Empirical Optimal Transport between Conditional Distributions

Given samples from two joint distributions, we consider the problem of Optimal Transportation (OT) between the corresponding distributions conditioned on a common variable. The objective of this work is to estimate the associated transport cost (Wasserstein distance) as well as the transport plan between the conditionals as a function of the conditioned value. Since matching conditional distributions is at the core of supervised training of discriminative models and (implicit) conditional-generative models, OT between conditionals has the potential to be employed in diverse machine learning applications. However, since the conditionals involved in OT are implicitly specified via the joint samples, it is challenging to formulate this problem, especially when (i) the variable conditioned on is continuous and (ii) the marginal of this variable in the two distributions is different. We overcome these challenges by employing a specific kernel MMD (Maximum Mean Discrepancy) based regularizer that ensures the marginals of our conditional transport plan are close to the conditionals specified via the given joint samples. Under mild conditions, we prove that our estimator for this regularized transport cost is statistically consistent and derive finite-sample bounds on the estimation error. Application-specific details for parameterizing our conditional transport plan are also presented. Furthermore, we empirically evaluate our methodology on benchmark datasets in applications like classification, prompt learning for few-shot classification, and conditional-generation in the context of predicting cell responses to cancer treatment

Co-regularized Alignment for Unsupervised Domain Adaptation

Deep neural networks, trained with large amount of labeled data, can fail to generalize well when tested with examples from a \emph{target domain} whose distribution differs from the training data distribution, referred as the \emph{source domain}. It can be expensive or even infeasible to obtain required amount of labeled data in all possible domains. Unsupervised domain adaptation sets out to address this problem, aiming to learn a good predictive model for the target domain using labeled examples from the source domain but only unlabeled examples from the target domain. Domain alignment approaches this problem by matching the source and target feature distributions, and has been used as a key component in many state-of-the-art domain adaptation methods. However, matching the marginal feature distributions does not guarantee that the corresponding class conditional distributions will be aligned across the two domains. We propose co-regularized domain alignment for unsupervised domain adaptation, which constructs multiple diverse feature spaces and aligns source and target distributions in each of them individually, while encouraging that alignments agree with each other with regard to the class predictions on the unlabeled target examples. The proposed method is generic and can be used to improve any domain adaptation method which uses domain alignment. We instantiate it in the context of a recent state-of-the-art method and observe that it provides significant performance improvements on several domain adaptation benchmarks.Comment: NIPS 2018 accepted versio