Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation

Optimal Transport (OT) distances such as Wasserstein have been used in several areas such as GANs and domain adaptation. OT, however, is very sensitive to outliers (samples with large noise) in the data since in its objective function, every sample, including outliers, is weighed similarly due to the marginal constraints. To remedy this issue, robust formulations of OT with unbalanced marginal constraints have previously been proposed. However, employing these methods in deep learning problems such as GANs and domain adaptation is challenging due to the instability of their dual optimization solvers. In this paper, we resolve these issues by deriving a computationally-efficient dual form of the robust OT optimization that is amenable to modern deep learning applications. We demonstrate the effectiveness of our formulation in two applications of GANs and domain adaptation. Our approach can train state-of-the-art GAN models on noisy datasets corrupted with outlier distributions. In particular, our optimization computes weights for training samples reflecting how difficult it is for those samples to be generated in the model. In domain adaptation, our robust OT formulation leads to improved accuracy compared to the standard adversarial adaptation methods. Our code is available at https://github.com/yogeshbalaji/robustOT . The OT sensitivity to outliers is undesirable, especially when we deal with large-scale datasets where the noise is inevitable. This sensitivity is a consequence of exactly satisfying the marginal constraints in OT's objective. Hence, to boost OT's robustness against outliers, we propose to utilize recent formulations of unbalanced optimal transport [6, 13] which relax OT's marginal constraints. The authors in [6, 13] provide an exact dual form for the unbalanced OT problem. However, we have found that using this dual optimization in large-scale deep learning applications such as GANs results in poor convergence and an unstable behaviour (see Section 3.1 and the appendix for details). To remedy this issue, in this work, we derive a computationally efficient dual form for the unbalanced OT optimization that is suited for practical deep learning applications. Our dual simplifies to a weighted OT objective, with low weights assigned to outlier samples. These instance weights can also be useful in interpreting the difficulty of input samples for learning a given task. We develop two solvers for this dual problem based on either a discrete formulation or a continuous stochastic relaxation. These solvers demonstrate high stability in large-scale deep learning applications. We show that, under mild assumptions, our robust OT measure (which is similar in form to the unbalanced OT) is upper bounded by a constant factor of the true OT distance (OT ignoring outliers) for any outlier distribution. Hence, our robust OT can be used for effectively handling outliers. This is visualized in Figure 1(c) , where couplings obtained by robust OT effectively ignores outlier samples, yielding a good estimate of the true OT distance. We demonstrate the effectiveness of the proposed robust OT formulation in two large-scale deep learning applications of generative modeling and domain adaptation. In generative modeling, we show how robust Wasserstein GANs can be trained using state-of-the-art GAN architectures to effectively ignore outliers in the generative distrubution. In domain adaptation, we utilize the robust OT framework for the challenging task of synthetic to real adaptation, where our approach improves adversarial adaptation techniques by ∼ 5%.