Papers

Choosing relevant predictors is central to the analysis of biomedical time-to-event data. Classical frequentist inference, however, presumes that the set of covariates is fixed in advance and does not account for data-driven variable selection. As a consequence, naive post-selection inference may be biased and misleading. In right-censored survival settings, these issues may be further exacerbated by the additional uncertainty induced by censoring. We investigate several inference procedures applied after variable selection for the coefficients of the Lasso and its extension, the adaptive Lasso, in the context of the Cox model. The methods considered include sample splitting, exact post-selection inference, and the debiased Lasso. Their performance is examined in a neutral simulation study reflecting realistic covariate structures and censoring rates commonly encountered in biomedical applications. To complement the simulation results, we illustrate the practical behavior of these procedures in an applied example using a publicly available survival dataset.

When multi-armed bandit (MAB) algorithms allocate pulls among competing arms, the resulting allocation can exhibit huge variation. This is particularly harmful in modern applications such as learning-enhanced platform operations and post-bandit statistical inference. Thus motivated, we introduce a new performance metric of MAB algorithms termed allocation variability, which is the largest (over arms) standard deviation of an arm's number of pulls. We establish a fundamental trade-off between allocation variability and regret, the canonical performance metric of reward maximization. In particular, for any algorithm, the worst-case regret $R_T$ and worst-case allocation variability $S_T$ must satisfy $R_T \cdot S_T=\Omega(T^{\frac{3}{2}})$ as $T\rightarrow\infty$, as long as $R_T=o(T)$. This indicates that any minimax regret-optimal algorithm must incur worst-case allocation variability $\Theta(T)$, the largest possible scale; while any algorithm with sublinear worst-case regret must necessarily incur ${S}_T= \omega(\sqrt{T})$. We further show that this lower bound is essentially tight, and that any point on the Pareto frontier $R_T \cdot S_T=\tilde{\Theta}(T^{3/2})$ can be achieved by a simple tunable algorithm UCB-f, a generalization of the classic UCB1. Finally, we discuss implications for platform operations and for statistical inference, when bandit algorithms are used. As a byproduct of our result, we resolve an open question of Praharaj and Khamaru (2025).

Multi-Regional Clinical Trials (MRCTs) play a central role in the development of new therapies by enabling the simultaneous evaluation of drug efficacy and safety across diverse global populations. Assessing the consistency of treatment effects across regions is a fundamental aspect of MRCTs. Existing methods typically focus on region-specific marginal treatment effects. However, when treatment effect heterogeneity arises due to effect-modifying baseline covariates, distributional differences in these covariates can lead to erroneous conclusions. In this paper, we explicitly account for this phenomenon in the consistency assessment by considering the conditional average treatment effect. We propose a two-step assessment strategy that complements existing methods and mitigates the impact of treatment effect heterogeneity. Results from numerical studies demonstrate the effectiveness of the proposed approach.

Stochastic processes are a flexible and widely used family of models for statistical modeling. While stochastic processes offer attractive properties such as inclusion of uncertainty properties, their inference is typically intractable, with the notable exception of Gaussian processes. Inference of models with non-Gaussian errors typically involves estimation of a high-dimensional latent variable. We propose two methods that use iterated posterior linearization followed by Hamiltonian Monte Carlo to sample the posterior distributions of such latent models with a particular focus on log-Gaussian gamma processes. The proposed methods are validated with two synthetic datasets generated from the log-Gaussian gamma process and a multiscale biocomposite stiffness model. In addition, we apply the methodology to an experimental Raman spectrum of argentopyrite.

Decision tree ensembles are widely used in critical domains, making robustness and sensitivity analysis essential to their trustworthiness. We study the feature sensitivity problem, which asks whether an ensemble is sensitive to a specified subset of features – such as protected attributes – whose manipulation can alter model predictions. Existing approaches often yield examples of sensitivity that lie far from the training distribution, limiting their interpretability and practical value. We propose a data-aware sensitivity framework that constrains the sensitive examples to remain close to the dataset, thereby producing realistic and interpretable evidence of model weaknesses. To this end, we develop novel techniques for data-aware search using a combination of mixed-integer linear programming (MILP) and satisfiability modulo theories (SMT) encodings. Our contributions are fourfold. First, we strengthen the NP-hardness result for sensitivity verification, showing it holds even for trees of depth 1. Second, we develop MILP-optimizations that significantly speed up sensitivity verification for single ensembles and for the first time can also handle multiclass tree ensembles. Third, we introduce a data-aware framework generating realistic examples close to the training distribution. Finally, we conduct an extensive experimental evaluation on large tree ensembles, demonstrating scalability to ensembles with up to 800 trees of depth 8, achieving substantial improvements over the state of the art. This framework provides a practical foundation for analyzing the reliability and fairness of tree-based models in high-stakes applications.

In adversarial multi-armed bandits, two performance measures are commonly used: static regret, which compares the learner to the best fixed arm, and dynamic regret, which compares it to the best sequence of arms. While optimal algorithms are known for each measure individually, there is no known algorithm achieving optimal bounds for both simultaneously. Marinov and Zimmert [2021] first showed that such simultaneous optimality is impossible against an adaptive adversary. Our work takes a first step to demonstrate its possibility against an oblivious adversary when losses are deterministic. First, we extend the impossibility result of Marinov and Zimmert [2021] to the case of deterministic losses. Then, we present an algorithm achieving optimal static and dynamic regret simultaneously against an oblivious adversary. Together, they reveal a fundamental separation between adaptive and oblivious adversaries when multiple regret benchmarks are considered simultaneously. It also provides new insight into the long open problem of simultaneously achieving optimal regret against switching benchmarks of different numbers of switches. Our algorithm uses negative static regret to compensate for the exploration overhead incurred when controlling dynamic regret, and leverages Blackwell approachability to jointly control both regrets. This yields a new model selection procedure for bandits that may be of independent interest.

In multi-armed trials, adaptive designs are a popular way to increase estimation efficiency or identify optimal treatments. Several recent papers have proposed adaptive variants of the classical Neyman allocation to assign treatments in sequential trials, with the goal of minimizing the error of a Horvitz-Thompson-style estimator. However, this approach may be inefficient, because it fails to borrow information across the treatment arms. In this paper, we consider adaptivity in a sequential trial with K active treatments and a control, and suggest the use of Stein-like shrinkage estimators to obtain the final causal estimates. These estimators share information across arms, yielding provable reductions in expected squared error loss relative to estimating each causal effect in isolation. Moreover, for each of our candidate shrinkers, the risk is the expectation of ratios of Gaussian quadratic forms, and can be computed efficiently via numerical integration. Hence, we suggest a simple algorithm for sequential adaptivity: assign treatments to each new arrival by choosing the arm that will minimize the estimated shrinker loss. Through simulations, we demonstrate that this approach can yield meaningful reductions in estimation error, especially in the low signal-to-noise regime. We also characterize how our adaptive algorithm assigns treatments differently than would a sequential Neyman allocation, and suggest a method for constructing shorter confidence intervals at the trial's conclusion.

In 1964, Aharonov, Bergmann, and Lebowitz introduced their well-known ABL rule with the intention of providing a time-symmetric formalism for computing novel kinds of conditional probabilities in quantum theory. Later papers attached additional significance to the ABL rule, including assertions that it supported violations of the uncertainty principle. The present work challenges these claims, as well as subsequent attempts to salvage the original interpretation of the ABL rule. Taking a broader view, this paper identifies a subtle category error at the heart of the ABL rule that consists of confusing observables that belong to a single system with emergent observables that arise only for physical ensembles. Along the way, this paper points out other problems and fallacious reasoning in the research literature surrounding the ABL rule, including the misuse of post-selection, a reliance on pattern matching to classical formulas, and a posture of measurementism that takes experimental data as providing answers to interpretational questions.

The survey experiment is widely used in economics and social sciences to evaluate the effects of treatments or programs. In a standard population-based survey experiment, the experimenter randomly draws experimental units from a target population of interest and then randomly assigns the sampled units to treatment or control conditions to explore the treatment effect of an intervention. Simple random sampling and treatment assignment can balance covariates on average. However, covariate imbalance often exists in finite samples. To address the imbalance issue, we study a stratified approach to balance covariates in a survey experiment. A stratified rejective sampling and rerandomization design is further proposed to enhance the covariate balance. We develop a design-based asymptotic theory for the widely used stratified difference-in-means estimator of the average treatment effect under the proposed design. In particular, we show that it is consistent and asymptotically a convolution of a normal distribution and two truncated normal distributions. This limiting distribution is more concentrated at the true average treatment effect than that under the existing experimental designs. Moreover, we propose a covariate adjustment method in the analysis stage, which can further improve the estimation efficiency. Numerical studies demonstrate the validity and improved efficiency of the proposed method.

The dynamics of gradient-based training in neural networks often exhibit nontrivial structures; hence, understanding them remains a central challenge in theoretical machine learning. In particular, a concept of feature unlearning, in which a neural network progressively loses previously learned features over long training, has gained attention. In this study, we consider the infinite-width limit of a two-layer neural network updated with a large-batch stochastic gradient, then derive differential equations with different time scales, revealing the mechanism and conditions for feature unlearning to occur. Specifically, we utilize the fast-slow dynamics: while an alignment of first-layer weights develops rapidly, the second-layer weights develop slowly. The direction of a flow on a critical manifold, determined by the slow dynamics, decides whether feature unlearning occurs. We give numerical validation of the result, and derive theoretical grounding and scaling laws of the feature unlearning. Our results yield the following insights: (i) the strength of the primary nonlinear term in data induces the feature unlearning, and (ii) an initial scale of the second-layer weights mitigates the feature unlearning. Technically, our analysis utilizes Tensor Programs and the singular perturbation theory.

We give new differentially private algorithms for the classic problems of learning decision lists and large-margin halfspaces in the PAC and online models. In the PAC model, we give a computationally efficient algorithm for learning decision lists with minimal sample overhead over the best non-private algorithms. In the online model, we give a private analog of the influential Winnow algorithm for learning halfspaces with mistake bound polylogarithmic in the dimension and inverse polynomial in the margin. As an application, we describe how to privately learn decision lists in the online model, qualitatively matching state-of-the art non-private guarantees.

Time series forecasting presents significant challenges in real-world applications across various domains. Building upon the decomposition of the time series, we enhance the architecture of machine learning models for better multivariate time series forecasting. To achieve this, we focus on the trend and seasonal components individually and investigate solutions to predict them with less errors. Recognizing that reversible instance normalization is effective only for the trend component, we take a different approach with the seasonal component by directly applying backbone models without any normalization or scaling procedures. Through these strategies, we successfully reduce error values of the existing state-of-the-art models and finally introduce dual-MLP models as more computationally efficient solutions. Furthermore, our approach consistently yields positive results with around 10% MSE average reduction across four state-of-the-art baselines on the benchmark datasets. We also evaluate our approach on a hydrological dataset extracted from the United States Geological Survey (USGS) river stations, where our models achieve significant improvements while maintaining linear time complexity, demonstrating real-world effectiveness. The source code is available at https://github.com/Sanjeev97/Time-Series-Decomposition

Proposition. Let $f$ be a predictor trained on a distribution $P$ and evaluated on a shifted distribution $Q$. Under verifiable regularity and complexity constraints, the excess risk under shift admits an explicit upper bound determined by a computable shift metric and model parameters. We develop a unified framework in which (i) risk under distribution shift is certified by explicit inequalities, (ii) verification of learned models is sound for nontrivial sizes, and (iii) interpretability is enforced through identifiability conditions rather than post hoc explanations. All claims are stated with explicit assumptions. Failure modes are isolated. Non-certifiable regimes are characterized.

Large language models (LLMs) are increasingly used in research as both tools and objects of study. Much of this work assumes that LLM performance under fixed conditions (identical model snapshot, hyperparameters, and prompt) is time-invariant, meaning that average output quality remains stable over time; otherwise, reliability and reproducibility would be compromised. To test the assumption of time invariance, we conducted a longitudinal study of GPT-4o's average performance under fixed conditions. The LLM was queried to solve the same physics task ten times every three hours over approximately three months. Spectral (Fourier) analysis of the resulting time series revealed substantial periodic variability, accounting for about 20% of total variance. The observed periodic patterns are consistent with interacting daily and weekly rhythms. These findings challenge the assumption of time invariance and carry important implications for research involving LLMs.

Background: High-dimensional genomic data exhibit strong group correlation structures that challenge conventional feature selection methods, which often assume feature independence or rely on pre-defined pathways and are sensitive to outliers and model misspecification. Methods: We propose the Dorfman screening framework, a multi-stage procedure that forms data-driven variable groups via hierarchical clustering, performs group and within-group hypothesis testing, and refines selection using elastic net or adaptive elastic net. Robust variants incorporate OGK-based covariance estimation, rank-based correlation, and Huber-weighted regression to handle contaminated and non-normal data. Results: In simulations, Dorfman-Sparse-Adaptive-EN performed best under normal conditions, while Robust-OGK-Dorfman-Adaptive-EN showed clear advantages under data contamination, outperforming classical Dorfman and competing methods. Applied to NSCLC gene expression data for trametinib response, robust Dorfman methods achieved the lowest prediction errors and enriched recovery of clinically relevant genes. Conclusions: The Dorfman framework provides an efficient and robust approach to genomic feature selection. Robust-OGK-Dorfman-Adaptive-EN offers strong performance under both ideal and contaminated conditions and scales to ultra-high-dimensional settings, making it well suited for modern genomic biomarker discovery.

Existing online change-point detection (CPD) methods rely on fixed-dimensional Euclidean summaries, implicitly assuming that distributional changes are well captured by moment-based or feature-based representations. They can obscure important changes in distributional shape or geometry. We propose an intrinsic distribution-valued CPD framework that treats streaming batch data as a stochastic process on the 2-Wasserstein space. Our method detects changes in the law of this process by mapping each empirical distribution to a tangent space relative to a pre-change Fréchet barycenter, yielding a reference-centered local linearization of 2-Wasserstein space. This representation enables sequential detectors by adapting classical multivariate monitoring statistics to tangent fields. We provide theoretical guarantees and demonstrate, via synthetic and real-world experiments, that our approach detects complex distributional shifts with reduced detection delay at matched $\mathrm{ARL}_0$ compared with moments-based and model-free baselines. The code is available at https://github.com/yyzeng43/IDD-icml .

In the study of heavy tail data, several models have been introduced. If the interest is in the tail of the distribution, block maxima or excess over thresholds are the typical approaches, wasting relevant information in the bulk of the data. To avoid this, mixture models for the body (below the threshold) and the tail (above the threshold) are proposed. In this paper, we exploit the richness of nonparametric mixture models to model heavy tail data. We specifically consider mixtures of shifted gamma-gamma distributions with four parameters and a Poisson-Dirichlet process as a mixing distribution. One of these parameters is associated with the tail. By studying the posterior distribution of the tail parameter, we are able to assess the tail heaviness for each component. We develop an efficient MCMC method with adapting Metropolis-Hastings steps to obtain posterior inference and illustrate with simulated and real datasets.

Adaptability has been regarded as a central feature in the foundation models, enabling them to effectively acclimate to unseen downstream tasks. Parameter-efficient fine-tuning methods such as celebrated LoRA facilitate efficient adaptation of large foundation models using labeled, high-quality and generally scarce task data. To mitigate data scarcity in fine-tuning of foundation models, we propose to leverage task similarity across multiple downstream users. Intuitively, users with similar tasks must be able to assist each other in boosting the effective fine-tuning data size. We propose Collaborative Low-Rank Adaptation, or CoLoRA, which exploits task similarity to collaboratively and efficiently fine-tune personalized foundation models. The main idea in CoLoRA is to train one shared adapter capturing underlying task similarities across all tasks, and personalized adapters tailored to user-specific tasks. We theoretically study CoLoRA on heterogeneous linear regression and provide provable guarantees for ground truth recovery. We also conduct several natural language experiments with varying task similarity, which further demonstrate that when trained together with similar tasks, individual performances are significantly boosted.

We study online learning in two-player uninformed Markov games, where the opponent's actions and policies are unobserved. In this setting, Tian et al. (2021) show that achieving no-external-regret is impossible without incurring an exponential dependence on the episode length $H$. They then turn to the weaker notion of Nash-value regret and propose a V-learning algorithm with regret $O(K^{2/3})$ after $K$ episodes. However, their algorithm and guarantee do not adapt to the difficulty of the problem: even in the case where the opponent follows a fixed policy and thus $O(\sqrt{K})$ external regret is well-known to be achievable, their result is still the worse rate $O(K^{2/3})$ on a weaker metric. In this work, we fully address both limitations. First, we introduce empirical Nash-value regret, a new regret notion that is strictly stronger than Nash-value regret and naturally reduces to external regret when the opponent follows a fixed policy. Moreover, under this new metric, we propose a parameter-free algorithm that achieves an $O(\min \{\sqrt{K} + (CK)^{1/3},\sqrt{LK}\})$ regret bound, where $C$ quantifies the variance of the opponent's policies and $L$ denotes the number of policy switches (both at most $O(K)$). Therefore, our results not only recover the two extremes – $O(\sqrt{K})$ external regret when the opponent is fixed and $O(K^{2/3})$ Nash-value regret in the worst case – but also smoothly interpolate between these extremes by automatically adapting to the opponent's non-stationarity. We achieve so by first providing a new analysis of the epoch-based V-learning algorithm by Mao et al. (2022), establishing an $O(\eta C + \sqrt{K/\eta})$ regret bound, where $\eta$ is the epoch incremental factor. Next, we show how to adaptively restart this algorithm with an appropriate $\eta$ in response to the potential non-stationarity of the opponent, eventually achieving our final results.

Route-level travel time reliability requires characterizing the distribution of total travel time across correlated segments – a problem where existing methods either assume independence (fast but miscalibrated) or model dependence via copulas and simulation (accurate but expensive). We propose a conjugate Bayesian dynamic Gamma model with a common random environment that resolves this trade-off. Each segment's travel time follows a Gamma distribution conditional on a shared latent environment process that evolves as a Markov chain, inducing cross-segment dependence while preserving conditional independence. A moment-matching approximation yields a closed-form $F$-distribution for route travel time, from which the Planning Time Index, Buffer Index, and on-time probability are computed instantly – at the same $O(1)$ cost as independence-based methods. The conjugate structure ensures that Bayesian posterior updates and the full predictive distribution are available in closed form as new sensor data arrives. Applied to 16 sensors spanning 8.26 miles on I-55 in Chicago, the model achieves 95.4% coverage of nominal 90% predictive intervals versus 34–37% for independence-based convolution, at identical computational cost.

We introduce an R package for Bayesian modeling and uncertainty quantification for problems involving count ratios. The modeling relies on the assumption that the quantity of interest is the ratio of Poisson means rather than the ratio of counts. We provide multiple different options for retrieval of this quantity for problems with and without spatial information included. Some added capability for uncertainty quantification for problems of the form $Z=(mT+z_0)^{p}$, where $Z$ is the intensity ratio and $T$ the quantity of interest, is included.

Standard attention stores keys/values losslessly but reads them via a per-head convex average, blocking channel-wise selection. We propose the Free Energy Mixer (FEM): a free-energy (log-sum-exp) read that applies a value-driven, per-channel log-linear tilt to a fast prior (e.g., from queries/keys in standard attention) over indices. Unlike methods that attempt to improve and enrich the $(q,k)$ scoring distribution, FEM treats it as a prior and yields a value-aware posterior read at unchanged complexity, smoothly moving from averaging to per-channel selection as the learnable inverse temperature increases, while still preserving parallelism and the original asymptotic complexity ($O(T^2)$ for softmax; $O(T)$ for linearizable variants). We instantiate a two-level gated FEM that is plug-and-play with standard and linear attention, linear RNNs and SSMs. It consistently outperforms strong baselines on NLP, vision, and time-series at matched parameter budgets.

We propose a framework for computing, optimizing and integrating with respect to a smooth marginal likelihood in statistical models that involve high-dimensional parameters/latent variables and continuous low-dimensional hyperparameters. The method requires samples from the posterior distribution of the parameters for different values of the hyperparameters on a simulation grid and returns inference on the marginal likelihood defined everywhere on its domain, and on its functionals. We show how the method relates to many of the methods that have been used in this context, including sequential Monte Carlo, Gibbs sampling, Monte Carlo maximum likelihood, and umbrella sampling. We establish the consistency of the proposed estimators as the sampling effort increases, both when the simulation grid is kept fixed and when it becomes dense in the domain. We showcase the approach on Gaussian process regression and classification and crossed effect models.

Bayesian optimization (BO) is a popular technique for sample-efficient optimization of black-box functions. In many applications, the parameters being tuned come with a carefully engineered default configuration, and practitioners only want to deviate from this default when necessary. Standard BO, however, does not aim to minimize deviation from the default and, in practice, often pushes weakly relevant parameters to the boundary of the search space. This makes it difficult to distinguish between important and spurious changes and increases the burden of vetting recommendations when the optimization objective omits relevant operational considerations. We introduce BONSAI, a default-aware BO policy that prunes low-impact deviations from a default configuration while explicitly controlling the loss in acquisition value. BONSAI is compatible with a variety of acquisition functions, including expected improvement and upper confidence bound (GP-UCB). We theoretically bound the regret incurred by BONSAI, showing that, under appropriate conditions, it retains the no-regret property of vanilla GP-UCB and removes irrelevant changes. Across many real-world applications, we empirically find that BONSAI substantially reduces the number of non-default parameters in recommended configurations while maintaining competitive optimization performance with little effect on wall time. Its candidate-generation cost averages only $1.5\times$ that of standard BO, compared with $7$-$34\times$ for prior sparse-BO methods.

Computation methods for solving entropy-regularized reward optimization – a class of problems widely used for fine-tuning generative models – have advanced rapidly. Among those, Adjoint Matching (AM, Domingo-Enrich et al., 2025) has proven highly effective in continuous state spaces with differentiable rewards. Transferring these practical successes to discrete generative modeling, however, remains particularly challenging and largely unexplored, mainly due to the drastic shift in generative model classes to discrete state spaces, which are nowhere differentiable. In this work, we propose Discrete Adjoint Matching (DAM) – a discrete variant of AM for fine-tuning discrete generative models characterized by Continuous-Time Markov Chains, such as diffusion-based large language models. The core of DAM is the introduction of discrete adjoint-an estimator of the optimal solution to the original problem but formulated on discrete domains-from which standard matching frameworks can be applied. This is derived via a purely statistical standpoint, in contrast to the control-theoretic viewpoint in AM, thereby opening up new algorithmic opportunities for general adjoint-based estimators. We showcase DAM's effectiveness on synthetic and mathematical reasoning tasks.

Zero-shot diffusion posterior sampling offers a flexible framework for inverse problems by accommodating arbitrary degradation operators at test time, but incurs high computational cost due to repeated likelihood-guided updates. In contrast, previous amortized diffusion approaches enable fast inference by replacing likelihood-based sampling with implicit inference models, but at the expense of robustness to unseen degradations. We introduce an amortization strategy for diffusion posterior sampling that preserves explicit likelihood guidance by amortizing the inner optimization problems arising in variational diffusion posterior sampling. This accelerates inference for in-distribution degradations while maintaining robustness to previously unseen operators, thereby improving the trade-off between efficiency and flexibility in diffusion-based inverse problems.

Modern Bayesian inference involves a mixture of computational methods for estimating, validating, and drawing conclusions from probabilistic models as part of principled workflows. An overarching motif of many Bayesian methods is that they are relatively slow, which often becomes prohibitive when fitting complex models to large data sets. Amortized Bayesian inference (ABI) offers a path to solving the computational challenges of Bayes. ABI trains neural networks on model simulations, rewarding users with rapid inference of any model-implied quantity, such as point estimates, likelihoods, or full posterior distributions. In this work, we present the Python library BayesFlow, Version 2.0, for general-purpose ABI. Along with direct posterior, likelihood, and ratio estimation, the software includes support for multiple popular deep learning backends, a rich collection of generative networks for sampling and density estimation, complete customization and high-level interfaces, as well as new capabilities for hyperparameter optimization, design optimization, and hierarchical modeling. Using a case study on dynamical system parameter estimation, combined with comparisons to similar software, we show that our streamlined, user-friendly workflow has strong potential to support broad adoption.

We investigate the existence and severity of local modes in posterior distributions from Bayesian analyses. These are known to occur in posterior tails resulting from heavy-tailed error models such as those used in robust regression. To understand this phenomenon clearly, we consider in detail location models with Student-$t$ errors in dimension $d$ with sample size $n$. For sufficiently heavy-tailed data-generating distributions, extreme observations become increasingly isolated as $n \to \infty$. We show that each such observation induces a unique local posterior mode with probability tending to $1$. We refer to such a local mode as a micromode. These micromodes are typically small in height but their domains of attraction are large and grow polynomially with $n$. We then connect this posterior geometry to computation. We establish an Arrhenius law for the time taken by one-dimensional piecewise deterministic Monte Carlo algorithms to exit these micromodes. Our analysis identifies a phase transition where a misspecified and overly underdispersed model causes exit times to increase sharply, leading to a pronounced deterioration in sampling performance.

We study off-policy reinforcement learning for controlling continuous-time Markov diffusion processes with discrete-time observations and actions. We consider model-free algorithms with function approximation that learn value and advantage functions directly from data, without unrealistic structural assumptions on the dynamics. Leveraging the ellipticity of the diffusions, we establish a new class of Hilbert-space positive definiteness and boundedness properties for the Bellman operators. Based on these properties, we propose the Sobolev-prox fitted $q$-learning algorithm, which learns value and advantage functions by iteratively solving least-squares regression problems. We derive oracle inequalities for the estimation error, governed by (i) the best approximation error of the function classes, (ii) their localized complexity, (iii) exponentially decaying optimization error, and (iv) numerical discretization error. These results identify ellipticity as a key structural property that renders reinforcement learning with function approximation for Markov diffusions no harder than supervised learning.

In this paper, we investigate several subsets of $n$-copulas and $n$-quasi-copulas from the perspective of convex-lineability and the recently introduced concept of convex-spaceability. Our purpose is to determine when such families contain extremely large algebraic structures, namely linearly independent sets of cardinality of the continuum whose convex hull, and in some cases a closed convex linearly independent subset, remain entirely inside the class under study. These include the families of asymmetric copulas, copulas with maximal asymmetric measure, and proper $n$-quasi-copulas, among others. In contrast, for several other natural classes of copulas we show that (maximal) convex lineability holds while convex spaceability remains an open problem.