Evolving Best Known Approximation to the QFunction Dao

Evolving Best Known Approximation to the Q-Function Dao Ngọc Phong, Nguyen Xuan Hoai*, Hanoi University (VN) Bob Mc. Kay, Seoul National University (Korea) Constantin Siriteanu, University of Kingston (Canada) Nguyen Quang Uy, Le Quy Don University (VN)

Contents n The Problem n n n Q-function. Why approximation? Previous human derived solutions. The need for (Meta) heuristics. The Method n TAG 3 P with local search. The results. n Conclusions & Future Work n

The Q function n Integrated tail of the Gaussian

Why Approximations? Q-function is immensely important as it is related to the Gaussian CDF. n In many fields, esp. in communications, the noise is assumed to be Gaussian. n 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! n Approximation by series (such as Taylor ’s series) would not work! (complicated, time consuming, low accuracy). n Good approximations to the Q-function in closed and simple forms are badly needed!

Why Approximations? Example 1: Evaluating performance averaged over the fading: n The instantaneous SNR varies due to multipath fading. Designers must be able to quickly compute the average Pe = f 1(Q(f 2(SNR))) over SNR distribution.

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

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

Human Derived Approximations Relative Absolute Error (RAE) in [0 -8], the interval of most concern (in communications), over 400 equidistance points. n 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. n OPBCS is the most accurate approximation (RAE is about 1. 7*E-3) but … n Accuracy is not the only objective. n n n 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. n 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? n

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

Heuristics Are Needed It proved (meta) heuristics such as GP could work for the problem. n Its accuracy is better than OPBCS (RAE = 8. 63 E-4) but … n It is rather complicated and does not ease the analyses and manipulations. n Ref. Dao Ngoc Phong, Nguyen Quang Uy, Nguyen Xuan Hoai, R. I. Mc. Kay, 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 … n Find the complexity and parameters of the models using GP, GA, and the likes. n 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, n R. I. Mc. Kay, 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 (TAG 3 P) with local search. n System Setup: n

The Method The Grammar for TAG 3 P and TAG 3 PL, where TL could be x, , 1, ERC in (0, 1). n

The Results TAG 3 PL was much better than TAG 3 P in finding good approximations for Q-function. n The best solution found (TAG-EXP): n

The Results TAG-EXP has RAE of 6. 189*E-4 – the most accurate approximation ever been published ! n Simple and easy for computations and analyses. n

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. n Heuristics, meta heuristics like GA, GP are expected to solve the problem better than human. n Our work has shown that GP could find solution that is better than any published solution by human experts so far. n

Conclusions and Future Work Future work includes: Strengthen GP solutions with meta heuristics techniques for parameter optimization (such as GA, CMA-ES) … n [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). ] n Find approximation in other forms (esp. Chiani’s form). n [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). ] n

Thank You !