A Tutorial on Mean Field and Re ned Mean Field...
Transcript of A Tutorial on Mean Field and Re ned Mean Field...
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A Tutorial on Mean Field and Refined Mean FieldApproximation
Nicolas Gast
Inria, Grenoble, France
ITC 31, August 2019, Budapest
Nicolas Gast – 1 / 47
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Good system design needs performance evaluationExample : caches
N contents
Which cache replacement policies?
LRU
2-LRU
RAND
RAND(m)
FIFO
We need models and methods to characterize emerging behavior startingfrom a stochastic model of interacting objects
We use simulation analytical methods and approximations.
Nicolas Gast – 2 / 47
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Good system design needs performance evaluationExample : caches
N contents
Which cache replacement policies?
LRU
2-LRU
RAND
RAND(m)
FIFO
We need models and methods to characterize emerging behavior startingfrom a stochastic model of interacting objects
We use simulation analytical methods and approximations.
Nicolas Gast – 2 / 47
![Page 4: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/4.jpg)
The main difficulty of probability : correlations
P [A,B] 6= P [A]P [B]
Problem: state space explosionS states per object, N objects ⇒ SN states
Nicolas Gast – 3 / 47
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“Mean field approximation” ≈ assume that all objects areindependentWhere has it been used?
Statistical mechanics, chemical reaction networks Gillepsie 92
Communication protocols ex: CSMA, Bianchi 00
Performance of caching algorithms ex: TTL-approximation, Fagins 77 +
many recent papers
Mean field games ex: evacuation, Mexican wave, Larsi-Lions 06
Understanding of Deep Neural Networks ex: Xiao et al. 2018
This talk1 What is mean field approximation
and how to apply it?
2 When is it valid?
3 How can we do better?
Nicolas Gast – 4 / 47
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“Mean field approximation” ≈ assume that all objects areindependentWhere has it been used?
Statistical mechanics, chemical reaction networks Gillepsie 92
Communication protocols ex: CSMA, Bianchi 00
Performance of caching algorithms ex: TTL-approximation, Fagins 77 +
many recent papers
Mean field games ex: evacuation, Mexican wave, Larsi-Lions 06
Understanding of Deep Neural Networks ex: Xiao et al. 2018
This talk1 What is mean field approximation
and how to apply it?
2 When is it valid?
3 How can we do better?
Nicolas Gast – 4 / 47
![Page 7: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/7.jpg)
Outline: Demystifying Mean Field Approximation
1 How to Construct a Mean Field Approximation (via an example)
2 How Accurate is the Classical Mean Field Approximation
3 We can Refine this Approximation
4 Conclusion and Open Questions
Nicolas Gast – 5 / 47
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Outline
1 How to Construct a Mean Field Approximation (via an example)Approach 1: Zoom on One Object and IndependenceApproach 2: Density Dependent Population ProcessesApproach 3: Discrete-Time Models
2 How Accurate is the Classical Mean Field ApproximationTransient AnalysisSteady-state Regime
3 We can Refine this ApproximationMain ResultsIdea of the Proof: System Size Expansion
4 Conclusion and Open Questions
Nicolas Gast – 6 / 47
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Analysis of Cache (Re)placement Policies
Popularity-oblivious policies (LRU, RANDOM1, TTL-caches2)
Popularity-aware policies / learning (LFU and variants3, network ofcaches4)
1started with [King 1971, Gelenbe 1973]2e.g., Fofack et al 2013, Berger et al. 20143Optimizing TTL Caches under Heavy-Tailed Demands (Ferragut et al. SIGMETRICS 2016)
4Adaptive Caching Networks with Optimality Guarantees (Ioannidis and Yeh,
SIGMETRICS16)Nicolas Gast – 7 / 47
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The RAND( ~m) policy 5
RANDOM: exchange the requested item with a random item
exchange cache
virtual
exchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
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The RAND( ~m) policy 5
RAND( ~m): exchange the requested item with an item from next list
cachevirtualexchange
exchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
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The RAND( ~m) policy 5
RAND( ~m): exchange the requested item with an item from next list
cachevirtualcopy
exchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
![Page 13: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/13.jpg)
The RAND( ~m) policy 5
RAND( ~m): exchange the requested item with an item from next list
cachevirtualcopy
exchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
![Page 14: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/14.jpg)
The RAND( ~m) policy 5
RAND( ~m): exchange the requested item with an item from next list
cachevirtualexchange
exchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
![Page 15: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/15.jpg)
The RAND( ~m) policy 5
RAND( ~m): exchange the requested item with an item from next list
cachevirtualexchange
exchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
![Page 16: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/16.jpg)
The RAND( ~m) policy 5
RAND( ~m): exchange the requested item with an item from next list
cachevirtualcopy
exchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
![Page 17: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/17.jpg)
The RAND( ~m) policy 5
RAND( ~m): exchange the requested item with an item from next list
cachevirtualexchange
exchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
![Page 18: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/18.jpg)
The RAND( ~m) policy 5
RAND( ~m): exchange the requested item with an item from next list
cachevirtualexchange
exchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
![Page 19: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/19.jpg)
The RAND( ~m) policy 5
RAND( ~m): exchange the requested item with an item from next list
cachevirtualexchange
5Can be generalized in LRU( ~m) or FIFO( ~m), see [ITC16,QUESTA16]Nicolas Gast – 8 / 47
![Page 20: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/20.jpg)
Our model is a heterogeneous population model
N items of identical sizes but different popularities.
The state of an item is the list in which it is {∅} ∪ {1, . . . , h}.The item k is requested at rate λk .
We write: Xk,i (t) = 1 if object k is in list i .
We want to find an approximation of E [Xk,i (t)]
(= probability that the object k is in state i).
Nicolas Gast – 9 / 47
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Our model is a heterogeneous population model
N items of identical sizes but different popularities.
The state of an item is the list in which it is {∅} ∪ {1, . . . , h}.The item k is requested at rate λk .
We write: Xk,i (t) = 1 if object k is in list i .
We want to find an approximation of E [Xk,i (t)]
(= probability that the object k is in state i).
Nicolas Gast – 9 / 47
![Page 22: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/22.jpg)
Construction of the mean-field approximation (RANDOM)
Out In
λi
1
m
∑j 6∈{cache}
λj
Out In
λi
1
m
n∑j=1
xjλj
Original model for RANDOM
MF approx: let xi (t) = P [i 6∈ {cache}].
(only one list)
If all objects are independent:
The “mean field” equations for the approximation model are:
xi = −λixi +1
m
n∑j=1
xj(t)λj(1− xi ).
Nicolas Gast – 10 / 47
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Construction of the mean-field approximation (RANDOM)
Out In
λi
1
m
∑j 6∈{cache}
λj
Out In
λi
1
m
n∑j=1
xjλj
Original model for RANDOM MF approx: let xi (t) = P [i 6∈ {cache}].(only one list) If all objects are independent:
The “mean field” equations for the approximation model are:
xi = −λixi +1
m
n∑j=1
xj(t)λj(1− xi ).
Nicolas Gast – 10 / 47
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Construction of the mean-field approximation (RANDOM)
Out In
λi
1
m
∑j 6∈{cache}
λj
Out In
λi
1
m
n∑j=1
xjλj
Original model for RANDOM MF approx: let xi (t) = P [i 6∈ {cache}].(only one list) If all objects are independent:
The “mean field” equations for the approximation model are:
xi = −λixi +1
m
n∑j=1
xj(t)λj(1− xi ).
Nicolas Gast – 10 / 47
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Steady-state analysis for RANDOM
The mean field equation xi = −λixi +1
m
n∑j=1
xj(t)λj(1− xi ). has a unique
fixed point that satisfies:
πi =z
z + λiwith z such that
n∑i=1
(1− πi ) = m.
Same equations as Fagins (77).
Nicolas Gast – 11 / 47
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Extension to the RAND(m) model (G, Van Houdt SIGMETRICS 2015)
Let Hi (t) be the popularity in list i .
If xk,i (t) is the probability that item k is in list i at time t, weapproximately have:
This approximation is of the form x = xQ(x).
Nicolas Gast – 12 / 47
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Extension to the RAND(m) model (G, Van Houdt SIGMETRICS 2015)
Let Hi (t) be the popularity in list i .
If xk,i (t) is the probability that item k is in list i at time t, weapproximately have:
This approximation is of the form x = xQ(x).Nicolas Gast – 12 / 47
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The mean field approximation is very accurate
n = 1000 objects with Zipf popularities.
0 2000 4000 6000 8000 10000number of requests
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
pro
babili
ty in c
ach
e
1 list (200)4 lists (50/50/50/50)ode aprox (1 list)ode approx (4 lists)
The popularities change Steady-state miss probabilitiesevery 2000 requests
Nicolas Gast – 13 / 47
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A population model
Consider now that we have K types of items with N items per type.
The state of an item is the list in which it is {∅} ∪ {1, . . . , h}.An item of type k is requested at rate λk .
We define Xk,i (t) has the fraction of type k items in list i .
The transitions are:
X 7→ X +1
N(−ek,i + ek,i+1 + e`,i − e`,i+1) at rate NλkXk,i
X`,i+1
mi+1
Nicolas Gast – 14 / 47
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A population model
Consider now that we have K types of items with N items per type.
The state of an item is the list in which it is {∅} ∪ {1, . . . , h}.An item of type k is requested at rate λk .
We define Xk,i (t) has the fraction of type k items in list i .
The transitions are:
X 7→ X +1
N(−ek,i + ek,i+1 + e`,i − e`,i+1) at rate NλkXk,i
X`,i+1
mi+1
Nicolas Gast – 14 / 47
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This is an example of a density dependent populationprocess (Kurtz, 70s)
A population process is a sequence of CTMCs XN(t) indexed by thepopulation size N, with state space EN ⊂ E and transitions (for ` ∈ L):
X 7→ X +`
Nat rate Nβ`(X ).
The Mean field approximation
The drift is f (x) =∑`
`β`(x) and the mean field
approximation is the solution of the ODE x = f (x).
Nicolas Gast – 15 / 47
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This is an example of a density dependent populationprocess (Kurtz, 70s)
A population process is a sequence of CTMCs XN(t) indexed by thepopulation size N, with state space EN ⊂ E and transitions (for ` ∈ L):
X 7→ X +`
Nat rate Nβ`(X ).
The Mean field approximation
The drift is f (x) =∑`
`β`(x) and the mean field
approximation is the solution of the ODE x = f (x).
Nicolas Gast – 15 / 47
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Other examples of density dependent population processesLoad Balancing and JSQ(2)
Xi (t) =1
N
N∑n=1
1{Sn(t)≥i} = fraction of queues with queue length ≥ i .
Arrival: x 7→ x +1
Nei.
Departures: x 7→ x − 1
Nei.
Pick two servers at random, what is the probability the least loaded hasi − 1 jobs?
x2i−1 − x2
i when picked with replacement
xi−1Nxi−1 − 1
N − 1− xi
Nxi − 1
N − 1when picked without replacement
Note: this becomes asymptotically the same as N goes to infinity.
Nicolas Gast – 16 / 47
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Other examples of density dependent population processesLoad Balancing and JSQ(2)
Xi (t) =1
N
N∑n=1
1{Sn(t)≥i} = fraction of queues with queue length ≥ i .
Arrival: x 7→ x +1
Nei.
Departures: x 7→ x − 1
Nei.
Pick two servers at random, what is the probability the least loaded hasi − 1 jobs?
x2i−1 − x2
i when picked with replacement
xi−1Nxi−1 − 1
N − 1− xi
Nxi − 1
N − 1when picked without replacement
Note: this becomes asymptotically the same as N goes to infinity.
Nicolas Gast – 16 / 47
![Page 35: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/35.jpg)
Other examples of density dependent population processesLoad Balancing and JSQ(2)
Xi (t) =1
N
N∑n=1
1{Sn(t)≥i} = fraction of queues with queue length ≥ i .
Arrival: x 7→ x +1
Nei.
Departures: x 7→ x − 1
Nei.
Pick two servers at random, what is the probability the least loaded hasi − 1 jobs?
x2i−1 − x2
i when picked with replacement
xi−1Nxi−1 − 1
N − 1− xi
Nxi − 1
N − 1when picked without replacement
Note: this becomes asymptotically the same as N goes to infinity.
Nicolas Gast – 16 / 47
![Page 36: A Tutorial on Mean Field and Re ned Mean Field Approximationpolaris.imag.fr/nicolas.gast/slides/Gast_ITC2019_meanField.pdf · 1 How to Construct a Mean Field Approximation (via an](https://reader033.fdocuments.net/reader033/viewer/2022042709/5f534e071b10085b6c6cdf0e/html5/thumbnails/36.jpg)
Transitions and Mean Field ApproximationState changes on x :
x 7→ x +1
Nei at rate Nρ(x2
i−1 − x2i )
x 7→ x − 1
Nei at rate N(xi − xi+1)
The mean field approximation is to consider the ODE associated with thedrift (average variation):
xi = ρ(x2i−1 − x2
i )︸ ︷︷ ︸Arrival
− (xi − xi+1)︸ ︷︷ ︸Departure
There is a unique fixed point (which also is an attractor):
πi = ρ2i−1
Nicolas Gast – 17 / 47
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Outline
1 How to Construct a Mean Field Approximation (via an example)Approach 1: Zoom on One Object and IndependenceApproach 2: Density Dependent Population ProcessesApproach 3: Discrete-Time Models
2 How Accurate is the Classical Mean Field ApproximationTransient AnalysisSteady-state Regime
3 We can Refine this ApproximationMain ResultsIdea of the Proof: System Size Expansion
4 Conclusion and Open Questions
Nicolas Gast – 18 / 47
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Benaım-Le Boudec’s model (PEVA 2007)Time is discrete.
Xi (k) = Proportion of object in state i at time step k
R(k) = State of the ”resource” at time k (discrete)
Assumptions:
Only O(1) objects change state at each time step and
f (x , r) =1
NE [X (k + 1)− X (k)|X (k) = x ,R(k) = r ] .
R evolves fast in a discrete state-space and:
P [R(k + 1) = j |X (k) = x ,R(k) = i ] = Pij(x).
For all x , P(x) is irreducible and has a unique stationary measure π(x , .).
Nicolas Gast – 19 / 47
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Benaım-Le Boudec’s model (PEVA 2007)Time is discrete.
Xi (k) = Proportion of object in state i at time step k
R(k) = State of the ”resource” at time k (discrete)
Assumptions:
Only O(1) objects change state at each time step and
f (x , r) =1
NE [X (k + 1)− X (k)|X (k) = x ,R(k) = r ] .
R evolves fast in a discrete state-space and:
P [R(k + 1) = j |X (k) = x ,R(k) = i ] = Pij(x).
For all x , P(x) is irreducible and has a unique stationary measure π(x , .).Nicolas Gast – 19 / 47
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Mean Field Approximation
Examples with resource: CSMA protocols, Opportunistic networks.
x =∑r
f (x , r)π(x , r),
where π(x , r) is the stationary measure of the resource given x .
The analysis of such models is done by considering stochasticapproximation algorithms. For example, without resource one has:
X (k + 1) = X (k) +1
N[f (X (k)) + M(k + 1)] ,
where M is some noise process.This is a noisy Euler discretization of an ordinary differential equation.
Nicolas Gast – 20 / 47
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Mean Field Approximation
Examples with resource: CSMA protocols, Opportunistic networks.
x =∑r
f (x , r)π(x , r),
where π(x , r) is the stationary measure of the resource given x .
The analysis of such models is done by considering stochasticapproximation algorithms. For example, without resource one has:
X (k + 1) = X (k) +1
N[f (X (k)) + M(k + 1)] ,
where M is some noise process.This is a noisy Euler discretization of an ordinary differential equation.
Nicolas Gast – 20 / 47
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Take-home message on this part
Three ways to construct mean fieldapproximation:
Independence assumption x = xQ(x).
Density dependent population process.
Discrete-time model with vanishing intensity.
In what follows, I will assume that X is a density dependent populationprocess (ex: SQ(d)). Analysis of other models are similar.
Nicolas Gast – 21 / 47
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Outline
1 How to Construct a Mean Field Approximation (via an example)Approach 1: Zoom on One Object and IndependenceApproach 2: Density Dependent Population ProcessesApproach 3: Discrete-Time Models
2 How Accurate is the Classical Mean Field ApproximationTransient AnalysisSteady-state Regime
3 We can Refine this ApproximationMain ResultsIdea of the Proof: System Size Expansion
4 Conclusion and Open Questions
Nicolas Gast – 22 / 47
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Convergence Result as N Goes to Infinity
Theorem (under technical conditions), for any density dependentpopulation process XN whose drift is Lipschitz-continuous, if XN(0)converges to x0, then for any finite T :
sup0≤t≤T
∥∥∥XN(t)− x(t)∥∥∥→ 0.
where x(t) is the unique solution of the ODE x = f (x).
Nicolas Gast – 23 / 47
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Illustration (cache model)
0 10 20 30 40 50time
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Fract
ion o
f popula
r it
em
s in
cach
e
odeExample. Fraction of popularitems in cache. m1 = N/4, m2 =N/4, N/2 popular object (popu-larity 4N/(5N), N/2 non-popularobjects (popularity N/(5N)).
Nicolas Gast – 24 / 47
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Illustration (cache model)
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r it
em
s in
cach
e
odeN= 5
Example. Fraction of popularitems in cache. m1 = N/4, m2 =N/4, N/2 popular object (popu-larity 4N/(5N), N/2 non-popularobjects (popularity N/(5N)).
Nicolas Gast – 24 / 47
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Illustration (cache model)
0 10 20 30 40 50time
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Fract
ion o
f popula
r it
em
s in
cach
e
odeN= 5
N= 50
Example. Fraction of popularitems in cache. m1 = N/4, m2 =N/4, N/2 popular object (popu-larity 4N/(5N), N/2 non-popularobjects (popularity N/(5N)).
Nicolas Gast – 24 / 47
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Illustration (cache model)
0 10 20 30 40 50time
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0.6
0.8
1.0
Fract
ion o
f popula
r it
em
s in
cach
e
odeN= 5
N= 50
N= 500
Example. Fraction of popularitems in cache. m1 = N/4, m2 =N/4, N/2 popular object (popu-larity 4N/(5N), N/2 non-popularobjects (popularity N/(5N)).
Nicolas Gast – 24 / 47
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The fixed point method
Transient regime
Stationary
Markov chain
p = pK
πK = 0
t →∞
Mean-field
x = xQ(x)
x∗Q(x∗) = 0fixed points
N →∞
N →∞
t →∞if yes
then yes
Theorem (Benaim Le Boudec 08)
If all trajectories of the ODE converges to the fixed points, the stationarydistribution πN concentrates on the fixed points
In that case, we also have:
limN→∞
P [S1 = i1 . . . Sk = ik ] = x∗1 . . . x∗k .
Nicolas Gast – 25 / 47
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The fixed point method
Transient regime
Stationary
Markov chain
p = pK
πK = 0
t →∞
Mean-field
x = xQ(x)
x∗Q(x∗) = 0fixed points
N →∞
?
N →∞
t →∞if yes
then yes
Method was used in many papers:
Bianchi 00, Performance analysis of the IEEE 802.11 distributed coordination function.
Ramaiyan et al. 08, Fixed point analys is of single cell IEEE 802.11e WLANs: Uniqueness, multistability.
Kwak et al. 05, Performance analysis of exponenetial backoff.
Kumar et al 08, New insights from a fixed-point analysis of single cell IEEE 802.11 WLANs.
Theorem (Benaim Le Boudec 08)
If all trajectories of the ODE converges to the fixed points, the stationarydistribution πN concentrates on the fixed points
In that case, we also have:
limN→∞
P [S1 = i1 . . . Sk = ik ] = x∗1 . . . x∗k .
Nicolas Gast – 25 / 47
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The fixed point method
Transient regime
Stationary
Markov chain
p = pK
πK = 0
t →∞
x∗Q(x∗) = 0
Mean-field
x = xQ(x)
x∗Q(x∗) = 0fixed points
N →∞
N →∞
t →∞
if yes
then yes
Theorem (Benaim Le Boudec 08)
If all trajectories of the ODE converges to the fixed points, the stationarydistribution πN concentrates on the fixed points
In that case, we also have:
limN→∞
P [S1 = i1 . . . Sk = ik ] = x∗1 . . . x∗k .
Nicolas Gast – 25 / 47
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The fixed point method
Transient regime
Stationary
Markov chain
p = pK
πK = 0
t →∞
x∗Q(x∗) = 0
Mean-field
x = xQ(x)
x∗Q(x∗) = 0fixed points
N →∞
N →∞
t →∞if yes
then yes
Theorem (Benaim Le Boudec 08)
If all trajectories of the ODE converges to the fixed points, the stationarydistribution πN concentrates on the fixed points
In that case, we also have:
limN→∞
P [S1 = i1 . . . Sk = ik ] = x∗1 . . . x∗k .
Nicolas Gast – 25 / 47
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The fixed point method does not always work
0.0
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1.0
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1.0
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Fixed pointtrue stationnary distribution
limit cycle
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Fixed point x ∗ = πN
The fixed point methods doesnot work for a = 0.1 It works for a = 0.3
Counter-examples for specific CSMA models (Cho, Le Boudec, Jiang 2011)Nicolas Gast – 26 / 47
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Fixed point method in practice
From the examples coming from queuing theory, many models have aunique attractor.
This holds for classical load balancing policies such as SQ(d),pull-push, JIQ,...
I Often comes from monotonicity
This holds in many cases in statistical physicsI Lyapunov methods (entropy, reversibility) (ex: G. 2016 for cache)
It does not always workI Theoretical biology / chemistryI Multi-stable models (ex: Kelly)I Counter-examples for specific CSMA models (Cho, Le Boudec, Jiang
2011)
Nicolas Gast – 27 / 47
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Outline
1 How to Construct a Mean Field Approximation (via an example)Approach 1: Zoom on One Object and IndependenceApproach 2: Density Dependent Population ProcessesApproach 3: Discrete-Time Models
2 How Accurate is the Classical Mean Field ApproximationTransient AnalysisSteady-state Regime
3 We can Refine this ApproximationMain ResultsIdea of the Proof: System Size Expansion
4 Conclusion and Open Questions
Nicolas Gast – 28 / 47
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For which system size N can mean-field be used?
Recall: XNi (t) = fraction of objects in state i .
Theorem (Kurtz (1970s), Ying (2016)):
If the drift f is Lipschitz-continuous:
XN(t) ≈ x(t) +1√NGt
If in addition the ODE has aunique attractor π:
E[XN(∞)− π
]= O(1/
√N)
0 10 20 30 40 50time
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0.8
1.0
Fract
ion o
f popula
r it
em
s in
cach
e
odeExample. Fraction of popularitems in cache. m1 = N/4, m2 =N/4, N/2 popular object (popu-larity 4, N/2 non-popular objects(popularity 1).
Nicolas Gast – 29 / 47
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For which system size N can mean-field be used?
Recall: XNi (t) = fraction of objects in state i .
Theorem (Kurtz (1970s), Ying (2016)):
If the drift f is Lipschitz-continuous:
XN(t) ≈ x(t) +1√NGt
If in addition the ODE has aunique attractor π:
E[XN(∞)− π
]= O(1/
√N)
0 10 20 30 40 50time
0.0
0.2
0.4
0.6
0.8
1.0
Fract
ion o
f popula
r it
em
s in
cach
e
odeN= 5
Example. Fraction of popularitems in cache. m1 = N/4, m2 =N/4, N/2 popular object (popu-larity 4, N/2 non-popular objects(popularity 1).
Nicolas Gast – 29 / 47
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For which system size N can mean-field be used?
Recall: XNi (t) = fraction of objects in state i .
Theorem (Kurtz (1970s), Ying (2016)):
If the drift f is Lipschitz-continuous:
XN(t) ≈ x(t) +1√NGt
If in addition the ODE has aunique attractor π:
E[XN(∞)− π
]= O(1/
√N)
0 10 20 30 40 50time
0.0
0.2
0.4
0.6
0.8
1.0
Fract
ion o
f popula
r it
em
s in
cach
e
odeN= 5
N= 50
Example. Fraction of popularitems in cache. m1 = N/4, m2 =N/4, N/2 popular object (popu-larity 4, N/2 non-popular objects(popularity 1).
Nicolas Gast – 29 / 47
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For which system size N can mean-field be used?
Recall: XNi (t) = fraction of objects in state i .
Theorem (Kurtz (1970s), Ying (2016)):
If the drift f is Lipschitz-continuous:
XN(t) ≈ x(t) +1√NGt
If in addition the ODE has aunique attractor π:
E[XN(∞)− π
]= O(1/
√N)
0 10 20 30 40 50time
0.0
0.2
0.4
0.6
0.8
1.0
Fract
ion o
f popula
r it
em
s in
cach
e
odeN= 5
N= 50
N= 500
Example. Fraction of popularitems in cache. m1 = N/4, m2 =N/4, N/2 popular object (popu-larity 4, N/2 non-popular objects(popularity 1).
Nicolas Gast – 29 / 47
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In practice, mean field works well even for “small” systemsHow small?
Cache size (N/2) 10 100 1000 +∞Proba popular∈cache 0.7604 0.7553 0.7548 0.7547
Error 0.0060 0.0006 <0.0001 –
Also works for load balancing policies (two-choice allocation):N-servers 10 100 1000 +∞
Average queue length (mN) 3.81 3.39 3.36 3.35
Error (mN −m∞) 0.45 0.039 0.004 0
Xi = fraction ofobjects in state i
xi = Proba(one ob-ject is in i) = E [Xi ]
xi (mean-fieldapprox)
O(1/√N)
O(1/√N) (CLT) ?
Nicolas Gast – 30 / 47
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In practice, mean field works well even for “small” systemsHow small?
Cache size (N/2) 10 100 1000 +∞Proba popular∈cache 0.7604 0.7553 0.7548 0.7547
Error 0.0060 0.0006 <0.0001 –
Also works for load balancing policies (two-choice allocation):N-servers 10 100 1000 +∞
Average queue length (mN) 3.81 3.39 3.36 3.35
Error (mN −m∞) 0.45 0.039 0.004 0
Xi = fraction ofobjects in state i
xi = Proba(one ob-ject is in i) = E [Xi ]
xi (mean-fieldapprox)
O(1/√N)
O(1/√N) (CLT) ?
Nicolas Gast – 30 / 47
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In practice, mean field works well even for “small” systemsHow small?
Cache size (N/2) 10 100 1000 +∞Proba popular∈cache 0.7604 0.7553 0.7548 0.7547
Error 0.0060 0.0006 <0.0001 –
Also works for load balancing policies (two-choice allocation):N-servers 10 100 1000 +∞
Average queue length (mN) 3.81 3.39 3.36 3.35
Error (mN −m∞) 0.45 0.039 0.004 0
Xi = fraction ofobjects in state i
xi = Proba(one ob-ject is in i) = E [Xi ]
xi (mean-fieldapprox)
O(1/√N)
O(1/√N) (CLT) ?
Nicolas Gast – 30 / 47
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Expected values estimated by mean field are 1/N-accurate
Theorem (Kolokoltsov 2012, G. 2017, G. and Van Houdt 2018). If thedrift f is C 2 and has a unique exponentially stable attractor, then for anyt ∈ [0,∞) ∪ {∞},
there exists a (deterministic) vector V (t) such that:
E[XN(t)
]= x(t) +
V (t)
N+ O(1/N2)
V (t) can be easily computed numerically
Nicolas Gast – 31 / 47
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The refined mean field approximation...... is defined as the classic mean field plus the 1/N correction term:
E[XN]
= x(t) +V (t)
N︸ ︷︷ ︸refined mean field
+O(1/N2),
where V (t) is computed analytically.
To compute V (t), we need:
Derivative of the drifts:
F ij (t) =
∂fi∂xj
(x(t)) and F ijk(t) =
∂2fi∂xj∂xk
(x(t))
A variance term:
Q(t) =∑`
`⊗ `β`(X (t))
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The refined mean field approximation...... is defined as the classic mean field plus the 1/N correction term:
E[XN]
= x(t) +V (t)
N︸ ︷︷ ︸refined mean field
+O(1/N2),
where V (t) is computed analytically.
To compute V (t), we need:
Derivative of the drifts:
F ij (t) =
∂fi∂xj
(x(t)) and F ijk(t) =
∂2fi∂xj∂xk
(x(t))
A variance term:
Q(t) =∑`
`⊗ `β`(X (t))
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Computational methodsTheorem (G, Van Houdt 2018, G. 2018) Given a density dependentprocess with twice-differentiable drift. Let h : E → R be atwice-differentiable function, then for t > 0:
E[XN(t)
]= x(t) +
1
NV (t) +
1
N2A(t) + O(
1
N3),
where
d
dtV i =
∑j
F ij V
j +∑jk
F ij ,kW
j ,k
d
dtW j ,k = Q jk +
∑m
F jmW
m,k +∑m
W j ,mF km
Theorem (G, Van Houdt 2018) The previous theorem also holds for thestationary regime (t = +∞) if the ODE has a unique exponentially stableattractor.
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Numerical example: caching and RANDOM(m) policy
Cache size (N/2) 10 20 30 50 100 +∞Simulation 0.7604 0.7575 0.7566 0.7559 0.7553 –Refined mf 0.7607 0.7576 0.7567 0.7558 0.7553 0.7547
Average fraction of most popular items in cache.
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Numerical example: caching and RANDOM(m) policy
Cache size (N/2) 10 20 30 50 100 +∞Simulation 0.7604 0.7575 0.7566 0.7559 0.7553 –Refined mf 0.7607 0.7576 0.7567 0.7558 0.7553 0.7547
Average fraction of most popular items in cache.
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The SQ(2)
Arrival at each server ρ.
Sample anotherqueue.
Allocate to theshortest of the two.
Service rate=1.
N = 10 N = 20 N = 50 N = 100
Mean Field 2.3527 2.3527 2.3527 2.35271/N-expansion 2.7513 2.5520 2.4324 2.3925
1/N2-expansion 2.8045 2.5653 2.4345 2.3930Simulation 2.8003 2.5662 2.4350 2.3931
SQ(2): Steady-state average queue length (ρ = 0.9).
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How does the expected queue length evolve with time?
0 10 20 30 40 50 60 70 80Time
2.4
2.5
2.6
2.7
2.8
Aver
age
queu
e le
ngth
Mean Field ApproximationSimulation (N = 1000)
Remark about computation time :
10min/1h (simulation N = 1000/N = 10), C++ code. Requires many simulations,confidence intervals,...
80ms (mean field), 700ms (1/N-expansion), 9s (1/N2-expansion), Python numpy
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How does the expected queue length evolve with time?
0 10 20 30 40 50 60 70 80Time
2.4
2.5
2.6
2.7
2.8
Aver
age
queu
e le
ngth
Mean Field ApproximationSimulation (N = 10)
Remark about computation time :
10min/1h (simulation N = 1000/N = 10), C++ code. Requires many simulations,confidence intervals,...
80ms (mean field), 700ms (1/N-expansion), 9s (1/N2-expansion), Python numpy
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How does the expected queue length evolve with time?
0 10 20 30 40 50 60 70 80Time
2.4
2.5
2.6
2.7
2.8
Aver
age
queu
e le
ngth
Mean Field Approximation1/N-expansion1/N2-expansionSimulation (N = 10)
Remark about computation time :
10min/1h (simulation N = 1000/N = 10), C++ code. Requires many simulations,confidence intervals,...
80ms (mean field), 700ms (1/N-expansion), 9s (1/N2-expansion), Python numpy
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Analysis of the computation time
For the numerical examples of SQ(2), I used a bounded queue size d .
100 200 300 400 500Number of dimensions (d)
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
Com
utat
ion
Tim
e (in
seco
nds)
10 20 30 40Number of dimensions (d)
0
5
10
15
20
Com
utat
ion
Tim
e (in
seco
nds)
Time to compute the 1/N-expansion Time to compute the 1/N2-expansionAnalysis of the computation time (Python numpy implementation)
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Does it always work?Can I always exchange the limits N → ∞, k → ∞, t → ∞?
E [X (t)] = x(t) +1
NV (t) +
1
N2A(t) + · · ·+ O(
1
Nk+1)
NO:
0 1 2 3 4 5Time t
1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
X A(t)
mean-field1/N-expansion1/N2-expansionsimulation (N = 200)
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Does it always work?Can I always exchange the limits N → ∞, k → ∞, t → ∞?
E [X (t)] = x(t) +1
NV (t) +
1
N2A(t) + · · ·+ O(
1
Nk+1)
NO:
0 1 2 3 4 5Time t
1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
X A(t)
mean-field1/N-expansion1/N2-expansionsimulation (N = 200)
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Outline
1 How to Construct a Mean Field Approximation (via an example)Approach 1: Zoom on One Object and IndependenceApproach 2: Density Dependent Population ProcessesApproach 3: Discrete-Time Models
2 How Accurate is the Classical Mean Field ApproximationTransient AnalysisSteady-state Regime
3 We can Refine this ApproximationMain ResultsIdea of the Proof: System Size Expansion
4 Conclusion and Open Questions
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Where does V /N come from? (SQ(2) example)Transitions on Xi : +
1
Nat rate N(x2
i−1 − x2i ) and − 1
Nat rate N(xi−xi+1).
Hence:
d
dtE [Xi (t)] = E
[X 2i−1(t)− X 2
i (t)− (Xi (t)− Xi+1(t))]
(exact)
= E[X 2i−1(t)
]− E
[X 2i (t)
]− E [Xi (t)] + E [Xi+1(t)]
≈E [Xi−1(t)]2 − E [Xi (t)]2 − E [Xi (t)] + E [Xi+1(t)] (mean field approx.)
If we now consider how E[X 2i
]evolves, we have:
d
dtE[X 2i
]= E
[(2Xi +
1
N)(X 2
i−1 − X 2i ) + (−2Xi +
1
N)(Xi − Xi+1)
]
= E
2XiX2i−1︸ ︷︷ ︸
E[XiX2i−1≈?]
+ . . . . . . . . . . . .
where we denote X instead of X (t) for simplicity.To close the equation: we assume that E
[(Xi − xi )
2(Xi−1 − xi−1)]≈ 0.
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Where does V /N come from? (SQ(2) example)Transitions on Xi : +
1
Nat rate N(x2
i−1 − x2i ) and − 1
Nat rate N(xi−xi+1).
Hence:
d
dtE [Xi (t)] = E
[X 2i−1(t)− X 2
i (t)− (Xi (t)− Xi+1(t))]
(exact)
= E[X 2i−1(t)
]− E
[X 2i (t)
]− E [Xi (t)] + E [Xi+1(t)]
≈E [Xi−1(t)]2 − E [Xi (t)]2 − E [Xi (t)] + E [Xi+1(t)] (mean field approx.)
If we now consider how E[X 2i
]evolves, we have:
d
dtE[X 2i
]= E
[(2Xi +
1
N)(X 2
i−1 − X 2i ) + (−2Xi +
1
N)(Xi − Xi+1)
]
= E
2XiX2i−1︸ ︷︷ ︸
E[XiX2i−1≈?]
+ . . . . . . . . . . . .
where we denote X instead of X (t) for simplicity.To close the equation: we assume that E
[(Xi − xi )
2(Xi−1 − xi−1)]≈ 0.
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System Size Expansion ApproachRecall that the transitions are X 7→ X +
`
Nat rate Nβ`(x).
d
dtE [X ] = E
[∑`
β`(X )`
]= E [f (X )] (Exact)
d
dtx = f (x) (Mean Field Approx.)
We can now look at the second moment:
E [(X − x)⊗ (X − x)] = E [(f (X )− f (x))⊗ (X − x)] (Exact)
+ E [(X − x)⊗ (f (X )− f (x))]
+1
NE
[∑`∈L
β`(X )`⊗ `
]... We can also look at higher order moments
E[(X − x)⊗3
]= 3SymE [(f (X )− f (x))⊗ (X − x)⊗ (X − x)]
+3
NSymE
[∑`∈L
β`(X )`⊗ `⊗ (X − x)
]+
1
NE
[∑`∈L
β`(X )`⊗ `⊗ `
]
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System Size Expansion ApproachRecall that the transitions are X 7→ X +
`
Nat rate Nβ`(x).
d
dtE [X ] = E
[∑`
β`(X )`
]= E [f (X )] (Exact)
d
dtx = f (x) (Mean Field Approx.)
We can now look at the second moment:
E [(X − x)⊗ (X − x)] = E [(f (X )− f (x))⊗ (X − x)] (Exact)
+ E [(X − x)⊗ (f (X )− f (x))]
+1
NE
[∑`∈L
β`(X )`⊗ `
]
... We can also look at higher order moments
E[(X − x)⊗3
]= 3SymE [(f (X )− f (x))⊗ (X − x)⊗ (X − x)]
+3
NSymE
[∑`∈L
β`(X )`⊗ `⊗ (X − x)
]+
1
NE
[∑`∈L
β`(X )`⊗ `⊗ `
]
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System Size Expansion ApproachRecall that the transitions are X 7→ X +
`
Nat rate Nβ`(x).
d
dtE [X ] = E
[∑`
β`(X )`
]= E [f (X )] (Exact)
d
dtx = f (x) (Mean Field Approx.)
We can now look at the second moment:
E [(X − x)⊗ (X − x)] = E [(f (X )− f (x))⊗ (X − x)] (Exact)
+ E [(X − x)⊗ (f (X )− f (x))]
+1
NE
[∑`∈L
β`(X )`⊗ `
]... We can also look at higher order moments
E[(X − x)⊗3
]= 3SymE [(f (X )− f (x))⊗ (X − x)⊗ (X − x)]
+3
NSymE
[∑`∈L
β`(X )`⊗ `⊗ (X − x)
]+
1
NE
[∑`∈L
β`(X )`⊗ `⊗ `
]Nicolas Gast – 41 / 47
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System Size Expansion and Moment Closure
Let x(t) be the mean field approximation and Y (t) = X (t)− x(t), andY (t)(k) = Y (t)⊗ · · · ⊗ Y (t)︸ ︷︷ ︸
k times
d
dtE[Y (t)(k)
]can be expressed as an exact
function of Y (t)(j) for j ∈ {0 . . . , k + 1}.
You can close the equations by assuming that Y (k) ≈ 0 for k ≥ K .
For K = 1, this gives the mean field approximation (1/N-accurate)
For K = 3, this gives the refined mean field (1/N2-accurate).
For K = 5, this gives a second order expansion (1/N3-accurate).
Limit: For a system of dimension d , Y (t)(k) has dk equations.
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System Size Expansion and Moment Closure
Let x(t) be the mean field approximation and Y (t) = X (t)− x(t), andY (t)(k) = Y (t)⊗ · · · ⊗ Y (t)︸ ︷︷ ︸
k times
d
dtE[Y (t)(k)
]can be expressed as an exact
function of Y (t)(j) for j ∈ {0 . . . , k + 1}.
You can close the equations by assuming that Y (k) ≈ 0 for k ≥ K .
For K = 1, this gives the mean field approximation (1/N-accurate)
For K = 3, this gives the refined mean field (1/N2-accurate).
For K = 5, this gives a second order expansion (1/N3-accurate).
Limit: For a system of dimension d , Y (t)(k) has dk equations.
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Outline
1 How to Construct a Mean Field Approximation (via an example)Approach 1: Zoom on One Object and IndependenceApproach 2: Density Dependent Population ProcessesApproach 3: Discrete-Time Models
2 How Accurate is the Classical Mean Field ApproximationTransient AnalysisSteady-state Regime
3 We can Refine this ApproximationMain ResultsIdea of the Proof: System Size Expansion
4 Conclusion and Open Questions
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Recap and extensions
For a mean field model with twice differentiable drift, then :
1 The accuracy of the classical mean field approximation is O(1/N).
2 We can use this to define a refined approximation.
3 The refined approximation is often accurate for N = 10.
Extensions:
Transient regime
Discrete-time (Synchronous)
Next expansion term in 1/N2.
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In many cases, the refined approximation is very accurate
“Truth” Refined mean field approximation Mean field approximation
E[XN]
π +V
Nπ (=fixed point)
6
6Ref : G., Van Houdt, 2018Nicolas Gast – 45 / 47
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Some open questions
Optimization (mean field games, reinforcement learning)
Non-exponential models (ex: general service time, TTL&LRU)
Heterogeneous models
Multi-timescale models
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Some References
http://mescal.imag.fr/membres/nicolas.gast
https://github.com/ngast/rmf_tool/
A Refined Mean Field Approximation by Gast and Van Houdt. SIGMETRICS 2018 (best paper award)
Size Expansions of Mean Field Approximation: Transient and Steady-State Analysis Gast, Bortolussi, Tribastone
Expected Values Estimated via Mean Field Approximation are O(1/N)-accurate by Gast. SIGMETRICS 2017.
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