*the*machine learning task: estimate an output for given inputs. Decades of (very successful) research have shown that the problem is, and will always be, how to generalize such learned input/output relations to previously unseen inputs. This problem gets severe when either very few training data is available, or the data itself is very noisy. When we wanted to learn the low-level control of the Bionic Handling Assistant, we had both problems. Very few (because relatively expensive), and very bad training data. Challenging.

Even based on such bad training data, the to-be-learned controller had to be

*reliable*. It should not go mad on unknown inputs, but should generalize smoothly and consistently. I wanted to have

*guarantees*, considering that the controller we needed should not just be a stand-alone show-case, but operate at the bottom of everything we wanted to do with the BHA.

We came up with the idea to use some

*prior knowledge*acting as constraint on the learning. In this case we had some prior knowledge. Such as: physics results in certain variables to have a strictly monotonous relation. Variables have ranges. These things we wanted to put into learning, and guarantee that the learner generalizes accordingly. Klaus (who just successfully defended his PhD, congratulations!) came up with a surprisingly flexible and general method to put such things into regression. Results just got published:

Neumann, K., M. Rolf, and J.J. Steil, "Reliable Integration of Continuous Constraints into Extreme Learning Machines", Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21(supp02), 12/2013. (pdf)

Let's dig into it...Abstract —The application of machine learning methods in the engineering of intelligent technical systems often requires the integration of continuous constraints like positivity, monotonicity, or bounded curvature in the learned function to guarantee a reliable performance. We show that the extreme learning machine is particularly well suited for this task. Constraints involving arbitrary derivatives of the learned function are effectively implemented through quadratic optimization because the learned function is linear in its parameters, and derivatives can be derived analytically. We further provide a constructive approach to verify that discretely sampled constraints are generalized to continuous regions and show how local violations of the constraint can be rectified by iterative relearning. We demonstrate the approach on a practical and challenging control problem from robotics, illustrating also how the proposed method enables learning from few data samples if additional prior knowledge about the problem is available.