We study convergence properties of the discrete-time Mean-Field Langevin Stochastic Gradient Descent-Ascent (MFL-SGDA) algorithm for solving distributional minimax optimization. These problems arise in various applications, such as zero-sum games, generative adversarial networks and distributionally robust learning. Despite the significance of MFL-SGDA in these contexts, the discrete-time convergence rate remains underexplored. To address this gap, we establish a last-iterate convergence rate of $O(\frac{1}{\epsilon}\log\frac{1}{\epsilon})$ for MFL-SGDA. This rate is nearly optimal when compared to the complexity lower bound of its Euclidean counterpart. This rate also matches the complexity of mean-field Langevin stochastic gradient descent for distributional minimization and the outer-loop iteration complexity of an existing double-loop algorithm for distributional minimax problems. By leveraging an elementary analysis framework that avoids PDE-based techniques, we overcome previous limitations and achieve a faster convergence rate.

Modeling human decision-making is central to applications such as recommendation, preference learning, and human-AI alignment. While many classic models assume context-independent choice behavior, a large body of behavioral research shows that preferences are often influenced by the composition of the choice set itself---a phenomenon known as the context effect or Halo effect. These effects can manifest as pairwise (first-order) or even higher-order interactions among the available alternatives. Recent models that attempt to capture such effects either focus on the featureless setting or, in the feature-based setting, rely on restrictive interaction structures or entangle interactions across all orders, which limits interpretability. In this work, we propose DeepHalo, a neural modeling framework that incorporates features while enabling explicit control over interaction order and principled interpretation of context effects. Our model enables systematic identification of interaction effects by order and serves as a universal approximator of context-dependent choice functions when specialized to a featureless setting. Experiments on synthetic and real-world datasets demonstrate strong predictive performance while providing greater transparency into the drivers of choice.

Discrete choice models are fundamental for predicting how individuals select among alternatives based on their preferences and the attributes of the options. Traditional Random Utility Models assume stochastic rationality, often failing to capture context-dependent behavior—where the presence of certain alternatives alters the perceived utility of others.

We present a new framework to address the non-convex robust hypothesis testing problem, wherein the goal is to seek the optimal detector that minimizes the maximum of worst-case type-land type-II risk functions. The distributional uncertainty sets are constructed to center around the empirical distribution derived from samples based on Sinkhorn discrepancy. Given that the objective involves non-convex, non-smooth probabilistic functions that are often intractable to optimize, existing methods resort to approximations rather than exact solutions. To tackle the challenge, we introduce an exact mixed-integer exponential conic reformulation of the problem, which can be solved into a global optimum with a moderate amount of input data. Subsequently, we propose a convex approximation, demonstrating its superiority over current state-of-the-art methodologies in literature. Furthermore, we establish connections between robust hypothesis testing and regularized formulations of non-robust risk functions, offering insightful interpretations.

Machine learning (ML) models can underperform on certain population groups due to choices made during model development and bias inherent in the data. We categorize sources of discrimination in the ML pipeline into two classes: aleatoric discrimination, which is inherent in the data distribution, and epistemic discrimination, which is due to decisions made during model development. We quantify aleatoric discrimination by determining the performance limits of a model under fairness constraints, assuming perfect knowledge of the data distribution. We demonstrate how to characterize aleatoric discrimination by applying Blackwell's results on comparing statistical experiments. We then quantify epistemic discrimination as the gap between a model's accuracy when fairness constraints are applied and the limit posed by aleatoric discrimination. We apply this approach to benchmark existing fairness interventions and investigate fairness risks in data with missing values. Our results indicate that state-of-the-art fairness interventions are effective at removing epistemic discrimination on standard (overused) tabular datasets. However, when data has missing values, there is still significant room for improvement in handling aleatoric discrimination.

Machine learning models trained by different optimization algorithms under different data distributions can exhibit distinct generalization behaviors. In this paper, we analyze the generalization of models trained by noisy iterative algorithms. We derive distribution-dependent generalization bounds by connecting noisy iterative algorithms to additive noise channels found in communication and information theory. Our generalization bounds shed light on several applications, including differentially private stochastic gradient descent (DP-SGD), federated learning, and stochastic gradient Langevin dynamics (SGLD). We demonstrate our bounds through numerical experiments, showing that they can help understand recent empirical observations of the generalization phenomena of neural networks.

We develop a kernel projected Wasserstein distance for the two-sample test, an essential building block in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. This method operates by finding the nonlinear mapping in the data space which maximizes the distance between projected distributions. In contrast to existing works about projected Wasserstein distance, the proposed method circumvents the curse of dimensionality more efficiently. We present practical algorithms for computing this distance function together with the non-asymptotic uncertainty quantification of empirical estimates. Numerical examples validate our theoretical results and demonstrate good performance of the proposed method.

Learning a robust classifier from a few samples remains a key challenge in machine learning. A major thrust of research has been focused on developing k-nearest neighbor (k-NN) based algorithms combined with metric learning that captures similarities between samples. When the samples are limited, robustness is especially crucial to ensure the generalization capability of the classifier. In this paper, we study a minimax distributionally robust formulation of weighted k-nearest neighbors, which aims to find the optimal weighted k-NN classifiers that hedge against feature uncertainties. We develop an algorithm, Dr.k-NN, that efficiently solves this functional optimization problem and features in assigning minimax optimal weights to training samples when performing classification. These weights are class-dependent, and are determined by the similarities of sample features under the least favorable scenarios. When the size of the uncertainty set is properly tuned, the robust classifier has a smaller Lipschitz norm than the vanilla k-NN, and thus improves the generalization capability. We also couple our framework with neural-network-based feature embedding. We demonstrate the competitive performance of our algorithm compared to the state-of-the-art in the few-training-sample setting with various real-data experiments.

We develop a projected Wasserstein distance for the two-sample test, a fundamental problem in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. In particular, we aim to circumvent the curse of dimensionality in Wasserstein distance: when the dimension is high, it has diminishing testing power, which is inherently due to the slow concentration property of Wasserstein metrics in the high dimension space. A key contribution is to couple optimal projection to find the low dimensional linear mapping to maximize the Wasserstein distance between projected probability distributions. We characterize theoretical properties of the two-sample convergence rate on IPMs and this new distance. Numerical examples validate our theoretical results.

(Distributionally) robust optimization has gained momentum in machine learning community recently, due to its promising applications in developing generalizable learning paradigms. In this paper, we derive generalization bounds for robust optimization and Wasserstein robust optimization for Lipschitz and piecewise Hölder smooth loss functions under both stochastic and adversarial setting, assuming that the underlying data distribution satisfies transportation-information inequalities. The proofs are built on new generalization bounds for variation regularization (such as Lipschitz or gradient regularization) and its connection with robustness.

Optimization is a key component for training machine learning models and has a strong impact on their generalization. In this paper, we consider a particular optimization method---the stochastic gradient Langevin dynamics (SGLD) algorithm---and investigate the generalization of models trained by SGLD. We derive a new generalization bound by connecting SGLD with Gaussian channels found in information and communication theory. Our bound can be computed from the training data and incorporates the variance of gradients for quantifying a particular kind of "sharpness" of the loss landscape. We also consider a closely related algorithm with SGLD, namely differentially private SGD (DP-SGD). We prove that the generalization capability of DP-SGD can be amplified by iteration. Specifically, our bound can be sharpened by including a time-decaying factor if the DP-SGD algorithm outputs the last iterate while keeping other iterates hidden. This decay factor enables the contribution of early iterations to our bound to reduce with time and is established by strong data processing inequalities---a fundamental tool in information theory. We demonstrate our bound through numerical experiments, showing that it can predict the behavior of the true generalization gap.

There are two types of deep generative models: explicit and implicit. The former defines an explicit density form that allows likelihood inference; while the latter targets a flexible transformation from random noise to generated samples. While the two classes of generative models have shown great power in many applications, both of them, when used alone, suffer from respective limitations and drawbacks. To take full advantages of both models and enable mutual compensation, we propose a novel joint training framework that bridges an explicit (unnormalized) density estimator and an implicit sample generator via Stein discrepancy. We show that our method 1) induces novel mutual regularization via kernel Sobolev norm penalization and Moreau-Yosida regularization, and 2) stabilizes the training dynamics. Empirically, we demonstrate that proposed method can facilitate the density estimator to more accurately identify data modes and guide the generator to output higher-quality samples, comparing with training a single counterpart. The new approach also shows promising results when the training samples are contaminated or limited.

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.