We introduce ScoreFusion, a theoretically grounded method for fusing multiple pre-trained diffusion models that are assumed to generate from auxiliary populations. ScoreFusion is particularly useful for enhancing the generative modeling of a target population with limited observed data. Our starting point considers the family of KL barycenters of the auxiliary populations, which is proven to be an optimal parametric class in the KL sense, but difficult to learn. Nevertheless, by recasting the learning problem as score matching in denoising diffusion, we obtain a tractable way of computing the optimal KL barycenter weights. We prove a dimension-free sample complexity bound in total variation distance, provided that the auxiliary models are well-fitted for their own task and the auxiliary tasks combined capture the target well. The sample efficiency of ScoreFusion is demonstrated by learning handwritten digits. We also provide a simple adaptation of a Stable Diffusion denoising pipeline that enables sampling from the KL barycenter of two auxiliary checkpoints; on a portrait generation task, our method produces faces that enhance population heterogeneity relative to the auxiliary distributions.

We explore the control of stochastic systems with potentially continuous state and action spaces, characterized by the state dynamics $X_{t+1} = f(X_t, A_t, W_t)$. Here, $X$, $A$, and $W$ represent the state, action, and exogenous random noise processes, respectively, with $f$ denoting a known function that describes state transitions. Traditionally, the noise process $(W_t)_{t \geq 0}$ is assumed to be independent and identically distributed, with a distribution that is either fully known or can be consistently estimated. However, the occurrence of distributional shifts, typical in engineering settings, necessitates the consideration of the robustness of the policy. This paper introduces a distributionally robust stochastic control paradigm that accommodates possibly adaptive adversarial perturbation to the noise distribution within a prescribed ambiguity set. We examine two adversary models: current-action-aware and current-action-unaware, leading to different dynamic programming equations. Furthermore, we characterize the optimal finite sample minimax rates for achieving uniform learning of the robust value function across continuum states under both adversary types, considering ambiguity sets defined by $f_k$-divergence and Wasserstein distance. Finally, we demonstrate the applicability of our framework across various real-world settings.

Wasserstein Distributionally Robust Optimization (WDRO) is a principled framework for robust estimation under distributional uncertainty. However, its standard formulation can be overly conservative, particularly in small-sample regimes. We propose a novel knowledge-guided WDRO (KG-WDRO) framework for transfer learning, which adaptively incorporates multiple sources of external knowledge to improve generalization accuracy. Our method constructs smaller Wasserstein ambiguity sets by controlling the transportation along directions informed by the source knowledge. This strategy can alleviate perturbations on the predictive projection of the covariates and protect against information loss. Theoretically, we establish the equivalence between our WDRO formulation and the knowledge-guided shrinkage estimation based on collinear similarity, ensuring tractability and geometrizing the feasible set. This also reveals a novel and general interpretation for recent shrinkage-based transfer learning approaches from the perspective of distributional robustness. In addition, our framework can adjust for scaling differences in the regression models between the source and target and accommodates general types of regularization such as lasso and ridge. Extensive simulations demonstrate the superior performance and adaptivity of KG-WDRO in enhancing small-sample transfer learning.

Motivated by practical applications where stable long-term performance is critical—such as robotics, operations research, and healthcare—we study the problem of distributionally robust (DR) average-reward reinforcement learning. We propose two algorithms that achieve near-optimal sample complexity. The first reduces the problem to a DR discounted Markov decision process (MDP), while the second, Anchored DR Average-Reward MDP, introduces an anchoring state to stabilize the controlled transition kernels within the uncertainty set. Assuming the nominal MDP is uniformly ergodic, we prove that both algorithms attain a sample complexity of $\widetilde{O}\left(|\mathbf{S}||\mathbf{A}| t_{\mathrm{mix}}^2\varepsilon^{-2}\right)$ for estimating the optimal policy as well as the robust average reward under KL and $f_k$-divergence-based uncertainty sets, provided the uncertainty radius is sufficiently small. Here, $\varepsilon$ is the target accuracy, $|\mathbf{S}|$ and $|\mathbf{A}|$ denote the sizes of the state and action spaces, and $t_{\mathrm{mix}}$ is the mixing time of the nominal MDP. This represents the first finite-sample convergence guarantee for DR average-reward reinforcement learning. We further validate the convergence rates of our algorithms through numerical experiments.

Dynamic decision-making under distributional shifts is of fundamental interest in theory and applications of reinforcement learning: The distribution of the environment in which the data is collected can differ from that of the environment in which the model is deployed. This paper presents two novel model-free algorithms, namely the distributionally robust Q-learning and its variance-reduced counterpart, that can effectively learn a robust policy despite distributional shifts. These algorithms are designed to efficiently approximate the $q$-function of an infinite-horizon $\gamma$-discounted robust Markov decision process with Kullback-Leibler ambiguity set to an entry-wise $\epsilon$-degree of precision. Further, the variance-reduced distributionally robust Q-learning combines the synchronous Q-learning with variance-reduction techniques to enhance its performance. Consequently, we establish that it attains a minimax sample complexity upper bound of $\tilde O(|\mathbf{S}||\mathbf{A}|(1-\gamma)^{-4}\epsilon^{-2})$, where $\mathbf{S}$ and $\mathbf{A}$ denote the state and action spaces. This is the first complexity result that is independent of the ambiguity size $\delta$, thereby providing new complexity theoretic insights. Additionally, a series of numerical experiments confirm the theoretical findings and the efficiency of the algorithms in handling distributional shifts.

We consider a reinforcement learning setting in which the deployment environment is different from the training environment. Applying a robust Markov decision processes formulation, we extend the distributionally robust Q-learning framework studied in [Liu et. al. 2022]. Further, we improve the design and analysis of their multi-level Monte Carlo estimator. Assuming access to a simulator, we prove that the worst-case expected sample complexity of our algorithm to learn the optimal robust Q-function within an $\epsilon$ error in the sup norm is upper bounded by $\tilde O(|S||A|(1-\gamma)^{-5}\epsilon^{-2}p_{\wedge}^{-6}\delta^{-4})$, where $\gamma$ is the discount rate, $p_{\wedge}$ is the non-zero minimal support probability of the transition kernels and $\delta$ is the uncertainty size. This is the first sample complexity result for the model-free robust RL problem. Simulation studies further validate our theoretical results.

We have developed a statistical testing framework to detect if a given machine learning classifier fails to satisfy a wide range of group fairness notions. Our test is a flexible, interpretable, and statistically rigorous tool for auditing whether exhibited biases are intrinsic to the algorithm or simply due to the randomness in the data. The statistical challenges, which may arise from multiple impact criteria that define group fairness and which are discontinuous on model parameters, are conveniently tackled by projecting the empirical measure to the set of group-fair probability models using optimal transport. This statistic is efficiently computed using linear programming, and its asymptotic distribution is explicitly obtained. The proposed framework can also be used to test for composite fairness hypotheses and fairness with multiple sensitive attributes. The optimal transport testing formulation improves interpretability by characterizing the minimal covariate perturbations that eliminate the bias observed in the audit.

We build a Bayesian contextual classification model using an optimistic score ratio for robust binary classification when there is limited information on the class-conditional, or contextual, distribution. The optimistic score searches for the distribution that is most plausible to explain the observed outcomes in the testing sample among all distributions belonging to the contextual ambiguity set which is prescribed using a limited structural constraint on the mean vector and the covariance matrix of the underlying contextual distribution. We show that the Bayesian classifier using the optimistic score ratio is conceptually attractive, delivers solid statistical guarantees and is computationally tractable. We showcase the power of the proposed optimistic score ratio classifier on both synthetic and empirical data.

Policy learning using historical observational data is an important problem that has found widespread applications. However, existing literature rests on the crucial assumption that the future environment where the learned policy will be deployed is the same as the past environment that has generated the data{–}an assumption that is often false or too coarse an approximation. In this paper, we lift this assumption and aim to learn a distributionally robust policy with bandit observational data. We propose a novel learning algorithm that is able to learn a robust policy to adversarial perturbations and unknown covariate shifts. We first present a policy evaluation procedure in the ambiguous environment and also give a heuristic algorithm to solve the distributionally robust policy learning problems efficiently. Additionally, we provide extensive simulations to demonstrate the robustness of our policy.

We consider the problem of estimating the Wasserstein distance between the empirical measure and a set of probability measures whose expectations over a class of functions (hypothesis class) are constrained. If this class is sufficiently rich to characterize a particular distribution (e.g., all Lipschitz functions), then our formulation recovers the Wasserstein distance to such a distribution. We establish a strong duality result that generalizes the celebrated Kantorovich-Rubinstein duality. We also show that our formulation can be used to beat the curse of dimensionality, which is well known to affect the rates of statistical convergence of the empirical Wasserstein distance. In particular, examples of infinite-dimensional hypothesis classes are presented, informed by a complex correlation structure, for which it is shown that the empirical Wasserstein distance to such classes converges to zero at the standard parametric rate. Our formulation provides insights that help clarify why, despite the curse of dimensionality, the Wasserstein distance enjoys favorable empirical performance across a wide range of statistical applications.