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The development and evaluation of Large Language Models (LLMs) have largely focused on individual capabilities. However, this overlooks the intersection of multiple abilities across different types of expertise that are often required for real-world tasks, which we term cross capabilities. To systematically explore this concept, we first define seven core individual capabilities and then pair them to form seven common cross capabilities, each supported by a manually constructed taxonomy. Building on these definitions, we introduce CrossEval, a benchmark comprising 1,400 human-annotated prompts, with 100 prompts for each individual and cross capability. To ensure reliable evaluation, we involve expert annotators to assess 4,200 model responses, gathering 8,400 human ratings with detailed explanations to serve as reference examples. Our findings reveal that, in both static evaluations and attempts to enhance specific abilities, current LLMs consistently exhibit the "Law of the Weakest Link," where cross-capability performance is significantly constrained by the weakest component. Specifically, across 58 cross-capability scores from 17 models, 38 scores are lower than all individual capabilities, while 20 fall between strong and weak, but closer to the weaker ability. These results highlight the under-performance of LLMs in cross-capability tasks, making the identification and improvement of the weakest capabilities a critical priority for future research to optimize performance in complex, multi-dimensional scenarios.

Large Language Models (LLMs) have drawn widespread attention and research due to their astounding performance in text generation and reasoning tasks. Derivative products, like ChatGPT, have been extensively deployed and highly sought after. Meanwhile, the evaluation and optimization of LLMs in software engineering tasks, such as code generation, have become a research focus. However, there is still a lack of systematic research on applying and evaluating LLMs in software engineering. Therefore, this paper comprehensively investigate and collate the research and products combining LLMs with software engineering, aiming to answer two questions: (1) What are the current integrations of LLMs with software engineering? (2) Can LLMs effectively handle software engineering tasks? To find the answers, we have collected related literature as extensively as possible from seven mainstream databases and selected 123 timely papers published starting from 2022 for analysis. We have categorized these papers in detail and reviewed the current research status of LLMs from the perspective of seven major software engineering tasks, hoping this will help researchers better grasp the research trends and address the issues when applying LLMs. Meanwhile, we have also organized and presented papers with evaluation content to reveal the performance and effectiveness of LLMs in various software engineering tasks, guiding researchers and developers to optimize.

The Shortest-Path Problem in Graph of Convex Sets (SPP in GCS) is a recently developed optimization framework that blends discrete and continuous decision making. Many relevant problems in robotics, such as collision-free motion planning, can be cast and solved as an SPP in GCS, yielding lower-cost solutions and faster runtimes than state-of-the-art algorithms. In this paper, we are motivated by motion planning of robot arms that must operate swiftly in static environments. We consider a multi-query extension of the SPP in GCS, where the goal is to efficiently precompute optimal paths between given sets of initial and target conditions. Our solution consists of two stages. Offline, we use semidefinite programming to compute a coarse lower bound on the problem's cost-to-go function. Then, online, this lower bound is used to incrementally generate feasible paths by solving short-horizon convex programs. For a robot arm with seven joints, our method designs higher quality trajectories up to two orders of magnitude faster than existing motion planners.

A extension of the Euler-Maclaurin (E-M) formula to near-singular functions is presented. This extension is derived based on earlier generalized E-M formulas for singular functions. The new E-M formulas consists of two components: a ``singular'' component that is a continuous extension of the earlier singular E-M formulas, and a ``jump'' component associated with the discontinuity of the integral with respect to a parameter that controls near singularity. The singular component of the new E-M formulas is an asymptotic series whose coefficients depend on the Hurwitz zeta function or the digamma function. Numerical examples of near-singular quadrature based on the extended E-M formula are presented, where accuracies of machine precision are achieved insensitive to the strength of the near singularity and with a very small number of quadrature nodes.

Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio and video generation as well as many more applications in science and beyond. The manifold hypothesis states that high-dimensional data often lie on lower-dimensional manifolds within the ambient space, and is widely believed to hold in provided examples. While recent results has provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models, making this a very fruitful research direction. In this work, we study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of learning the score. In terms of sampling, we obtain rates independent of the ambient dimension w.r.t. the Kullback-Leibler divergence, and $O(\sqrt{D})$ w.r.t. the Wasserstein distance. We do this by developing a new framework connecting diffusion models to the well-studied theory of extrema of Gaussian Processes.

ABCDE is a sophisticated technique for evaluating differences between very large clusterings. Its main metric that characterizes the magnitude of the difference between two clusterings is the JaccardDistance, which is a true distance metric in the space of all clusterings of a fixed set of (weighted) items. The JaccardIndex is the complementary metric that characterizes the similarity of two clusterings. Its relationship with the JaccardDistance is simple: JaccardDistance + JaccardIndex = 1. This paper decomposes the JaccardDistance and the JaccardIndex further. In each case, the decomposition yields Impact and Quality metrics. The Impact metrics measure aspects of the magnitude of the clustering diff, while Quality metrics use human judgements to measure how much the clustering diff improves the quality of the clustering. The decompositions of this paper offer more and deeper insight into a clustering change. They also unlock new techniques for debugging and exploring the nature of the clustering diff. The new metrics are mathematically well-behaved and they are interrelated via simple equations. While the work can be seen as an alternative formal framework for ABCDE, we prefer to view it as complementary. It certainly offers a different perspective on the magnitude and the quality of a clustering change, and users can use whatever they want from each approach to gain more insight into a change.

MAX NAE-SAT is a natural optimization problem, closely related to its better-known relative MAX SAT. The approximability status of MAX NAE-SAT is almost completely understood if all clauses have the same size $k$, for some $k\ge 2$. We refer to this problem as MAX NAE-$\{k\}$-SAT. For $k=2$, it is essentially the celebrated MAX CUT problem. For $k=3$, it is related to the MAX CUT problem in graphs that can be fractionally covered by triangles. For $k\ge 4$, it is known that an approximation ratio of $1-\frac{1}{2^{k-1}}$, obtained by choosing a random assignment, is optimal, assuming $P\ne NP$. For every $k\ge 2$, an approximation ratio of at least $\frac{7}{8}$ can be obtained for MAX NAE-$\{k\}$-SAT. There was some hope, therefore, that there is also a $\frac{7}{8}$-approximation algorithm for MAX NAE-SAT, where clauses of all sizes are allowed simultaneously. Our main result is that there is no $\frac{7}{8}$-approximation algorithm for MAX NAE-SAT, assuming the unique games conjecture (UGC). In fact, even for almost satisfiable instances of MAX NAE-$\{3,5\}$-SAT (i.e., MAX NAE-SAT where all clauses have size $3$ or $5$), the best approximation ratio that can be achieved, assuming UGC, is at most $\frac{3(\sqrt{21}-4)}{2}\approx 0.8739$. Using calculus of variations, we extend the analysis of O'Donnell and Wu for MAX CUT to MAX NAE-$\{3\}$-SAT. We obtain an optimal algorithm, assuming UGC, for MAX NAE-$\{3\}$-SAT, slightly improving on previous algorithms. The approximation ratio of the new algorithm is $\approx 0.9089$. We complement our theoretical results with some experimental results. We describe an approximation algorithm for almost satisfiable instances of MAX NAE-$\{3,5\}$-SAT with a conjectured approximation ratio of 0.8728, and an approximation algorithm for almost satisfiable instances of MAX NAE-SAT with a conjectured approximation ratio of 0.8698.

Big models have achieved revolutionary breakthroughs in the field of AI, but they might also pose potential concerns. Addressing such concerns, alignment technologies were introduced to make these models conform to human preferences and values. Despite considerable advancements in the past year, various challenges lie in establishing the optimal alignment strategy, such as data cost and scalable oversight, and how to align remains an open question. In this survey paper, we comprehensively investigate value alignment approaches. We first unpack the historical context of alignment tracing back to the 1920s (where it comes from), then delve into the mathematical essence of alignment (what it is), shedding light on the inherent challenges. Following this foundation, we provide a detailed examination of existing alignment methods, which fall into three categories: Reinforcement Learning, Supervised Fine-Tuning, and In-context Learning, and demonstrate their intrinsic connections, strengths, and limitations, helping readers better understand this research area. In addition, two emerging topics, personal alignment, and multimodal alignment, are also discussed as novel frontiers in this field. Looking forward, we discuss potential alignment paradigms and how they could handle remaining challenges, prospecting where future alignment will go.

Recently, Mutual Information (MI) has attracted attention in bounding the generalization error of Deep Neural Networks (DNNs). However, it is intractable to accurately estimate the MI in DNNs, thus most previous works have to relax the MI bound, which in turn weakens the information theoretic explanation for generalization. To address the limitation, this paper introduces a probabilistic representation of DNNs for accurately estimating the MI. Leveraging the proposed MI estimator, we validate the information theoretic explanation for generalization, and derive a tighter generalization bound than the state-of-the-art relaxations.

Graph Neural Networks (GNNs) have been studied from the lens of expressive power and generalization. However, their optimization properties are less well understood. We take the first step towards analyzing GNN training by studying the gradient dynamics of GNNs. First, we analyze linearized GNNs and prove that despite the non-convexity of training, convergence to a global minimum at a linear rate is guaranteed under mild assumptions that we validate on real-world graphs. Second, we study what may affect the GNNs' training speed. Our results show that the training of GNNs is implicitly accelerated by skip connections, more depth, and/or a good label distribution. Empirical results confirm that our theoretical results for linearized GNNs align with the training behavior of nonlinear GNNs. Our results provide the first theoretical support for the success of GNNs with skip connections in terms of optimization, and suggest that deep GNNs with skip connections would be promising in practice.