Co-evolutionary Modular Neural Networks for Automatic Problem Decomposition Vineet R. Khare, Xin Yao Bernhard Sendhoff, Yaochu Jin, Heiko Wersing Natural Computation Group Honda Research Institute Europe GmbH, School of Computer Science Carl-Legien-Strasse 30 Edgbaston, Birmingham B15 2TT D-63073 Offenbach / Main, Germany {V.R.Khare, X.Yao}@cs.bham.ac.uk

{bs, yaochu.jin, heiko.wersing}@honda-ri.de

Overview • Automatic Problem Decomposition – Modular Neural Networks (MNNs) & Co-evolution

• Types of Decompositions – A Test Problem and candidate solutions

• Co-evolutionary Problem Decomposition • Experimental Results & Discussion • Conclusion

Automatic Problem Decomposition • Divide-and-conquer : MNNs & Co-evolution • With least amount of domain knowledge! – No manual crafting – Novel Solutions

• In the context of learning: – Parallel Decomposition – Sequential Decomposition

• Within Parallel Decomposition (contd.)

Types of Decompositions • One instance – one sub-task [2, 5] r r r r f ( X , Y ) = f1 ( X ) OR f 2 ( Y ) t

• Subtasks on separate outputs [1, 3] r r r r r f ( X , Y ) = { f1 ( X ), f 2 ( Y ) }

• Combination of subtasks at one output [4] r r r r f ( X , Y ) = g ( f1 ( X ), f 2 ( Y ) )

Artificial Time-series Mixture Problem r r r r f ( X , Y ) = g ( f1 ( X ), f 2 ( Y ) ) Mackey-Glass Time-series Prediction

Lorenz - z Time-series Prediction

A Few Candidate Solutions Sequential Decomposition

Parallel Decomposition

Sub-task Specialization

Co-evolutionary Problem Decomposition • Co-evolving modular neural networks alongside their constituent modules

Co-evolutionary Problem Decomposition • System Fitness • Module Fitness

F(si) = 1/ (nrmsevalid + c) 1. f ( M i ) =

∑ f (S

j∈topSys

ij

)

2. f ( M i ) = freq i ( in last 10 gen. )

Co-evolutionary Problem Decomposition

Co-evolutionary Model : Stage 1 • Only parallel decomposition • 2 Modules • AVERAGING problem • Function ‘g’ known! • Complimentarity constraint

Co-evolutionary Model : Stage 2 • Parallel & Sequential decomposition • 2 Modules • PRODUCT problem • Function ‘g’ unknown!

Parameter Values

Results & Discussion • Stage 1 : All 30 runs produce pure-modular structure. • Stage 2: Out of 10 runs, – 5 pure-modular, 2 incomplete pure modular and 3

imbalanced incomplete

Results & Discussion • Stage 1 : Feature decomposition, easily achieved • Stage 2 : Sequential and parallel decomposition is difficult – – – –

Much bigger search space Other structures close to pure-modular New structures discovered Ensemble effect

Conclusions •

A two-level co-evolutionary model to design and optimize modular neural networks with sub-task specialization.



Evolutionary pressure to increase the overall fitness of the two populations provides the needed stimulus for the emergence of the subtask specific modules.



Emergence of other good decompositions besides the intuitive ones.



Generic model can be applied to variety of problems ranging from feature decomposition and feature selection in neural network ensembles to problems which require pre-processing.

Extensions • Making the model adaptive in terms of – Number of modules per network – Structure of combining module

• Alternative uses of modularity.

References [1] Michael Husken, Christian Igel, and Marc Toussaint. Task-dependent evolution of modularity in neural networks. Connection Science, 14:219–229, 2002. [2] Robert A. Jacobs, Michael I. Jordan, and Andrew G.Barto. Task Decomposition Through Competition in a Modular Connectionist Architecture: The What and Where Vision Tasks. Cognitive Science, 15:219–250, 1991. [3] Robert A. Jacobs, Michael I. Jordan, Steven J. Nowlan, and Geoffrey E. Hinton. Adaptive Mixtures of Local Experts. Neural Computation, 3(1):79–87, 1991. [4] Yuansong Liao and John Moody. Constructing heterogeneous committees using input feature grouping: Application to economic forecasting. Advances in Neural Information Processing Systems, 12:921–927, 1999. [5] Bao-Liang Lu and Masami Ito. Task decomposition and module combination based on class relations: A modular neural network for pattern classification. IEEE Transactions on Neural Networks, 10:1244– 1256, 1999.

Co-evolutionary Modular Neural Networks for ...

Co-evolutionary Model : Stage 1. • Only parallel decomposition. • 2 Modules. • AVERAGING problem. • Function 'g' known! • Complimentarity constraint ...

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