Smart portable applications increasingly rely on edge computing due to privacy andlatencyconcerns.

The Weighted First-Order Model Counting Problem (WFOMC) asks to compute the weighted sum of models of a given first-order logic sentence over a given domain. Conditioning WFOMC on evidence -- fixing the truth values of a set of ground literals -- has been shown impossible in time polynomial in the domain size (unless $\mathsf{\#P \subseteq FP}$) even for fragments of logic that are otherwise tractable for WFOMC without evidence. In this work, we address the barrier by restricting the binary evidence to the case where the underlying Gaifman graph has bounded treewidth. We present a polynomial-time algorithm in the domain size for computing WFOMC for the two-variable fragments $\text{FO}^2$ and $\text{C}^2$ conditioned on such binary evidence. Furthermore, we show the applicability of our algorithm in combinatorial problems by solving the stable seating arrangement problem on bounded-treewidth graphs of bounded degree, which was an open problem. We also conducted experiments to show the scalability of our algorithm compared to the existing model counting solvers.

Probabilistic Circuits (PCs) offer a computationally scalable framework for generative modeling, supporting exact and efficient inference of a wide range of probabilistic queries. While recent advances have significantly improved the expressiveness and scalability of PCs, effectively training their parameters remains a challenge. In particular, a widely used optimization method, full-batch Expectation-Maximization (EM), requires processing the entire dataset before performing a single update, making it ineffective for large datasets. Although empirical extensions to the mini-batch setting, as well as gradient-based mini-batch algorithms, converge faster than full-batch EM, they generally underperform in terms of final likelihood. We investigate this gap by establishing a novel theoretical connection between these practical algorithms and the general EM objective. Our analysis reveals a fundamental issue that existing mini-batch EM and gradient-based methods fail to properly regularize distribution changes, causing each update to effectively ``overfit'' the current mini-batch. Motivated by this insight, we introduce anemone, a new mini-batch EM algorithm for PCs. Anemone applies an implicit adaptive learning rate to each parameter, scaled by how much it contributes to the likelihood of the current batch. Across extensive experiments on language, image, and DNA datasets, anemone consistently outperforms existing optimizers in both convergence speed and final performance.

Efficient probabilistic inference by variable elimination in graphical models requires an optimal elimination order. However, finding an optimal order is a challenging combinatorial optimisation problem for models with a large number of random variables. Most recently, a reinforcement learning approach has been proposed to find efficient contraction orders in tensor networks. Due to the duality between graphical models and tensor networks, we adapt this approach to probabilistic inference in graphical models. Furthermore, we incorporate structure exploitation into the process of finding an optimal order. Currently, the agent's cost function is formulated in terms of intermediate result sizes which are exponential in the number of indices (i.e., random variables). We show that leveraging specific structures during inference allows for introducing compact encodings of intermediate results which can be significantly smaller. By considering the compact encoding sizes for the cost function instead, we enable the agent to explore more efficient contraction orders. The structure we consider in this work is the presence of local symmetries (i.e., symmetries within a model's factors).

Feature attribution methods have become essential for explaining machine learning models. Many popular approaches, such as SHAP and Banzhaf values, are grounded in power indices from cooperative game theory, which measure the contribution of features to model predictions. This work studies the computational complexity of power indices beyond SHAP, addressing the conditions under which they can be computed efficiently. We identify a simple condition on power indices that ensures that computation is polynomially equivalent to evaluating expected values, extending known results for SHAP. We also introduce Bernoulli power indices, showing that their computation can be simplified to a constant number of expected value evaluations. Furthermore, we explore interaction power indices that quantify the importance of feature subsets, proving that their computation complexity mirrors that of individual features.

Probabilistic circuits (PCs) is a unifying representation for probabilistic models that support tractable inference. Numerous applications of PCs like controllable text generation depend on the ability to efficiently multiply two circuits. Existing multiplication algorithms require that the circuits respect the same structure, i.e. variable scopes decomposes according to the same vtree. In this work, we propose and study the task of restructuring structured(-decomposable) PCs, that is, transforming a structured PC such that it conforms to a target vtree. We propose a generic approach for this problem and show that it leads to novel polynomial-time algorithms for multiplying circuits respecting different vtrees, as well as a practical depth-reduction algorithm that preserves structured decomposibility. Our work opens up new avenues for tractable PC inference, suggesting the possibility of training with less restrictive PC structures while enabling efficient inference by changing their structures at inference time.

This paper establishes a rigorous connection between circuit representations and tensor factorizations, two seemingly distinct yet fundamentally related areas. By connecting these fields, we highlight a series of opportunities that can benefit both communities. Our work generalizes popular tensor factorizations within the circuit language, and unifies various circuit learning algorithms under a single, generalized hierarchical factorization framework. Specifically, we introduce a modular "Lego block" approach to build tensorized circuit architectures. This, in turn, allows us to systematically construct and explore various circuit and tensor factorization models while maintaining tractability. This connection not only clarifies similarities and differences in existing models, but also enables the development of a comprehensive pipeline for building and optimizing new circuit/tensor factorization architectures. We show the effectiveness of our framework through extensive empirical evaluations, and highlight new research opportunities for tensor factorizations in probabilistic modeling.

The limitations of purely neural learning have sparked an interest in probabilistic neurosymbolic models, which combine neural networks with probabilistic logical reasoning. As these neurosymbolic models are trained with gradient descent, we study the complexity of differentiating probabilistic reasoning. We prove that although approximating these gradients is intractable in general, it becomes tractable during training. Furthermore, we introduce WeightME, an unbiased gradient estimator based on model sampling. Under mild assumptions, WeightME approximates the gradient with probabilistic guarantees using a logarithmic number of calls to a SAT solver. Lastly, we evaluate the necessity of these guarantees on the gradient. Our experiments indicate that the existing biased approximations indeed struggle to optimize even when exact solving is still feasible.