EE 685 presentation

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EE 685 presentation

• Optimization Flow Control,
• I Basic Algorithm and Convergence
• By Steven Low and David Lapsley
• Asynchronous Distributed Algorithm Proof

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Objective of the paper

• Propose an optimization approach for flow control
  on a network whose resources are shared by a set
  of S of sources
• Maximization of aggregate source utility over
  transmission rates is aimed
• Sources select transmission rates that maximize
  their benefit (utility bandwidth cost)
• Synchronous and asynchronous distributed
  algorithms for converging optimal behavior in
  static environment is presented

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Problem Framework

• The problem is formulated for
• A network that consists of a set L of
  unidirectional links of capacities cl, where l is
  element of L.
• The network is shared by a set S of sources,
  where source s is characterized by a utility
  function Us(xs) that is concave increasing in its
  transmission rate xs
• .
• The goal is to calculate source rates that
  maximize the sum of the utilities ?s ? S Us(xs)
  over xs subject to capacity constraints.

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Problem Framework
DESTINATION NODES
link l4 S(l4)s1,s3
l3
l5
l2
L(s1)l1,l2,l3,l4
l1
l6
S
..........
SOURCE NODES
s1
s2
s3
ss
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Centralized optimization Why not

• Centralized optimization of source rates is
  possible in theory but not feasible and practical
  in real networks as
• Knowledge of all utility functions are required
• Resource usage is coupled due to shared links. So
  all the sources should be coordinated
  simultaneously
• Therefore a distributed and decentralized
  approach is needed.

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The value of the optimization frame-work presented

• It is not always critical or feasible to attain
  exact optimality in a flow control problem
• However the optimal framework acts as a guideline
  to shape the network dynamics to a desirable
  operating point where source utilities and
  resource costs are taken into consideration
• Optimization frameworks may be used to refine and
  ameliorate practical flow control schemes

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The optimization problem Primal problem
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The optimization problem Lagrangian for primal
problem

• pl represents Lagrange multipliers utilized in
  standard convex optimization method
• By using this approach coupled link capacity
  constraints are integrated to the objective
  function
• Notice separability in terms of xs so maximizing
  lagrangian function as aggregate of different xs
  related terms means gives the same result as
  summing up maximum of each individual xs related
  term. Therefore we have

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The optimization problem Dual problem

• Here pl is the price per unit bandwidth at link
  l.
• ps is the total price per unit bandwidth for all
  links in the path of s
• The dual problem has been defined as minimization
  of D(p) (upper bound of Lagrangian function) for
  non-negative bandwidth prices.
• Each source can independently solve maximization
  problem in (3) for a given p

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The optimization problem Concavity and duality
gap

• For each p, a unique maximizer denoted by xs(p)
  exists since Us (source utility function) is
  strictly concave
• Concavity of Us and linear constraints for the
  primal problem guarantees that there is no
  duality gap and dual optimal prices exist in the
  form of Lagrange Multipliers
• Once p is obtained by solving the dual problem,
  the primal optimal source rates xx(p) can be
  computed by individual sources s by solving (3)
• Therefore given p individual sources can solve
  (3) without any coordination (key concept for
  distributed algorithm)
• So p acts as a coordination signal that aligns
  individual and joint optimality for flow control
  problem

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Notations and assumptions
s number of flows/sources
R11 R12 ... R1s
R21 R22 ... R2s
.... .... ... ...
Rl1 Rl2 ... Rls
l number of links
R

• R is the routing matrix where Rls1 if l ? L(s)
  or s ? L(s)
• For each source s, pTR is the path bandwidth
  price that source s faces which is equal to ps
• Let xs(p) be the unique maximizer for (3) then
  xs(p) could be written as follows

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Routing matrix
DESTINATION NODES
s1 s2
l1 1 0
l2 1 0
l3 1 0
l4 1 1
l5 0 1
l6 0 1
link l4
R
l3
l5
l2
L(s1)l1,l2,l3,l4
l1
l6
S
..........
SOURCE NODES
s1
s2
s3
ss
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Concave utility function its derivative and
inverse
TYPICAL CONCAVE UTILITY FUNCTION
Inverse of derivative of utility rate x
derivative of utility U(x)
U(x)
rate x
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Source rate as demand function

• The above figure depict xs(p) as a possible
  solution of (6)
• Similar to inverse of U figure in previous slide
  the rate is obtained as a decreasing function of
  U-1(rate)
• This means that xs(p) acts as a demand function
  seen in Microeconomics.

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Synchronous Distributed Algorithmbased on
gradient projection applied to dual problem

• The dual problem is solved via gradient
  projection method where link prices are adjusted
  in the opposite direction of gradient of D(p)

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Synchronous Distributed Algorithm based on
gradient projection appliedd to dual problem

• Equation (9) shows that the price of a link l is
  updated based on how much demand exceeds supply.

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Synchronous Distributed AlgorithmGeneric outline
of the algorithm

• Given aggregate source rate that goes through
  link l, the adjustment algorithm (9) is
  completely distributed
• Therefore network links l and sources s could be
  treated as processors in a distributed
  computation system to solve the dual problem at
  (5)
• In each iteration sources s solve (3)
  independently and communicate their results xs(p)
  to links on their path (L(s)).
• Links l then update their prices pl according to
  (9) and communicate their new prices to sources s
• The cycle repeats goes back to 1 with updated p
  values
• It is possible to prove that under C1 and C2
  conditions this algorithm converges to a stable
  and optimal x (optimal source rates) and p
  (optimal bandwidth prices) for static network
  conditions (THEOREM 1)

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Synchronous Distributed Algorithm
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THEOREM 1
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THEOREM 1 (proof)

• LEMMA 1 Under C1, D(p) is convex, lower bounded
  and continuously differentiable

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THEOREM 1 (proof)

• LEMMA 2 Under C1, The Hessian of D is given by
  ?2D(p)RB(p)RT

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THEOREM 1 (proof)
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THEOREM 1 (proof)

• LEMMA 3 Under C1-C2, ?D is Lipschitz continuous
  with
• ? ?D(q) - ?D(p) ?2 aLS ?q-p ? 2 for all p,q
  0

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THEOREM 1 (proof)

• LEMMA 3 Under C1-C2, ?D is Lipschitz continous
  with
• ? ?D(q) - ?(p) ?2 aLS ?q-p ? 2 for all p,q
  0

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THEOREM 1 (proof)
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Fundamental assumptions C1,C2 and C3
assumptions for the utility functions

• Here ms and Ms are minimum and maximum
  transmission rates for source s

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Asynchronous Distributed Algorithm why
asynchronous model is needed

• The synchronous model of the last section
  assumes that updates at the sources and the links
  are synchronized
• In realistic large network scenarios, synchronous
  updates might not be possible as
• Sources may be located at different distances
  from the network links
• .Network states (prices in our case) may be
  probed by different sources at different rates,
  e.g., the Resource Management
• Feedbacks may reach different sources after
  different, and variable, delays.
• These complications make our distributed
  computation system consisting of links and
  sources asynchronous.
• The communication delays may be substantial and
  time-varying.

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Asynchronous Distributed Algorithm Generic
outline of the algorithm

• The main approach of interdependent update of
  source rates and bandwidth prices iteratively is
  followed in asynchronous version of the algorithm
  as well
• For bandwidth price updates the links use an
  estimate of the gradient based on past source
  rates at link l
• Two type of policies are applied
• Latest data only Only the last received rate is
  used
• Latest average Only the average of latest k
  received rates is used
• The convergence of both synchronous and
  asyncronous algorithms depend on sufficiently
  small step size for (7) and (9)
• Convergence for asynchronous version of the
  algorithm could be proven as long as assumptions
  C1 and C3 hold.

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Asynchronous Distributed Algorithm
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Asynchronous Distributed AlgorithmProof
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Asynchronous Distributed AlgorithmProof (lemma 4)
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Asynchronous Distributed AlgorithmProof (lemma 4)

• Lemma 4 a) comes from applying the projection
  theorem to the scalar pl(t1) pl(t)-??l(t)
• Lemma 4 b) follows from

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Asynchronous Distributed AlgorithmProof (lemma 4)
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Asynchronous Distributed AlgorithmProof (lemma 4)
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Asynchronous Distributed AlgorithmProof (lemma 4)

• The next lemma (LEMMA 5) bounds the error in
  gradient estimation in terms of the successive
  price change p(t) p(t1)-p(t).

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Asynchronous Distributed AlgorithmProof (lemma 5)
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Asynchronous Distributed AlgorithmProof (lemma 5)
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Asynchronous Distributed AlgorithmProof (lemma 5)
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Asynchronous Distributed AlgorithmProof (lemma 6)
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Asynchronous Distributed AlgorithmProof (lemma 6)
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Asynchronous Distributed AlgorithmProof (lemma 6)
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Asynchronous Distributed AlgorithmTheorem 2
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Asynchronous Distributed AlgorithmTheorem 2
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Asynchronous Distributed AlgorithmTheorem 2