Two Examples ofSubmodularity in WirelessCommunications
Ni Ding
13 June 2017
www.data61.csiro.au
Outline
IntroductionSubmodular FunctionTarski Fixed Point TheoremSubmodular (Set) Function Minimization
Adaptive Modulation in Network-coded Two-way Relay ChannelSystemTwo-player Game ModelPure Strategy Nash EquilibriumCournot Tatonnement
Communication for OmniscienceSystemMinimum Sum-ratePrincipal Sequence of PartitionsModified Decomposition AlgorithmExtensions: Secret Capacity, Clustering and Data Compression
Conclusion
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Submodularity
• a property of functions defined on lattice [1]1 [2]2.
I lattice: fundamental algebraic structure on partial order.
• applications: economics, machine learning, operations research.
Study on Machine Learning in [3]3:
Submodularity imposes a structure which allows much strongermathematical results than we would be able to achieve without it.
• submodularity on
I vector lattice: discrete convexity and comparative staticsI set lattice: combinatorial optimization, e.g., in graph theory
1Topkis 20012Murota 20053Vondrak 2007
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Lattice
Poset
For a set L and a binary order , (L,) is a poset (partially ordered set)if either a b or a 6 b,∀a, b ∈ L.
examples: (RN ,≤), (1, . . . , 4N ,≤), (2V ,⊆) and (1, 1, 2, 3,⊆)
Lattice
A poset (L,) is a lattice with notation (L,∨,∧) ifa ∨ b = supa, b ∈ L and a ∧ b = infa, b ∈ L,∀a, b ∈ L with sup andinf w.r.t.
• maximum∨L = supL and minimum
∧L = inf L exist;
examples: (RN ,∨,∧), (1, . . . , 4N ,∨,∧) with r ∨ r′ = (maxri , r ′i : i ∈ 1, . . . ,N)and r ∧ r′ = (minri , r ′i : i ∈ 1, . . . ,N); (2V ,∪,∩) and (1, 1, 2,∪,∩)
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Submodular Function
Submodularity
f : (L,∨,∧) 7→ R is submodular if
f (a) + f (b) ≥ f (a ∨ b) + f (a ∧ b), ∀a, b ∈ L.
f is supermodular if −f is submodular.
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Tarski Fixed Point TheoremN-player game model Ω = N , Ai , ci (a)i∈N witha ∈ A = ×i∈NAi ⊆ RN :
• pure strategy Nash equilibrium (PSNE): best response functionψ : A 7→ A with ai ∈ Ai , a−i ∈ ×i ′∈N\iAi ′ and
ψi (a−i ) ∈ arg minci (ai , a−i ) : ai ∈ Ai
, ∀i ∈ N .
a∗ is an PSNE if a∗ = ψ(a∗), i.e., a∗ is a fixed point of ψ.
• question: PSNE exists for discrete A? supermodular game withstrategic complements: ψ : (A,∨,∧) 7→ (A,∨,∧) is non-decreasingif ci is submodolar ∀i .
Tarski Fixed Point Theorem [4]4
The fixed points of non-decreasing ψ : (A,∨,∧) 7→ (A,∨,∧) form a(nonempty) lattice.
4Tarski et. al 1955
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Submodular (Set) FunctionMinimization
For f : (2V ,∪,∩) 7→ R, consider
minf (X ) : X ⊆ V (1)
combinatorial optimization: NP-complete or NP-hard in general
SFM (submodular function minimization) algorithm
If f is submodular, i.e.,
f (X ) + f (Y ) ≥ f (X ∪ Y ) + f (X ∩ Y ), ∀X ,Y ⊆ V ,
(1) can be solved in polynomial time and the minimizers form a lattice:⋃argminf (X ) : X ⊆ V and
⋂argminf (X ) : X ⊆ V exist [5]5.
5Fujishige 2005
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Adaptive Modulation inNetwork-coded Two-way RelayChannel
network-coded two-way relay channel (NC-TWRC): two userscommunicate via a center node, relay ‘R’.
user 1wireless
Rwireless
user 2
physical-layer network coding (PNC): messages x1 and x2 transmittedsimultaneously in phase I, the superposition z broadcast in phase II.
user 1x1
Rx2
user 2
phase I: multiple access (MAC)
user 1z
Rz
user 2
phase II: amplify and forward (AF)
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Adaptive Modulation inNC-TWRC
assumption: m-quadrature amplitude modulation (m-QAM) adopted byeach user:
• constellation size mi = 2ai of user i with ai ∈ 0, 1, . . . , thenumber of bits/symbol, determined by user i
Strategic Complements
increasing best response:
• spectral efficiency: one tends to transmit while the other does so
• equal share of the channel: one increases ai while the other−i = 1, 2 \ i does so
proposal: two-player game model parameterized by user-to-user channelsignal-to-noise ratios (SNRs)
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Two-player Game ModelΩγ = N , Γ, Ai , ci (γi , a)i∈N :• N = 1, 2;• γ = (γ1, γ2) ∈ Γ = Γ1 × Γ2 with γi ∈ Γi being SNR of user i-to-user−i channel determined by PNC scheme;
• a = (a1, a2) ∈ A = A1 ×A2 = 0, 1, . . . ,Am2 with Am being themaximum number of bits/symbol
• cost function: ci : Γi ×A 7→ R+
ci (γi , a) = ce(γi , ai ) + cr (a)
with the cost associated with transmission error rate
ce(γi , ai ) =− ln(5Pb)(2ai − 1)
1.5γi
and the cost associated with spectral efficiency and fairness
cr (a) =a−i + 1
ai + 1
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Pure Strategy Nash Equilibrium(PSNE)
Submodularity
ci : (A,∨,∧) 7→ R+ is submodular, i.e.,
ci (γi , a) + ci (γi , a′) ≥ ci (γi , a ∨ a′) + ci (γi , a ∧ a′),
for all a, a′ ∈ (A,∨,∧), i ∈ N and γ ∈ Γ.
Tarski Fixed Point Theorem =⇒ Existence of PSNE
Pure strategy θ : Γ 7→ A, where θ(γ) = (θ1(γ), θ2(γ)) with θi (γ) ∈ Ai
being the pure strategy of user i when SNRs are γ = (γ1, γ2):
• PSNE θ∗ exists
• The largest PSNE θ∗
and the smallest PSNEs θ∗ exist
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Cournot Tatonnementdetermine extremal PSNEs: Cournot tatonnement [6]6
• Let ψ : Γ×A 7→ A and ψ : Γ×A 7→ A be the maximal andminimal best response functions, respectively, with
ψi (γ, a−i ) =∨
arg minai∈Ai
ci (γi , ai , a−i )
ψi(γ, a−i ) =
∧arg min
ai∈Ai
ci (γi , ai , a−i )
• recursions with θ(0)
(γ) =∨A and θ(0)(γ) =
∧A:
θ(γ) := ψ(γ,θ(γ))
θ(γ) := ψ(γ,θ(γ))
Convergence
θ(k)(γ) and θ(k)(γ) converge monotonically downward and upward
to θ∗(γ) and θ∗(γ), respectively, for all γ.
6Vives 1990
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Experiment Iperformance of extremal PSNEs:A = 0, 1, . . . , 92 and simulation lasts for 104 symbol durations:
extremal PSNEs θ∗
and θ∗ are compared to the single-agent adaptivemodulation (Single-AM) and 2-QAM scheme.
−6 −4 −2 0 2 4 610−5
10−4
10−3
10−2
10−1
100
γ(dB)
biterrorrate
(BER)
θ∗
θ∗
Single-AM2-QAM
(a) bit error rate (BER)
−6 −4 −2 0 2 4 60
2
4
6
8
10
γ(dB)
bits
per
symbol
duration
θ∗
θ∗
Single-AM2-QAM
(b) spectral efficiency
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Experiment IIexample of Cournot tatonnement
A = 0, 1, . . . , 92 and the sequences θ(k)
1 (γ) and θ(k)1 (γ) generated
by Cournot tatonnement for certain γ
1 2 3 40
2
4
6
8
iteration index k
strategy
ofschedu
ler1 θ
(k)1 (γ)
θ∗1(γ)
θ(k)1 (γ)θ∗1(γ)
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Communication for Omniscience
indices of users: a finite ground set V with |V | > 1discrete correlated random source: ZV = (Zi : i ∈ V )
• user i observes an i.i.d. n-sequence Zni of Zi in private
communication for omniscience (CO) [7]7:
• users exchange Zi s directly over noiseless broadcast channels
• goal: attain omniscience, the state that each user recovers ZnV
Minimum Sum-rate Problem
how to attain omniscience with RCO(V ), the minimum total number oftransmissions: value of RCO(V ) and an optimal rate vectorr∗V = (r∗i : i ∈ V )
7Csiszar et. al 2004: CO formulated based on the study on secret capacity
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Example: Coded CooperativeData Exchange (CCDE)
client 1
Z1 = [Wa,Wb,Wc ,Wd ,We ]ᵀ
client 2
Z2 = [Wa,Wb,Wf ]ᵀ
client 3
Z3 = [Wc ,Wd ,Wf ]ᵀ
3-mobile clients in V = 1, 2, 3; Zi : partial observation of a packet setwith Wj denoting a packet
Solutions to Minimum Sum-rate Problem
RCO(V ) = 72 and r∗V = (r∗1 , r
∗2 , r∗3 ) = ( 5
2 ,12 ,
12 ): by packet-splitting
Wj =⇒W(1)j ,W
(2)j ; transmit (r1, r2, r3) = (5, 1, 1) with ri denote the
number of linear combinations of packet chunks W(k)j .
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Omniscience-achievabilityFor X ⊆ V : r(X ) =
∑i∈X ri for rV = (ri : i ∈ V )
H(X ): the amount of randomness in ZX measured by Shannon entropy
Omniscience-achievability [7]8
An omniscience-achievable rV satisfies the Slepian-Wolf (SW) constraint:r(X ) ≥ H(X |V \ X ) = H(V )− H(V \ X ),∀X ( V .
achievable rate vector set:
R(V ) = rV ∈ R|V | : r(X ) ≥ H(X |V \ X ),∀X ( V
minimum sum-rate:
RCO(V ) = minr(V ) : rV ∈ R(V )
constant sum-rate set: Rα(V ) = rV ∈ R(V ) : r(V ) = αoptimal rate vector set: RRCO(V )(V )
8Csiszar et. al 2004
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Nonemptiness of BasePolyhedron
For α ∈ R+, let
fα(X ) =
H(X |V \ X ) X ( V
α X = V.
polyhedron: P(fα,≥) = rV ∈ R|V | : r(X ) ≥ fα(X ),∀X ⊆ V base polyhedron: B(fα,≥) = rV ∈ P(fα,≥) : r(V ) = fα(V ) = α
• B(fα,≥) = Rα(V ) 6= ∅ ⇐⇒ ∃ achievable rV with r(V ) = α
dual set function: f #α (X ) = fα(V )− fα(V \ X )
• B(fα,≥) = B(f #α ,≤) [5]9;
why consider B(f #α ,≤)? f #
α is intersecting submodular, i.e.,
f #α (X ) + f #
α (Y ) ≥ f #α (X ∪ Y ) + f #
α (X ∩ Y ), ∀X ,Y : X ∩ Y 6= ∅.9Fujishige 2005
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Minimum Sum-rateΠ(V ): the set of all partitions of V and Π′(V ) = Π(V ) \ V achievability of α: B(f #
α ,≤) 6= ∅ iff α = minP∈Π(V )
∑C∈P f #
α (C ) [5]10
Minimum Sum-rate
RCO(V ) = maxP∈Π′(V ) φ(P) with the finest maximizer P∗. Here,
φ(P) =∑C∈P
H(V \ C |C )
|P| − 1.
interpretation: ∀C ∈ P, the cut C ,V \ C imposes SW constraintr(V \ C ) ≥ H(V \ C |C ) so that∑
C∈P
r(V \ C ) = (|P| − 1)r(V ) ≥∑C∈P
H(V \ C |C )
A multi-way cut P ∈ Π′(V ) imposes r(V ) ≥ φ(P).
10Fujishige 2005: minP∈Π(V )
∑C∈P f #
α (C) is called the Dilworth truncation of f #α .
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Principal Sequence of Partitions
Principal Sequence of Partitions (PSP) [8]11:
minP∈Π(V )
∑C∈P f
#α (C ) is a piecewise linear increasing curve in α
that is fully characterized by p ≤ |V | − 1 critical points
H(V ) = α0 > α1 > α2 > . . . > αp ≥ 0.
Let Pj be the finest minimizer of minP∈Π(V )
∑C∈P f
#α (C ).
P0 P1 P2 . . . Pp
where P P ′ denotes P ′ is strictly finer than P.
The first critical point determines the solutions to the minimumsum-rate problem: RCO(V ) = α1,P∗ = P1.
11Nagano et. al 2010
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Example of PSP
client 1
Z1 = [Wa,Wb,Wc ,Wd ,We ]ᵀ
client 2
Z2 = [Wa,Wb,Wf ]ᵀ
client 3
Z3 = [Wc ,Wd ,Wf ]ᵀ
0 1 2 3 4 5 6
−5
0
5
10
α1 = 72,P1 = 1, 2, 3
α0 = 0,P0 = 1, 2, 3
α
minP∈Π
(V)
∑ C∈P
f# α(C
)
minP∈Π(V )
∑C∈P f
#α (C )
α
PSP results:α0 > α1 and P0 P1:
• No omniscience-achievablerV if α < α1, because α 6=minP∈Π(V )
∑C∈P f #
α (C );
• RCO(V ) = α1 andP∗ = 1, 2, 3 = P1
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Properties of φ(P) in PSP
αj and Pj :
• If j = 1, αj = φ(Pj);
• When j > 1, let α = φ(Pj) and Pj′ be the finest minimizer ofminP∈Π(V )
∑C∈P f #
α (C ). Then,
αj < α < α1
j ′ < j =⇒ Pj′ Pj
Suggestion: A Recursive Algorithm
• iteratively updates α and P, the estimation of α1 = RCO(V ) andP∗ = P1;
• terminate when α = φ(P).
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Modified DecompositionAlgorithm
recursion in modified decomposition algorithm (MDA):
α := φ(P)
P(n) is the finest minimizer of minP∈Π(V )
∑C∈P f #
α(n) (C ) and
P(0) = i : i ∈ V
Optimality of the MDA algorithm
α(n) and P(n) converge monotonically towards α1 = RCO(V ) and
P1 = P∗, respectively. Also returns rV ∈ B(f #RCO(V ),≤) = RRCO(V )(V )
• minP∈Π(V )
∑C∈P f #
α(n) (C ) reduces to⋂argminf #
α(n) (X )− r(X ) : i ∈ X ⊆ V ,∀i ∈ V , SFM due to the
intersecting submodularity of f #α(n)
• complexity: O(|V |2 · SFM(|V |))12
12SFM(|V |): the complexity of minimizing submodular function f : 2V 7→ R.
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ExperimentV = 1, . . . , 5: Wm is an independent uniformly distributed random bit:
Z1 = (Wb,Wc ,Wd ,Wh,Wi ),
Z2 = (We ,Wf ,Wh,Wi ),
Z3 = (Wb,Wc ,We ,Wj ),
Z4 = (Wa,Wb,Wc ,Wd ,Wf ,Wg ,Wi ,Wj ),
Z5 = (Wa,Wb,Wc ,Wf ,Wi ,Wj ),
0 1 2 3 4 5 6 7 8 9 10
−20
−10
0
10
P3 = 1, 2, 3, 4, 5
P2 = 4, 5, 1, 2, 3
P1 = 1, 4, 5, 2, 3 = P∗P0 = 1, 2, 3, 4, 5
α
minP∈Π
(V)
∑ C∈P
f# α(C
)
minP∈Π(V )
∑C∈P f
#α (C )
α
(c) PSP
0 1 2 3
5.8
6
6.2
6.4
6.6
iteration index
α
α(n)RCO(V )
(d) α(n) by MDA algorithm
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Extensions of CO: SecretCapacity
secret capacity CS(V ): the maximum rate at which the secret key can begenerated by the users in V with results in [7]13:
• dual relationship: RCO(V ) = H(V )− CS(V )
• mutual dependence upper bound on CS(V ):
CS(V ) ≤ I (V ) = minP∈Π′(V )
∑C∈P H(C )− H(V )
|P| − 1︸ ︷︷ ︸mutual dependence in ZV
tightness [9]14: CS(V ) = I (V ) = H(V )− RCO(V )question: how to achieve CS(V )? with interactive communication rater(V ) = RCO(V )? silly! CS(V ) can be attained with r(V ) ≤ RCO(V ) [7]
13Csiszar et. al 200414Chan et. al 2015
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Extensions of CO: Clustering
Inspired by the name ‘mutual dependence’
I (V ) = minP∈Π′(V )
∑C∈P H(C )− H(V )
|P| − 1
is proposed in [9]15 as a generalization of Shannon’s mutual informationto multivariate case: I (V ) = H(1) + H(2)− H(1, 2) whenV = 1, 2.
• realization: I (V ) is the similarity measure of more than two rvs.
I limitation in existing clustering algorithms: pairwisesimilarity/dissimilarity measure
I agglomerative clustering result given by PSP determined inO(|V |2 · SFM(|V |)) time
question: can PSP clustering frame work provide more objective overviewof the dataset?
15Chan et. al 2015
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Extensions of CO (Digiscape):Source Coding with SideInformation
1
Zn1
2
Zn2
. . . . . . |V |
Zn|V |
T
sensor nodes i ∈ V = 1, . . . , |V | reveal all information to sink T .
• for lossless data compression/aggregation, SW constraints:
r(X ) ≥ H(X |V \ X ),∀X ( V , r(V ) = H(V ) =⇒ RH(V )(V )
• an extreme, one of the unfairest, rV ∈ RH(V )(V ) can be determinedin O(|V |) time
• question: how to find a fair rate allocation in RH(V )(V ) efficiently?27 | Two Examples of Submodularity in Wireless Communications | Ni Ding
Conclusion
two examples of submodularity in wireless communications:
• vector lattice: the existence of PSNEs in a game modeledadaptive modulation problem in NC-TWRC
• set lattice: polynomial time algorithm for solving CO problem
future:
• vector lattice: more applications of discrete convexity, e.g.,the energy-delay trade-off in data aggregation tree inDigiscape, and monotone comparative statics
• set lattice: improving efficiency for determining PSPI less call of SFM algorithmI improving complexity SFM(|V |): SFM belongs to worst
polynomial algorithm category, e.g., |V |5 to |V |8
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Bibliography I
D. M. Topkis, Supermodularity and complementarity. Princeton: PrincetonUniversity Press, 2001.
K. Murota, “Note on multimodularity and l-convexity,” Math. Oper. Res.,vol. 30, no. 3, pp. 658–661, Aug. 2005.
J. Vondrak, “Submodularity in combinatorial optimization,” Ph.D. dissertation,Dept. Appl. Math., Charles Univ., Prague, 2007.
A. Tarski et al., “A lattice-theoretical fixpoint theorem and its applications,”Pacific J. Math., vol. 5, no. 2, pp. 285–309, 1955.
S. Fujishige, Submodular functions and optimization, 2nd ed. Amsterdam, TheNetherlands: Elsevier, 2005.
X. Vives, “Nash equilibrium with strategic complementarities,” J. Math. Econ.,vol. 19, no. 3, pp. 305 – 321, 1990.
I. Csiszar and P. Narayan, “Secrecy capacities for multiple terminals,” IEEETrans. Inf. Theory, vol. 50, no. 12, pp. 3047–3061, Dec. 2004.
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Bibliography II
K. Nagano, Y. Kawahara, and S. Iwata, “Minimum average cost clustering,” inProc. Advances in Neural Inf. Process. Syst., Vancouver, Candada, 2010, pp.1759–1767.
C. Chan, A. Al-Bashabsheh, J. Ebrahimi, T. Kaced, and T. Liu, “Multivariatemutual information inspired by secret-key agreement,” Proc. IEEE, vol. 103,no. 10, pp. 1883–1913, Oct. 2015.
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