{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.