Wasserstein Dependency Measure for Representation Learning -
Self-Supervised Learning is a broad category of approaches whose goal is to learn useful representations by asking networks to perform constructed tasks that only use the content of a dataset itself, and not external labels. The idea with these tasks is to design tasks such that solving them requires the network to have learned useful Some examples of this approach include predicting the rotation of rotated images, reconstructing color from greyscale, and, the topic of this paper, maximizing mutual information between different areas of the image. The hope behind this last approach is that if two areas of an image are generated by the same set of underlying factors (in the case of a human face: they're parts of the same person's face), then a representation that correctly captures those factors for one area will give you a lot of information about the representation of the other area. Historically, this conceptual desire for representations that are mutually informative has been captured by mutual information. If we define the representation distribution over the data of area 1 as p(x) and area 2 as q(x), the mutual information is the KL divergence between the joint distribution of these two distributions and the product of their marginals. This is an old statistical intuition: the closer the joint is to the product of marginals, the closer the variables are to independent; the farther away, the closer they are to informationally identical. This paper argues that the presence of the KL divergence in this mutual information formulation impedes the ability of networks to learn useful representations. This argument is theoretically based on a result from a recent paper (which for the moment I'll just take as foundation, without reading it myself) that empirical lower-bound measurements of mutual information, of the kind used in these settings, are upper bounded by log(n) where n is datapoints. Our hope in maximizing a lower bound to any quantity is that the bound is fairly tight, since that means that optimizing a network to push upward a lower bound actually has the effect of pushing the actual value up as well. If the lower bound we can estimate is constrained to be far below the actual lower bound in the data, then pushing it upward doesn't actually require the value to move upward. The authors identify this as a particular problem in areas where the underlying mutual information of the data is high, such as in videos where one frame is very predictive of the next, since in those cases the constraint imposed by the dataset size will be small relative to the actual possible maximum mutual information you could push your network to achieve. Taking a leaf out of the GAN literature, the authors suggest keeping replacing the KL divergence component of mutual information and replacing it with the Wasserstein Distance; otherwise known as the "earth-mover distance", the Wasserstein distance measures the cost of the least costly way to move probability mass from one distribution to another, assuming you're moving that mass along some metric space. A nice property of the Wasserstein distance, in both GANs and in this application) is that they don't saturate quite as quickly: the value of a KL divergence can shoot up if the distributions are even somewhat different, making it unable to differentiate between distributions that are somewhat and very far away, whereas a Wasserstein distance continues to have more meaningful signal in that regime. In the context of the swap for mutual information, the authors come up with the "Wasserstein Dependency Measure", which is just the Wasserstein Distance between the joint distributions and the product of the marginals. In practice, they use the dual formulation of the Wasserstein distance, which amounts to applying a (neural network) function f(x) to values from both distributions, optimizing f(x) so that the values are far apart, and using that distance as your training signal. Crucially, this function has to be relatively smooth in order for the dual formulation to work: in particular it has to have a small Lipschitz value (meaning its derivatives are bounded by some value). Intuitively, this has the effect of restricting the capacity of the network, which is hoped to incentivize it to use its limited capacity to represent true factors of variation, which are assumed to be the most compact way to represent the data. Empirically, the authors found that their proposed Wasserstein Dependency Measure (with a slight variation applied to reduce variance) does have the predicted property of performing better for situations where the native mutual information between two areas is high. I found the theoretical points of this paper interesting, and liked the generalization of the idea of Wasserstein distances from GANs to a new area. That said, I wish I had a better mechanical sense for how it ground out in actual neural network losses: this is partially just my own lack of familiarity with how e.g. mutual information losses are actually formulated as network objectives, but I would have appreciated an appendix that did a bit more of that mapping between mathematical intuition and practical network reality.