networks: distributed control and emerging phenomena (formerly: networks, algorithms...
TRANSCRIPT
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Networks: distributed control and emerging phenomena
(formerly: Networks, algorithms and probability)
Laurent Massoulié [email protected]
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How to assign bandwidth in networks Understanding TCP, the protocol regulating most Internet
trafficConvex optimization theory & dynamical systems
Distributed control for data transport
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Distributed control for data transport
How to manage collisions (i.e. lost transmissions because of interference) between wireless transmitters
Aloha and Ethernet protocols
Markov chains and criteria for ergodicity
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Distributed control for data transport
Crossbar switchwith input queues:
How to schedule transmissions in switches, and multi-hop wireless networksMax-weight & backpressure algorithms
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Network epidemics
Spread of a picture on facebook
https://www.facebookstories.com/stories/2200/
Spread of “CodeRed” Internet worm, 2001
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What makes an epidemic potent or weakrandom graphs, branching processes and phase transitions
What features of network topology affect epidemic outbreakgraph topology descriptors, comparison of Markov chains by “coupling”
How to maximize size of outbreaksubmodular functions and greedy maximization
Network epidemics
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Network epidemics
What is a “small world” networkAnd how to search for information in it
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Network epidemics
Why are most networks “scale-free” (a.k.a. power-law)martingales, coupling and “concentration inequalities”
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Network epidemics
How to find community structure and recommend contacts in a social network spectra of random graphs and spectral methods
Political blogs:
Republican vs Democrats
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Network resource allocation: principles and algorithms
Convex optimization model
A “primal” algorithm
Reverse-engineering TCP
Lagrangian, duality and Lagrange multipliers
A “dual” algorithm
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TCP in one slide
Source dynamics: Maintain Nb of (sent¬ acked pkts)=cwnd (congestion window) Update cwnd cwnd+1/cwnd upon receipt of pkt ack cwnd/2 upon detection of pkt loss“Congestion avoidance” alg introduced in 1993After Internet congestion collapse
source
dest
buffer
LinkData
packets
acknowledgements
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Outline
Convex optimization model
A ”primal” algorithm
Reverse-engineering TCP
Lagrangian, duality and multipliers
A ”dual” algorithm
Laurent Massoulie Network resource allocation
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Network model
Resources, or links, ` ∈ L, each with capacity C` > 0
Users, or transmissions, or flows, s ∈ SUser s uses same rate at all ` ∈ s (s ↔ subset of L)
Feasible rates:variables xs ≥ 0, s ∈ Ssuch that ∀` ∈ L,
∑s3` xs ≤ C`
Potential applications
Links on single path from source to destination
Links on tree of transmission from source to set of receivers
Laurent Massoulie Network resource allocation
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Network model
Resources, or links, ` ∈ L, each with capacity C` > 0
Users, or transmissions, or flows, s ∈ SUser s uses same rate at all ` ∈ s (s ↔ subset of L)
Feasible rates:variables xs ≥ 0, s ∈ Ssuch that ∀` ∈ L,
∑s3` xs ≤ C`
Potential applications
Links on single path from source to destination
Links on tree of transmission from source to set of receivers
Laurent Massoulie Network resource allocation
![Page 15: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/15.jpg)
Network model
Resources, or links, ` ∈ L, each with capacity C` > 0
Users, or transmissions, or flows, s ∈ SUser s uses same rate at all ` ∈ s (s ↔ subset of L)
Feasible rates:variables xs ≥ 0, s ∈ Ssuch that ∀` ∈ L,
∑s3` xs ≤ C`
Potential applications
Links on single path from source to destination
Links on tree of transmission from source to set of receivers
Laurent Massoulie Network resource allocation
![Page 16: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/16.jpg)
Allocation principles 1
max-min fairness: feasible xmm such that∀s ∈ S, ∃` ∈ s with
∑t3` x
mmt = C` and xmm
s = maxt3` xmmt
(“no envy”: each s can find competing t at least as poor as s)
Proportional fairness: feasible xpf such that
for all feasible y ,∑
sys−xpfsxpfs
≤ 0
Alternative characterization:Unique maximizer of
∑s log(xs) among feasible x
Laurent Massoulie Network resource allocation
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Allocation principles 1
max-min fairness: feasible xmm such that∀s ∈ S, ∃` ∈ s with
∑t3` x
mmt = C` and xmm
s = maxt3` xmmt
(“no envy”: each s can find competing t at least as poor as s)
Proportional fairness: feasible xpf such that
for all feasible y ,∑
sys−xpfsxpfs
≤ 0
Alternative characterization:Unique maximizer of
∑s log(xs) among feasible x
Laurent Massoulie Network resource allocation
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Allocation principles 1
max-min fairness: feasible xmm such that∀s ∈ S, ∃` ∈ s with
∑t3` x
mmt = C` and xmm
s = maxt3` xmmt
(“no envy”: each s can find competing t at least as poor as s)
Proportional fairness: feasible xpf such that
for all feasible y ,∑
sys−xpfsxpfs
≤ 0
Alternative characterization:Unique maximizer of
∑s log(xs) among feasible x
Laurent Massoulie Network resource allocation
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Allocation principles 1
Alternative characterization: Nash’s bargaining solution
i.e. unique vector φ(C) in feasible convex set C ⊂ RS+s.t.
Pareto efficiency: φ(C) ≤ x ∈ C ⇒ x = φ(C)
independence of irrelevant alternatives:φ(C) ∈ C′ ⊂ C ⇒ φ(C) = φ(C′)symmetry: C symmetric ⇒ φ(C)i ≡ φ(C)1
scale invariance: for diagonal D with Dii ≥ 0,φ(DC) = Dφ(C)
Laurent Massoulie Network resource allocation
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Allocation principles 2
Network Utility Maximization x∗: solution of
Max∑
s Us(xs)
Over xs ≥ 0 (P)
Such that ∀`,∑
s3` xs ≤ C`
for concave, increasing utility functions Us : R+ → R
⇒ A concave optimization program
ExamplesProportional fair xpf : Us = log
For w , α > 0, (w , α)-fair x = x(w , α): Us(xs) = wsx1−αs
1−α
[Exercise: limα→1 x(1, α) = xpf and limα→+∞ x(1, α) = xmm]
Laurent Massoulie Network resource allocation
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Allocation principles 2
Network Utility Maximization x∗: solution of
Max∑
s Us(xs)
Over xs ≥ 0 (P)
Such that ∀`,∑
s3` xs ≤ C`
for concave, increasing utility functions Us : R+ → R
⇒ A concave optimization program
ExamplesProportional fair xpf : Us = log
For w , α > 0, (w , α)-fair x = x(w , α): Us(xs) = wsx1−αs
1−α
[Exercise: limα→1 x(1, α) = xpf and limα→+∞ x(1, α) = xmm]
Laurent Massoulie Network resource allocation
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Allocation principles 2
Network Utility Maximization x∗: solution of
Max∑
s Us(xs)
Over xs ≥ 0 (P)
Such that ∀`,∑
s3` xs ≤ C`
for concave, increasing utility functions Us : R+ → R
⇒ A concave optimization program
ExamplesProportional fair xpf : Us = log
For w , α > 0, (w , α)-fair x = x(w , α): Us(xs) = wsx1−αs
1−α
[Exercise: limα→1 x(1, α) = xpf and limα→+∞ x(1, α) = xmm]
Laurent Massoulie Network resource allocation
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Allocation principles 2
Network Utility Maximization x∗: solution of
Max∑
s Us(xs)
Over xs ≥ 0 (P)
Such that ∀`,∑
s3` xs ≤ C`
for concave, increasing utility functions Us : R+ → R
⇒ A concave optimization program
ExamplesProportional fair xpf : Us = log
For w , α > 0, (w , α)-fair x = x(w , α): Us(xs) = wsx1−αs
1−α
[Exercise: limα→1 x(1, α) = xpf and limα→+∞ x(1, α) = xmm]
Laurent Massoulie Network resource allocation
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Allocation principles 2
Network Utility Maximization x∗: solution of
Max∑
s Us(xs)
Over xs ≥ 0 (P)
Such that ∀`,∑
s3` xs ≤ C`
for concave, increasing utility functions Us : R+ → R
⇒ A concave optimization program
ExamplesProportional fair xpf : Us = log
For w , α > 0, (w , α)-fair x = x(w , α): Us(xs) = wsx1−αs
1−α
[Exercise: limα→1 x(1, α) = xpf and limα→+∞ x(1, α) = xmm]
Laurent Massoulie Network resource allocation
![Page 25: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/25.jpg)
Relaxed constraints and a “primal” algorithm
Relaxed problem: Max∑s
Us(xs)−∑`
C`(y`)
Over xs ≥ 0 (RP)
with y` =∑s3`
xs
for concave increasing utility functions Us and convex increasingcost functions C`
primal algorithm: for Us and C` differentiable, let
d
dtxs = κs(xs)
(U ′s(xs)−
∑`∈s
C ′`(y`)
)“gradient ascent”
→ Implementable in a distributed fashion
Laurent Massoulie Network resource allocation
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Relaxed constraints and a “primal” algorithm
Relaxed problem: Max∑s
Us(xs)−∑`
C`(y`)
Over xs ≥ 0 (RP)
with y` =∑s3`
xs
for concave increasing utility functions Us and convex increasingcost functions C`primal algorithm: for Us and C` differentiable, let
d
dtxs = κs(xs)
(U ′s(xs)−
∑`∈s
C ′`(y`)
)“gradient ascent”
→ Implementable in a distributed fashion
Laurent Massoulie Network resource allocation
![Page 27: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/27.jpg)
Relaxed constraints and a “primal” algorithm
Relaxed problem: Max∑s
Us(xs)−∑`
C`(y`)
Over xs ≥ 0 (RP)
with y` =∑s3`
xs
for concave increasing utility functions Us and convex increasingcost functions C`primal algorithm: for Us and C` differentiable, let
d
dtxs = κs(xs)
(U ′s(xs)−
∑`∈s
C ′`(y`)
)“gradient ascent”
→ Implementable in a distributed fashion
Laurent Massoulie Network resource allocation
![Page 28: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/28.jpg)
Stability via Lyapunov functions
Criterion for convergence of ODE x = F (x) with trajectories inO ⊂ Rn
Theorem
Assume F continuous on O, and ∃V : O → R such that:
(i) V continuously differentiable
(ii) ∀a ≤ A, {x ∈ O : V (x) ≤ A} and {x ∈ O : V (x) ∈ [a,A]}either compact or empty
(iii) ∀x ∈ O \ B, ∇V (x) · F (x) < 0, where B = argminx∈O{V (x)}
Then limt→∞ V (x(t)) = infx∈O V (x), limt→∞ d(x(t),B) = 0.
If B = {x∗} then limt→∞ x(t) = x∗.
Laurent Massoulie Network resource allocation
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Application to gradient ascent / descent dynamics
d
dtxs = κs(xs)
(U ′s(xs)−
∑`∈s
C ′`(y`)
)Let W (x) =
∑s Us(xs)−
∑` C`(y`) (system welfare)
and V (x) = −W (x)
Then: ∇V (x) · F (x) = −∑
s κs(xs)[∂∂xs
W (x)]2
Theorem
For Us strictly concave with U ′s(0+) = +∞,C` convex, continuously differentiable,[ ⇒ strict concavity and continuous differentiability of W ]κs > 0, continuous [ ⇒continuity of F ]∃xs > 0 s.t. U ′s(xs) <
∑`∈s C
′`(xs)
[ ⇒ Max of W achieved at single point x∗ ∈ O := (0,∞)S ]Then “primal” dynamics converge to unique maximizer x∗ of W
Laurent Massoulie Network resource allocation
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Application to gradient ascent / descent dynamics
d
dtxs = κs(xs)
(U ′s(xs)−
∑`∈s
C ′`(y`)
)Let W (x) =
∑s Us(xs)−
∑` C`(y`) (system welfare)
and V (x) = −W (x)
Then: ∇V (x) · F (x) = −∑
s κs(xs)[∂∂xs
W (x)]2
Theorem
For Us strictly concave with U ′s(0+) = +∞,C` convex, continuously differentiable,[ ⇒ strict concavity and continuous differentiability of W ]κs > 0, continuous [ ⇒continuity of F ]∃xs > 0 s.t. U ′s(xs) <
∑`∈s C
′`(xs)
[ ⇒ Max of W achieved at single point x∗ ∈ O := (0,∞)S ]Then “primal” dynamics converge to unique maximizer x∗ of W
Laurent Massoulie Network resource allocation
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Application to gradient ascent / descent dynamics
d
dtxs = κs(xs)
(U ′s(xs)−
∑`∈s
C ′`(y`)
)Let W (x) =
∑s Us(xs)−
∑` C`(y`) (system welfare)
and V (x) = −W (x)
Then: ∇V (x) · F (x) = −∑
s κs(xs)[∂∂xs
W (x)]2
Theorem
For Us strictly concave with U ′s(0+) = +∞,C` convex, continuously differentiable,[ ⇒ strict concavity and continuous differentiability of W ]κs > 0, continuous [ ⇒continuity of F ]∃xs > 0 s.t. U ′s(xs) <
∑`∈s C
′`(xs)
[ ⇒ Max of W achieved at single point x∗ ∈ O := (0,∞)S ]Then “primal” dynamics converge to unique maximizer x∗ of W
Laurent Massoulie Network resource allocation
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Reverse engineering TCP
Approx. xs ≈ cwnds/Ts where Ts : packet round-trip time
Approx. ddt cwnds ≈ xs(1/cwnds)− p(s)[cwnds/2]
where p(s): packet loss probability along path of s
Approx. p(s) ≈∑
`∈s p`(y`) for link packet loss prob. p`(y)[e.g. p`(y) = max(0, 1− C`/y)]
⇒ xs =
(x2s
2
)[2
(xsTs)2−∑`∈s
p`(y`)
]
TCP implicitly runs primal alg. with utility function:Us(x) = wsx
1−α/(1− α) with α = 2, ws = 2/T 2s
→ Leads to (w , α)-fairness with suitable parametersCan tweak congestion avoidance alg. if want e.g. proportionalfairness (α = 1) instead
Laurent Massoulie Network resource allocation
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Reverse engineering TCP
Approx. xs ≈ cwnds/Ts where Ts : packet round-trip time
Approx. ddt cwnds ≈ xs(1/cwnds)− p(s)[cwnds/2]
where p(s): packet loss probability along path of s
Approx. p(s) ≈∑
`∈s p`(y`) for link packet loss prob. p`(y)[e.g. p`(y) = max(0, 1− C`/y)]
⇒ xs =
(x2s
2
)[2
(xsTs)2−∑`∈s
p`(y`)
]
TCP implicitly runs primal alg. with utility function:Us(x) = wsx
1−α/(1− α) with α = 2, ws = 2/T 2s
→ Leads to (w , α)-fairness with suitable parametersCan tweak congestion avoidance alg. if want e.g. proportionalfairness (α = 1) instead
Laurent Massoulie Network resource allocation
![Page 34: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/34.jpg)
Reverse engineering TCP
Approx. xs ≈ cwnds/Ts where Ts : packet round-trip time
Approx. ddt cwnds ≈ xs(1/cwnds)− p(s)[cwnds/2]
where p(s): packet loss probability along path of s
Approx. p(s) ≈∑
`∈s p`(y`) for link packet loss prob. p`(y)[e.g. p`(y) = max(0, 1− C`/y)]
⇒ xs =
(x2s
2
)[2
(xsTs)2−∑`∈s
p`(y`)
]
TCP implicitly runs primal alg. with utility function:Us(x) = wsx
1−α/(1− α) with α = 2, ws = 2/T 2s
→ Leads to (w , α)-fairness with suitable parametersCan tweak congestion avoidance alg. if want e.g. proportionalfairness (α = 1) instead
Laurent Massoulie Network resource allocation
![Page 35: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/35.jpg)
Reverse engineering TCP
Approx. xs ≈ cwnds/Ts where Ts : packet round-trip time
Approx. ddt cwnds ≈ xs(1/cwnds)− p(s)[cwnds/2]
where p(s): packet loss probability along path of s
Approx. p(s) ≈∑
`∈s p`(y`) for link packet loss prob. p`(y)[e.g. p`(y) = max(0, 1− C`/y)]
⇒ xs =
(x2s
2
)[2
(xsTs)2−∑`∈s
p`(y`)
]
TCP implicitly runs primal alg. with utility function:Us(x) = wsx
1−α/(1− α) with α = 2, ws = 2/T 2s
→ Leads to (w , α)-fairness with suitable parametersCan tweak congestion avoidance alg. if want e.g. proportionalfairness (α = 1) instead
Laurent Massoulie Network resource allocation
![Page 36: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/36.jpg)
Reverse engineering TCP
Approx. xs ≈ cwnds/Ts where Ts : packet round-trip time
Approx. ddt cwnds ≈ xs(1/cwnds)− p(s)[cwnds/2]
where p(s): packet loss probability along path of s
Approx. p(s) ≈∑
`∈s p`(y`) for link packet loss prob. p`(y)[e.g. p`(y) = max(0, 1− C`/y)]
⇒ xs =
(x2s
2
)[2
(xsTs)2−∑`∈s
p`(y`)
]
TCP implicitly runs primal alg. with utility function:Us(x) = wsx
1−α/(1− α) with α = 2, ws = 2/T 2s
→ Leads to (w , α)-fairness with suitable parametersCan tweak congestion avoidance alg. if want e.g. proportionalfairness (α = 1) instead
Laurent Massoulie Network resource allocation
![Page 37: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/37.jpg)
Reverse engineering TCP
Approx. xs ≈ cwnds/Ts where Ts : packet round-trip time
Approx. ddt cwnds ≈ xs(1/cwnds)− p(s)[cwnds/2]
where p(s): packet loss probability along path of s
Approx. p(s) ≈∑
`∈s p`(y`) for link packet loss prob. p`(y)[e.g. p`(y) = max(0, 1− C`/y)]
⇒ xs =
(x2s
2
)[2
(xsTs)2−∑`∈s
p`(y`)
]
TCP implicitly runs primal alg. with utility function:Us(x) = wsx
1−α/(1− α) with α = 2, ws = 2/T 2s
→ Leads to (w , α)-fairness with suitable parameters
Can tweak congestion avoidance alg. if want e.g. proportionalfairness (α = 1) instead
Laurent Massoulie Network resource allocation
![Page 38: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/38.jpg)
Reverse engineering TCP
Approx. xs ≈ cwnds/Ts where Ts : packet round-trip time
Approx. ddt cwnds ≈ xs(1/cwnds)− p(s)[cwnds/2]
where p(s): packet loss probability along path of s
Approx. p(s) ≈∑
`∈s p`(y`) for link packet loss prob. p`(y)[e.g. p`(y) = max(0, 1− C`/y)]
⇒ xs =
(x2s
2
)[2
(xsTs)2−∑`∈s
p`(y`)
]
TCP implicitly runs primal alg. with utility function:Us(x) = wsx
1−α/(1− α) with α = 2, ws = 2/T 2s
→ Leads to (w , α)-fairness with suitable parametersCan tweak congestion avoidance alg. if want e.g. proportionalfairness (α = 1) instead
Laurent Massoulie Network resource allocation
![Page 39: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/39.jpg)
Convex optimization: Lagrangian, duality, multipliers
Generic convex optimization programFor convex set C0, convex functions J, f` : C0 → R,
Min J(x)
Over x ∈ C0 (P)
Such that ∀` ∈ L, f`(x) ≤ 0
Associated Lagrangian L(x , λ) := J(x) +∑
` λ`f`(x),
x ∈ C0, λ ≥ 0
λ: Lagrange multipliers of (P)’s constraints
Dual problem (D): Max D(λ) Over λ ≥ 0where D(λ) := infx∈C0 L(x , λ)
Laurent Massoulie Network resource allocation
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Convex optimization: Lagrangian, duality, multipliers
Generic convex optimization programFor convex set C0, convex functions J, f` : C0 → R,
Min J(x)
Over x ∈ C0 (P)
Such that ∀` ∈ L, f`(x) ≤ 0
Associated Lagrangian L(x , λ) := J(x) +∑
` λ`f`(x),
x ∈ C0, λ ≥ 0
λ: Lagrange multipliers of (P)’s constraints
Dual problem (D): Max D(λ) Over λ ≥ 0where D(λ) := infx∈C0 L(x , λ)
Laurent Massoulie Network resource allocation
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Convex optimization: Lagrangian, duality, multipliers
Generic convex optimization programFor convex set C0, convex functions J, f` : C0 → R,
Min J(x)
Over x ∈ C0 (P)
Such that ∀` ∈ L, f`(x) ≤ 0
Associated Lagrangian L(x , λ) := J(x) +∑
` λ`f`(x),
x ∈ C0, λ ≥ 0
λ: Lagrange multipliers of (P)’s constraints
Dual problem (D): Max D(λ) Over λ ≥ 0where D(λ) := infx∈C0 L(x , λ)
Laurent Massoulie Network resource allocation
![Page 42: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/42.jpg)
Convex optimization: Lagrangian, duality, multipliers
Generic convex optimization programFor convex set C0, convex functions J, f` : C0 → R,
Min J(x)
Over x ∈ C0 (P)
Such that ∀` ∈ L, f`(x) ≤ 0
Associated Lagrangian L(x , λ) := J(x) +∑
` λ`f`(x),
x ∈ C0, λ ≥ 0
λ: Lagrange multipliers of (P)’s constraints
Dual problem (D): Max D(λ) Over λ ≥ 0where D(λ) := infx∈C0 L(x , λ)
Laurent Massoulie Network resource allocation
![Page 43: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/43.jpg)
Kuhn-Tucker theorem and strong duality
Def: λ∗ ≥ 0 a Kuhn-Tucker vector iff ∀x ∈ C0, L(x , λ∗) ≥ J∗
where J∗: optimal value of (P).
Remark: J∗ ≥ D∗ where D∗ optimal value of (D)
Theorem
Assume there exists λ∗ a Kuhn-Tucker vector. Then(i) λ∗ solves (D), and J∗ = D∗ (a.k.a. strong duality)(ii) x∗ ∈ C0 if optimal for (P) then achieves minx∈C0 L(x , λ∗)(iii) For x∗int(C0) an optimum of (P) at which ∃∇J,∇f`, then
∀`, λ∗` f`(x∗) = 0 (complementarity)∇J(x∗) +
∑` λ∗` f`(x
∗) = 0 (stationarity)
Reciprocally assume stationarity + complementarityfor some λ∗ ≥ 0 and some x∗ feasible for (P),Then λ∗: Kuhn-Tucker and x∗ optimal for (P)
Laurent Massoulie Network resource allocation
![Page 44: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/44.jpg)
Kuhn-Tucker theorem and strong duality
Def: λ∗ ≥ 0 a Kuhn-Tucker vector iff ∀x ∈ C0, L(x , λ∗) ≥ J∗
where J∗: optimal value of (P).Remark: J∗ ≥ D∗ where D∗ optimal value of (D)
Theorem
Assume there exists λ∗ a Kuhn-Tucker vector. Then(i) λ∗ solves (D), and J∗ = D∗ (a.k.a. strong duality)(ii) x∗ ∈ C0 if optimal for (P) then achieves minx∈C0 L(x , λ∗)(iii) For x∗int(C0) an optimum of (P) at which ∃∇J,∇f`, then
∀`, λ∗` f`(x∗) = 0 (complementarity)∇J(x∗) +
∑` λ∗` f`(x
∗) = 0 (stationarity)
Reciprocally assume stationarity + complementarityfor some λ∗ ≥ 0 and some x∗ feasible for (P),Then λ∗: Kuhn-Tucker and x∗ optimal for (P)
Laurent Massoulie Network resource allocation
![Page 45: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/45.jpg)
Kuhn-Tucker theorem and strong duality
Def: λ∗ ≥ 0 a Kuhn-Tucker vector iff ∀x ∈ C0, L(x , λ∗) ≥ J∗
where J∗: optimal value of (P).Remark: J∗ ≥ D∗ where D∗ optimal value of (D)
Theorem
Assume there exists λ∗ a Kuhn-Tucker vector. Then(i) λ∗ solves (D), and J∗ = D∗ (a.k.a. strong duality)(ii) x∗ ∈ C0 if optimal for (P) then achieves minx∈C0 L(x , λ∗)(iii) For x∗int(C0) an optimum of (P) at which ∃∇J,∇f`, then
∀`, λ∗` f`(x∗) = 0 (complementarity)∇J(x∗) +
∑` λ∗` f`(x
∗) = 0 (stationarity)
Reciprocally assume stationarity + complementarityfor some λ∗ ≥ 0 and some x∗ feasible for (P),Then λ∗: Kuhn-Tucker and x∗ optimal for (P)
Laurent Massoulie Network resource allocation
![Page 46: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/46.jpg)
Sufficient conditions for applying Kuhn-Tucker
Lemma
Assume J∗ > −∞ and ∃x ∈ C0 such that ∀`, f`(x) < 0.Then a Kuhn-Tucker vector λ∗ exists.
In practice: verify Lemma’s conditions + existence of optimumx∗ ∈ int(C0) at which ∃∇J, ∇f`.Then find x∗ that verifies complementarity + stationarity (nowguaranteed to exist)
Laurent Massoulie Network resource allocation
![Page 47: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/47.jpg)
Solving original problem: dual algorithm
Lagrangian: L(x , λ) =∑
s Us(xs) +∑
` λ`[C` −∑
s3` xs ]
Dual: D(λ) =∑
s Us(gs(λs)) +∑
` λ`[C` −∑
s3` gs(λs)]
where λs :=∑
`∈s λs and gs := (U ′s)−1
⇒ ∂∂λs
D(λ) = C` −∑
s3` gs(λs)
Dual algorithm: xs ≡ gs(λs),
λ` = κ`[∑
s3` xs − C`]+λ`
where [a]+b = a if b > 0, max(a, 0) if b ≤ 0
Laurent Massoulie Network resource allocation
![Page 48: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/48.jpg)
Solving original problem: dual algorithm
Lagrangian: L(x , λ) =∑
s Us(xs) +∑
` λ`[C` −∑
s3` xs ]
Dual: D(λ) =∑
s Us(gs(λs)) +∑
` λ`[C` −∑
s3` gs(λs)]
where λs :=∑
`∈s λs and gs := (U ′s)−1
⇒ ∂∂λs
D(λ) = C` −∑
s3` gs(λs)
Dual algorithm: xs ≡ gs(λs),
λ` = κ`[∑
s3` xs − C`]+λ`
where [a]+b = a if b > 0, max(a, 0) if b ≤ 0
Laurent Massoulie Network resource allocation
![Page 49: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/49.jpg)
Solving original problem: dual algorithm
Lagrangian: L(x , λ) =∑
s Us(xs) +∑
` λ`[C` −∑
s3` xs ]
Dual: D(λ) =∑
s Us(gs(λs)) +∑
` λ`[C` −∑
s3` gs(λs)]
where λs :=∑
`∈s λs and gs := (U ′s)−1
⇒ ∂∂λs
D(λ) = C` −∑
s3` gs(λs)
Dual algorithm: xs ≡ gs(λs),
λ` = κ`[∑
s3` xs − C`]+λ`
where [a]+b = a if b > 0, max(a, 0) if b ≤ 0
Laurent Massoulie Network resource allocation
![Page 50: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/50.jpg)
Solving original problem: dual algorithm
Lagrangian: L(x , λ) =∑
s Us(xs) +∑
` λ`[C` −∑
s3` xs ]
Dual: D(λ) =∑
s Us(gs(λs)) +∑
` λ`[C` −∑
s3` gs(λs)]
where λs :=∑
`∈s λs and gs := (U ′s)−1
⇒ ∂∂λs
D(λ) = C` −∑
s3` gs(λs)
Dual algorithm: xs ≡ gs(λs),
λ` = κ`[∑
s3` xs − C`]+λ`
where [a]+b = a if b > 0, max(a, 0) if b ≤ 0
Laurent Massoulie Network resource allocation
![Page 51: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/51.jpg)
Solving original problem: dual algorithm
Theorem
Under suitable conditions(Us strictly concave, twice differentiable, U ′s(0+) = +∞,U ′s(+∞) = 0)Trajectories xs of dual algorithm converge to unique maximizer x∗
of primal problem.
[Proof: involved. Quasiproof: Lyapunov function argument]
Potential implementation: multiplier dynamics ≡ queue dynamics⇒ Let λ` = queueing delay of packets and instantaneously let xsto gs(λs)⇒ Principle underlying TCP-Vegas, an alternative to default TCP( TCP Reno)
Laurent Massoulie Network resource allocation
![Page 52: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/52.jpg)
Solving original problem: dual algorithm
Theorem
Under suitable conditions(Us strictly concave, twice differentiable, U ′s(0+) = +∞,U ′s(+∞) = 0)Trajectories xs of dual algorithm converge to unique maximizer x∗
of primal problem.
[Proof: involved. Quasiproof: Lyapunov function argument]
Potential implementation: multiplier dynamics ≡ queue dynamics⇒ Let λ` = queueing delay of packets and instantaneously let xsto gs(λs)⇒ Principle underlying TCP-Vegas, an alternative to default TCP( TCP Reno)
Laurent Massoulie Network resource allocation
![Page 53: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/53.jpg)
Solving original problem: dual algorithm
Theorem
Under suitable conditions(Us strictly concave, twice differentiable, U ′s(0+) = +∞,U ′s(+∞) = 0)Trajectories xs of dual algorithm converge to unique maximizer x∗
of primal problem.
[Proof: involved. Quasiproof: Lyapunov function argument]
Potential implementation: multiplier dynamics ≡ queue dynamics
⇒ Let λ` = queueing delay of packets and instantaneously let xsto gs(λs)⇒ Principle underlying TCP-Vegas, an alternative to default TCP( TCP Reno)
Laurent Massoulie Network resource allocation
![Page 54: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/54.jpg)
Solving original problem: dual algorithm
Theorem
Under suitable conditions(Us strictly concave, twice differentiable, U ′s(0+) = +∞,U ′s(+∞) = 0)Trajectories xs of dual algorithm converge to unique maximizer x∗
of primal problem.
[Proof: involved. Quasiproof: Lyapunov function argument]
Potential implementation: multiplier dynamics ≡ queue dynamics⇒ Let λ` = queueing delay of packets and instantaneously let xsto gs(λs)
⇒ Principle underlying TCP-Vegas, an alternative to default TCP( TCP Reno)
Laurent Massoulie Network resource allocation
![Page 55: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/55.jpg)
Solving original problem: dual algorithm
Theorem
Under suitable conditions(Us strictly concave, twice differentiable, U ′s(0+) = +∞,U ′s(+∞) = 0)Trajectories xs of dual algorithm converge to unique maximizer x∗
of primal problem.
[Proof: involved. Quasiproof: Lyapunov function argument]
Potential implementation: multiplier dynamics ≡ queue dynamics⇒ Let λ` = queueing delay of packets and instantaneously let xsto gs(λs)⇒ Principle underlying TCP-Vegas, an alternative to default TCP( TCP Reno)
Laurent Massoulie Network resource allocation
![Page 56: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/56.jpg)
Takeaway messages
For unconstrained convex minimization, gradient descentconverges to optimizer [Lyapunov stability]
Admits distributed implementation in network optimizationsetting
TCP implicitly achieves (w , α)-fair allocation by runninggradient descent
Kuhn-Tucker Theorem: Complementarity + Stationaritycharacterization of (P)’s optima
Queue dynamics implicitly perform gradient descent formultipliers of constrained program
Remaining question: How to discriminate between allocationobjectives?
Laurent Massoulie Network resource allocation
![Page 57: Networks: distributed control and emerging phenomena (formerly: Networks, algorithms ...lelarge/X15/cours/slides1.pdf · 2015. 9. 10. · TCP in one slide Source dynamics: Maintain](https://reader035.vdocuments.us/reader035/viewer/2022081603/6126a2e606d335580c1d85ff/html5/thumbnails/57.jpg)
Takeaway messages
For unconstrained convex minimization, gradient descentconverges to optimizer [Lyapunov stability]
Admits distributed implementation in network optimizationsetting
TCP implicitly achieves (w , α)-fair allocation by runninggradient descent
Kuhn-Tucker Theorem: Complementarity + Stationaritycharacterization of (P)’s optima
Queue dynamics implicitly perform gradient descent formultipliers of constrained program
Remaining question: How to discriminate between allocationobjectives?
Laurent Massoulie Network resource allocation