For an experimental testbed, we used the ``Arm2D'' problem described
by Cohn [1994]. 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.