On the Optimality of Treating Interference as Noise in Competitive Scenarios

Transcription

1 On the Optimality of Treating Interference as Noise in Competitive Scenarios A. DYTSO, D. TUNINETTI, N. DEVROYE WORK PARTIALLY FUNDED BY NSF UNDER AWARD

2 OUTLINE CHANNEL MODEL AND PAST WORK ADVANTAGES OF MIXED INPUTS (MI) MUTUAL INFO. LOWER BOUND WITH MI CONNECTION TO SUM-SET THEORY MAIN RESULT: TIN IS GDOF OPTIMAL APPLICATION TO BLOCK ASYNCHRONOUS AND CODEBOOK OBLIVIOUS GAUSSIAN IC

3 MAIN CONTRIBUTION SIMPLE ACHIEVABLE SCHEME: IID INPUTS AND TREAT INTERFERENCE AS NOISE (TIN) GAUSSIAN INPUTS ARE BAD IN TIN REGION SOMETIME OUTPERFORMED BY OTHER INPUTS BY CAREFULLY MIXING DISCRETE AND GAUSSIAN INPUTS WE SURPRISINGLY SHOW GDOF OPTIMALITY OF TIN

4 MAIN TAKEAWAY DON T THINK GAUSSIAN

5 CHANNEL MODEL Z 1 X n 1 h 11 W 1 Enc. Dec. Ŵ 1 h 21 h 12 Y1 n USERS COMPETE FOR RESOURCES W 2 Enc. X n 2 h 22 Y n 2 Dec. Ŵ 2 SYMMETRIC: h 11 2 = h 22 2 =S h 21 2 = h 12 2 =I Z 2 1 n POWER CONSTRAINT nx t=1 X 2 j,t apple 1, j =1, 2

6 KNOWN RESULTS CAPACITY REGION C IC =lim n!1 co S P X n 1 Xn 2 =P X n 1 P X n 2 (R 1,R 2 ): R 1 apple 1 n I(Xn 1 ; Y n R 2 apple 1 n I(Xn 2 ; Y n 1 ) 2 ) R. Ahlswede, Multi-way communication channels, in Proc. IEEE Int. Symp. Inf. Theory, Tsahkadsor, Mar. 1973, pp NOT CLEAR HOW TO COMPUTE IT; MULTIVARIATE GAUSSIAN IN NOT OPTIMAL. R. Cheng and S. Verdu, On limiting characterizations of memoryless multiuser capacity regions, IEEE Trans. Inf. Theory, vol. 39, no. 2, pp , 1993.

7 KNOWN RESULTS CAPACITY REGION IN STRONG INTERFERENCE h 11 2 apple h 21 2 and h 22 2 apple h 12 2 Sato, H., "The capacity of the Gaussian interference channel under strong interference," IEEE Trans. Inf. Theory, vol. 27, no. 6, pp. 786,788, Nov THE STRATEGY IS TO DECODE INTERFERENCE FIRST

8 KNOWN RESULTS SUM-CAPACITY IN VERY WEAK INTERFERENCE C IC = {max(r 1 + R 2 ) (R 1,R 2 ) 2 C IC } I.I.D. GAUSSIAN OPTIMAL FOR r S I (1 + I) apple 1 2 Annapureddy, V.S.; Veeravalli, V.V., "Gaussian Interference Networks: Sum Capacity in the Low-Interference Regime and New Outer Bounds on the Capacity Region," Information Theory, IEEE Transactions on, vol.55, no. 7, pp.3032,3050, July 2009 X. Shang, G. Kramer, and B. Chen, A new outer bound and the noisy-interference sum rate capacity for Gaussian interference channels, IEEE Trans. Inf. Theory, vol. 55, no. 2, pp , Motahari, A.S.; Khandani, A.K., "Capacity Bounds for the Gaussian Interference Channel," Information Theory, IEEE Transactions on, vol.55, no.2, pp.620,643, Feb. 2009

9 KNOWN RESULTS CAPACITY REGION TO WITHIN 1/2 BIT R. Etkin, D. Tse, and H. Wang, Gaussian interference channel capacity to within one bit, IEEE Trans. Inf. Theory, vol. 54, no. 12, pp , Dec USED HAN AND KOBAYASHI STRATEGY

10 SUM-CAPACITY SUM-CAPACITY C IC =sup n 1 max P X n P 1 X n 2 n 2X u=1 I(X n u ; Y n u ) INNER BOUND WITH I.I.D. INPUTS R L = max I(X 1 ; Y 1 )+I(X 2 ; Y 2 ) P X1,X 2 =P X1 P X2 TREATING INTERFERENCE AS NOISE (TIN). GAUSSIAN INPUTS ARE NOT OPTIMAL. E. Abbe and L. Zheng, A coordinate system for gaussian networks, IEEE Trans. Inf. Theory, vol. 58, no. 2, pp WHAT IS MAXIMIZING INPUT DISTRIBUTION?

11 SUM GDOF GENERALIZED DEGREES OF FREEDOM (GDOF) = log(i) log(s) () I=S LEVEL OF INTERFERENCE d( ) := lim S!1 1 2 C IC log(s) :I=S DOF IS SPECIAL CASE OF GDOF

12 SUM GDOF GENERALIZED DEGREES OF FREEDOM (GDOF) d W ( ) R. Etkin, D. Tse, and H. Wang, Gaussian interference channel capacity to within one bit, IEEE Trans. Inf. Theory, vol. 54, no. 12, pp , Dec /3 1 TIN G H+K IS GDOF OPTIMAL 1/2 2/3 1 2

13 PROPERTIES OF GAUSSIAN INPUT WORST NOISE LEMMA I(X; hx + Z G ) I(X G ; hx G + Z) BEST INPUT WORST NOISE I(X G ; hx G + Z G ) I(X G ; hx G + Z G ) NOTE: UNDER SECOND MOMENT CONSTRAINS

14 WHY GAUSSIAN IS NOT OPTIMAL? EVALUATE R L = I(X 1 ; Y 1 )+I(X 2 ; Y 2 ) MAXIMIZES MINIMIZES R L = I(X 1 ; Y 1 )+I(X 2 ; Y 2 ) MINIMIZES MAXIMIZES X 1G X 2G

15 MIXED INPUTS INPUT X i,mix = p ix G,i + p 1 ix D,i : i apple 1 X G,i N (0, 1), X D,I PAM(N i ),i2 1, 2 EVALUATE R L = I(X 1 ; Y 1 )+I(X 2 ; Y 2 )

16 MIXED INPUTS DESIRABLE PROPERTIES: I(X D ;hx D + Z G ) log(n) GOOD INPUT I(X G ;hx G + X D + Z G ) I(X G ;hx G + Z G ) GOOD INTERFERENCE Dytso, A.; Tuninetti, D.; Devroye, N., "On discrete alphabets for the two-user Gaussian interference channel with one receiver lacking knowledge of the interfering codebook," ITA, 2014, vol., no., pp.1,8, 9-14 Feb. 2014

17 BEHAVIOR OF MIXED INPUT EVALUATE R L = I(X 1 ; Y 1 )+I(X 2 ; Y 2 ) R L = I(X 1 ; Y 1 )+I(X 2 ; Y 2 ) MAXIMIZES X 1,MIX APPROX. NO EFFECT MAXIMIZES APPROX. NO EFFECT X 2,MIX NOTE:WITH PROPERLY CHOSEN PARAMETERS

18 NEW LOWER BOUND Theorem 1.Let X D be a discrete r.v. with N distinct masses and minimum distance d min, Z G N (0, 1). Then, for any constant h I(X D ; hx D + Z G ) apple log(n) 1 e 2 log 2 log 1+(N 1)e h 2 d 2 min 4 + where d 2 min := min s i,s j 2supp(X D ):i6=j s i s j 2. HOLDS FOR ALL H (SNR) AND ALL INPUT DISTRIBUTIONS

19 SUM SETS: CARDINALITY p p Y 1 = h 11 1X G,1 + h X D,1 p p + h 12 2X G,1 + h X D,2 + Z 1 CARDINALITY???? MINIMUM DISTANCE???? CARDINALITY One can show show that h 11 X D,1 + h 12 X D,2 = X D,1 X D,2 almost everywhere

20 SUM SETS: DMIN h 11 X 1 h 11 X 1 + h 12 X 2 SCENARIO 1 d min d min h 11 X 1 h 11 X 1 + h 12 X 2 SCENARIO 2 d min

21 SUM SETS: DMIN SCENARIO 1 Lemma : d min(h11 X D,1 +h 12 X D,2 ) =min( h 11 d min(x1 ), h 12 d min(x2 )) if X D,2 h 12 d min(xd,2 ) apple h 11 d min(xd,1 ), (1) or if X D,1 h 11 d min(xd,1 ) apple h 12 d min(xd,2 ). (2) SCENARIO 2 Lemma : d min(h11 X 1 +h 12 X 2 ) min(1, )min( h 11 d min(xd,1 ), h 12 d min(xd,2 )) for all (h 11,h 12 ) 2 E R 2 where the complement of E has measure smaller that 2, for any > 0.

22 SELECTING NUMBER OF POINTS POINT-TO-POINT AWGN C = 1 2 log(1 + SNR) I(X D ;hx D + Z G ) log(n) CHOOSE N = b p 1 + SNRc FOR IC: LOG(N) = RATE COMMON MESSAGE IN HK SCHEME SNR := h 2

23 RESULTS HIGH SNR: DGOF Theorem 1: Mixed Inputs achieve d W ( ) up to an outage set of measure 2 for any > 0. FINITE SNR RESULT: GAP Theorem 2: Mixed inputs achieve sum capacity to with in O (log log(snr)) up to an outage set of measure 2, for any > 0

24 APPLICATIONS BLOCK ASYNCHRONOUS IC Y 1,i = h 11 X 1,i + h 12 X 2,i D2 + Z 1,i Y 2,i = h 21 X 1,i D1 + h 22 X 2,i + Z 2,i E. Calvo, J. Fonollosa, and J. Vidal, On the totally asynchronous interference channel with single-user receivers, in Proc. IEEE Int. Symp. Inf. Theory, 2009, pp IC WITH NO CODEBOOK KNOWLEDGE C = max I(X 1 ; Y 1 )+I(X 2 ; Y 2 ) P X n 1,X 2 n =P X 1 P X2 O. Simeone, E. Erkip, and S. Shamai, On codebook information for interference relay channels with out-of-band relaying, IEEE Trans. Inf. Theory, vol. 57, no. 5, pp , May 2011.

25 CONCLUSIONS TREAT INTERFERENCE AS NOISE IS GDOF OPTIMAL IF YOU CHOOSE YOUR INPUTS SMARTLY CURRENT WORK: EXTENSION TO - WHOLE CAPACITY REGION - K-USER, K > 2

26 MORE DETAILS Dytso, A.; Tuninetti, D.; Devroye, N., "On the Two-user Interference Channel with Partial Codebook Knowledge at one Receiver, submitted to IT, April Dytso, A.; Tuninetti, D.; Devroye, N., "On the optimality of TIN in competitive scenarios," to be submitted to IT THANK YOU