Learning discontinuities for switching between local models

This page contains some supplementary materials to a paper submitted to IJCAI 05: Marc Toussaint & Sethu Vijayakumar: Learning discontinuities for switching between local models. These are mainly larger graphs and more experimental results than the limited length of the paper permitted. Please refer to this paper for more details.

1D test function - data and learned switching model

The following graph displays noisy data from a 1D test function. The dashed line is the underlying test function without noist. E.g., in the interval [-.5,-.2] one can see a significant deviation between the true test function and the function learned by our model. Clearly, the reason is that the noise hardly allows to infer that the true test function has an extra step. In total, the test function is composed of 10 pieces and 1000 data points were used.

1D test function - blended switching model and LWPR

The next graph displayes the same 1D test function (dashed line) and a learned switching model (cont. line). But here, the output of the learned model is given as the weighted average of the outputs of the family members ‐, weighted by the coefficient beta. Strictly speaking, this is not conform with the probabilistic framework, which says that beta is a probability associated with a model, and not an averaging coefficient. Still, the graph allows to see the sigmoids that are actually behind the switching.

The dash-dotted curve displays the function learned with LWPR, which allocated 15 kernels to represent this function. Clearly, LWPR is not designed to learn discontinuities.

Family and classification errors for 2D, 5D, and 10D test functions

The following graphs display what we call the family error and the classification error. The family error only evaluates the quality of the family of models independent of how well the second level of the algorithm (the product of sigmoids) can predict which model gives the best output. For every data point, it simply evalues the MSE of the best fitting eligible model within the family. This is averaged over a whole test data set. In contrast, the classification error indicates the quality of the second level of the algorithm by counting how often the product of sigmoids do indeed predict the correct model as being the best for a given (input) datum. The error is given as a percentage over the a test data set. E.g., a family error of .01 and a classification error of 4% means that in 96% of the test data points, the model chose correctly the best fitting model from the family, which has, on average, an MSE of .01 (which is optimal given the noise level).

The following two graphs display 10 runs over random 2D test functions with training data set size 1000. The bold lines are the averages over the 10 independent runs.

The following two graphs display 10 runs over random 5D test functions with training data set size 10000. The bold lines are the averages over the 10 independent runs.

The following two graphs display 10 runs over random 10D test functions with training data set size 10000. The bold lines are the averages over the 10 independent runs.