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In many problems, it is desirable to optimize an objective function while imposing constraints on some other aspect of the problem. A Constrained Partially Observable Markov Decision Process (C-POMDP) allows modelling of such problems while subject to transition uncertainty and partial observability. Typically, the constraints in C-POMDPs enforce a threshold on expected cumulative costs starting from an initial state distribution. In this work, we first show that optimal C-POMDP policies may violate Bellman's principle of optimality and thus may exhibit pathological behaviors, which can be undesirable for many applications. To address this drawback, we introduce a new formulation, the Recursively-Constrained POMDP (RC-POMDP), that imposes additional history dependent cost constraints on the C-POMDP. We show that, unlike C-POMDPs, RC-POMDPs always have deterministic optimal policies, and that optimal policies obey Bellman's principle of optimality. We also present a point-based dynamic programming algorithm that synthesizes optimal policies for RC-POMDPs. In our evaluations, we show that policies for RC-POMDPs produce more desirable behavior than policies for C-POMDPs and demonstrate the efficacy of our algorithm across a set of benchmark problems.

Optimal algorithms are developed for robust detection of changes in non-stationary processes. These are processes in which the distribution of the data after change varies with time. The decision-maker does not have access to precise information on the post-change distribution. It is shown that if the post-change non-stationary family has a distribution that is least favorable in a well-defined sense, then the algorithms designed using the least favorable distributions are robust and optimal. Non-stationary processes are encountered in public health monitoring and space and military applications. The robust algorithms are applied to real and simulated data to show their effectiveness.

Synthesizing inductive loop invariants is fundamental to automating program verification. In this work, we observe that Large Language Models (such as gpt-3.5 or gpt-4) are capable of synthesizing loop invariants for a class of programs in a 0-shot setting, yet require several samples to generate the correct invariants. This can lead to a large number of calls to a program verifier to establish an invariant. To address this issue, we propose a {\it re-ranking} approach for the generated results of LLMs. We have designed a ranker that can distinguish between correct inductive invariants and incorrect attempts based on the problem definition. The ranker is optimized as a contrastive ranker. Experimental results demonstrate that this re-ranking mechanism significantly improves the ranking of correct invariants among the generated candidates, leading to a notable reduction in the number of calls to a verifier.

Self-normalized processes arise naturally in many statistical tasks. While self-normalized concentration has been extensively studied for scalar-valued processes, there is less work on multidimensional processes outside of the sub-Gaussian setting. In this work, we construct a general, self-normalized inequality for $\mathbb{R}^d$-valued processes that satisfy a simple yet broad "sub-$\psi$" tail condition, which generalizes assumptions based on cumulant generating functions. From this general inequality, we derive an upper law of the iterated logarithm for sub-$\psi$ vector-valued processes, which is tight up to small constants. We demonstrate applications in prototypical statistical tasks, such as parameter estimation in online linear regression and auto-regressive modeling, and bounded mean estimation via a new (multivariate) empirical Bernstein concentration inequality.

We focus on learning adversarially robust classifiers under a cost-sensitive scenario, where the potential harm of different classwise adversarial transformations is encoded in a binary cost matrix. Existing methods are either empirical that cannot certify robustness or suffer from inherent scalability issues. In this work, we study whether randomized smoothing, a more scalable robustness certification framework, can be leveraged to certify cost-sensitive robustness. Built upon a notion of cost-sensitive certified radius, we show how to adapt the standard randomized smoothing certification pipeline to produce tight robustness guarantees for any cost matrix. In addition, with fine-grained certified radius optimization schemes specifically designed for different data subgroups, we propose an algorithm to train smoothed classifiers that are optimized for cost-sensitive robustness. Extensive experiments on image benchmarks and a real-world medical dataset demonstrate the superiority of our method in achieving significantly improved performance of certified cost-sensitive robustness while having a negligible impact on overall accuracy.

Representing and rendering dynamic scenes has been an important but challenging task. Especially, to accurately model complex motions, high efficiency is usually hard to maintain. We introduce the 4D Gaussian Splatting (4D-GS) to achieve real-time dynamic scene rendering while also enjoying high training and storage efficiency. An efficient deformation field is constructed to model both Gaussian motions and shape deformations. Different adjacent Gaussians are connected via a HexPlane to produce more accurate position and shape deformations. Our 4D-GS method achieves real-time rendering under high resolutions, 70 FPS at a 800$\times$800 resolution on an RTX 3090 GPU, while maintaining comparable or higher quality than previous state-of-the-art methods. More demos and code are available at //guanjunwu.github.io/4dgs/.

Out-of-distribution (OOD) detection is essential for reliable and trustworthy machine learning. Recent multi-modal OOD detection leverages textual information from in-distribution (ID) class names for visual OOD detection, yet it currently neglects the rich contextual information of ID classes. Large language models (LLMs) encode a wealth of world knowledge and can be prompted to generate descriptive features for each class. Indiscriminately using such knowledge causes catastrophic damage to OOD detection due to LLMs' hallucinations, as is observed by our analysis. In this paper, we propose to apply world knowledge to enhance OOD detection performance through selective generation from LLMs. Specifically, we introduce a consistency-based uncertainty calibration method to estimate the confidence score of each generation. We further extract visual objects from each image to fully capitalize on the aforementioned world knowledge. Extensive experiments demonstrate that our method consistently outperforms the state-of-the-art.

Matrix-variate distributions are a recent addition to the model-based clustering field, thereby making it possible to analyze data in matrix form with complex structure such as images and time series. Due to its recent appearance, there is limited literature on matrix-variate data, with even less on dealing with outliers in these models. An approach for clustering matrix-variate normal data with outliers is discussed. The approach, which uses the distribution of subset log-likelihoods, extends the OCLUST algorithm to matrix-variate normal data and uses an iterative approach to detect and trim outliers.

Federated Learning (FL) is a decentralized machine-learning paradigm, in which a global server iteratively averages the model parameters of local users without accessing their data. User heterogeneity has imposed significant challenges to FL, which can incur drifted global models that are slow to converge. Knowledge Distillation has recently emerged to tackle this issue, by refining the server model using aggregated knowledge from heterogeneous users, other than directly averaging their model parameters. This approach, however, depends on a proxy dataset, making it impractical unless such a prerequisite is satisfied. Moreover, the ensemble knowledge is not fully utilized to guide local model learning, which may in turn affect the quality of the aggregated model. Inspired by the prior art, we propose a data-free knowledge distillation} approach to address heterogeneous FL, where the server learns a lightweight generator to ensemble user information in a data-free manner, which is then broadcasted to users, regulating local training using the learned knowledge as an inductive bias. Empirical studies powered by theoretical implications show that, our approach facilitates FL with better generalization performance using fewer communication rounds, compared with the state-of-the-art.

Knowledge graph embedding, which aims to represent entities and relations as low dimensional vectors (or matrices, tensors, etc.), has been shown to be a powerful technique for predicting missing links in knowledge graphs. Existing knowledge graph embedding models mainly focus on modeling relation patterns such as symmetry/antisymmetry, inversion, and composition. However, many existing approaches fail to model semantic hierarchies, which are common in real-world applications. To address this challenge, we propose a novel knowledge graph embedding model---namely, Hierarchy-Aware Knowledge Graph Embedding (HAKE)---which maps entities into the polar coordinate system. HAKE is inspired by the fact that concentric circles in the polar coordinate system can naturally reflect the hierarchy. Specifically, the radial coordinate aims to model entities at different levels of the hierarchy, and entities with smaller radii are expected to be at higher levels; the angular coordinate aims to distinguish entities at the same level of the hierarchy, and these entities are expected to have roughly the same radii but different angles. Experiments demonstrate that HAKE can effectively model the semantic hierarchies in knowledge graphs, and significantly outperforms existing state-of-the-art methods on benchmark datasets for the link prediction task.