On the hardness of offline multi-objective optimization

TLDR: It is shown that the convergence rate of all comparison-based multi-objective algorithms, for the Hausdorff distance, is not much better than the converge rate of the random search under certain conditions.

Abstract: It has been empirically established that multiobjective evolutionary algorithms do not scale well with the number of conflicting objectives. This paper shows that the convergence rate of all comparison-based multi-objective algorithms, for the Hausdorff distance, is not much better than the convergence rate of the random search under certain conditions. The number of objectives must be very moderate and the framework should hold the following assumptions: the objectives are conflicting and the computational cost is lower bounded by the number of comparisons is a good model. Our conclusions are: (i) the number of conflicting objectives is relevant (ii) the criteria based on comparisons with random-search for multi-objective optimization is also relevant (iii) having more than 3-objectives optimization is very hard. Furthermore, we provide some insight into cross-over operators.

Figures

Fig. 1. 50 points are randomly drawn, i.e. Pn is a set of 50 points uniformly drawn in D. Only the ones with fitness(xi) 6= 0 are presented. X is the part dominated by the target Pareto-front; Xn is the part dominated by the empirical Pareto-front (Xn = X (Pn) in the notations of algorithms 1, 2, 3, 4 and 5).

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On the hardness of oine multiobjective optimization
Olivier Teytaud
To cite this version:
Olivier Teytaud. On the hardness of oine multiobjective optimization. Evolutionary Computation,
Massachusetts Institute of Technology Press (MIT Press), 2007. �inria-00173239�

On the hardness of offline multi-objective
optimization
Olivier Teytaud
TAO-inria, LRI, UMR 8623(CNRS - Universite Paris-Sud ),
bat 490 Universite Paris-Sud 91405 Orsay Cedex France
Abstract. It is empirically established that multiobjective evolutionary
algorithms do not scale well with the number of conflicting objectives.
We here show that the convergence rate of all comparison-based multi-
objective algorithms, for the Hausdorff distance, is not much better than
the convergence rate of the random search, unless the numb er of objec-
tives is very moderate, in a framework in which the stronger assumptions
are (i) that the objectives are conflicting (ii) that lower bounding the
computational cost by the number of comparisons is a good model. Our
conclusions are (i) the relevance of the number of confl icting objectives
(ii) the relevance of criteria based on comparisons with random-search
for multi-objective optimization (iii) the very-hardness of more than 3-
objectives optimization (iv) some hints about cross-over operators.
1 Introduction
Many evolutionary algorithms ar e comparison-based, in the sense that the only
information coming from the objective function and used by the algorithm is the
results of binary comparisons between fitness-values. [15] has shown that this lim-
itation has non-trivial co ns e quences in continuous mono-objective optimization.
We here apply similar techniques in order to show that comparison-bas e d MOO
has strong limitations in terms of convergence rates, when it is applied to contin-
uous problems in which (i) the computational cost is well approximated by the
number of comparisons (ii) only binary comparisons are used (iii) all o bjectives
are conflicting (iv) the number of objectives is high. This is not only a general
negative result, as it emphasizes some tricks for avoiding these limitations, such
as removing non-conflicting objectives as in [3] or using non-binary comparisons
as done in [17] through ”informed” cross-over.
Consider fitness = (fitness
1
, . . . , f itness
d
) some real-valued objective func-
tions (to be maximized) on a same do main D. A point y ∈ D dominates (or

Pareto-dominates) a point x ∈ D if for all i ∈ [[1, d]], fitness
i
(y) ≥ f itness
i
(x),
and for at least one i
0
, fitness
i
0
(y) > fitness
i
0
(x) (i.e. y is at least as good as
x for each objective and y is better than x for at least one objective). This is
denoted by y ≻ x. We denote by y  x the fact that ∀i ∈ [[1, d]], fi t ness
i
(y) ≥
fitness
i
(x). We use the same notation for p oints in the so-called fitness-space:
fitness(x) ≻ fitness(y) (resp. ) if and only if x ≻ y (resp. x  y). Also, we
say that a set A dominates a set B if ∀b ∈ B, ∃a ∈ A; a  b; This is denoted by
A  B. A point in D is said Pareto-optimal if it is not Pareto-dominated by any
other point in D. Multi-objective optimization (MOO,[2, 14 , 5]) is the research
of the set of non-dominated points, i.e.
{x ∈ D; ∄y ∈ D, y ≻ x}. (1)
This set is called the Pareto set.
Offline MOO is the research of the whole Pareto s e t, which is, a fter the
optimization (offline), studied by the user. On the other hand, on-line MOO
is the interactive research of an interesting point in the Pareto set; typically,
such programs use a weighted average of the various objectives, and the weights
are updated by the user depending on his preferences, during the run of the
MOO. On-line MOO is in some sense easier than offline MOO: the user provides
some information during the run of the MOO and this information simplifies the
problem by restricting the optimization to a part of the Pareto set chosen by
the user. Offline MOO a nd online MOO are compared in algos 1 and 2.
Algorithm 1 Offline MOO.
Init n = 0.
while stopping criterion not met do
Modify the population (mutation, selection, crossover, . . . ) (if n > 0) or initialize
it (if n = 0).
for Each newly visited point x do
n ← n + 1
P
n
← P
n
∪ {x}
end for
end while
Output P
n
(or possibly only the non-dominated points in P
n
, or any oth er subset of
P
n
).
A main tool for studying families of objective functions is the notion of
conflicting objectives ([16, 3]). Consider F a family of objective functions and

Algorithm 2 Online MOO (interactive MOO). This case is not studied in this
paper; we focus on offline algorithms as in algo. 1.
Init n = 0.
while stopping criterion not met do
Evaluate the population.
Modify the population (mutation, selection, crossover, . . . ) (if n > 0) or initialize
it (if n = 0).
for Each newly visited point x do
n ← n + 1
P
n
← P
n
∪ {x}
end for
Possibly: show information about P
n
(possibly the full P
n
or only the non-
dominated points in P
n
) t o the user and update some information about the
preferences of the user (p ossibly a simple weighting of t he preferences).
end while
Output P
n
(or possibly only the non-dominated points in P
n
, or any oth er subset of
P
n
).
δ > 0. We say that x ≻
δ
F
y if, ∀ f ∈ F, fi tness(x) ≥ fitness(y) + δ. Given two
sets F
1
and F
2
of objective functions, we say that F
1
and F
2
are δ-non-conflicting
if
∀(x, y) ∈ D, x ≻
δ
F
1
y ⇒ x≻
F
2
y
∀(x, y) ∈ D, x ≻
δ
F
2
y ⇒ x≻
F
1
y
A set of objectives F is δ-minimal wrt F if
There exists no F
′
⊂ F, F
′
6= F such that F
′
is δ-non-conflicting with F . (2)
The case δ = 0 can also be considered; a set of objectives is minimal if it
is 0-minimal. Mainly, minimal sets of objective functions are sets of o bjective
functions that can not be replaced by smaller sets of functions. We will here
study minimal sets of objective functions.
The analysis of performa nce of evolutionary algorithms is typically the study
of the co mputation time required by the algorithm for reaching a given preci-
sion for all problems of a given family of problems. Upper bounds show that
this computation time is smaller than a given quantity for a given algorithm.
Lower bo unds show that this computation time is larger tha n a given quantity
for a given algorithm, or in some cases for all algorithms of a given family of
algorithms.

In the stochastic case (stochastic algorithms), and if we only have to reach
the target within a given precision, then the writing is a bit more tedious. Let’s
consider a family P of problems. An upper bound is therefore of the form:
Upper bound: there is an algo rithm A, such that ∀ problem ∈ P, the
computation time required by algorithm A for solving it with precision ǫ and
with probability at least 1 − δ is at mos t UpperBound(ǫ, δ).
and a lower bound is of the form:
Lower bound: ∀ algorithm ∈ a given family, ∃ a problem in P such that the
computation time required for solving it w ith pre cision ǫ and with probability
at least 1 − δ is at least LowerBound(ǫ, δ).
If UpperBound is close to LowerBound, the complexity problem is solved.
Here, we will consider lower bounds for all algorithms that are based on
binary comparisons (see the formalization of this assumption in alg o. 4; this non-
negligible assumption is discussed in 4), and upper bounds for a simple naive
algorithm. The problems in this paper are a family of problems with s mooth
Pareto sets; mainly, the assumptions underlying the family of problems P (used
in the lower bound, in theorem 2) are that (i) it includes all possible Lipschitzian
Pareto sets with some given bound o n the Lipschitz coefficient (ii) there’s no
possible reduction of the number of objectives (the set of objectives is minimal).
Roughly, our results are as follows:
Upper bound: there is an algo rithm A, namely random se arch as in eq. 3,
such that ∀ problem ∈ P, the computation time required by algorithm A for
solving it with precision ǫ and with probability at least 1 − δ is a t most
Up perBound(ǫ, δ).
and:
Lower bound: ∀ algorithm which is based on binary comparisons (all
algorithms as in algo. 4), ∃ a problem in P such that the computation time
required for solving it with precision ǫ and with probability at least 1 − δ is at
least LowerBound(ǫ, δ).
Both LowerBound and U pperBound depend on the dimension d, and interest-
ingly they are close to each other when d is large. Our conclusion is therefore
that, at least for the family P of problems, and when the dimension d is larg e ,
all algorithms are roughly (at best) equivalent to random s earch - at least for
the criteria tha t we have defined (see discussion for more information about the
non-negligible effect of the assumptions in theorems 1 and 2).

Frequently asked questions

Q1. What have the authors contributed in "On the hardness of offline multiobjective optimization" ?

The authors here show that the convergence rate of all comparison-based multiobjective algorithms, for the Hausdorff distance, is not much better than the convergence rate of the random search, unless the number of objectives is very moderate, in a framework in which the stronger assumptions are ( i ) that the objectives are conflicting ( ii ) that lower bounding the computational cost by the number of comparisons is a good model. 

Q2. what is the upper bound for a very poor random search?

To strenghten the results, the upper bound is proved for a very poor random search (see the poor distribution used in the random search of theorem 1) and the lower bound is proved for a very small family of fitness functions (see theorem 2; the upper bound holds a fortiori for larger families of problems). 

Q3. What are the assumptions underlying the family of problems P?

The problems in this paper are a family of problems with smooth Pareto sets; mainly, the assumptions underlying the family of problems P (used in the lower bound, in theorem 2) are that (i) it includes all possible Lipschitzian Pareto sets with some given bound on the Lipschitz coefficient (ii) there’s no possible reduction of the number of objectives (the set of objectives is minimal). 

Q4. What is the way to deal with non-comparison-based methods?

This is not an artificial way of dealing with non-comparison-based methods; for example, in the mono-objective case, limits on the convergence rate of comparison-based algorithms derived through entropy theorems ([15]) do also hold in practice for gradient-based techniques, as the gradient is computed with a finite precision; as well as comparison-based EA, Newton’s method is only linear when dimensionality is sufficient to see the effects of the finite precision; this is an already known fact (see e.g. [21]). 

Q5. What is the effect of reducing the set of assumptions?

If the authors reduce the set of assumptions, the packing numbers increase - the lower bound remains essentially the same, and the proximity between the upper and the lower bound is preserved. 

Q6. What is the way to calculate the fitness of a random search?

Assume that ∀x, 0 ≤ fitness(x) ≤d(Xn,X ) ≤ K d √ en/q, where en = O (d log(n) − log(δ)) /n.This is a very poor random search, with a distribution uniform in the fitness space. 

Q7. What is the definition of the term 'analysis of performance of evolutionary algorithms'?

The analysis of performance of evolutionary algorithms is typically the study of the computation time required by the algorithm for reaching a given precision for all problems of a given family of problems. 

Q8. What is the definition of a pareto-optimal algorithm?

A point in D is said Pareto-optimal if it is not Pareto-dominated by any other point in D. Multi-objective optimization (MOO,[2, 14, 5]) is the research of the set of non-dominated points, i.e.{x ∈ D; ∄y ∈ D, y ≻ x}. 

Q9. What is the main difference between the two types of MOO?

The authors here apply similar techniques in order to show that comparison-based MOO has strong limitations in terms of convergence rates, when it is applied to continuous problems in which (i) the computational cost is well approximated by the number of comparisons (ii) only binary comparisons are used (iii) all objectives are conflicting (iv) the number of objectives is high.