Drought poses a pervasive environmental challenge in Bangladesh, impacting agriculture, socio-economic stability, and food security due to its unique geographic and anthropogenic vulnerabilities. Traditional drought indices, such as the Standardized Precipitation Index (SPI) and Palmer Drought Severity Index (PDSI), often overlook crucial factors like soil moisture and temperature, limiting their resolution. Moreover, current machine learning models applied to drought prediction have been underexplored in the context of Bangladesh, lacking a comprehensive integration of satellite data across multiple districts. To address these gaps, we propose a satellite data-driven machine learning framework to classify drought across 38 districts of Bangladesh. Using unsupervised algorithms like K-means and Bayesian Gaussian Mixture for clustering, followed by classification models such as KNN, Random Forest, Decision Tree, and Naive Bayes, the framework integrates weather data (humidity, soil moisture, temperature) from 2012-2024. This approach successfully classifies drought severity into different levels. However, it shows significant variabilities in drought vulnerabilities across regions which highlights the aptitude of machine learning models in terms of identifying and predicting drought conditions.

Energy-based models (EBMs) estimate unnormalized densities in an elegant framework, but they are generally difficult to train. Recent work has linked EBMs to generative adversarial networks, by noting that they can be trained through a minimax game using a variational lower bound. To avoid the instabilities caused by minimizing a lower bound, we propose to instead work with bidirectional bounds, meaning that we maximize a lower bound and minimize an upper bound when training the EBM. We investigate four different bounds on the log-likelihood derived from different perspectives. We derive lower bounds based on the singular values of the generator Jacobian and on mutual information. To upper bound the negative log-likelihood, we consider a gradient penalty-like bound, as well as one based on diffusion processes. In all cases, we provide algorithms for evaluating the bounds. We compare the different bounds to investigate, the pros and cons of the different approaches. Finally, we demonstrate that the use of bidirectional bounds stabilizes EBM training and yields high-quality density estimation and sample generation.

Physics-Informed Neural Networks (PINNs) have been widely used to obtain solutions to various physical phenomena modeled as Differential Equations. As PINNs are not naturally equipped with mechanisms for Uncertainty Quantification, some work has been done to quantify the different uncertainties that arise when dealing with PINNs. In this paper, we use a two-step procedure to train Bayesian Neural Networks that provide uncertainties over the solutions to differential equation systems provided by PINNs. We use available error bounds over PINNs to formulate a heteroscedastic variance that improves the uncertainty estimation. Furthermore, we solve forward problems and utilize the obtained uncertainties when doing parameter estimation in inverse problems in cosmology.

While theory and practice are often seen as separate domains, this article shows that theoretical insight is essential for overcoming real-world engineering barriers. We begin with a practical challenge: training a cross-morphology embodied AI policy that generalizes across diverse robot morphologies. We formalize this as the Heterogeneous Embodied Agent Training (HEAT) problem and prove it reduces to a structured Partially Observable Markov Decision Process (POMDP) that is PSPACE-complete. This result explains why current reinforcement learning pipelines break down under morphological diversity, due to sequential training constraints, memory-policy coupling, and data incompatibility. We further explore Collective Adaptation, a distributed learning alternative inspired by biological systems. Though NEXP-complete in theory, it offers meaningful scalability and deployment benefits in practice. This work illustrates how computational theory can illuminate system design trade-offs and guide the development of more robust, scalable embodied AI. For practitioners and researchers to explore this problem, the implementation code of this work has been made publicly available at https://github.com/airs-admin/HEAT

The Bayes Error Rate (BER) is the fundamental limit on the achievable generalizable classification accuracy of any machine learning model due to inherent uncertainty within the data. BER estimators offer insight into the difficulty of any classification problem and set expectations for optimal classification performance. In order to be useful, the estimators must also be accurate with a limited number of samples on multivariate problems with unknown class distributions. To determine which estimators meet the minimum requirements for "usefulness", an in-depth examination of their accuracy is conducted using Monte Carlo simulations with synthetic data in order to obtain their confidence bounds for binary classification. To examine the usability of the estimators on real-world applications, new test scenarios are introduced upon which 2500 Monte Carlo simulations per scenario are run over a wide range of BER values. In a comparison of k-Nearest Neighbor (kNN), Generalized Henze-Penrose (GHP) divergence and Kernel Density Estimation (KDE) techniques, results show that kNN is overwhelmingly the more accurate non-parametric estimator. In order to reach the target of an under 5 percent range for the 95 percent confidence bounds, the minimum number of required samples per class is 1000. As more features are added, more samples are needed, so that 2500 samples per class are required at only 4 features. Other estimators do become more accurate than kNN as more features are added, but continuously fail to meet the target range.

We propose a framework for hypothesis testing on conditional probability distributions, which we then use to construct conditional two-sample statistical tests. These tests identify the inputs -- called covariates in this context -- where two conditional expectations differ with high probability. Our key idea is to transform confidence bounds of a learning method into a conditional two-sample test, and we instantiate this principle for kernel ridge regression (KRR) and conditional kernel mean embeddings. We generalize existing pointwise-in-time or time-uniform confidence bounds for KRR to previously-inaccessible yet essential cases such as infinite-dimensional outputs with non-trace-class kernels. These bounds enable circumventing the need for independent data in our statistical tests, since they allow online sampling. We also introduce bootstrapping schemes leveraging the parametric form of testing thresholds identified in theory to avoid tuning inaccessible parameters, making our method readily applicable in practice. Such conditional two-sample tests are especially relevant in applications where data arrive sequentially or non-independently, or when output distributions vary with operational parameters. We demonstrate their utility through examples in process monitoring and comparison of dynamical systems. Overall, our results establish a comprehensive foundation for conditional two-sample testing, from theoretical guarantees to practical implementation, and advance the state-of-the-art on the concentration of vector-valued least squares estimation.

Due to their intuitive appeal, Bayesian methods of modeling and uncertainty quantification have become popular in modern machine and deep learning. When providing a prior distribution over the parameter space, it is straightforward to obtain a distribution over the parameters that is conventionally interpreted as uncertainty quantification of the model. We challenge the validity of such Bayesian uncertainty quantification by discussing the equivalent optimization-based representation of Bayesian updating, provide an alternative interpretation that is coherent with the optimization-based perspective, propose measures of the quality of the Bayesian inferential stage, and suggest directions for future work.

Geometry optimization of atomic structures is a common and crucial task in computational chemistry and materials design. Following the learning to optimize paradigm, we propose a new multi-agent reinforcement learning method called Multi-Agent Crystal Structure optimization (MACS) to address periodic crystal structure optimization. MACS treats geometry optimization as a partially observable Markov game in which atoms are agents that adjust their positions to collectively discover a stable configuration. We train MACS across various compositions of reported crystalline materials to obtain a policy that successfully optimizes structures from the training compositions as well as structures of larger sizes and unseen compositions, confirming its excellent scalability and zero-shot transferability. We benchmark our approach against a broad range of state-of-the-art optimization methods and demonstrate that MACS optimizes periodic crystal structures significantly faster, with fewer energy calculations, and the lowest failure rate.

Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs) are known to be NEXP-Complete and intractable to solve. However, for problems such as cooperative navigation, obstacle avoidance, and formation control, basic assumptions can be made about local visibility and local dependencies. The work DeWeese and Qu 2024 formalized these assumptions in the construction of the Locally Interdependent Multi-Agent MDP. In this setting, it establishes three closed-form policies that are tractable to compute in various situations and are exponentially close to optimal with respect to visibility. However, it is also shown that these solutions can have poor performance when the visibility is small and fixed, often getting stuck during simulations due to the so called "Penalty Jittering" phenomenon. In this work, we establish the Extended Cutoff Policy Class which is, to the best of our knowledge, the first non-trivial class of near optimal closed-form partially observable policies that are exponentially close to optimal with respect to the visibility for any Locally Interdependent Multi-Agent MDP. These policies are able to remember agents beyond their visibilities which allows them to perform significantly better in many small and fixed visibility settings, resolve Penalty Jittering occurrences, and under certain circumstances guarantee fully observable joint optimal behavior despite the partial observability. We also propose a generalized form of the Locally Interdependent Multi-Agent MDP that allows for transition dependence and extended reward dependence, then replicate our theoretical results in this setting.

Large language models (LLMs) have notably progressed in multi-step and long-chain reasoning. However, extending their reasoning capabilities to encompass deep interactions with search remains a non-trivial challenge, as models often fail to identify optimal reasoning-search interaction trajectories, resulting in suboptimal responses. We propose R-Search, a novel reinforcement learning framework for Reasoning-Search integration, designed to enable LLMs to autonomously execute multi-step reasoning with deep search interaction, and learn optimal reasoning search interaction trajectories via multi-reward signals, improving response quality in complex logic- and knowledge-intensive tasks. R-Search guides the LLM to dynamically decide when to retrieve or reason, while globally integrating key evidence to enhance deep knowledge interaction between reasoning and search. During RL training, R-Search provides multi-stage, multi-type rewards to jointly optimize the reasoning-search trajectory. Experiments on seven datasets show that R-Search outperforms advanced RAG baselines by up to 32.2% (in-domain) and 25.1% (out-of-domain). The code and data are available at https://github.com/QingFei1/R-Search.