When some parameters of a constrained optimization problem (COP) are uncertain, this gives rise to a predict-then-optimize (PtO) problem, comprising two stages: the prediction of the unknown parameters from contextual information and the subsequent optimization using those predicted parameters. Decision-focused learning (DFL) implements the first stage by training a machine learning (ML) model to optimize the quality of the decisions made using the predicted parameters. When the predicted parameters occur in the constraints, they can lead to infeasible solutions. Therefore, it is important to simultaneously manage both feasibility and decision quality. We develop a DFL framework for predicting constraint parameters in a generic COP. While prior works typically assume that the underlying optimization problem is a linear program (LP) or integer LP (ILP), our approach makes no such assumption. We derive two novel loss functions based on maximum likelihood estimation (MLE): the first one penalizes infeasibility (by penalizing predicted parameters that lead to infeasible solutions), while the second one penalizes suboptimal decisions (by penalizing predicted parameters that make the true optimal solution infeasible). We introduce a single tunable parameter to form a weighted average of the two losses, allowing decision-makers to balance suboptimality and feasibility. We experimentally demonstrate that adjusting this parameter provides decision-makers control over this trade-off. Moreover, across several COP instances, we show that adjusting the tunable parameter allows a decision-maker to prioritize either suboptimality or feasibility, outperforming the performance of existing baselines in either objective.

Contemporary theories model language processing as integrating both top-down expectations and bottom-up inputs. One major prediction of such models is that the quality of the bottom-up inputs modulates ease of processing -- noisy inputs should lead to difficult and effortful comprehension. We test this prediction in the domain of reading. First, we propose an information-theoretic operationalization for the "quality" of bottom-up information as the mutual information (MI) between visual information and word identity. We formalize this prediction in a mathematical model of reading as a Bayesian update. Second, we test our operationalization by comparing participants' reading times in conditions where words' information quality has been reduced, either by occluding their top or bottom half, with full words. We collect data in English and Chinese. We then use multimodal language models to estimate the mutual information between visual inputs and words. We use these data to estimate the specific effect of reduced information quality on reading times. Finally, we compare how information is distributed across visual forms. In English and Chinese, the upper half contains more information about word identity than the lower half. However, the asymmetry is more pronounced in English, a pattern which is reflected in the reading times.

Ordering-based approaches to causal discovery identify topological orders of causal graphs, providing scalable alternatives to combinatorial search methods. Under the Additive Noise Model (ANM) assumption, recent causal ordering methods based on score matching require an accurate estimation of the Hessian diagonal of the log-densities. In this paper, we aim to improve the approximation of the Hessian diagonal of the log-densities, thereby enhancing the performance of ordering-based causal discovery algorithms. Existing approaches that rely on Stein gradient estimators are computationally expensive and memory-intensive, while diffusion-model-based methods remain unstable due to the second-order derivatives of score models. To alleviate these problems, we propose Score-informed Neural Operator (SciNO), a probabilistic generative model in smooth function spaces designed to stably approximate the Hessian diagonal and to preserve structural information during the score modeling. Empirical results show that SciNO reduces order divergence by 42.7% on synthetic graphs and by 31.5% on real-world datasets on average compared to DiffAN, while maintaining memory efficiency and scalability. Furthermore, we propose a probabilistic control algorithm for causal reasoning with autoregressive models that integrates SciNO's probability estimates with autoregressive model priors, enabling reliable data-driven causal ordering informed by semantic information. Consequently, the proposed method enhances causal reasoning abilities of LLMs without additional fine-tuning or prompt engineering.

We study in-context learning problems where a Transformer is pretrained on tasks drawn from a mixture distribution $π=\sum_{α\in\mathcal{A}} λ_α π_α$, called the pretraining prior, in which each mixture component $π_α$ is a distribution on tasks of a specific difficulty level indexed by $α$. Our goal is to understand the performance of the pretrained Transformer when evaluated on a different test distribution $μ$, consisting of tasks of fixed difficulty $β\in\mathcal{A}$, and with potential distribution shift relative to $π_β$, subject to the chi-squared divergence $χ^2(μ,π_β)$ being at most $κ$. In particular, we consider nonparametric regression problems with random smoothness, and multi-index models with random smoothness as well as random effective dimension. We prove that a large Transformer pretrained on sufficient data achieves the optimal rate of convergence corresponding to the difficulty level $β$, uniformly over test distributions $μ$ in the chi-squared divergence ball. Thus, the pretrained Transformer is able to achieve faster rates of convergence on easier tasks and is robust to distribution shift at test time. Finally, we prove that even if an estimator had access to the test distribution $μ$, the convergence rate of its expected risk over $μ$ could not be faster than that of our pretrained Transformers, thereby providing a more appropriate optimality guarantee than minimax lower bounds.

Single-cell RNA sequencing (scRNA-seq) enables the study of cellular heterogeneity. Y et, clustering accuracy, and with it downstream analyses based on cell labels, remain challenging due to measurement noise and biological variability. In standard latent spaces (e.g., obtained through PCA), data from different cell types can be projected close together, making accurate clustering difficult. We introduce a latent plug-and-play diffusion framework that separates the observation and de-noising space. This separation is operationalized through a novel Gibbs sampling procedure: the learned diffusion prior is applied in a low-dimensional latent space to perform denoising, while to steer this process, noise is reintroduced into the original high-dimensional observation space. This unique "input-space steering" ensures the denoising trajectory remains faithful to the original data structure. Our approach offers three key advantages: (1) adaptive noise handling via a tunable balance between prior and observed data; (2) uncertainty quantification through principled uncertainty estimates for downstream analysis; and (3) generalizable denoising by leveraging clean reference data to denoise noisier datasets, and via averaging, improve quality beyond the training set. We evaluate robustness on both synthetic and real single-cell genomics data. Our method improves clustering accuracy on synthetic data across varied noise levels and dataset shifts. On real-world single-cell data, our method demonstrates improved biological coherence in the resulting cell clusters, with cluster boundaries that better align with known cell type markers and developmental trajectories. Single-cell RNA sequencing (scRNA-seq) has revolutionized biomedical research by enabling high-resolution profiling of cellular heterogeneity (Park et al., 2020; Miragaia et al., 2019), with large-scale initiatives like the Human Cell Atlas providing foundational references for cell type annotation (Regev et al., 2017; Lindeboom et al., 2021; Elmentaite et al., 2022; Stuart et al., 2019; Lopez et al., 2018).

The Hierarchical Reasoning Model (HRM) has impressive reasoning abilities given its small size, but has only been applied to supervised, static, fully-observable problems. One of HRM's strengths is its ability to adapt its computational effort to the difficulty of the problem. However, in its current form it cannot integrate and reuse computation from previous time-steps if the problem is dynamic, uncertain or partially observable, or be applied where the correct action is undefined, characteristics of many real-world problems. This paper presents HRM-Agent, a variant of HRM trained using only reinforcement learning. We show that HRM can learn to navigate to goals in dynamic and uncertain maze environments. Recent work suggests that HRM's reasoning abilities stem from its recurrent inference process. We explore the dynamics of the recurrent inference process and find evidence that it is successfully reusing computation from earlier environment time-steps.

Decomposing prediction uncertainty into its aleatoric (irreducible) and epistemic (reducible) components is critical for the development and deployment of machine learning systems. A popular, principled measure for epistemic uncertainty is the mutual information between the response variable and model parameters. However, evaluating this measure requires access to the posterior distribution of the model parameters, which is challenging to compute. In view of this, we introduce a frequentist measure of epistemic uncertainty based on the bootstrap. Our main theoretical contribution is a novel asymptotic expansion that reveals that our proposed (frequentist) measure and the (Bayesian) mutual information are asymptotically equivalent. This provides frequentist interpretations to mutual information and new computational strategies for approximating it. Moreover, we link our proposed approach to the widely-used heuristic approach of deep ensembles, giving added perspective on their practical success.

Causal discovery from observational data is a fundamental task in artificial intelligence, with far-reaching implications for decision-making, predictions, and interventions. Despite significant advances, existing methods can be broadly categorized as constraint-based or score-based approaches. Constraint-based methods offer rigorous causal discovery but are often hindered by small sample sizes, while score-based methods provide flexible optimization but typically forgo explicit conditional independence testing. This work explores a third avenue: developing differentiable $d$-separation scores, obtained through a percolation theory using soft logic. This enables the implementation of a new type of causal discovery method: gradient-based optimization of conditional independence constraints. Empirical evaluations demonstrate the robust performance of our approach in low-sample regimes, surpassing traditional constraint-based and score-based baselines on a real-world dataset. Code and data of the proposed method are publicly available at https://github$.$com/PurdueMINDS/DAGPA.

We study online learning in finite-horizon episodic Markov decision processes (MDPs) under the challenging aggregate bandit feedback model, where the learner observes only the cumulative loss incurred in each episode, rather than individual losses at each state-action pair. While prior work in this setting has focused exclusively on worst-case analysis, we initiate the study of best-of-both-worlds (BOBW) algorithms that achieve low regret in both stochastic and adversarial environments. We propose the first BOBW algorithms for episodic tabular MDPs with aggregate bandit feedback. In the case of known transitions, our algorithms achieve $O(\log T)$ regret in stochastic settings and ${O}(\sqrt{T})$ regret in adversarial ones. Importantly, we also establish matching lower bounds, showing the optimality of our algorithms in this setting. We further extend our approach to unknown-transition settings by incorporating confidence-based techniques. Our results rely on a combination of FTRL over occupancy measures, self-bounding techniques, and new loss estimators inspired by recent advances in online shortest path problems. Along the way, we also provide the first individual-gap-dependent lower bounds and demonstrate near-optimal BOBW algorithms for shortest path problems with bandit feedback.

Performative predictions are forecasts which influence the outcomes they aim to predict, undermining the existence of correct forecasts and standard methods of elicitation and estimation. We show that conditioning forecasts on covariates that separate them from the outcome renders the target distribution forecast-invariant, guaranteeing well-posedness of the forecasting problem. However, even under this condition, classical proper scoring rules fail to elicit correct forecasts. We prove a general impossibility result and identify two solutions: (i) in decision-theoretic settings, elicitation of correct and incentive-compatible forecasts is possible if forecasts are separating; (ii) scoring with unbiased estimates of the divergence between the forecast and the induced distribution of the target variable yields correct forecasts. Applying these insights to parameter estimation, conditional forecasts and proper scoring rules enable performatively stable estimation of performatively correct parameters, resolving the issues raised by Perdomo et al. (2020). Our results expose fundamental limits of classical forecast evaluation and offer new tools for reliable and accurate forecasting in performative settings.