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Chih-Ming Chen, Student Member, IEEE, Ying-ping Chen, Member, IEEE,Tzu-Ching Shen, and John K. Zao, Senior Member, IEEE

Evolutionary Computation (CEC), 2010 IEEE Congress on

On the Optimization of Degree Distributions in LT Code with Covariance Matrix Adaptation Evolution StrategyOutlineIntroductionOptimization methodDecision VariablesObjectivesExperiments and resultsIntroductionLT codesAn appropriate degree distribution : soliton distributionResearchers 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.Raptor codesIntegrating LT code with a pre-coding layerRequiring a degree distribution, called weakened LTSeveral 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.

Introduction[9] A. Shokrollahi,Raptor codes, IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2551-2567, 2006In 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 promisingLT code : Soliton distribution

After k processing step, the source data can be ideally recovered.

The overhead = K/k denotes the performance of LT codek : the number of source symbolsK: the number of encoding symbols received by receivers

LT code : Robust soliton distribution

Robust soliton distribution can ensure that only encoding symbols are required with a successful probability at least

LT code

Optimization method

Evolution strategies (ES)To evolve strategic parameters as well as decision variablesWell-known to be quite capable of dealing with continuous optimization problems.

Using natural problem-dependent representations, and primarily mutation andselection, as search operators.An iteration of the loop is called a generation. The sequence of generations is continued until a termination criterion is met.

Evolution strategies use natural problem-dependent representations, and primarily mutation andselection, as search operators. In common withevolutionary algorithms, the operators are applied in a loop. An iteration of the loop is called a generation. The sequence of generations is continued until a termination criterion is met.(Evolutionary Strategies,ES)I.RechenbergH.P.Sehwefel1963ES ()(:VBANormalRand)ESES(1+1)-ES(1+1)-ES(+)-ES(,)-ES +[(+)-ES(,)-ES ]Evolution strategies(ES) arestochastic,derivative-free methods fornumerical optimizationof non-linearor non-convexcontinuous optimizationproblems.

9ES

Repeated interplay of variation (via mutation and recombination) and selectionIn each generation (iteration) new individuals (candidate solutions, denoted as x) are generated by variationAnd then some individuals are selected for the next generation based on their fitness orobjective functionvalue Like this, over the generation sequence, individuals with better and better -values are generated.

(1+1)-ES

In an evolution strategy, new candidate solutions are sampled according to a multivariate normal distribution 10Covariance Matrix Adaptation Evolution StrategyIn anevolution strategy, new candidate solutions are sampled according to amultivariate normal distribution.Pairwise dependencies between the variables in multivariate normal distribution are represented by a covariance matrix. The covariance matrix adaptation (CMA) is a method to update thecovariance matrixof this distribution.

Fewer assumptions on the nature of the underlying objective function are made.

CMA-ESCMA-ES

As the generations progress, the algorithm approaches the global optimum while simultaneously directing the search along the path to the global optimum. Image based on work by Nikolaus Hansen and others.

12Decision Variables

Using the degree distribution to form a real-number vectorIn 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 degreeWe choose some degrees called tags to form the vector v(i) of decision variables

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 denotes the expected rate of overhead to transmit data.This objective is to obtain some degree distribution for a specific k with the smallest .

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.

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 dataMost 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 .Experiments and results

Tags are encoded as an individual : v(i)Initial values of tags are set as 1/|v| uniformlyApplying 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 decodingThe average number of source symbols that cannot be recovered for a constant = 1.1 is considered

OverheadWe minimize the overhead for different k sizes

The expected overhead of robust soliton distribution

Failure rateWe 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 . =1.1

ConclusionAlgorithmically optimize the degree distribution adopted in LT codeEvolutionary computationCMA-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 overheadTo reduce the decoding failure rate.

The optimized overhead was decreased as least 10%The results of failure rate minimization were also remarkably promising