We prove part the first part of the proposition (weak duality) by induction. It is well-known that, by the value iteration algorithm's convergence, Q Consider a state s S and a feasible action a A (s). We use an induction proof. B (w), which follows by the convergence of value iteration.A.2 Proof of Theorem 1 Proof. Now we state the following lemma.

A.2 Proof of Theorem 1 We write ϕ

The graphical Lasso (GLASSO) is a widely used algorithm for learning high-dimensional undirected Gaussian graphical models (GGM). Given i.i.d. observations from a multivariate normal distribution, GLASSO estimates the precision matrix by maximizing the log-likelihood with an \ell_1-penalty on the off-diagonal entries. However, selecting an optimal regularization parameter λin this unsupervised setting remains a significant challenge. A well-known issue is that existing methods, such as out-of-sample likelihood maximization, select a single global λand do not account for heterogeneity in variable scaling or partial variances. Standardizing the data to unit variances, although a common workaround, has been shown to negatively affect graph recovery. Addressing the problem of nodewise adaptive tuning in graph estimation is crucial for applications like computational neuroscience, where brain networks are constructed from highly heterogeneous, region-specific fMRI data. In this work, we develop Locally Adaptive Regularization for Graph Estimation (LARGE), an approach to adaptively learn nodewise tuning parameters to improve graph estimation and selection. In each block coordinate descent step of GLASSO, we augment the nodewise Lasso regression to jointly estimate the regression coefficients and error variance, which in turn guides the adaptive learning of nodewise penalties. In simulations, LARGE consistently outperforms benchmark methods in graph recovery, demonstrates greater stability across replications, and achieves the best estimation accuracy in the most difficult simulation settings. We demonstrate the practical utility of our method by estimating brain functional connectivity from a real fMRI data set.

Images generated by diffusion models like Stable Diffusion are increasingly widespread. Recent works and even lawsuits have shown that these models are prone to replicating their training data, unbeknownst to the user. In this paper, we first analyze this memorization problem in text-to-image diffusion models. While it is widely believed that duplicated images in the training set are responsible for content replication at inference time, we observe that the text conditioning of the model plays a similarly important role. In fact, we see in our experiments that data replication often does not happen for unconditional models, while it is common in the text-conditional case. Motivated by our findings, we then propose several techniques for reducing data replication at both training and inference time by randomizing and augmenting image captions in the training set. Code is available at https://github.com/somepago/DCR.

Accurate and efficient surrogate modeling is essential for modern computational science, and there are a staggering number of emulation methods to choose from. With new methods being developed all the time, comparing the relative strengths and weaknesses of different methods remains a challenge due to inconsistent benchmarking practices and (sometimes) limited reproducibility and transparency. In this work, we present a large-scale, fully reproducible comparison of $29$ distinct emulators across $60$ canonical test functions and $40$ real emulation datasets. To facilitate rigorous, apples-to-apples comparisons, we introduce the R package \texttt{duqling}, which streamlines reproducible simulation studies using a consistent, simple syntax, and automatic internal scaling of inputs. This framework allows researchers to compare emulators in a unified environment and makes it possible to replicate or extend previous studies with minimal effort, even across different publications. Our results provide detailed empirical insight into the strengths and weaknesses of state-of-the-art emulators and offer guidance for both method developers and practitioners selecting a surrogate for new data. We discuss best practices for emulator comparison and highlight how \texttt{duqling} can accelerate research in emulator design and application.

We introduce SOCK, a benchmark command line interface (CLI) that measures large language models' (LLMs) ability to self-replicate without human intervention. In this benchmark, self-replication is defined not only as an LLM's ability to create a functioning and running copy of itself, but also the ability for that self-replication to persist and occur across different computational contexts. Accordingly, we've developed a system to categorize LLMs based on broad self-replication capabilities in two general classes, Replication-Capability Levels (RCL) and Persistence-Capability Levels (PCL). Using a five-task suite based on practically manipulable modern CLI utilities and computer processes, experiments are orchestrated in a controlled environment with an LLM acting agentically. The performance of the LLM on agent tasks is then computed to produce an R-score (a quantitative evaluation of overall self-replication ability) and data used to categorize LLMs into specific RCL-PCL matrices. SOCK offers two primary contributions: (1) Provides the first formalized definitions and benchmark suite for evaluating LLM self-replication, with the goal of establishing a standard for future research, to our knowledge; (2) Allows the industry to track the effectiveness of future multi-agent systems and mitigate potential self-replication threat vectors within them. The results compiled from evaluating a variety of open-weight and proprietary frontier models reveal significant obstacles to persistent self-replication and multi-agent systems, including context retention and multi-agent decision-making. We propose future research directions to safely reduce the severity of these obstacles, potentially lowering future risk of more functional multi-agent systems.

The rapid expansion of cross-border e-commerce (CBEC) has created significant opportunities for small- and medium-sized sellers, yet financing remains a critical challenge due to their limited credit histories. Third-party logistics (3PL)-led supply chain finance (SCF) has emerged as a promising solution, leveraging in-transit inventory as collateral. We propose an advanced credit risk management framework tailored for 3PL-led SCF, addressing the dual challenges of credit risk assessment and loan size determination. Specifically, we leverage conditional generative modeling of sales distributions through Quantile-Regression-based Generative Metamodeling (QRGMM) as the foundation for risk measures estimation. We propose a unified framework that enables flexible estimation of multiple risk measures while introducing a functional risk measure formulation that systematically captures the relationship between these risk measures and varying loan levels, supported by theoretical guarantees. To capture complex covariate interactions in e-commerce sales data, we integrate QRGMM with Deep Factorization Machines (DeepFM). Extensive experiments on synthetic and real-world data validate the efficacy of our model for credit risk assessment and loan size determination. This study explores the use of generative models in CBEC SCF risk management, illustrating their potential to strengthen credit assessment and support financing for small- and medium-sized sellers.