Evaluating machine learning models requires evaluation data.

We study the problem of efficiently estimating the mean of a $k$-class random variable, $Y$, using a limited number of labels, $N$, in settings where the analyst has access to auxiliary information (i.e.: covariates) $X$ that may be informative about $Y$. We propose an active learning algorithm ("PartiBandits") to estimate $\mathbb{E}[Y]$. The algorithm yields an estimate, $\widehatμ_{\text{PB}}$, such that $\left( \widehatμ_{\text{PB}} - \mathbb{E}[Y]\right)^2$ is $\tilde{\mathcal{O}}\left( \frac{ν+ \exp(c \cdot (-N/\log(N))) }{N} \right)$, where $c > 0$ is a constant and $ν$ is the risk of the Bayes-optimal classifier. PartiBandits is essentially a two-stage algorithm. In the first stage, it learns a partition of the unlabeled data that shrinks the average conditional variance of $Y$. In the second stage it uses a UCB-style subroutine ("WarmStart-UCB") to request labels from each stratum round-by-round. Both the main algorithm's and the subroutine's convergence rates are minimax optimal in classical settings. PartiBandits bridges the UCB and disagreement-based approaches to active learning despite these two approaches being designed to tackle very different tasks. We illustrate our methods through simulation using nationwide electronic health records. Our methods can be implemented using the PartiBandits package in R.

Axioms are a feature of the Planning Domain Definition Language PDDL that can be considered as a generalization of database query languages such as Datalog. The PDDL standard restricts negative occurrences of predicates in axiom bodies to predicates that are directly set by actions and not derived by axioms. In the literature, authors often deviate from this limitation and only require that the set of axioms is stratifiable. Both variants can express exactly the same queries as least fixed-point logic, indicating that negative occurrences of derived predicates can be eliminated. We present the corresponding transformation.

Evaluating machine learning models requires evaluation data.

Large language model (LLM) agents increasingly rely on external tools such as search engines to solve complex, multi-step problems, and reinforcement learning (RL) has become a key paradigm for training them. However, the trajectories of search agents are structurally heterogeneous, where variations in the number, placement, and outcomes of search calls lead to fundamentally different answer directions and reward distributions. Standard policy gradient methods, which use a single global baseline, suffer from what we identify and formalize as cross-stratum bias-an "apples-to-oranges" comparison of heterogeneous trajectories. This cross-stratum bias distorts credit assignment and hinders exploration of complex, multi-step search strategies. To address this, we propose Stratified GRPO, whose central component, Stratified Advantage Normalization (SAN), partitions trajectories into homogeneous strata based on their structural properties and computes advantages locally within each stratum. This ensures that trajectories are evaluated only against their true peers. Our analysis proves that SAN eliminates cross-stratum bias, yields conditionally unbiased unit-variance estimates inside each stratum, and retains the global unbiasedness and unit-variance properties enjoyed by standard normalization, resulting in a more pure and scale-stable learning signal. To improve practical stability under finite-sample regimes, we further linearly blend SAN with the global estimator. Extensive experiments on diverse single-hop and multi-hop question-answering benchmarks demonstrate that Stratified GRPO consistently and substantially outperforms GRPO by up to 11.3 points, achieving higher training rewards, greater training stability, and more effective search policies. These results establish stratification as a principled remedy for structural heterogeneity in RL for LLM search agents.

In applications across agriculture, ecology, and human development, machine learning with satellite imagery (SatML) is limited by the sparsity of labeled training data. While satellite data cover the globe, labeled training datasets for SatML are often small, spatially clustered, and collected for other purposes (e.g., administrative surveys or field measurements). Despite the pervasiveness of this issue in practice, past SatML research has largely focused on new model architectures and training algorithms to handle scarce training data, rather than modeling data conditions directly. This leaves scientists and policymakers who wish to use SatML for large-scale monitoring uncertain about whether and how to collect additional data to maximize performance. Here, we present the first problem formulation for the optimization of spatial training data in the presence of heterogeneous data collection costs and realistic budget constraints, as well as novel methods for addressing this problem. In experiments simulating different problem settings across three continents and four tasks, our strategies reveal substantial gains from sample optimization. Further experiments delineate settings for which optimized sampling is particularly effective. The problem formulation and methods we introduce are designed to generalize across application domains for SatML; we put special emphasis on a specific problem setting where our coauthors can immediately use our findings to augment clustered agricultural surveys for SatML monitoring in Togo.

We consider the problem of propagating the uncertainty from a possibly large number of random inputs through a computationally expensive model. Stratified sampling is a well-known variance reduction strategy, but its application, thus far, has focused on models with a limited number of inputs due to the challenges of creating uniform partitions in high dimensions. To overcome these challenges, we perform stratification with respect to the uniform distribution defined over the unit interval, and then derive the corresponding strata in the original space using nonlinear dimensionality reduction. We show that our approach is effective in high dimensions and can be used to further reduce the variance of multifidelity Monte Carlo estimators.

However, real-world data often exhibit complex local structures that can be challenging for single-model approaches with a smooth global manifold in the embedding space to unravel. In this work, we conjecture that in the latent space of these large language models, the embeddings live in a local manifold structure with different dimensions depending on the perplexities and domains of the input data, commonly referred to as a Stratified Manifold structure, which in combination form a structured space known as a Stratified Space. To investigate the validity of this structural claim, we propose an analysis framework based on a Mixture-of-Experts (MoE) model where each expert is implemented with a simple dictionary learning algorithm at varying sparsity levels. By incorporating an attention-based soft-gating network, we verify that our model learns specialized sub-manifolds for an ensemble of input data sources, reflecting the semantic stratification in LLM embedding space. We further analyze the intrinsic dimensions of these stratified sub-manifolds and present extensive statistics on expert assignments, gating entropy, and inter-expert distances. Our experimental results demonstrate that our method not only validates the claim of a stratified manifold structure in the LLM embedding space, but also provides interpretable clusters that align with the intrinsic semantic variations of the input data.

Effective fall risk assessment is critical for post-stroke patients. The present study proposes a novel, data-informed fall risk assessment method based on the instrumented Timed Up and Go (ITUG) test data, bringing in many mobility measures that traditional clinical scales fail to capture. IFRA, which stands for Instrumented Fall Risk Assessment, has been developed using a two-step process: first, features with the highest predictive power among those collected in a ITUG test have been identified using machine learning techniques; then, a strategy is proposed to stratify patients into low, medium, or high-risk strata. The dataset used in our analysis consists of 142 participants, out of which 93 were used for training (15 synthetically generated), 17 for validation and 32 to test the resulting IFRA scale (22 non-fallers and 10 fallers). Features considered in the IFRA scale include gait speed, vertical acceleration during sit-to-walk transition, and turning angular velocity, which align well with established literature on the risk of fall in neurological patients. In a comparison with traditional clinical scales such as the traditional Timed Up & Go and the Mini-BESTest, IFRA demonstrates competitive performance, being the only scale to correctly assign more than half of the fallers to the high-risk stratum (Fischer's Exact test p = 0.004). Despite the dataset's limited size, this is the first proof-of-concept study to pave the way for future evidence regarding the use of IFRA tool for continuous patient monitoring and fall prevention both in clinical stroke rehabilitation and at home post-discharge. Keywords: Fall Risk, Stroke Rehabilitation, Machine Learning, Mobility Impairment, Instrumented Timed Up and Go test, Inertial Measurement Units, Feature Selection 1 1.

There has been widespread use of causal inference methods for the rigorous analysis of observational studies and to identify policy evaluations. In this article, we consider coarsened exact matching, developed in Iacus et al. (2011). While they developed some statistical properties, in this article, we study the approach using asymptotics based on a superpopulation inferential framework. This methodology is generalized to what we termed as coarsened confounding, for which we propose two new algorithms. We develop asymptotic results for the average causal effect estimator as well as providing conditions for consistency. In addition, we provide an asymptotic justification for the variance formulae in Iacus et al. (2011). A bias correction technique is proposed, and we apply the proposed methodology to data from two well-known observational studies.