Recent advances in self-supervised learning and neural network scaling have enabled the creation of large models, known as foundation models, which can be easily adapted to a wide range of downstream tasks. The current paradigm for comparing foundation models involves evaluating them with aggregate metrics on various benchmark datasets. This method of model comparison is heavily dependent on the chosen evaluation metric, which makes it unsuitable for situations where the ideal metric is either not obvious or unavailable. In this work, we present a methodology for directly comparing the embedding space geometry of foundation models, which facilitates model comparison without the need for an explicit evaluation metric. Our methodology is grounded in random graph theory and enables valid hypothesis testing of embedding similarity on a per-datum basis. Further, we demonstrate how our methodology can be extended to facilitate population level model comparison. In particular, we show how our framework can induce a manifold of models equipped with a distance function that correlates strongly with several downstream metrics. We remark on the utility of this population level model comparison as a first step towards a taxonomic science of foundation models.

The rapid advancement of artificial intelligence (AI) techniques has opened up new opportunities to revolutionize various fields, including operations research (OR). This survey paper explores the integration of AI within the OR process (AI4OR) to enhance its effectiveness and efficiency across multiple stages, such as parameter generation, model formulation, and model optimization. By providing a comprehensive overview of the state-of-the-art and examining the potential of AI to transform OR, this paper aims to inspire further research and innovation in the development of AI-enhanced OR methods and tools. The synergy between AI and OR is poised to drive significant advancements and novel solutions in a multitude of domains, ultimately leading to more effective and efficient decision-making.

Hybrid model predictive control with both continuous and discrete variables is widely applicable to robotic control tasks, especially those involving contact with the environment. Due to the combinatorial complexity, the solving speed of hybrid MPC can be insufficient for real-time applications. In this paper, we proposed a hybrid MPC solver based on Generalized Benders Decomposition (GBD). The algorithm enumerates and stores cutting planes online inside a finite buffer. After a short cold-start phase, the stored cuts provide warm-starts for the new problem instances to enhance the solving speed. Despite the disturbance and randomly changing environment, the solving speed maintains. Leveraging on the sparsity of feasibility cuts, we also propose a fast algorithm for Benders master problems. Our solver is validated through controlling a cart-pole system with randomly moving soft contact walls, and a free-flying robot navigating around obstacles. The results show that with significantly less data than previous works, the solver reaches competitive speeds to the off-the-shelf solver Gurobi despite the Python overhead.

The line coverage problem involves finding efficient routes for the coverage of linear features by one or more resource-constrained robots. Linear features model environments like road networks, power lines, and oil and gas pipelines. Two modes of travel are defined for robots: servicing and deadheading. A robot services a feature if it performs task-specific actions, such as taking images, as it traverses the feature; otherwise, it is deadheading. Traversing the environment incurs costs (e.g., travel time) and demands on resources (e.g., battery life). Servicing and deadheading can have different cost and demand functions, which can be direction-dependent. The environment is modeled as a graph, and an integer linear program is provided. As the problem is NP-hard, we design a fast and efficient heuristic algorithm, Merge-Embed-Merge (MEM). Exploiting the constructive property of the MEM algorithm, algorithms for line coverage of large graphs with multiple depots are developed. Furthermore, turning costs and nonholonomic constraints are efficiently incorporated into the algorithm. The algorithms are benchmarked on road networks and demonstrated in experiments with aerial robots.

In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.

The effectiveness of advertising in e-commerce largely depends on the ability of merchants to bid on and win impressions for their targeted users. The bidding procedure is highly complex due to various factors such as market competition, user behavior, and the diverse objectives of advertisers. In this paper we consider the problem at the level of user timelines instead of individual bid requests, manipulating full policies (i.e. pre-defined bidding strategies) and not bid values. In order to optimally allocate policies to users, typical multiple treatments allocation methods solve knapsack-like problems which aim at maximizing an expected value under constraints. In the industrial contexts such as online advertising, we argue that optimizing for the probability of success is a more suited objective than expected value maximization, and we introduce the SuccessProbaMax algorithm that aims at finding the policy allocation which is the most likely to outperform a fixed reference policy. Finally, we conduct comprehensive experiments both on synthetic and real-world data to evaluate its performance. The results demonstrate that our proposed algorithm outperforms conventional expected-value maximization algorithms in terms of success rate.

While ensuring stability for linear systems is well understood, it remains a major challenge for nonlinear systems. A general approach in such cases is to compute a combination of a Lyapunov function and an associated control policy. However, finding Lyapunov functions for general nonlinear systems is a challenging task. To address this challenge, several methods have been proposed that represent Lyapunov functions using neural networks. However, such approaches either focus on continuous-time systems, or highly restricted classes of nonlinear dynamics. We propose the first approach for learning neural Lyapunov control in a broad class of discrete-time systems. Three key ingredients enable us to effectively learn provably stable control policies. The first is a novel mixed-integer linear programming approach for verifying the discrete-time Lyapunov stability conditions, leveraging the particular structure of these conditions. The second is a novel approach for computing verified sublevel sets. The third is a heuristic gradient-based method for quickly finding counterexamples to significantly speed up Lyapunov function learning. Our experiments on four standard benchmarks demonstrate that our approach significantly outperforms state-of-the-art baselines. For example, on the path tracking benchmark, we outperform recent neural Lyapunov control baselines by an order of magnitude in both running time and the size of the region of attraction, and on two of the four benchmarks (cartpole and PVTOL), ours is the first automated approach to return a provably stable controller. Our code is available at: https://github.com/jlwu002/nlc_discrete.

Flight Trajectory Prediction (FTP) is an essential task in Air Traffic Control (ATC), which can assist air traffic controllers in managing airspace more safely and efficiently. Existing approaches generally perform multi-horizon FTP tasks in an autoregressive manner, thereby suffering from error accumulation and low-efficiency problems. In this paper, a novel framework, called FlightBERT++, is proposed to i) forecast multi-horizon flight trajectories directly in a non-autoregressive way, and ii) improve the limitation of the binary encoding (BE) representation in the FlightBERT. Specifically, the FlightBERT++ is implemented by a generalized encoder-decoder architecture, in which the encoder learns the temporal-spatial patterns from historical observations and the decoder predicts the flight status for the future horizons. Compared with conventional architecture, an innovative horizon-aware contexts generator is dedicatedly designed to consider the prior horizon information, which further enables non-autoregressive multi-horizon prediction. Moreover, a differential prompted decoder is proposed to enhance the capability of the differential predictions by leveraging the stationarity of the differential sequence. The experimental results on a real-world dataset demonstrated that the FlightBERT++ outperformed the competitive baselines in both FTP performance and computational efficiency.

The dynamic Schr\"odinger bridge problem seeks a stochastic process that defines a transport between two target probability measures, while optimally satisfying the criteria of being closest, in terms of Kullback-Leibler divergence, to a reference process. We propose a novel sampling-based iterative algorithm, the iterated diffusion bridge mixture (IDBM) procedure, aimed at solving the dynamic Schr\"odinger bridge problem. The IDBM procedure exhibits the attractive property of realizing a valid transport between the target probability measures at each iteration. We perform an initial theoretical investigation of the IDBM procedure, establishing its convergence properties. The theoretical findings are complemented by numerical experiments illustrating the competitive performance of the IDBM procedure. Recent advancements in generative modeling employ the time-reversal of a diffusion process to define a generative process that approximately transports a simple distribution to the data distribution. As an alternative, we propose utilizing the first iteration of the IDBM procedure as an approximation-free method for realizing this transport. This approach offers greater flexibility in selecting the generative process dynamics and exhibits accelerated training and superior sample quality over larger discretization intervals. In terms of implementation, the necessary modifications are minimally intrusive, being limited to the training loss definition.

We analyze a general problem in a crowd-sourced setting where one user asks a question (also called item) and other users return answers (also called labels) for this question. Different from existing crowd sourcing work which focuses on finding the most appropriate label for the question (the "truth"), our problem is to determine a ranking of the users based on their ability to answer questions. We call this problem "ability discovery" to emphasize the connection to and duality with the more well-studied problem of "truth discovery". To model items and their labels in a principled way, we draw upon Item Response Theory (IRT) which is the widely accepted theory behind standardized tests such as SAT and GRE. We start from an idealized setting where the relative performance of users is consistent across items and better users choose better fitting labels for each item. We posit that a principled algorithmic solution to our more general problem should solve this ideal setting correctly and observe that the response matrices in this setting obey the Consecutive Ones Property (C1P). While C1P is well understood algorithmically with various discrete algorithms, we devise a novel variant of the HITS algorithm which we call "HITSNDIFFS" (or HND), and prove that it can recover the ideal C1P-permutation in case it exists. Unlike fast combinatorial algorithms for finding the consecutive ones permutation (if it exists), HND also returns an ordering when such a permutation does not exist. Thus it provides a principled heuristic for our problem that is guaranteed to return the correct answer in the ideal setting. Our experiments show that HND produces user rankings with robustly high accuracy compared to state-of-the-art truth discovery methods. We also show that our novel variant of HITS scales better in the number of users than ABH, the only prior spectral C1P reconstruction algorithm.