Transcript Document

Evolving Best Known
Approximation to the Q-Function
Dao Ngọc Phong, Nguyen Xuan Hoai*,
Hanoi University (VN)
Bob McKay,
Seoul National University (Korea)
Constantin Siriteanu,
University of Kingston (Canada)
Nguyen Quang Uy,
Le Quy Don University (VN)
Contents

The Problem





Q-function.
Why approximation?
Previous human derived solutions.
The need for (Meta) heuristics.
The Method

TAG3P with local search.
The results.
 Conclusions & Future Work

The Q function

Integrated tail of the Gaussian
Why Approximations?
Q-function is immensely important as it is related to
the Gaussian CDF.
 In many fields, esp. in communications, the noise is
assumed to be Gaussian.
 In communications, many problems require the use
of Q-function in a closed and simple form for the
various calculations and analyses.
… but no closed form of Q-function is known!
 Approximation by series (such as Taylor’s series)
would not work! (complicated, time consuming, low
accuracy).

Good approximations to the Q-function in
closed and simple forms are badly needed!
Why Approximations?
Example 1: Evaluating performance averaged
over the fading:

The instantaneous SNR varies due to multipath fading.
Designers must be able to quickly compute the
average Pe = f1(Q(f2(SNR))) over SNR distribution.
Why Approximations?
Example 2: Power control for link adaptation
in wireless communications

Rx must compute quickly and accurately the error
probability for the current SNR and inform Tx to
increase or decrease power in order to meet
performance requirements.
Why Approximations?
Example 3: Rate control for link adaptation in
wireless networks:

Rx must compute quickly and accurately the error
probability for the current M and inform Tx to increase
or decrease M in order to meet performance
requirements.
Human Derived Approximations
P. Borjesson and C. Sundberg. Simple
Approximations of the Error Function q(x) for
Communications Applications, IEEE
Transactions on Communications, 27: 639–
643, 1979.

PBCS:
OPBCS:
Human Derived Approximations
M. Chiani, D. Dardari, and M. K. Simon. New
Exponential Bounds and Approximations for the
Computation of Error Probability in Fading Channels,
IEEE Transactions on Wireless Communications, 2(4) :
840–845, 2003.

CDS:
Human Derived Approximations
A. Karagiannidis and A. Lioumpas. An improved
Approximation for the Gaussian Q-function. IEEE
Communication Letters, 11:644–646, 2007.

GKAL:
Human Derived Approximations
M. Benitez and F. Casadevall. Versatile, Accurate, and
Analytically Tractable Approximation for the Gaussian
Q-function, IEEE Transactions on Communications,
59(4) : 917–922, 2011.

EXP:
Human Derived Approximations
Relative Absolute Error (RAE) in [0-8], the interval of
most concern (in communications), over 400 equidistance points.

Name
RAE
PBCS
0.0346417
OPBCS 0.0017471
CDS
0.2437469
GKAL
0.0614184
EXP
0.0348177
Human Derived Approximations
Exponential function is common in these
approximations.
 OPBCS is the most accurate approximation
(RAE is about 1.7*E-3) but …
 Accuracy is not the only objective.



Fast computation.
Ease for analyses and manipulations (e.g
integrability)
Heuristics Are Needed
Approximations with better accuracy, ease
for analyses, fast in computation are still
needed.
 Heuristics could help to find new
approximations or to optimize coefficients by
using the power of computers (or super
computers).
-> Heuristics like GA, GP are welcome! But …
Could they beat the human experts?

Heuristics Are Needed
Our first result using GP with an improved crossover
operator.

Heuristics Are Needed
It proved (meta) heuristics such as GP could
work for the problem.
 Its accuracy is better than OPBCS (RAE =
8.63E-4) but …
 It is rather complicated and does not ease
the analyses and manipulations.

Ref. Dao Ngoc Phong, Nguyen Quang Uy, Nguyen Xuan Hoai,
R.I. McKay, Evolving Approximations for the Gaussian Q-function
by Genetic Programming with Semantic Based Crossover, in
Proceedings of IEEE World Congress on Evolutionary
Computation (CEC'2012), 2012.
The Method
Based on human’s forms of function and …
 Find the complexity and parameters of the
models using GP, GA, and the likes.
 In this work, we find approximations,
inspired by Benitez and Casadevall’ 2011 IEEE
Trans Comms paper, in the form of
e^f(x)
Where f(x) is a polynomial.
Ref. Dao Ngoc Phong, Nguyen Xuan Hoai, Constantin Siriteanu,

R.I. McKay,and Nguyen Quang Uy, Evolving a Best Known
Approximation to the Q Function, In the Proceedings of ACMSIGEVO Genetic and Evolutionary Algorithms (GECCO'2012),
2012.
The Method
The system: Tree Adjoining Grammar Guided
Genetic Programming (TAG3P) with local
search.
 System Setup:

The Method
The Grammar for TAG3P and TAG3PL, where
TL could be x, , 1, ERC in (0,1).

The Results
TAG3PL was much better than TAG3P in
finding good approximations for Q-function.
 The best solution found (TAG-EXP):

The Results
TAG-EXP has RAE of 6.189*E-4 – the most
accurate approximation ever been published !
 Simple and easy for computations and
analyses.

The Results
Validation for the usefulness of TAG-EXP:
Computing Pe for Evaluating performance
averaged over the fading (example 1)
Conclusions and Future Work
Finding good Q-function approximation is
important in many areas especially in
communications.
 Heuristics, meta heuristics like GA, GP are
expected to solve the problem better than
human.
 Our work has shown that GP could find
solution that is better than any published
solution by human experts so far.

Conclusions and Future Work
Future work includes:
Strengthen GP solutions with meta heuristics
techniques for parameter optimization (such as GA,
CMA-ES) …

[Our confession 1:
We have obtained even better coefficients for TAG-EXP with the
help of CMA-ES (we are checking it for publication in the near
future).]

Find approximation in other forms (esp. Chiani’s
form).

[Our confession 2:
We have obtained a very good approximation in Chiani’s form
with the help of CMA-ES (we are checking it for publication in the
near future).]

Thank You !