Evolving Best Known Approximation to the Q-Function

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Evolving Best Known Approximation to the
Q-Function
Dao Ng?c Phong, Nguyen Xuan Hoai, Hanoi
University (VN)Bob McKay,Seoul National
University (Korea)Constantin Siriteanu,Universit
y of Kingston (Canada)Nguyen Quang Uy,Le Quy
Don University (VN)
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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

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The Q function

• Integrated tail of the Gaussian

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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 Taylors
  series) would not work! (complicated, time
  consuming, low accuracy).
• Good approximations to the Q-function in closed
  and simple forms are badly needed!

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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.

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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.

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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.

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Human Derived Approximations

• P. Borjesson and C. Sundberg. Simple
  Approximations of the Error Function q(x) for
  Communications Applications, IEEE Transactions on
  Communications, 27 639643, 1979.
• PBCS
• OPBCS

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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) 840845, 2003.
• CDS

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Human Derived Approximations

• A. Karagiannidis and A. Lioumpas. An improved
  Approximation for the Gaussian Q-function. IEEE
  Communication Letters, 11644646, 2007.
• GKAL

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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) 917922,
  2011.
• EXP

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Human Derived Approximations

• Relative Absolute Error (RAE) in 0-8, the
  interval of most concern (in communications),
  over 400 equi-distance points.

Name RAE
PBCS 0.0346417
OPBCS 0.0017471
CDS 0.2437469
GKAL 0.0614184
EXP 0.0348177
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Human Derived Approximations

• Exponential function is common in these
  approximations.
• OPBCS is the most accurate approximation (RAE is
  about 1.7E-3) but
• Accuracy is not the only objective.
• Fast computation.
• Ease for analyses and manipulations (e.g
  integrability)

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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).
• -gt Heuristics like GA, GP are welcome! But
• Could they beat the human experts?

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Heuristics Are Needed

• Our first result using GP with an improved
  crossover operator.

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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.

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The Method

• Based on humans 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
• ef(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 ACM-SIGEVO
  Genetic and Evolutionary Algorithms (GECCO'2012),
  2012.

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The Method

• The system Tree Adjoining Grammar Guided
  Genetic Programming (TAG3P) with local search.
• System Setup

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The Method

• The Grammar for TAG3P and TAG3PL, where TL could
  be x, ?, 1, ERC in (0,1).

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The Results

• TAG3PL was much better than TAG3P in finding
  good approximations for Q-function.
• The best solution found (TAG-EXP)

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The Results

• TAG-EXP has RAE of 6.189E-4 the most accurate
  approximation ever been published !
• Simple and easy for computations and analyses.

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The Results

• Validation for the usefulness of TAG-EXP
• Computing Pe for Evaluating performance averaged
  over the fading (example 1)

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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.

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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. Chianis
  form).
• Our confession 2
• We have obtained a very good approximation in
  Chianis form with the help of CMA-ES (we are
  checking it for publication in the near future).

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Thank You !