Regression via Kirszbraun Extension with Applications to Imitation Learning

Learning by demonstration is a versatile and rapid mechanism for transferring motor skills from a teacher to a learner. A particular challenge in imitation learning is the so-called correspondence problem, which involves mapping actions between a teacher and a learner having substantially different embodiments (say, human to robot). We present a general, model free and non-parametric imitation learning algorithm based on regression between two Hilbert spaces. We accomplish this via Kirszbraun's extension theorem --- apparently the first application of this technique to supervised learning --- and analyze its statistical and computational aspects. We begin by formulating the correspondence problem in terms of quadratically constrained quadratic program (QCQP) regression. Then we describe a procedure for smoothing the training data, which amounts to regularizing hypothesis complexity via its Lipschitz constant. The Lipschitz constant is tuned via a Structural Risk Minimization (SRM) procedure, based on the covering-number risk bounds we derive. We apply our technique to a static posture imitation task between two robotic manipulators with different embodiments, and report promising results.