In many safety-critical settings, probabilistic ML systems have to make predictions subject to algebraic constraints, e.g., predicting the most likely trajectory that does not cross obstacles. These real-world constraints are rarely convex, nor the densities considered are (log-)concave. This makes computing this constrained maximum a posteriori (MAP) prediction efficiently and reliably extremely challenging. In this paper, we first investigate under which conditions we can perform constrained MAP inference over continuous variables exactly and efficiently and devise a scalable message-passing algorithm for this tractable fragment. Then, we devise a general constrained MAP strategy that interleaves partitioning the domain into convex feasible regions with numerical constrained optimization. We evaluate both methods on synthetic and real-world benchmarks, showing our approaches outperform constraint-agnostic baselines, and scale to complex densities intractable for SoTA exact solvers.

We introduce WildCat, a high-accuracy, low-cost approach to compressing the attention mechanism in neural networks. While attention is a staple of modern network architectures, it is also notoriously expensive to deploy due to resource requirements that scale quadratically with the input sequence length $n$. WildCat avoids these quadratic costs by only attending over a small weighted coreset. Crucially, we select the coreset using a fast but spectrally-accurate subsampling algorithm -- randomly pivoted Cholesky -- and weight the elements optimally to minimise reconstruction error. Remarkably, given bounded inputs, WildCat approximates exact attention with super-polynomial $O(n^{-\sqrt{\log(\log(n))}})$ error decay while running in near-linear $O(n^{1+o(1)})$ time. In contrast, prior practical approximations either lack error guarantees or require quadratic runtime to guarantee such high fidelity. We couple this advance with a GPU-optimized PyTorch implementation and a suite of benchmark experiments demonstrating the benefits of WildCat for image generation, image classification, and language model KV cache compression.

Spectral clustering is a popular and effective algorithm designed to find $k$ clusters in a graph $G$.In the classical spectral clustering algorithm, the vertices of $G$ are embedded into $\mathbb{R}^k$ using $k$ eigenvectors of the graph Laplacian matrix.However, computing this embedding is computationally expensive and dominates the running time of the algorithm.In this paper, we present a simple spectral clustering algorithm based on a vertex embedding with $O(\log(k))$ vectors computed by the power method.The vertex embedding is computed in nearly-linear time with respect to the size of the graph, andthe algorithm provably recovers the ground truth clusters under natural assumptions on the input graph.We evaluate the new algorithm on several synthetic and real-world datasets, finding that it is significantly faster than alternative clustering algorithms,while producing results with approximately the same clustering accuracy.

Meticulously analysing the empirical strengths and weaknesses of reinforcement learning methods in hard (challenging) environments is essential to inspire innovations and assess progress in the field. In tabular reinforcement learning, there is no well-established standard selection of environments to conduct such analysis, which is partially due to the lack of a widespread understanding of the rich theory of hardness of environments. The goal of this paper is to unlock the practical usefulness of this theory through four main contributions. First, we present a systematic survey of the theory of hardness, which also identifies promising research directions. Second, we introduce $\texttt{Colosseum}$, a pioneering package that enables empirical hardness analysis and implements a principled benchmark composed of environments that are diverse with respect to different measures of hardness. Third, we present an empirical analysis that provides new insights into computable measures. Finally, we benchmark five tabular agents in our newly proposed benchmark. While advancing the theoretical understanding of hardness in non-tabular reinforcement learning remains essential, our contributions in the tabular setting are intended as solid steps towards a principled non-tabular benchmark. Accordingly, we benchmark four agents in non-tabular versions of $\texttt{Colosseum}$ environments, obtaining results that demonstrate the generality of tabular hardness measures.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper proposes several approaches to sample from a Gibbs distribution over a discrete space by solving randomly perturbed combinatorial optimization problems (MAP inference) over the same space. The starting point is a known result [5] that allows to do sampling (in principle, using high dimensional perturbations with exponential complexity) by solving a single optimization problem. In this paper they propose to 1) use more efficient low-dimensional random perturbations to do approximate sampling (with probabilistic accuracy guarantees on tree structured models) 2) estimate (conditional) marginals using ratios of partition function estimates, and sequentially sample variables. They propose a clever rejection strategy based on self reduction that guarantees unbiasedness of the samples.

We thank the reviewers for their detailed and thoughtful comments. All minor comments and corrections will be addressed in the final version. In the following, we address each reviewer's comments in detail one by one. Q1: Natural images may not have low-rank structures. A1: We did not model natural images by low-rank structures.

Answer Set Programming (ASP) is often hindered by the grounding bottleneck: large Herbrand universes generate ground programs so large that solving becomes difficult. Many methods employ ad-hoc heuristics to improve grounding performance, motivating the need for a more formal and generalizable strategy. We introduce the notion of diminution, defined as a selected subset of the Herbrand universe used to generate a reduced ground program before solving. We give a formal definition of diminution, analyze its key properties, and study the complexity of identifying it. We use a specific encoding that enables off-the-shelf ASP solver to evaluate candidate subsets. Our approach integrates seamlessly with existing grounders via domain predicates. In extensive experiments on five benchmarks, applying diminutions selected by our strategy yields significant performance improvements, reducing grounding time by up to 70% on average and decreasing the size of grounding files by up to 85%. These results demonstrate that leveraging diminutions constitutes a robust and general-purpose approach for alleviating the grounding bottleneck in ASP.

Answer Set Programming (ASP) provides a powerful declarative paradigm for knowledge representation and reasoning. Recently, counting answer sets has emerged as an important computational problem with applications in probabilistic reasoning, network reliability analysis, and other domains. This has motivated significant research into designing efficient ASP counters. While substantial progress has been made for normal logic programs, the development of practical counters for disjunctive logic programs remains challenging. We present SharpASP-SR, a novel framework for counting answer sets of disjunctive logic programs based on subtractive reduction to projected propositional model counting. Our approach introduces an alternative characterization of answer sets that enables efficient reduction while ensuring that intermediate representations remain of polynomial size. This allows SharpASP-SR to leverage recent advances in projected model counting technology. Through extensive experimental evaluation on diverse benchmarks, we demonstrate that SharpASP-SR significantly outperforms existing counters on instances with large answer set counts. Building on these results, we develop a hybrid counting approach that combines enumeration techniques with SharpASP-SR to achieve state-of-the-art performance across the full spectrum of disjunctive programs.