@article{fournier:inria-00452791,
hal_id = {inria-00452791},
url = {http://hal.inria.fr/inria-00452791},
title = {{Lower Bounds for Comparison Based Evolution Strategies using VC-dimension and Sign Patterns}},
author = {Fournier, Herv{\'e} and Teytaud, Olivier},
abstract = {{We derive lower bounds on the convergence rate of comparison based or selection based algorithms, improving existing results in the continuous setting, and extending them to non-trivial results in the discrete case. This is achieved by considering the VC-dimension of the level sets of the fitness functions; results are then obtained through the use of the shatter function lemma. In the special case of optimization of the sphere function, improved lower bounds are obtained by an argument based on the number of sign patterns.}},
keywords = {Evolutionary Algorithms;Parallel Optimization;Comparison-based algorithms;VC-dimension;Sign patterns;Complexity},
language = {Anglais},
affiliation = {Parall{\'e}lisme, R{\'e}seaux, Syst{\`e}mes d'information, Mod{\'e}lisation - PRISM , Laboratoire de Recherche en Informatique - LRI , TAO - INRIA Saclay - Ile de France},
publisher = {Springer},
journal = {Algorithmica},
audience = {internationale },
year = {2010},
pdf = {http://hal.inria.fr/inria-00452791/PDF/evolution.pdf},
}
@incollection{teytaud:inria-00593179,
hal_id = {inria-00593179},
url = {http://hal.inria.fr/inria-00593179},
title = {{Lower Bounds for Evolution Strategies}},
author = {Teytaud, Olivier},
abstract = {{The mathematical analysis of optimization algorithms involves upper and lower bounds; we here focus on the second case. Whereas other chap- ters will consider black box complexity, we will here consider complexity based on the key assumption that the only information available on the fitness values is the rank of individuals - we will not make use of the exact fitness values. Such a reduced information is known efficient in terms of ro- bustness (Gelly et al., 2007), what gives a solid theoretical foundation to the robustness of evolution strategies, which is often argued without mathemat- ical rigor - and we here show the implications of this reduced information on convergence rates. In particular, our bounds are proved without infi- nite dimension assumption, and they have been used since that time for designing algorithms with better performance in the parallel setting.}},
language = {Anglais},
affiliation = {Laboratoire de Recherche en Informatique - LRI , TAO - INRIA Saclay - Ile de France},
booktitle = {{Theory of Randomized Search Heuristics}},
publisher = {World Scientific},
pages = {327-354},
volume = {1},
editor = {Anne Auger, Benjamin Doerr },
series = {Series on Theoretical Computer Science },
audience = {internationale },
year = {2011},
month = May,
pdf = {http://hal.inria.fr/inria-00593179/PDF/ws-book9x6.pdf},
}
Complexity bounds for comparison-based optimization and parallel optimization
1. Complexity bounds in parallel
evolution
A. Auger, H. Fournier,
N. Hansen, P. Rolet,
F. Teytaud, O. Teytaud
Paris, 2010
Tao, Inria Saclay Ile-De-France,
LRI (Université Paris Sud, France),
UMR CNRS 8623, I&A team, Digiteo,
Pascal Network of Excellence.
3. Outline
Introduction
- What is optimization ?
- What are comparison-based optimization
algorithms ?
- Why we are interested in cp-based opt ?
- Why we consider parallel machines ?
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 3
4. Introduction: what is optimization ?
Consider
f: X --> R
We look for x* such that
x,f(x*) ≤ f(x) w random
variable
f is randomly drawn; f(x) = f(x,w).
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 4
5. Introduction: what is optimization ?
Quality of “Opt” quantified as follows:
(to be minimized)
w random
variable
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 5
6. Introduction: what is optimization ?
Consider
f: X --> R
We look for x* such that
x,f(x*) ≤ f(x)
==> Quasi-Newton, random search,
Newton, Simplex, Interior points...
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 6
7. Comparison-based optimization
is comparison-based if
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 7
8. The main rules for step-size adaptation
While ( I have time )
{
Generate points (x1,...,x) distributed as N(x,)
Evaluate the fitness at x1,...,x
Update x, update
}
Main trouble: choosing
Cumulative step-size adaptation
Mutative self-adaptation
Estimation of Multivariate Normal Algorithm
9. Example 1: Estimation of Multivariate Normal Algorithm
While ( I have time )
{
Generate points (x1,...,x) distributed as N(x,)
Evaluate the fitness at x1,...,x
X= mean best points
= standard deviation of best points
}
I have a Gaussian...
10. Example 1: Estimation of Multivariate Normal Algorithm
While ( I have time )
{
Generate points (x1,...,x) distributed as N(x,)
Evaluate the fitness at x1,...,x
X= mean best points
= standard deviation of best points
}
I generate 6 points
11. Example 1: Estimation of Multivariate Normal Algorithm
While ( I have time )
{
Generate points (x1,...,x) distributed as N(x,)
Evaluate the fitness at x1,...,x
X= mean best points
= standard deviation of best points
}
I select the three best
12. Example 1: Estimation of Multivariate Normal Algorithm
While ( I have time )
{
Generate points (x1,...,x) distributed as N(x,)
Evaluate the fitness at x1,...,x
X= mean best points
= standard deviation of best points
}
I update the Gaussian
13. Example 1: Estimation of Multivariate Normal Algorithm
While ( I have time )
{
Generate points (x1,...,x) distributed as N(x,)
Evaluate the fitness at x1,...,x
X= mean best points
= standard deviation of best points
}
Obviously 6-parallel
14. Example 2: Mutative self-adaptation
= / 4
While ( I have time )
{
Generate points (1,...,) as x exp(- k.N)
Generate points (x1,...,x) distributed as N(x,i)
Select the best points
Update x (=mean), update (=log. mean)
}
15. Plenty of comparison-based algorithms
EMNA and other EDA
Self-adaptive algorithms
Cumulative step-size adaptation
Pattern Search Methods ...
16. Families of comparison-based algorithms
Main parameter = = number of
evaluations per iteration = parallelism
Full-Ranking vs Selection-Based (param )
FR: we know the ranking of the best
SB: we just know which are the best
Elitist or not
Elitist: comparison with all visited points
Non-elitist: only within current offspring
17. EMNA ? Self-adaptation ?
Main parameter = = number of
evaluations per iteration = parallelism
Full-Ranking vs Selection-Based
FR: we know the ranking of all visited points
SB: we just know which are the best
Elitist or not
Elitist: comparison with all visited points
Non-elitist: only within current offspring
==> yet, they work quite well
18. Comparison-based algorithms are robust
Consider
f: X --> R
We look for x* such that
x,f(x*) ≤ f(x)
==> what if we see g o f (g increasing) ?
==> x* is the same, but xn might change
==> then, comparison-based methods are
optimal
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 18
19. Robustness of comparison-based algorithms: formal
statement
this does not depend on g for a
comparison-based algorithm
a comparison-based algorithm is optimal
for
(I don't give a proof here, but I promise it's true)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 19
20. Introduction: I like large
● Grid5000 = 5 000 cores (increasing)
● Submitting jobs ==> grouping runs
==> much bigger than number of cores.
● Next generations of computers: tenths,
hundreds, thousands of cores.
● Evolutionary algorithms are population
based but they have a bad speed-up.
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 20
21. Introduction: I like large
● Grid5000 = 5 000 cores (increasing)
● Submitting jobs ==> grouping runs
==> much bigger than
number of cores.
● Next generations of computers: tenths,
hundreds, thousands of cores.
● Evolutionary algorithms are population
based but they have a bad speed-up.
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 21
22. Introduction: I like large
● Grid5000 = 5 000 cores (increasing)
● Submitting jobs ==> grouping runs
==> much bigger
than number of cores.
● Next generations of computers: tenths,
hundreds, thousands of cores.
● Evolutionary algorithms are population
based but they have a bad speed-up.
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 22
23. Introduction: I like large
● Grid5000 = 5 000 cores (increasing)
● Submitting jobs ==> grouping runs
==> much bigger than number of cores.
● Next generations of computers: tenths,
hundreds, thousands of cores.
● Evolutionary algorithms are population
based but they have a bad speed-up.
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 23
24. Introduction: concluding :-)
● Optimization = finding minima
● Many algorithms are comparison-based
● ==> good idea for robustness
● Parallel case interesting
●
==> now we can have fun with bounds
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 24
25. Outline
Introduction On a given domain D
On a space F of objective
Complexity bounds functions such that
{x*(f);f∈F}=D
Branching Factor
Automatic Parallelization
Real-world algorithms
Log() corrections
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 25
26. Complexity bounds (N = dimension)
= nb of fitness evaluations for precision
with probability at least ½ for all f
N() = cov. number of the search space
Exp ( - Convergence ratio ) = Convergence rate
Convergence ratio ~ 1 / computational cost
==> more convenient for speed-ups
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 26
27. Complexity bounds ½
= nb of fitness evaluations for precision
with probability at least ½ for all f
N() = cov. number of the search space
Exp ( - Convergence ratio ) = Convergence rate
Convergence ratio ~ 1 / computational cost
==> more convenient for speed-ups
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 27
28. Complexity bounds: basic technique
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number n of iterations should verify
Kn ≥ N ( )
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 28
29. Complexity bounds: basic technique
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number n of iterations should verify
Kn ≥ N ( )
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 29
30. Complexity bounds: basic technique
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number n of iterations should verify
Kn ≥ N ( )
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 30
31. Complexity bounds: basic technique
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number n of iterations should verify
Kn ≥ N ( )
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 31
32. Complexity bounds: -balls
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number n of iterations should verify
Kn ≥ N ( )
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 32
33. Complexity bounds: -balls
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number n of iterations should verify
Kn ≥ N ( )
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 33
34. Complexity bounds: -balls
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number n of iterations should verify
Kn ≥ N ( )
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 34
35. Complexity bounds: basic technique
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number n of iterations should verify
Kn ≥ N ( )
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 35
36. Complexity bounds on the convergence ratio
FR: full ranking (selected points are ranked)
SB: selection-based (selected points are not ranked)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 36
37. Complexity bounds on the convergence ratio
Linear in ?
FR: full ranking (selected points are ranked)
SB: selection-based (selected points are not ranked)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 37
38. Linear speed-up ? My bound is
tight,
I've proved it!
Bounds:
On a given domain D
On a space F of objective
functions such that
{x*(f);f∈F}=D
==> very strange F possible!
==> much easier than
F={||x-x*|| ; x*∈ D }
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 38
39. Linear speed-up ? My bound is
Ok, tight
bound. tight,
But what I've proved it!
happens with
a
better model ?
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 39
40. Complexity bounds on the convergence ratio
- Comparison-based optimization
(or opt. with limited precision numbers)
- We have developped bounds based on:
Branching factor: finitely many possible
informations on the problem per time step
(→ communication. compl)
Packing number (lower bound on number of
possible outcomes)
Adding assumptions ==> better bounds ?
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 40
41. Complexity bounds: improved technique
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number nMany of these K verify
of iterations should
Kn ≥ Nbranches are
( )
very unlikely !
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 41
42. Complexity bounds: improved technique
We want to know how many iterations we need for reaching precision
in an evolutionary algorithm.
Key observation: (most) evolutionary algorithms are comparison-based
Let's consider (for simplicity) a deterministic selection-based non-elitist
algorithm
First idea: how many different branches we have in a run ?
We select points among
Therefore, at most K = ! / ( ! ( - )!) different branches
Second idea: how many different answers should we able to give ?
Use packing numbers: at least N() different possible answers
Conclusion: the number n of iterations should verify
Many of these K
n
K ≥ N( )
branches are We'll use...
… VC-dimension !
very unlikely !
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 42
43. (these slides “shattering + VC-dim”
extracted from Xue Mei's talk
at ENEE698A)
Definition of shattering:
A set S of points is shattered by a set H of
sets if for every dichotomy of S there is a
consistent hypothesis in H
48. VC-dimension
VC-dimension( set of sets ) =
maximum cardinal of a shattered set
VC-dimension (set of functions ) =
VC-dimension ( level sets)
Known (as a function of the dimension)
for many sets of functions
In particular, quadratic for ellipsoids,
linear for homotheties of a fixed ellipsoid
linear for circles...
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 48
49. VC-dimension
VC-dimension( set of sets ) =
maximum cardinal of a shattered set
VC-dimension (set of functions ) =
VC-dimension ( level sets)
Known (as a function of the dimension)
for many sets of functions
In particular, quadratic for ellipsoids,
linear for homotheties of a fixed ellipsoid
linear for circles...
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 49
50. VC-dimension
VC-dimension( set of sets ) =
maximum cardinal of a shattered set
VC-dimension (set of functions ) =
VC-dimension ( level sets)
Known (as a function of the dimension)
for many sets of functions
In particular, quadratic for ellipsoids,
linear for homotheties of a fixed ellipsoid
linear for circles...
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 50
51. VC-dimension
VC-dimension( set of sets ) =
maximum cardinal of a shattered set
VC-dimension (set of functions ) =
VC-dimension ( sublevel sets)
Known (as a function of the dimension)
for many sets of functions
In particular, quadratic for ellipsoids,
linear for homotheties of a fixed ellipsoid
linear for circles...
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 51
52. VC-dimension: the link with optimization ?
Sauer's lemma:
number of subsets of V points consistent
V
with a set of VC-dim V at most
So what ?
number of possible selections at most
V
K≤
==> instead of K = ! / ( ! ( - )!)
(V at least 3, otherwise a few details change...)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 52
53. Complexity bounds on the convergence ratio
FR: full ranking (selected points are ranked)
SB: selection-based (selected points are not ranked)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 53
54. Complexity bounds on the convergence ratio
Should not be
linear in !
FR: full ranking (selected points are ranked)
SB: selection-based (selected points are not ranked)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 54
55. Complexity bounds on the convergence ratio
Something
remains!
FR: full ranking (selected points are ranked)
SB: selection-based (selected points are not ranked)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 55
61. Branching factor K (more in Gelly06; Fournier08)
Rewrite your evolutionary algorithm as follows:
g has values in a finite set of cardinal K:
- e.g. subsets of {1,2,...,} of size (K=! / (!(-)!) )
- or ordered subsets (K=! / (-)! ).
- ...
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 61
62. Outline
Upper bounds for the
Introduction dependency in
Complexity bounds
Branching Factor
Automatic Parallelization
Real-world algorithms
Log() corrections
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 62
67. Speculative parallelization with branching factor
3
Parallel version for D=2.
Population = union of all pops for 2 iterations.
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 67
69. Real world algorithms
Define:
Necessary condition for log() speed-up:
- E log( * ) ~ log()
But for many algorithms,
- E log( * ) = O(1) ==> constant speed-up
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 69
70. One-fifth rule: - E log( * ) = O(1)
= proportion of mutated points better than x
While ( I have time )
{
Generate points (x1,...,x) distributed as N(x,)
Evaluate the fitness at x1,...,x
Update x = mean
Update
By 1/5th rule
}
or
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 70
71. One-fifth rule: - E log( * ) = O(1)
= proportion of mutated points better than x
Consider e.g.
Or consider e.g.
In both cases * is lower-bounded
independently of
==> parameters should
strongly depend on !
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 71
72. Self-adaptation, cumulative step-size adaptation
In many cases, the same result:
with parameters depending on the
dimension only (and not depending on ),
the speed-up is limited by a constant!
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 72
74. The starting point of this work
●We have shown tight bounds.
●Usual algorithms don't reach the bounds
for large.
●
●Trouble: the algorithms we propose are
boring (too complicated), people prefer usual
algorithms.
●
● A simple patch for these algorithms?
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 74
75. Log() corrections
● In the discrete case (XPs): automatic
parallelization surprisingly efficient.
● Simple trick in the continuous case:
- E log( *) should be linear in log()
(this provides corrections which
work for SA, EMNA and CSA)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 75
76. Example 1: Estimation of Multivariate Normal Algorithm
While ( I have time )
{
Generate points (x1,...,x) distributed as N(x,)
Evaluate the fitness at x1,...,x
X= mean best points
= standard deviation of best points
/= log( / 7)1 / d
}
I select the three best
77. Ex 2: Log(lambda) correction for mutative self-adapt.
= / 4 ==> min( /4,d)
While ( I have time )
{
Generate points (1,...,) as x exp(- k.N)
Generate points (x1,...,x) distributed as N(x,i)
Select the best points
Update x (=mean), update (=log. mean)
}
78. Log() corrections (SA, dim 3)
● In the discrete case (XPs): automatic
parallelization surprisingly efficient.
● Simple trick in the continuous case
- E log( *) should be linear in log()
(this provides corrections which
work for SA and CSA)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 78
79. Log() corrections
● In the discrete case (XPs): automatic
parallelization surprisingly efficient.
● Simple trick in the continuous case
- E log( *) should be linear in log()
(this provides corrections which
work for SA and CSA)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 79
80. Conclusion
The case of large population size is not well
handled by usual algorithms.
We proposed
(I) theoretical bounds
(II) an automatic parallelization
matching the bound, and
which works well in the discrete case.
(III) a necessary condition for the
continuous case, which provides
useful hints.
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 80
81. Main limitation (of the application to the design of algo)
All this is about a logarithmic speed-up.
The computational
power is like this ==>
<== and the result is like that.
==> much better speed-up for noisy
optimization.
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 81
82. Further work 1
Apply VC-bounds for considering only
“reasonnable” branches in the automatic
parallelization.
Theoretically easy, but provides extremely
complicated algorithms.
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 82
83. Further work 2
We have:
- proofs for complicated algorithms
- efficient (unproved) hints for usual
algorithms
Proofs for the versions with the “trick” ?
Nb: the discrete case is moral: the best
algorithm is the proved one :-)
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 83
84. Further work 3
What if the optimum is not a point but a
subset with topological dimension
N' < N ?
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 84
85. Further work 4
Parallel bandits ?
Experimentally, parallel UCT >> seq. UCT.
with speed-up depending on nb of arms.
Theory ? Perhaps not very hard, but not
done yet.
Auger, Fournier, Hansen, Rolet, Teytaud, Teytaud parallel evolution 85
Notes de l'éditeur
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse
I am Frederic Lemoine, PhD student at the University Paris Sud. I will present you my work on GenoQuery, a new querying module adapted to a functional genomics warehouse