IEEE Computational Intelligence Magazine - November 2021 - 23

set to 1.0 in [40], respectively. Notably,
MTO-DRA postulates that the IoI vector
can serve as a performance indicator for
each optimization task, where the tasks
with smaller improvements are considered
in a stagnated situation, and the tasks with
larger improvements are regarded as hard tasks that deserve
more computational efforts [40]. Equipped with the IoI vector,
the resource allocation strategy in MTO-DRA follows
the algorithm procedure, as illustrated in Algorithm 1. Specifically,
in MTO-DRA, the resource allocation operation is
conducted after all the tasks have been evaluated for GD generation
rounds. During the resource allocation step as depicted
in line 7 to line 11, the tasks with larger IoI values possess
a higher probability to attain extra GD generation rounds.
B. Theoretical Analysis on MTO-DRA
Given the backbone of MTO-DRA in Algorithm 1, herein we
discuss its motivation and behavior in an analytical approach.
1) Resource Allocation Intensity
Notice that, the resource allocation phase in Algorithm 1 is
triggered in a randomized way, where items of the IoI vector
serve as the probability thresholds. Hence, to investigate the
randomized behavior of resource allocation, we should take
into account its long-term behavior. First, we define the
resource allocation intensity (RAI) as equation (5):
RAI=
eval
eval + eval
ae
a
(5)
where evala and evale represent the actively allocated generation
rounds as shown in line 11 of Algorithm 1 and the equally
allocated generation rounds as shown in line 4, respectively.
Assuming that the task size in EMTO problems is K and the
total algorithm loops amount is N, then apparently evale equals
NK GD
. Besides, evala should be computed from a probability
perspective, as equation (6):
eval ()NE resource
a = )
(6)
where E(resource) stands for the expected value of actively
resource allocation in each algorithm loop. Since the actively
allocated resources are controlled by the IoI vector as illustrated
in line 9 of Algorithm 1, and the IoI items have been normalized
by softmax function as indicated in equation (4), the
expected actively allocated resources E(resource) can be computed
as equation (7):
K
EresourceE resource
PresourceG
IoIG G
() ()
()
=
=
==k )DD
k=1
/
/
/
k
k=1
K
k ) D
k=1
K
(7)
2 In the original MTO-DRA paper [40], this function is written as f(x(t)), and in this
paper we will reformulate it as g(t) for simplicity.
3 The readers can refer to Section S.I in the supplementary material for the proof of
this lemma.
... it is urgent for EMTO to allocate different
computational resources towards all the tasks
based on their properties.
where () represents the probability of obtaining GD
Presourcek
generations for task k. Therefore, according to the definition of
RAI in equation (5), the exact value of RAI for MTO-DRA
should be as equation (8):
RAI=
NG NK G
NG
DD
D
+
=
1 K
1
+
(8)
Notably, the exact value of RAI for MTO-DRA increases in
inverse proportion to the task size K, which means that the
resource allocation intensity of MTO-DRA cannot be adjusted
by users flexibly. However, this allocation mechanism is improper
for EMTO, since when more tasks are encountered, the
resource allocation component should behave more actively for
better capturing attributes of various incoming tasks, rather than
becoming more inactive as is the case with MTO-DRA.
2) Implicit Objective Function
It is claimed that the resource allocation strategy in MTODRA
is intended to solve an implicit objective function. In this
subsection, a corresponding proof is provided for explicitly
explaining such a latent property of MTO-DRA.
Definition. The current best function2 (( ))fx t is a function of t,
which refers to the best fitness function value of a given solver in the
first t generations.
Lemma. Under the assumption that the current best function is a
convex function, the relative improvement value of sub-population i, i~ ,
can be concerned as a weighted approximation for the sub-gradient of
the i-th normalized current best function.3
Algorithm 1 Resource Allocation of MTO-DRA.
1 while termination condition is not met do
2
3
4
5
6
7
8
9
10
11
12
13
14 end
/*Evaluate tasks equally*/
foreach task i do
Evaluate task i for GD iterations
end
/*Prepare for allocation*/
Calculate the IoI vector according to equation (3) (4)
foreach task i do
if rand (, ) 1 then
01 IoIi
/*Conduct resource allocation*/
Evaluate task i for GD iterations
end
end
NOVEMBER 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 23
IEEE Computational Intelligence Magazine - November 2021

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