IEEE Computational Intelligence Magazine - May 2022 - 19

the negated expectation of the log-likelihood of the sample x is
calculated. The second term uses a Kullback-Leibler (KL)
divergence to regularize the posterior latent distribution with a
simple prior. The KL divergence measures the proximity of the
posterior distribution to the prior distribution. The third term
uses the expected mean squared error (MSE) of property prediction
to further regularize the posterior distribution of latent
codes. Properties in the property predictor are in original scales,
i.e., unnormalized. Previous studies unveil that VAE can easily
fail on modeling text data because of the training imbalance
between the reconstruction error (difficult to reduce once the
KL divergence becomes very small) and the KL divergence
(easy to diminish to zero). Thus, proper trade-off between the
reconstruction error and KL divergence through VAE-b
[39]
is vital in text generation and molecular generation [40], [41].
In practice, the value of b should be smaller than 1. To obtain a
suitable value of
b , we design a versatile function, called
b-function ,
b = minmax$
()
$
ta lu , ,,. .
ek`j-1 T
t
! 12 f
(3)
where T represents the total number of epochs, {, ,, }tT
indicates the current index of epoch, k controls the incremental
speed, a defines the amplitude, l and u serve as lower and upper
bounds respectively for the value of
b . With different settings, a
variety of curves of this function are shown in Figure S1 (see
Supplementary Materials). Using this function, we can generate
constant or annealing b values, allowing flexibility to control
the trade-off between the reconstruction error and the KL divergence
in DGMs. The value of a can be set similarly in FragVAE.
Finally, given S training molecules {( ,),,(, )},
xy xy
11
f
l .; /2 i =1
p xz
zi}i ss
s =1
,,
1 / e
S
S
-+ -- +
+
loglogvn vii i
ss m
b
aSE f} zy
22 2
I
SS
empirical loss is calculated as
() ()
(( ), ),
1
(4)
where I is the latent vector length, i.e., length ();z
z ;i and SE stands for squared error.
2
n i and i
v
are respectively the posterior mean and variance of the i-th
latent variable
B. Non-Domination Rank and Crowding Distance
To get non-domination rank and crowding distance of a feasible
solution for guiding sample selection (Section III-C) and
population merging (Section III-D), the fast non-dominated
sort and crowding comparison methods are adopted from the
classic NSGA-II algorithm for multi-objective optimization
[42]. The properties in molecular design are treated as
objectives. In an optimization problem with K objectives
() {( ), ,( )},zz z
ff fK
=
ff and 7f! {, ,}: ff
kk
() () kK1
zz
12
#
1
f
'
kk
() ().zz Using
12
this concept of domination, all feasible solutions in a collection
can be sorted to form Pareto frontiers (or fronts, ranks)
the
C. Evolutionary Operations
The evolutionary operations include parent selection, recombination,
and mutation to produce new offspring in the evolutionary
process. Binary tournament selection [44] is applied to
select one out of two randomly drawn samples from the current
population. In such a selection process, let us suppose z1
and z2
Sample z1
1 p .s
-
are randomly taken from the population and zz .12n
will be selected with selection probability ps
'
which
is close to one, and z2
will be selected with a small chance
This selection process is repeated M times to thus find
M parents where M is the fixed population size. A pair of such
parents will produce two children through recombination and
mutation operations. Given the two parents' latent representations
zp1
and z ,p2
t
there are two recombination options, blending
linear and discrete methods [44], [45], to produce their
new children z1
and
aa +
12
z .2
t
(BLX), ()zz zzrpp p
t =+ -
11 12 1
For blending linear crossover
t =+ -
where () ,( ),
,(,).01Uniform
++ =- ++
rd 12dr dd d
11 022
and ()zz zzrpp p
12
21 22 1
=- aa $ , and
When d 0= , only interpolation is
allowed, implying exploitation only. When d 02 , extrapolation
is also allowed to further explore the space. According to
the general suggestion in [44] and [45], we set
d 025=
.
11 2
pp
such that [[ :],[ :]]
0.01) of getting mutation. For
If 1
z ,m
t
in our
dominate z2 (denoted by zz ),12 6f! {, ,}: ({ ,, })mM1 f!
1
feasible solution z1 is said to zz zll L11 [[ :],[ :]].
if kK1
22 1
from Unifor (, ).01m
experiments. For discrete method, supposing a latent representation
vector is of length L, an integer l is randomly drawn
from {, ,}L11f -
pp
zz zll L11t =+ and
t =+ After crossover, a new sample zm
t
will have a small mutation probability pm
domly selected from {, ,}L1 f
(say
a random value r is drawn
rp ,m then a random integer l is ransuch
that the l-th position of
MAY 2022 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 19
FF F12
= {, ,}. Samples in the same frontier do not dominate
each other. Frontier Fi
f
dominates Fj
for ji .2 Thus, we
define function ()zF to retrieve the rank (i.e., frontier index) of
any feasible solution z in the population.
The crowding distance of a feasible solution is computed as
the normalized perimeter of the cuboid formed by its immediate
neighbors along all objective axes. To compute the crowding
distance of ,zi
neighbors above (denoted by )za
ki ka kb k
the normalized distance between its nearest
and below (denoted by )zb
df ff fmax
min
zz z=- - k
), where fmax and fmin
k
k
ii k
() = R =1 ki
zz:dd z
K
it
with respect to the k-th objective axis is calculated using
() (( )( ))/(
are respectively the maximal and minimal values of the k-th
objective. Then, these individual results are summed up to form
the crowding distance of
(). The concept
of crowding distance measures the density of the area around a
feasible solution.
Using the two concepts, partial order can be defined. One
' if either (1) zz FF
12 dd
says zzn12
FF
zz
=
12
() () and () ().zz When two solutions have
12
2
same rank, the one with larger crowding distance is preferred as
it helps maintain a diverse population. Note that, the above
multi-objective sorting method is used as a prototype in our
framework. Weighted average of ranks and other ranking methods
[43] can be adopted for the purpose as well.
' (i.e., () ()),zz or (2)
12
1
IEEE Computational Intelligence Magazine - May 2022

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - May 2022
