Polynomial Semantics of Tractable Probabilistic Circuits

Probabilistic circuits compute multilinear polynomials that represent probability distributions. They are tractable models that support efficient marginal inference. However, various polynomial semantics have been considered in the literature (e.g., network polynomials, likelihood polynomials, generating functions, Fourier transforms, and characteristic polynomials). The relationships between these polynomial encodings of distributions is largely unknown. In this paper, we prove that for binary distributions, each of these probabilistic circuit models is equivalent in the sense that any circuit for one of them can be transformed into a circuit for any of the others with only a polynomial increase in size. They are therefore all tractable for marginal inference on the same class of distributions. Finally, we explore the natural extension of one such polynomial semantics, called probabilistic generating circuits, to categorical random variables, and establish that marginal inference becomes #P-hard.

Tractable Control for Autoregressive Language Generation

Despite the success of autoregressive large language models in text generation, it remains a major challenge to generate text that satisfies complex constraints: sampling from the conditional distribution $\Pr(\text{text} | \alpha)$ is intractable for even the simplest lexical constraints $\alpha$. To overcome this challenge, we propose to use tractable probabilistic models to impose lexical constraints in autoregressive text generation, which we refer to as GeLaTo. To demonstrate the effectiveness of this framework, we use distilled hidden Markov models to control autoregressive generation from GPT2. GeLaTo achieves state-of-the-art performance on CommonGen, a challenging benchmark for constrained text generation, beating a wide range of strong baselines by a large margin. Our work not only opens up new avenues for controlling large language models but also motivates the development of more expressive tractable probabilistic models.

Mixtures of All Trees

Tree-shaped graphical models are widely used for their tractability. However, they unfortunately lack expressive power as they require committing to a particular sparse dependency structure. We propose a novel class of generative models called mixtures of all trees: that is, a mixture over all possible ($n^{n-2}$) tree-shaped graphical models over $n$ variables. We show that it is possible to parameterize this Mixture of All Trees (MoAT) model compactly (using a polynomial-size representation) in a way that allows for tractable likelihood computation and optimization via stochastic gradient descent. Furthermore, by leveraging the tractability of tree-shaped models, we devise fast-converging conditional sampling algorithms for approximate inference, even though our theoretical analysis suggests that exact computation of marginals in the MoAT model is NP-hard. Empirically, MoAT achieves state-of-the-art performance on density estimation benchmarks when compared against powerful probabilistic models including hidden Chow-Liu Trees.

Scaling Up Probabilistic Circuits by Latent Variable Distillation

Probabilistic Circuits (PCs) are a unified framework for tractable probabilistic models that support efficient computation of various probabilistic queries (e.g., marginal probabilities). One key challenge is to scale PCs to model large and high-dimensional real-world datasets: we observe that as the number of parameters in PCs increases, their performance immediately plateaus. This phenomenon suggests that the existing optimizers fail to exploit the full expressive power of large PCs. We propose to overcome such bottleneck by latent variable distillation: we leverage the less tractable but more expressive deep generative models to provide extra supervision over the latent variables of PCs. Specifically, we extract information from Transformer-based generative models to assign values to latent variables of PCs, providing guidance to PC optimizers. Experiments on both image and language modeling benchmarks (e.g., ImageNet and WikiText-2) show that latent variable distillation substantially boosts the performance of large PCs compared to their counterparts without latent variable distillation. In particular, on the image modeling benchmarks, PCs achieve competitive performance against some of the widely-used deep generative models, including variational autoencoders and flow-based models, opening up new avenues for tractable generative modeling.

On the Paradox of Learning to Reason from Data

Logical reasoning is needed in a wide range of NLP tasks. Can a BERT model be trained end-to-end to solve logical reasoning problems presented in natural language? We attempt to answer this question in a confined problem space where there exists a set of parameters that perfectly simulates logical reasoning. We make observations that seem to contradict each other: BERT attains near-perfect accuracy on in-distribution test examples while failing to generalize to other data distributions over the exact same problem space. Our study provides an explanation for this paradox: instead of learning to emulate the correct reasoning function, BERT has in fact learned statistical features that inherently exist in logical reasoning problems. We also show that it is infeasible to jointly remove statistical features from data, illustrating the difficulty of learning to reason in general. Our result naturally extends to other neural models and unveils the fundamental difference between learning to reason and learning to achieve high performance on NLP benchmarks using statistical features.

Probabilistic Generating Circuits

Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs strictly subsume many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework that unifies vastly different existing models, but also show huge potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks. We also highlight PGCs' connection to the theory of strongly Rayleigh distributions.

On the Relationship Between Probabilistic Circuits and Determinantal Point Processes

Scaling probabilistic models to large realistic problems and datasets is a key challenge in machine learning. Central to this effort is the development of tractable probabilistic models (TPMs): models whose structure guarantees efficient probabilistic inference algorithms. The current landscape of TPMs is fragmented: there exist various kinds of TPMs with different strengths and weaknesses. Two of the most prominent classes of TPMs are determinantal point processes (DPPs) and probabilistic circuits (PCs). This paper provides the first systematic study of their relationship. We propose a unified analysis and shared language for discussing DPPs and PCs. Then we establish theoretical barriers for the unification of these two families, and prove that there are cases where DPPs have no compact representation as a class of PCs. We close with a perspective on the central problem of unifying these tractable models.