evolving best known approximation to the q-function dao ngọc phong, nguyen xuan hoai*, hanoi...

Evolving Best Known Evolving Best Known Approximation to the Q-Approximation to the Q-

FunctionFunction

Dao Ngọc Phong, Nguyen Xuan Hoai*, Dao Ngọc Phong, Nguyen Xuan Hoai*, Hanoi University (VN)Hanoi University (VN)

Bob McKay,Bob McKay,Seoul National University (Korea)Seoul National University (Korea)

Constantin Siriteanu,Constantin Siriteanu,University of Kingston (Canada)University of Kingston (Canada)

Nguyen Quang Uy,Nguyen Quang Uy,Le Quy Don University (VN)Le Quy Don University (VN)

ContentsContents The ProblemThe Problem

Q-function.Q-function. Why approximation?Why approximation? Previous human derived solutions.Previous human derived solutions. The need for (Meta) heuristics.The need for (Meta) heuristics.

The MethodThe Method TAG3P with local search.TAG3P with local search.

The results.The results. Conclusions & Future WorkConclusions & Future Work

The Q functionThe Q function Integrated tail of the GaussianIntegrated tail of the Gaussian

Why Approximations?Why Approximations? Q-function is immensely important as it is Q-function is immensely important as it is related to the Gaussian CDF.related to the Gaussian CDF. In many fields, esp. in communications, the In many fields, esp. in communications, the noise is assumed to be Gaussian. noise is assumed to be Gaussian. In communications, many problems require In communications, many problems require the use of Q-function in a closed and simple the use of Q-function in a closed and simple form for the various calculations and analyses.form for the various calculations and analyses.

… … but no closed form of Q-function is known!but no closed form of Q-function is known! Approximation by series (such as Taylor’s Approximation by series (such as Taylor’s series) would not work! (complicated, time series) would not work! (complicated, time consuming, low accuracy).consuming, low accuracy).

Good approximations to the Q-function in Good approximations to the Q-function in closed and simple forms are badly closed and simple forms are badly needed!needed!

Why Approximations?Why Approximations? Example 1Example 1: Evaluating performance : Evaluating performance averaged over the fading:averaged over the fading:The instantaneous SNR varies due to multipath The instantaneous SNR varies due to multipath fading. Designers must be able to quickly fading. Designers must be able to quickly compute the average Pcompute the average Pee = f = f11(Q(f(Q(f22(SNR))) over (SNR))) over SNR distribution.SNR distribution.

Why Approximations?Why Approximations? Example 2Example 2: Power control for link : Power control for link adaptation in wireless communicationsadaptation in wireless communicationsRx must compute quickly and accurately the Rx must compute quickly and accurately the error probability for the current SNR and inform error probability for the current SNR and inform Tx to increase or decrease power in order to Tx to increase or decrease power in order to meet performance requirements.meet performance requirements.

Why Approximations?Why Approximations? Example 3Example 3: Rate control for link : Rate control for link adaptation in wireless networks: adaptation in wireless networks: Rx must compute quickly and accurately the Rx must compute quickly and accurately the error probability for the current M and inform Tx error probability for the current M and inform Tx to increase or decrease M in order to meet to increase or decrease M in order to meet performance requirements.performance requirements.

Human Derived Human Derived ApproximationsApproximations

P. Borjesson and C. Sundberg. Simple P. Borjesson and C. Sundberg. Simple Approximations of the Error Function q(x) Approximations of the Error Function q(x) for Communications Applications, for Communications Applications, IEEE IEEE Transactions on CommunicationsTransactions on Communications, 27: , 27: 639–643, 1979.639–643, 1979.

PBCS: PBCS:

OPBCS:OPBCS:


M. Chiani, D. Dardari, and M. K. Simon. New M. Chiani, D. Dardari, and M. K. Simon. New Exponential Bounds and Approximations for the Exponential Bounds and Approximations for the Computation of Error Probability in Fading Computation of Error Probability in Fading Channels,Channels,

IEEE Transactions on Wireless CommunicationsIEEE Transactions on Wireless Communications, , 2(4) : 840–845, 2003.2(4) : 840–845, 2003.

CDS:CDS:


A. Karagiannidis and A. Lioumpas. An A. Karagiannidis and A. Lioumpas. An improved Approximation for the Gaussian Q-improved Approximation for the Gaussian Q-function. IEEE Communication Letters, 11:644–function. IEEE Communication Letters, 11:644–646, 2007.646, 2007.

GKAL: GKAL:


M. Benitez and F. Casadevall. Versatile, M. Benitez and F. Casadevall. Versatile, Accurate, and Analytically Tractable Accurate, and Analytically Tractable Approximation for the Gaussian Q-function, Approximation for the Gaussian Q-function, IEEE IEEE Transactions on CommunicationsTransactions on Communications, 59(4) : 917–, 59(4) : 917–922, 2011.922, 2011.

EXP: EXP:


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

NameName RAERAE

PBCSPBCS 0.0346410.03464177

OPBCOPBCSS

0.0017470.00174711

CDSCDS 0.2437460.24374699

GKALGKAL 0.0614180.06141844

EXPEXP 0.0348170.03481777


Exponential function is common in Exponential function is common in these approximations. these approximations. OPBCS is the most accurate OPBCS is the most accurate approximation (RAE is about 1.7*E-3) but approximation (RAE is about 1.7*E-3) but …… Accuracy is not the only objective.Accuracy is not the only objective.

Fast computation.Fast computation. Ease for analyses and manipulations (e.g Ease for analyses and manipulations (e.g

integrability) integrability)

Heuristics Are NeededHeuristics Are Needed

Approximations with better accuracy, Approximations with better accuracy, ease for analyses, fast in computation ease for analyses, fast in computation are still needed. are still needed. Heuristics could help to find new Heuristics could help to find new approximations or to optimize approximations or to optimize coefficients by using the power of coefficients by using the power of computers (or super computers).computers (or super computers). -> Heuristics like GA, GP are welcome!-> Heuristics like GA, GP are welcome! But …But …

Could they beat the human experts?Could they beat the human experts?

Heuristics Are NeededHeuristics Are Needed Our first result using GP with an improved Our first result using GP with an improved crossover operator.crossover operator.

Heuristics Are NeededHeuristics Are Needed It proved (meta) heuristics such as GP It proved (meta) heuristics such as GP could work for the problem. could work for the problem. Its accuracy is better than OPBCS (RAE Its accuracy is better than OPBCS (RAE = 8.63E-4) but …= 8.63E-4) but … It is rather complicated and does not It is rather complicated and does not ease the analyses and manipulations.ease the analyses and manipulations.

Ref. Ref. Dao Ngoc Phong, Nguyen Quang Uy, Nguyen Xuan Dao Ngoc Phong, Nguyen Quang Uy, Nguyen Xuan Hoai, R.I. McKay, Evolving Approximations for the Hoai, R.I. McKay, Evolving Approximations for the Gaussian Q-function by Genetic Programming with Gaussian Q-function by Genetic Programming with Semantic Based Crossover, in Semantic Based Crossover, in Proceedings of IEEE World Proceedings of IEEE World Congress on Evolutionary ComputationCongress on Evolutionary Computation (CEC'2012), 2012. (CEC'2012), 2012.

The MethodThe Method Based on human’s forms of function Based on human’s forms of function and …and … Find the complexity and parameters of Find the complexity and parameters of the models using GP, GA, and the likes.the models using GP, GA, and the likes. In this work, we find approximations, In this work, we find approximations, inspired by Benitez and Casadevall’ 2011 inspired by Benitez and Casadevall’ 2011 IEEE Trans Comms paper, in the form ofIEEE Trans Comms paper, in the form of

e^f(x) e^f(x)

Where f(x) is a polynomial.Where f(x) is a polynomial.Ref. Ref. Dao Ngoc Phong, Nguyen Xuan Hoai, Constantin Dao Ngoc Phong, Nguyen Xuan Hoai, Constantin Siriteanu, R.I. McKay,and Nguyen Quang Uy, Evolving a Siriteanu, R.I. McKay,and Nguyen Quang Uy, Evolving a Best Known Approximation to the Q Function, In Best Known Approximation to the Q Function, In the the Proceedings of ACM-SIGEVO Genetic and Evolutionary Proceedings of ACM-SIGEVO Genetic and Evolutionary Algorithms (GECCO'2012Algorithms (GECCO'2012), 2012.), 2012.

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

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

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

The ResultsThe Results TAG-EXP has RAE of 6.189*E-4 – TAG-EXP has RAE of 6.189*E-4 – the the most accurate approximation ever been most accurate approximation ever been publishedpublished !! Simple and easy for computations and Simple and easy for computations and analyses.analyses.

The ResultsThe ResultsValidation for the usefulness of TAG-EXP:Validation for the usefulness of TAG-EXP:

Computing PComputing Pee for Evaluating performance for Evaluating performance averaged over the fading (example 1)averaged over the fading (example 1)

Conclusions and Future Conclusions and Future WorkWork

Finding good Q-function approximation Finding good Q-function approximation is important in many areas especially in is important in many areas especially in communications.communications. Heuristics, meta heuristics like GA, GP Heuristics, meta heuristics like GA, GP are expected to solve the problem better are expected to solve the problem better than human.than human. Our work has shown that GP could find Our work has shown that GP could find solution that is better than any published solution that is better than any published solution by human experts so far.solution by human experts so far.

Conclusions and Future Conclusions and Future WorkWork

Future work includes:Future work includes: Strengthen GP solutions with meta heuristics Strengthen GP solutions with meta heuristics techniques for parameter optimization (such as techniques for parameter optimization (such as GA, CMA-ES) …GA, CMA-ES) …

[[Our confession 1:Our confession 1: We have obtained even better coefficients for TAG-EXP We have obtained even better coefficients for TAG-EXP with the help of CMA-ES (we are checking it for with the help of CMA-ES (we are checking it for publication in the near future).]publication in the near future).] Find approximation in other forms (esp. Find approximation in other forms (esp. Chiani’s form).Chiani’s form).

[Our confession 2:[Our confession 2: We have obtained a very good approximation in We have obtained a very good approximation in Chiani’s form with the help of CMA-ES (we are checking it Chiani’s form with the help of CMA-ES (we are checking it for publication in the near future).]for publication in the near future).]

Thank You !Thank You !

evolving best known approximation to the q-function dao ngọc phong, nguyen xuan hoai*, hanoi...

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