1. 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)

2. Contents The Problem The 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 Method The Method TAG3P with local search. TAG3P with local search. The results. The results. Conclusions & Future Work Conclusions & Future Work

3. The Q function Integrated tail of the Gaussian Integrated tail of the Gaussian

4. Why Approximations? Q-function is immensely important as it is related to the Gaussian CDF. 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 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. 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! … 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). 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!

5. Why Approximations? Example 1: Evaluating performance averaged over the fading: 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 P e = f 1 (Q(f 2 (SNR))) over SNR distribution.

6. Why Approximations? Example 2: Power control for link adaptation in wireless communications 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.

7. Why Approximations? Example 3: Rate control for link adaptation in wireless networks: 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.

8. 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. 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: PBCS: OPBCS: OPBCS:

9. 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, 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:

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

11. 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. 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:

12. Human Derived Approximations Relative Absolute Error (RAE) in [0-8], the interval of most concern (in communications), over 400 equi- distance points. Relative Absolute Error (RAE) in [0-8], the interval of most concern (in communications), over 400 equi- distance points. NameRAE PBCS0.0346417 OPBCS0.0017471 CDS0.2437469 GKAL0.0614184 EXP0.0348177

13. Human Derived Approximations Exponential function is common in these approximations. Exponential function is common in these approximations. OPBCS is the most accurate approximation (RAE is about 1.7*E-3) but … OPBCS is the most accurate 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 integrability) Ease for analyses and manipulations (e.g integrability)

14. Heuristics Are Needed Approximations with better accuracy, ease for analyses, fast in computation are still 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 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 … -> Heuristics like GA, GP are welcome! But … Could they beat the human experts?

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

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

17. The Method Based on human’s forms of function and … Based on human’s forms of function and … Find the complexity and parameters of the models using GP, GA, and the likes. 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 In this work, we find approximations, inspired by Benitez and Casadevall’ 2011 IEEE Trans Comms paper, in the form of e^f(x) 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 ACM- SIGEVO Genetic and Evolutionary Algorithms (GECCO'2012), 2012.

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

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

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

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

22. The Results Validation for the usefulness of TAG-EXP: Computing P e for Evaluating performance averaged over the fading (example 1)

23. Conclusions and Future Work Finding good Q-function approximation is important in many areas especially in communications. 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. 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. Our work has shown that GP could find solution that is better than any published solution by human experts so far.

24. Conclusions and Future Work Future work includes: Strengthen GP solutions with meta heuristics techniques for parameter optimization (such as GA, CMA-ES) … Strengthen GP solutions with meta heuristics techniques for parameter optimization (such as GA, CMA-ES) … [ Our confession 1: [ 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).] 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). 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).] 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).]

25. Thank You !