Cramer Type Distances for Learning Gaussian Mixture Models by Gradient
  Descent
Ruichong Zhang
The learning of Gaussian Mixture Models (also referred to simply as GMMs)
plays an important role in machine learning. Known for their expressiveness and
interpretability, Gaussian mixture models have a wide range of applications,
from statistics, computer vision to distributional reinforcement learning.
However, as of today, few known algorithms can fit or learn these models, some
of which include Expectation-Maximization algorithms and Sliced Wasserstein
Distance. Even fewer algorithms are compatible with gradient descent, the
common learning process for neural networks.
  In this paper, we derive a closed formula of two GMMs in the univariate,
one-dimensional case, then propose a distance function called Sliced Cram\'er
2-distance for learning general multivariate GMMs. Our approach has several
advantages over many previous methods. First, it has a closed-form expression
for the univariate case and is easy to compute and implement using common
machine learning libraries (e.g., PyTorch and TensorFlow). Second, it is
compatible with gradient descent, which enables us to integrate GMMs with
neural networks seamlessly. Third, it can fit a GMM not only to a set of data
points, but also to another GMM directly, without sampling from the target
model. And fourth, it has some theoretical guarantees like global gradient
boundedness and unbiased sampling gradient. These features are especially
useful for distributional reinforcement learning and Deep Q Networks, where the
goal is to learn a distribution over future rewards. We will also construct a
Gaussian Mixture Distributional Deep Q Network as a toy example to demonstrate
its effectiveness. Compared with previous models, this model is parameter
efficient in terms of representing a distribution and possesses better
interpretability.