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Title: The Cramer Distance as a Solution to Biased Wasserstein Gradients

Paper content:
The Wasserstein probability metric has received much attention from the machine learning community.
Unlike the Kullback-Leibler divergence, which strictly measures change in probability, the Wasserstein metric reflects the underlying geometry between outcomes.
The value of being sensitive to this geometry has been demonstrated, among others, in ordinal regression and generative modelling, and most recently in reinforcement learning.
In this paper we describe three natural properties of probability divergences that we believe reflect requirements from machine learning: sum invariance, scale sensitivity, and unbiased sample gradients.
The Wasserstein metric possesses the first two properties but, unlike the Kullback-Leibler divergence, does not possess the third.
We provide empirical evidence suggesting this is a serious issue in practice.
Leveraging insights from probabilistic forecasting we propose an alternative to the Wasserstein metric, the Cramér distance.
We show that the Cramér distance possesses all three desired properties, combining the best of the Wasserstein and Kullback-Leibler divergences.
We give empirical results on a number of domains comparing these three divergences.
To illustrate the practical relevance of the Cramér distance we design a new algorithm, the Cramér Generative Adversarial Network (GAN), and show that it has a number of desirable properties over the related Wasserstein GAN.
In machine learning, the Kullback-Leibler (KL) divergence is perhaps the most common way of assessing how well a probabilistic model explains observed data.
Among the reasons for its popularity is that it is directly related to maximum likelihood estimation and is easily optimized.
However, the KL divergence suffers from a significant limitation: it does not take into account how close two outcomes might be, but only their relative probability.
This closeness can matter a great deal: in image modelling, for example, perceptual similarity is key (Rubner et al., 2000; BID13 .
Put another way, the KL divergence cannot reward a model that "gets it almost right".To
address this limitation, researchers have turned to the Wasserstein metric, which does incorporate the underlying geometry between outcomes. The
Wasserstein metric can be applied to distributions with non-overlapping supports, and has good out-of-sample performance BID11 . Yet
, practical applications of the Wasserstein distance, especially in deep learning, remain tentative. In
this paper we provide a clue as to why that might be: estimating the Wasserstein metric from samples yields biased gradients, and may actually lead to the wrong minimum. This
precludes using stochastic gradient descent (SGD) and SGD-like methods, whose fundamental mode of operation is sample-based, when optimizing for this metric.As a replacement we propose the Cramér distance (Székely, 2002; Rizzo & Székely, 2016) , also known as the continuous ranked probability score in the probabilistic forecasting literature BID14 . The
Cramér distance, like the Wasserstein metric, respects the underlying geometry but also has unbiased sample gradients. To
underscore our theoretical findings, we demonstrate a significant quantitative difference between the two metrics when employed in typical machine learning scenarios: categorical distribution estimation, regression, and finally image generation. In
the latter case, we use a multivariate generalization of the Cramér distance, the energy distance (Székely, 2002) , itself an instantiation of the MMD family of metrics BID16 .
There are many situations in which the KL divergence, which is commonly used as a loss function in machine learning, is not suitable.
The desirable alternatives, as we have explored, are the divergences that are ideal and allow for unbiased estimators: they allow geometric information to be incorporated into the optimization problem; because they are scale-sensitive and sum-invariant, they possess the convergence properties we require for efficient learning; and the correctness of their sample gradients means we can deploy them in large-scale optimization problems.
Among open questions, we mention deriving an unbiased estimator that minimizes the Wasserstein distance, and variance analysis and reduction of the Cramér distance gradient estimate.

assistant: text: The Wasserstein distance is hard to minimize with stochastic gradient descent, while the Cramer distance can be optimized easily and works just as well.