Chih -Ming Chen, Student Member, IEEE, Ying-ping Chen, Member, IEEE,

1. On the Optimization of Degree Distributions in LT Code with Covariance Matrix Adaptation Evolution Strategy Chih-Ming Chen, Student Member, IEEE, Ying-ping Chen, Member, IEEE, Tzu-ChingShen, and John K. Zao, Senior Member, IEEE Evolutionary Computation (CEC), 2010 IEEE Congress on

2. Outline • Introduction • Optimization method • Decision Variables • Objectives • Experiments and results

3. Introduction • LT codes • An appropriate degree distribution : soliton distribution • Researchers started to optimize the degree distribution [5] [6] • Only focus on the parameters of soliton distribution • We directly consider the degree distribution itself as our decision variables [5] E. A. Bodine and M. K. Cheng, “Characterization of luby transform codes with small message size for low-latency decoding,” in IEEE International Conference on Communications (ICC ‘08), 2008, pp. 1195-1199. [6] E. Hyytia, T. Tirronen, and J. Virtamo, “Optimal degree distribution for LT codes with small message length,” in Proceedings of the 26th IEEE International Conference on Computer Communications (INFOCOM 2007), 2007, pp. 2576-V2580.

4. Introduction • Raptor codes • Integrating LT code with a pre-coding layer • Requiring a degree distribution, called weakened LT • Several instances were given in [9] for certain particular sizes of source symbols. • We demonstrate the use of optimization techniques proposed in evolutionary computation for generating degree distributions of different , desired properties. [9] A. Shokrollahi, ’’Raptor codes, ’’ IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2551-2567, 2006

5. In this paper • Utilizing evolutionary computation techniques to optimize the degree distribution for LT code . • Demonstrating the feasibility of customizing degree distributions for different purposes. • Particularly, we adopt the covariance matrix adaptation evolution strategy (CMA-ES) [10] • To directly optimize degree distributions : • Reducing the overhead • Lowering the failure rate. • The experimental results are remarkably promising

6. LT code : Soliton distribution • After k processing step, the source data can be ideally recovered. • The overhead = K/k denotes the performance of LT code • k : the number of source symbols • K: the number of encoding symbols received by receivers

7. LT code : Robust soliton distribution • Robust soliton distribution can ensure that only encoding symbols are required with a successful probability at least

8. LT code

9. Optimization method • Evolution strategies (ES) • To evolve strategic parameters as well as decision variables • Well-known to be quite capable of dealing with continuous optimization problems. • Using natural problem-dependent representations, and primarily mutation and selection, as search operators. • An iteration of the loop is called a generation. • The sequence of generations is continued until a termination criterion is met.

10. ES • Repeated interplay of variation (via mutation and recombination) and selection • In each generation (iteration) new individuals (candidate solutions, denoted as x) are generated by variation • And then some individuals are selected for the next generation based on their fitness or objective function value • Like this, over the generation sequence, individuals with better and better  -values are generated. • (1+1)-ES

11. CMA-ES • Covariance Matrix Adaptation Evolution Strategy • In an evolution strategy, new candidate solutions are sampled according to a multivariate normal distribution.   • Pairwisedependencies between the variables in multivariate normal distribution are represented by a covariance matrix. • The covariance matrix adaptation (CMA) is a method to update the covariance matrix of this distribution. • Fewer assumptions on the nature of the underlying objective function are made.

12. CMA-ES

13. Decision Variables • Using the degree distribution to form a real-number vector • In the evaluation phase , a real-number vector of arbitrary values can be interpreted as a probability distribution. • We usually do not need a non-zero probability on every single degree • We choose some degrees called tags to form the vector v(i) of decision variables

14. Objectives • We try to use two indicators to evaluate degree distributions for LT code • The efficiency of the LT code with the optimized degree distribution • ε denotes the expected rate of overhead to transmit data. • This objective is to obtain some degree distribution for a specific k with the smallest ε. • We provide infinite encoding symbols, in the form of a stream of encoding symbols, to simulate the decoding process until all source data are recovered.

15. Objectives • The amount of source symbols that cannot be recovered when a constant ratio of encoding symbols are received. • In raptor codes, Low-density-paritycheck (LDPC) [15] is introduced as a second layer pre-coding into LT code. • LDPC can fix errors of data • Most of source symbols can be recovered with a small overhead is sufficient. • We try to minimize the number of un-recovered source symbols given a constant overhead ε.

16. Experiments and results • Tags are encoded as an individual : v(i) • Initial values of tags are set as 1/|v| uniformly • Applying CMA-ES without any customization or modification • One hundred independent runs of simulation for each function evaluation. • Two experiments: • Minimizing the expected number of encoding symbols for full decoding • The average number of source symbols that cannot be recovered for a constant ε = 1.1 is considered

17. Overhead • We minimize the overhead ε for different ksizes

18. The expected overhead of robust soliton distribution

21. Failure rate • We are concerned with how many source symbols can be recovered in the second set of experiments. • The objective value is the average number of source symbols that cannot be recovered with a constant overhead ε.

22. ε=1.1

24. Conclusion • Algorithmically optimize the degree distribution adopted in LT code • Evolutionary computation • CMA-ES was indeed capable of finding good degree distributions for different purposes without any guideline or human intervention. • Two sets of experiments: • To minimize the overhead • To reduce the decoding failure rate. • The optimized overhead was decreased as least 10% • The results of failure rate minimization were also remarkably promising