A Compositional Atlas for Algebraic Circuits

Circuits based on sum-product structure have become a ubiquitous representation to compactly encode knowledge, from Boolean functions to probability distributions. By imposing constraints on the structure of such circuits, certain inference queries become tractable, such as model counting and most probable configuration. Recent works have explored analyzing probabilistic and causal inference queries as compositions of basic operators to derive tractability conditions. In this paper, we take an algebraic perspective for compositional inference, and show that a large class of queries - including marginal MAP, probabilistic answer set programming inference, and causal backdoor adjustment - correspond to a combination of basic operators over semirings: aggregation, product, and elementwise mapping. Using this framework, we uncover simple and general sufficient conditions for tractable composition of these operators, in terms of circuit properties (e.g., marginal determinism, compatibility) and conditions on the elementwise mappings. Applying our analysis, we derive novel tractability conditions for many such compositional queries. Our results unify tractability conditions for existing problems on circuits, while providing a blueprint for analysing novel compositional inference queries.

Restructuring Tractable Probabilistic Circuits

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.

Controllable Generation via Locally Constrained Resampling

Autoregressive models have demonstrated an unprecedented ability at modeling the intricacies of natural language. However, they continue to struggle with generating complex outputs that adhere to logical constraints. Sampling from a fully-independent distribution subject to a constraint is hard. Sampling from an autoregressive distribution subject to a constraint is doubly hard: We have to contend not only with the hardness of the constraint but also the distribution's lack of structure. We propose a tractable probabilistic approach that performs Bayesian conditioning to draw samples subject to a constraint. Our approach considers the entire sequence, leading to a more globally optimal constrained generation than current greedy methods. Starting from a model sample, we induce a local, factorized distribution which we can tractably condition on the constraint. To generate samples that satisfy the constraint, we sample from the conditional distribution, correct for biases in the samples and resample. The resulting samples closely approximate the target distribution and are guaranteed to satisfy the constraints. We evaluate our approach on several tasks, including LLM detoxification and solving Sudoku puzzles. We show that by disallowing a list of toxic expressions our approach is able to steer the model's outputs away from toxic generations, outperforming similar approaches to detoxification. We conclude by showing that our approach achieves a perfect accuracy on Sudoku compared to <50% for GPT4-o and Gemini 1.5.

Discrete Copula Diffusion

Discrete diffusion models have recently shown significant progress in modeling complex data, such as natural languages and DNA sequences. However, unlike diffusion models for continuous data, which can generate high-quality samples in just a few denoising steps, modern discrete diffusion models still require hundreds or even thousands of denoising steps to perform well. In this paper, we identify a fundamental limitation that prevents discrete diffusion models from achieving strong performance with fewer steps -- they fail to capture dependencies between output variables at each denoising step. To address this issue, we provide a formal explanation and introduce a general approach to supplement the missing dependency information by incorporating another deep generative model, termed the copula model. Our method does not require fine-tuning either the diffusion model or the copula model, yet it enables high-quality sample generation with significantly fewer denoising steps. When we apply this approach to autoregressive copula models, the combined model outperforms both models individually in unconditional and conditional text generation. Specifically, the hybrid model achieves better (un)conditional text generation using 8 to 32 times fewer denoising steps than the diffusion model alone. In addition to presenting an effective discrete diffusion generation algorithm, this paper emphasizes the importance of modeling inter-variable dependencies in discrete diffusion.

Where is the signal in tokenization space?

Large Language Models (LLMs) are typically shipped with tokenizers that deterministically encode text into so-called canonical token sequences, to which the LLMs assign probability values. One common assumption is that the probability of a piece of text is the probability of its canonical token sequence. However, the tokenization of a string is not unique: e.g., the Llama2 tokenizer encodes Tokens as [Tok,ens], but [Tok,en,s] also represents the same text. In this paper, we study non-canonical tokenizations. We prove that, given a string, it is computationally hard to find the most likely tokenization for an autoregressive LLM, as well as to compute the marginal probability over all possible tokenizations. We then show how the marginal is, in most cases, indistinguishable from the canonical probability. Surprisingly, we then empirically demonstrate the existence of a significant amount of signal hidden within tokenization space. Notably, by simply aggregating the probabilities of non-canonical tokenizations, we achieve improvements across a range of LLM evaluation benchmarks for a variety of architectures, including transformers and state space models.

Probabilistic Circuits for Cumulative Distribution Functions

A probabilistic circuit (PC) succinctly expresses a function that represents a multivariate probability distribution and, given sufficient structural properties of the circuit, supports efficient probabilistic inference. Typically a PC computes the probability mass (or density) function (PMF or PDF) of the distribution. We consider PCs instead computing the cumulative distribution function (CDF). We show that for distributions over binary random variables these representations (PMF and CDF) are essentially equivalent, in the sense that one can be transformed to the other in polynomial time. We then show how a similar equivalence holds for distributions over finite discrete variables using a modification of the standard encoding with binary variables that aligns with the CDF semantics. Finally we show that for continuous variables, smooth, decomposable PCs computing PDFs and CDFs can be efficiently transformed to each other by modifying only the leaves of the circuit.

On the Relationship Between Monotone and Squared Probabilistic Circuits

Probabilistic circuits are a unifying representation of functions as computation graphs of weighted sums and products. Their primary application is in probabilistic modeling, where circuits with non-negative weights (monotone circuits) can be used to represent and learn density/mass functions, with tractable marginal inference. Recently, it was proposed to instead represent densities as the square of the circuit function (squared circuits); this allows the use of negative weights while retaining tractability, and can be exponentially more compact than monotone circuits. Unfortunately, we show the reverse also holds, meaning that monotone circuits and squared circuits are incomparable in general. This raises the question of whether we can reconcile, and indeed improve upon the two modeling approaches. We answer in the positive by proposing InceptionPCs, a novel type of circuit that naturally encompasses both monotone circuits and squared circuits as special cases, and employs complex parameters. Empirically, we validate that InceptionPCs can outperform both monotone and squared circuits on image datasets.

Adaptable Logical Control for Large Language Models

Despite the success of Large Language Models (LLMs) on various tasks following human instructions, controlling model generation at inference time poses a persistent challenge. In this paper, we introduce Ctrl-G, an adaptable framework that facilitates tractable and flexible control of LLM generation to reliably follow logical constraints. Ctrl-G combines any production-ready LLM with a Hidden Markov Model, enabling LLM outputs to adhere to logical constraints represented as deterministic finite automata. We show that Ctrl-G, when applied to a TULU2-7B model, outperforms GPT3.5 and GPT4 on the task of interactive text editing: specifically, for the task of generating text insertions/continuations following logical constraints, Ctrl-G achieves over 30% higher satisfaction rate in human evaluation compared to GPT4. When applied to medium-size language models (e.g., GPT2-large), Ctrl-G also beats its counterparts for constrained generation by large margins on standard benchmarks. Additionally, as a proof-of-concept study, we experiment Ctrl-G on the Grade School Math benchmark to assist LLM reasoning, foreshadowing the application of Ctrl-G, as well as other constrained generation approaches, beyond traditional language generation tasks.

Scaling Tractable Probabilistic Circuits: A Systems Perspective

Probabilistic Circuits (PCs) are a general framework for tractable deep generative models, which support exact and efficient probabilistic inference on their learned distributions. Recent modeling and training advancements have enabled their application to complex real-world tasks. However, the time and memory inefficiency of existing PC implementations hinders further scaling up. This paper proposes PyJuice, a general GPU implementation design for PCs that improves prior art in several regards. Specifically, PyJuice is 1-2 orders of magnitude faster than existing systems (including very recent ones) at training large-scale PCs. Moreover, PyJuice consumes 2-5x less GPU memory, which enables us to train larger models. At the core of our system is a compilation process that converts a PC into a compact representation amenable to efficient block-based parallelization, which significantly reduces IO and makes it possible to leverage Tensor Cores available in modern GPUs. Empirically, PyJuice can be used to improve state-of-the-art PCs trained on image (e.g., ImageNet32) and language (e.g., WikiText, CommonGen) datasets. We further establish a new set of baselines on natural image and language datasets by benchmarking existing PC structures but with much larger sizes and more training epochs, with the hope of incentivizing future research. Code is available at https://github.com/Tractables/pyjuice.

Learning to Discretize Denoising Diffusion ODEs

Diffusion Probabilistic Models (DPMs) are powerful generative models showing competitive performance in various domains, including image synthesis and 3D point cloud generation. However, sampling from pre-trained DPMs involves multiple neural function evaluations (NFE) to transform Gaussian noise samples into images, resulting in higher computational costs compared to single-step generative models such as GANs or VAEs. Therefore, a crucial problem is to reduce NFE while preserving generation quality. To this end, we propose LD3, a lightweight framework for learning time discretization while sampling from the diffusion ODE encapsulated by DPMs. LD3 can be combined with various diffusion ODE solvers and consistently improves performance without retraining resource-intensive neural networks. We demonstrate analytically and empirically that LD3 enhances sampling efficiency compared to distillation-based methods, without the extensive computational overhead. We evaluate our method with extensive experiments on 5 datasets, covering unconditional and conditional sampling in both pixel-space and latent-space DPMs. For example, in about 5 minutes of training on a single GPU, our method reduces the FID score from 6.63 to 2.68 on CIFAR10 (7 NFE), and in around 20 minutes, decreases the FID from 8.51 to 5.03 on class-conditional ImageNet-256 (5 NFE). LD3 complements distillation methods, offering a more efficient approach to sampling from pre-trained diffusion models.