Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality

Wasserstein distributionally robust optimization (DRO) aims to find robust and generalizable solutions by hedging against data perturbations in Wasserstein distance. Despite its recent empirical success in operations research and machine learning, existing performance guarantees for generic loss functions are either overly conservative due to the curse of dimensionality, or plausible only in large sample asymptotics. In this paper, we develop a non-asymptotic framework for analyzing the out-of-sample performance for Wasserstein robust learning and the generalization bound for its related Lipschitz and gradient regularization problems. To the best of our knowledge, this gives the first finite-sample guarantee for generic Wasserstein DRO problems without suffering from the curse of dimensionality. Our results highlight the bias-variation trade-off intrinsic in the Wasserstein DRO, which automatically balances between the empirical mean of the loss and the variation of the loss, measured by the Lipschitz norm or the gradient norm of the loss. Our analysis is based on two novel methodological developments which are of independent interest: 1) a new concentration inequality characterizing the decay rate of large deviation probabilities by the variation of the loss and, 2) a localized Rademacher complexity theory based on the variation of the loss.

Distributionally Robust $k$-Nearest Neighbors for Few-Shot Learning

Learning a robust classifier from a few samples remains a key challenge in machine learning. A major thrust of research in few-shot classification has been based on metric learning to capture similarities between samples and then perform the $k$-nearest neighbor algorithm. To make such an algorithm more robust, in this paper, we propose a distributionally robust $k$-nearest neighbor algorithm Dr.k-NN, which features assigning minimax optimal weights to training samples when performing classification. We also couple it with neural-network-based feature embedding. We demonstrate the competitive performance of our algorithm comparing to the state-of-the-art in the few-shot learning setting with various real-data experiments.

Remove Appearance Shift for Ultrasound Image Segmentation via Fast and Universal Style Transfer

Deep Neural Networks (DNNs) suffer from the performance degradation when image appearance shift occurs, especially in ultrasound (US) image segmentation. In this paper, we propose a novel and intuitive framework to remove the appearance shift, and hence improve the generalization ability of DNNs. Our work has three highlights. First, we follow the spirit of universal style transfer to remove appearance shifts, which was not explored before for US images. Without sacrificing image structure details, it enables the arbitrary style-content transfer. Second, accelerated with Adaptive Instance Normalization block, our framework achieved real-time speed required in the clinical US scanning. Third, an efficient and effective style image selection strategy is proposed to ensure the target-style US image and testing content US image properly match each other. Experiments on two large US datasets demonstrate that our methods are superior to state-of-the-art methods on making DNNs robust against various appearance shifts.

Stein Bridging: Enabling Mutual Reinforcement between Explicit and Implicit Generative Models

Deep generative models are generally categorized into explicit models and implicit models. The former defines an explicit density form, whose normalizing constant is often unknown; while the latter, including generative adversarial networks (GANs), generates samples without explicitly defining a density function. In spite of substantial recent advances demonstrating the power of the two classes of generative models in many applications, both of them, when used alone, suffer from respective limitations and drawbacks. To mitigate these issues, we propose Stein Bridging, a novel joint training framework that connects an explicit density estimator and an implicit sample generator with Stein discrepancy. We show that the Stein Bridge induces new regularization schemes for both explicit and implicit models. Convergence analysis and extensive experiments demonstrate that the Stein Bridging i) improves the stability and sample quality of the GAN training, and ii) facilitates the density estimator to seek more modes in data and alleviate the mode-collapse issue. Additionally, we discuss several applications of Stein Bridging and useful tricks in practical implementation used in our experiments.

Robust Hypothesis Testing Using Wasserstein Uncertainty Sets

We develop a novel computationally efficient and general framework for robust hypothesis testing. The new framework features a new way to construct uncertainty sets under the null and the alternative distributions, which are sets centered around the empirical distribution defined via Wasserstein metric, thus our approach is data-driven and free of distributional assumptions. We develop a convex safe approximation of the minimax formulation and show that such approximation renders a nearly-optimal detector among the family of all possible tests. By exploiting the structure of the least favorable distribution, we also develop a tractable reformulation of such approximation, with complexity independent of the dimension of observation space and can be nearly sample-size-independent in general. Real-data example using human activity data demonstrated the excellent performance of the new robust detector.

Wasserstein Distributional Robustness and Regularization in Statistical Learning

A central question in statistical learning is to design algorithms that not only perform well on training data, but also generalize to new and unseen data. In this paper, we tackle this question by formulating a distributionally robust stochastic optimization (DRSO) problem, which seeks a solution that minimizes the worst-case expected loss over a family of distributions that are close to the empirical distribution in Wasserstein distances. We establish a connection between such Wasserstein DRSO and regularization. More precisely, we identify a broad class of loss functions, for which the Wasserstein DRSO is asymptotically equivalent to a regularization problem with a gradient-norm penalty. Such relation provides new interpretations for problems involving regularization, including a great number of statistical learning problems and discrete choice models (e.g. multinomial logit). The connection suggests a principled way to regularize high-dimensional, non-convex problems. This is demonstrated through the training of Wasserstein generative adversarial networks in deep learning.

Multi-Focus Image Fusion Via Coupled Sparse Representation and Dictionary Learning

We address the multi-focus image fusion problem, where multiple images captured with different focal settings are to be fused into an all-in-focus image of higher quality. Algorithms for this problem necessarily admit the source image characteristics along with focused and blurred feature. However, most sparsity-based approaches use a single dictionary in focused feature space to describe multi-focus images, and ignore the representations in blurred feature space. Here, we propose a multi-focus image fusion approach based on coupled sparse representation. The approach exploits the facts that (i) the patches in given training set can be sparsely represented by a couple of overcomplete dictionaries related to the focused and blurred categories of images; and (ii) merging such representations leads to a more flexible and therefore better fusion strategy than the one based on just selecting the sparsest representation in the original image estimate. By jointly learning the coupled dictionary, we enforce the similarity of sparse representations in the focused and blurred feature spaces, and then introduce a fusion approach to combine these representations for generating an all-in-focus image. We also discuss the advantages of the fusion approach based on coupled sparse representation and present an efficient algorithm for learning the coupled dictionary. Extensive experimental comparisons with state-of-the-art multi-focus image fusion algorithms validate the effectiveness of the proposed approach.

Image Fusion With Cosparse Analysis Operator

The paper addresses the image fusion problem, where multiple images captured with different focus distances are to be combined into a higher quality all-in-focus image. Most current approaches for image fusion strongly rely on the unrealistic noise-free assumption used during the image acquisition, and then yield limited robustness in fusion processing. In our approach, we formulate the multi-focus image fusion problem in terms of an analysis sparse model, and simultaneously perform the restoration and fusion of multi-focus images. Based on this model, we propose an analysis operator learning, and define a novel fusion function to generate an all-in-focus image. Experimental evaluations confirm the effectiveness of the proposed fusion approach both visually and quantitatively, and show that our approach outperforms state-of-the-art fusion methods.

Sensing Subjective Well-being from Social Media

Subjective Well-being(SWB), which refers to how people experience the quality of their lives, is of great use to public policy-makers as well as economic, sociological research, etc. Traditionally, the measurement of SWB relies on time-consuming and costly self-report questionnaires. Nowadays, people are motivated to share their experiences and feelings on social media, so we propose to sense SWB from the vast user generated data on social media. By utilizing 1785 users' social media data with SWB labels, we train machine learning models that are able to "sense" individual SWB from users' social media. Our model, which attains the state-by-art prediction accuracy, can then be used to identify SWB of large population of social media users in time with very low cost.