For an experimental testbed, we used the ``Arm2D'' problem described by Cohn . The task is to learn the kinematics of a toy 2-degree-of-freedom robot arm (see Figure 4). The inputs are joint angles , and the outputs are the Cartesian coordinates of the tip . One of the implicit assumptions of both models described here is that the noise is Gaussian in the output dimensions. To test the robustness of the algorithm to this assumption, we ran experiments using no noise, using additive Gaussian noise in the outputs, and using additive Gaussian noise in the inputs. The results of each were comparable; we report here the results using additive Gaussian noise in the inputs. Gaussian input noise corresponds to the case where the arm effectors or joint angle sensors are noisy, and results in non-Gaussian errors in the learner's outputs. The input distribution is assumed to be uniform.
Figure 4: The arm kinematics problem. The learner attempts to predict tip position given a set of joint angles .
We compared the performance of the variance-minimizing criterion by comparing the learning curves of a learner using the criterion with that of one learning from random samples. The learning curves plot the mean squared error and variance of the learner as its training set size increases. The curves are created by starting with an initial sample, measuring the learner's mean squared error or estimated variance on a set of ``reference'' points (independent of the training set), selecting and adding a new example to the training set, retraining the learner on the augmented set, and repeating.
On each step, the variance-minimizing learner chose a set of 64 unlabeled reference points drawn from input distribution . It then selected a query that it estimated would minimize over the reference set. In the experiments reported here, the best was selected from another set of 64 ``candidate'' points drawn at random on each iteration.