Optimal transport is widely used to learn distributions, enforce distributional constraints, and model uncertainty. In applications, transport losses are often computed from samples through tractable representations, such as one-dimensional sorting formulas or sliced Wasserstein costs, making them practical components in training pipelines. We study parameterized objectives defined by sampled transport costs and prove graphical convergence of their subdifferentials to the subdifferential of the population objective. In particular, this ensures that standard subgradient methods consistently approach stationary points of the population-level problem. We illustrate the results in several settings, including risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems. Our analysis highlights that smooth parameterizations provide a favorable interface between statistical consistency and optimization. By contrast, transport objectives with nonsmooth costs and models may exhibit unstable derivatives in the large-sample limit.

Classical mixture models (MMs) are widely used tractable proposals for approximate inference settings such as variational inference (VI) and importance sampling (IS). Recently, mixture models with negative coefficients, called subtractive mixture models (SMMs), have been proposed as a potentially more expressive alternative. However, how to effectively use SMMs for VI and IS is still an open question as they do not provide latent variable semantics and therefore cannot use sampling schemes for classical MMs. In this work, we study how to circumvent this issue by designing several expectation estimators for IS and learning schemes for VI with SMMs, and we empirically evaluate them for distribution approximation. Finally, we discuss the additional challenges in estimation stability and learning efficiency that they carry and propose ways to overcome them. Code is available at: https://github.com/april-tools/delta-vi.

We study a generalization of boosting to the multiclass setting. We introduce a weak learning condition for multiclass classification that captures the original notion ofweak learnability asbeing "slightly better than random guessing".

We study offline reinforcement learning under $Q^\star$-approximation and partial coverage, a setting that motivates practical algorithms such as Conservative $Q$-Learning (CQL; Kumar et al., 2020) but has received limited theoretical attention. Our work is inspired by the following open question: "Are $Q^\star$-realizability and Bellman completeness sufficient for sample-efficient offline RL under partial coverage?" We answer in the negative by establishing an information-theoretic lower bound. Going substantially beyond this, we introduce a general framework that characterizes the intrinsic complexity of a given $Q^\star$ function class, inspired by model-free decision-estimation coefficients (DEC) for online RL (Foster et al., 2023b; Liu et al., 2025b). This complexity recovers and improves the quantities underlying the guarantees of Chen and Jiang (2022) and Uehara et al. (2023), and extends to broader settings. Our decision-estimation decomposition can be combined with a wide range of $Q^\star$ estimation procedures, modularizing and generalizing existing approaches. Beyond the general framework, we make further contributions: By developing a novel second-order performance difference lemma, we obtain the first $ε^{-2}$ sample complexity under partial coverage for soft $Q$-learning, improving the $ε^{-4}$ bound of Uehara et al. (2023). We remove Chen and Jiang's (2022) need for additional online interaction when the value gap of $Q^\star$ is unknown. We also give the first characterization of offline learnability for general low-Bellman-rank MDPs without Bellman completeness (Jiang et al., 2017; Du et al., 2021; Jin et al., 2021), a canonical setting in online RL that remains unexplored in offline RL except for special cases. Finally, we provide the first analysis for CQL under $Q^\star$-realizability and Bellman completeness beyond the tabular case.