ) is the spectral radius of the matrix S max , and the elements of the matrix S max are ( 0 if i = j, [S max ]ij = |hij |2 max h∈H |hii |2 , else. We study this condition further. From here on wards, we use the notation max{g1 , . . . , gn2 } γ= . min{h1 , . . . , hn1 }

Lemma 1. ρ(S max ) < 1 if and only if γ
0. We find conditions for ˜ +h Fǫn (P) = HP V I(A, Fǫn ) to be strongly monotone. ˜ is positive semidefinite, V I(A, Fǫ ) is a Theorem 1. If H n strongly monotone V I, for ǫn > 0. T

Proof. (Fǫn (P) − Fǫn (V)) (P − V) ˜ + ǫn P − HV ˜ − ǫn V)T (P − V) = (HP ˜ T (P − V) + ǫn (P − V)T (P − V) = (P − V)T H

≥ ǫn k(P − V)k2 .

Thus, V I(A, Fǫn ) is a strongly monotone V I. Using (9), we can find a solution of V I(A, Fǫn ). It is shown in [18] that as ǫn → 0, the solution of V I(A, Fǫn ) converges to that of V I(A, F ). Thus, we can apply (9) to solve V I(A, Fǫn ) for sufficiently small ǫn > 0, to get a close ˜ is positive semidefinite. approximation of a NE whenever H We refer to this as our regularization algorithm. ˜ is positive definite, V I(A, F ) is a strictly monotone If H V I. A strictly monotone V I admits at most one solution ([18], page 156). Since existence of a solution of V I(A, F ) ˜ is follows from existence of a NE of our game, when H positive definite this solution is in fact unique. We show in the following lemma that our convergence condition is weaker than the existing condition. ˆ < 1 then H ˜ is positive definite matrix. Lemma 3. If ρ(H) ˆ < 1, then all eigenvalues λ1 , . . . , λN of H ˆ are Proof. If ρ(H) in unit circle. Thus, the eigenvalues 1 + λi , i = 1, . . . , N of ˜ = I +H ˆ have positive real parts and hence H ˜ is positive H definite. The other-way implication is not true. For example, consider a 3-user interference channel with Hd = {0.3, 0.6} ˆ > 1 but H ˜ is and Hc = {0.2, 0.1}. It can be seen that ρ(H) positive definite. Thus we can find the NE using (9).

˜ is positive semidefinite is a much The condition that H ˆ < 1. The former condition weaker condition than ρ(H) ˜ to lie in the right half plane, requires the eigenvalues of H whereas the latter requires the eigenvalues to lie in a unit circle with (1, 0) as center. ˜ VI. C ONDITIONS FOR P OSITIVE SEMIDEFINITE H In this section we provide sufficient conditions un˜ is positive semidefinite for the 3-user interder which H ˜ is a block diagonal matrix given by ference channel. H o n ˜ ˜ ˆ = I+H(h) where diag H(h), h ∈ H with each block H(h) ˜ is positive semidefinite if I is an N × N identity matrix. H ˜ and only if each diagonal block H(h) for h ∈ H is positive semidefinite. To derive a sufficient condition and a necessary ˜ condition for H(h) to be positive semidefinite, we use the fact that a matrix A is positive semidefinite if and only if its symmetric part 12 (A + AT ) is positive semidefinite. ˜ to In Section V, we provided a sufficient condition for H be positive semidefinite for the 2-user interference channel. ˜ to be positive We first give a necessary condition for H semidefinite in the case of the N-user interference channel, which reduces to the above sufficient condition for the 2-user interference channel. ˜ is positive Lemma 4. For an N-user interference channel, if H semidefinite then γ ≤ 1. Proof. We prove this result using the fact that, if a real n × n symmetric matrix A = [aij ] is positive semidefinite, then aii ajj ≥ a2ij ([21], p. 398). We apply this fact to the symmetric ˜ matrix, B = 1 (H(h)+ ˜ ˜ T (h)), part of a diagonal block of H H 2 given by ( 1 if i = j,  (11) [B]ij = 1  |hij |2 |hji |2 else. 2 |hii |2 + |hjj |2 , In the matrix B, all diagonal elements are equal to 1 and hence if B is positive semidefinite, we have   |hji |2 1 |hij |2 ≤ 1. (12) + 2 |hii |2 |hjj |2

The maximum value of left hand side of (12) occurs in a channel state with hij = max{g1 , . . . , gn2 } and hii = min{h1 , . . . , hn1 } and (12) is true for each h ∈ H if and only if γ ≤ 1. ˜ to be positive We now give a sufficient condition for H semidefinite for a 3-user interference channel.

˜ is positive Lemma 5. In a 3-user interference channel, H semidefinite if and only if 1 γ≤√ . (13) 3 Proof. We consider the symmetric part of a diagonal block of ˜ matrix, B = 1 (H(h) ˜ ˜ T (h)) given by H +H 2 ( 1 if i = j, [B]ij = bij else,

  |hji |2 |h |2 where bij = 12 |hij + . Let λ be an eigenvalue of B. 2 2 |hjj | ii | Then the characteristic polynomial of B is (1 − λ)3 − (b212 + b213 + b223 )(1 − λ) + 2b12 b13 b23 = 0. Let x = 1 − λ and define

The function g(x) has three real roots as a symmetric matrix has real eigenvalues. Let x1 , x2 and x3 be the roots of g(x). Here, g(x) must have atleast one positive root. For, if all the three roots of g(x) are non-positive, then we can write g(x) = (x − x1 )(x − x2 )(x − x3 ) and all the coefficients in the polynomial g(x) should be non-negative. This is not the case. Assume without loss of generality that x3 > 0 is the maximum among the three roots. Rewriting g(x) as g(x) = x(x2 − (b212 + b213 + b223 )) + 2b12 b13 b23 , we see that g(x3 ) = 0 and x3 > 0 implying x23 < (b212 + b213 + b223 ). Thus, to guarantee that all eigenvalues λ are nonnegative, we should have x3 ≤ 1, and this happens if (14)

It can be shown that the polynomial g(x) has a local minimum q b212 +b213 +b223 ∗ ˜ is positive at x = and x3 > x∗ . Thus if H 3

semidefinite, then x∗ < 1 and hence b212 + b213 + b223 < 3. This ˜ to be positive semidefinite. Now is a necessary condition for H observe that there exists a channel state in H with bij = γ, for i 6= j. To satisfy condition (14), it is necessary and sufficient to have γ ≤ √13 . This is a weaker sufficient condition than (8) for N = 3. VII. PARETO O PTIMAL S OLUTIONS

In this section, we consider Pareto optimal solutions to the game G. A power allocation P∗ is Pareto optimal if there does not exist a power allocation P such that ri (Pi , P−i ) ≥ ri (P∗i , P∗−i ) for all i = 1, . . . , N with at least one strict inequality. It is well-known that the solution of a weightedsum optimization of the utility functions is Pareto optimal, i.e., the solution of the following optimization problem, max

N X i=1

i=1

wi ri (Pi , P−i )+ +c

N X i=1

X i

λi (P i −

(P i −

X

X

π(h)Pi (h))

h∈H

π(h)Pi (h))2 .

h∈H

We denote the gradient of L(P, λ) with respect to power

g(x) = x3 − (b212 + b213 + b223 )x + 2b12 b13 b23 .

(b212 + b213 + b223 ) ≤ 1.

L(P, λ) =

N X

wi ri (Pi , P−i ), such that Pi ∈ Ai for all i, (15)

with wi > 0, is Pareto optimal. Thus, since A is compact and ri are continuous, a Pareto point exists for our problem. We apply the weighted-sum optimization (15) to the game G to find a Pareto-optimal power allocation. To solve the non-convex optimization problem in a distributed way, we employ augmented Lagrangian method [22] and solve for the stationary points using the algorithm in [19]. We present the resulting algorithm to find the Pareto power allocation in Algorithm 1. Define the augmented Lagrangian as

Algorithm 1 Augmented Lagrangian method to find Pareto optimal power allocation (1)

(0)

Initialize λi , Pi for all i = 1, . . . , N . for n = 1 → ∞ do P(n) = Steepest Ascent(λ(n) , P(n−1) ) P (n) if |P i − h π(h)Pi (h)| < ǫ for all i = 1, . . . , N then break else P (n) (n+1) (n) λi = λi − α(P i − h π(h)Pi (h)) n=n+1 end if end for function S TEEPEST A SCENT(λ, P) Fix δ, ǫ Initialize t = 1, P(t) = P. loop for i = 1 → N do player i updates his power variables as (t) (t) (t) Qi = Pi + δ▽i L(Pi , P−i , λ) end for Choose P(t+1) as (t) (t) (t) i∗ = arg maxi L(Qi , P−i , λ) − L(Pi , P−i , λ) (t) P(t+1) = (Qi∗ , P−i∗ ) t = t + 1. (t) (t) Till k▽i L(Pi , P−i , λ)k2 < ǫ for each i. end loop return P(t) end function variables of player i by ▽i L(Pi , P−i , λ). In Algorithm 1, the step sizes α, δ are chosen sufficiently small. Convergence of the steepest ascent function in Algorithm 1 is proved in [19]. Since this is a nonconvex optimization problem, Algorithm 1 converges to a local Pareto point ([20]) depending on the initial power allocation. We can get better local Pareto points by initializing the algorithm from different power allocations and choosing the Pareto point which gives the best sum rate among the ones obtained. We consider this in our illustrative examples. VIII. N UMERICAL E XAMPLES In this section we compare the sum rate achieved at a Nash equilibrium and a Pareto optimal point obtained by the algorithm provided above. In all our numerical computations we use τ = 0.1 to find a NE and we choose δ = 0.1, α = 0.25 in computations of Pareto points. We choose a 3-user interference channel. For Example 1: Hd = {3, 1.5} and Hc = {0.1, 0.5} and for Example 2:

larization to find the NE of the proposed power allocation game. The sufficient conditions obtained for convergence of the algorithm based on VI are weaker than those of iterative water-filling. We have also presented a distributed algorithm to find local Pareto optimal solutions. This algorithm converges under general conditions and provides more efficient solutions than the NE.

10 9

Sum Rate

8 7 6 5 4 3 0

Sum rate at Pareto points Sum rate at NE Sum rate at Pareto points using Algorithm in [11] 5

10

15

Average transmit SNR constraint

20

Fig. 1. Sum rate comparison at Pareto points and NE for Example 1. 8

Sum Rate

7

Sum rate at Pareto points Sum rate at NE Sum rate at Pareto points using Algorithm in [11]

6 5 4 3 2 0

5

10

15

Average transmit SNR constraint

20

Fig. 2. Sum rate comparison at Pareto points and NE for Example 2.

Hd = {0.3, 1} and Hc = {0.2, 0.1}. Here, we assume that all elements of Hd , Hc occur with equal probability, i.e., with probability 0.5. ˆ = 0.6667, hence water-filling In Example 1, ρ(H) function is a contraction and iterative water-filling converges ˆ = 1.3333, but H ˜ is a to the unique NE. In Example 2, ρ(H) positive definite matrix since each diagonal block of the block ˜ is positive definite, even though the sufficient diagonal H condition (13) is not satisfied. Thus it has a unique NE. In Example 2, iterative water-filling does not converge but we can use the regularization algorithm to find the NE. To find Pareto optimal points, in both examples, we choose weights wi equal to 1, c = 105 and we use Algorithm 1. We initialize Algorithm 1 from 10 different initial power allocations chosen at random. The best Pareto point among the 10 Pareto points is chosen and plotted in Figure 1. We compare the sum rates for the NE and the Pareto point in Figure 1 for Example 1 and in Figure 2 for Example 2. In Figures 1, 2, we also compare the sum rate at the Pareto point achieved using our distributed algorithm for the setup in [11] where we decode the strong and very strong interference instead of treating them as noise. The two Pareto optimal curves in Figures 1, 2 almost coincide, since in both examples all the channel states have weak interference alone, and this interference is treated as noise. We notice here that Pareto optimal points are more efficient in terms of sum rate than NE. IX. C ONCLUSIONS We have considered a channel shared by multiple users. We presented a variational inequality approach using regu-

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