KV cache compression promises increased throughput and efficiency with negligible loss in performance.While the gains in throughput are indisputable and recent literature has indeed shown minimal degradation on particular benchmarks, in general the consequences of compression in realistic scenarios such as multiinstruction prompting have been insufficiently studied.In this paper, we identify several pitfalls that practitioners should be aware of when deploying KV cache compressed LLMs.We evaluate five KV cache compression methods (StreamingLLM, SnapKV, TOVA, H2O, and K-Norm) on Llama3 8B and Qwen2.5 14B under multi-instruction prompting with IFEval.Importantly, we show that certain instructions degrade much more rapidly with compression, effectively causing them to be completely ignored by the LLM.As a practical example, we highlight system prompt leakage as a case study, empirically demonstrating the impact of compression on leakage and general instruction-following.We identify several factors that contribute to system prompt leakage: compression method, instruction order, and KV eviction bias.We then propose simple changes to KV cache eviction policies that can reduce the impact of these factors and improve the overall performance in multi-instruction tasks.alexluchen/pitfalls-of-kv-cache-compression

Probabilistic Circuits (PCs) models are emerging popular tractable probabilistic models. Their internal connections are represented in the form of directed acyclic graphs (DAGs) with sum nodes and product nodes, ensuring their internal parameter efficiency and model expressiveness in terms of probabilistic inference. Despite these algorithmic advantages, executing PC still faces graph structure deployment issues. PyJuice on GPU with the block-sparse parallel computation methods causes a parallelism-sparsity gap, while DAG-style processing does not take advantage of the repetitive characteristics of PC internal nodes, resulting in low throughput. To address this challenge, this work proposes the ESTroM, an efficient architecture that provides novel graph-element (nodes/edges) parallelism with sparsity-aware compilation. Through analysis of the sum/product node computational requirements, ESTroM core uses compressed matrices for sum/product nodes DAG representations, edge-based dataflow for product node processing, and node-based dataflow for sum node processing. With intra-core rewind and intercore multicast optimizations, we develop a prototype ESTrom chip and a demonstrative system for a PC-based neural lossless compression application. Our ablation experiments show ESTrom offers a speed improvement of$2.11 \sim 3.79 \times$compared to the state-of-the-art DAG processing unit (DPU)-v2 with the same computing resources. Under various typical PC structures, ESTrom achieves a speedup of$18.7 \times$compared to DPU-v2 and$3.9 \times$compared to NVIDIA RTX 4090 GPU with PyJuice framework. In terms of neural lossless compression, ESTroM demonstrates a$1.39 \times$improvement in compression ratio compared to the industrial-standard Z-standard (Zstd) algorithms with the highest compression level, while offering$16.3 \sim 65.2 \times$improvement in compression speed compared to Zstd on Intel Xeon Gold 6230. In a nutshell, this work develops novel graph element parallelism and element-flow architecture theory with practical prototype chips and systems, revealing a new hardware-perspective path for the “scaling law” of emerging tractable probabilistic models.

Most research on query optimization has centered on binary join algorithms like hash join and sort-merge join. However, recent years have seen growing interest in theoretically optimal algorithms, notably Yannakakis' algorithm. These algorithms rely on join trees, which differ from the operator trees for binary joins and require new optimization techniques. We propose three approaches to constructing join trees for acyclic queries. First, we give an algorithm to enumerate all join trees of an alpha-acyclic query by edits with amortized constant delay, which forms the basis of a cost-based optimizer for acyclic joins. Second, we show that the Maximum Cardinality Search algorithm by Tarjan and Yannakakis constructs a unique shallowest join tree, rooted at any relation, for a Berge-acyclic query; this tree enables parallel execution of large join queries. Finally, we prove that any connected left-deep linear plan for a gamma-acyclic query can be converted into a join tree by a simple algorithm, allowing reuse of optimization infrastructure developed for binary joins.

We introduce Tree Decision Diagrams (TDD) as a model for Boolean functions that generalizes OBDD. They can be seen as a restriction of structured d-DNNF; that is, d-DNNF that respect a vtree T. We show that TDDs enjoy the same tractability properties as OBDD, such as model counting, enumeration, conditioning, and apply, and are more succinct. In particular, we show that CNF formulas of treewidth k can be represented by TDDs of FPT size, which is known to be impossible for OBDD. We study the complexity of compiling CNF formulas into deterministic TDDs via bottom-up compilation and relate the complexity of this approach with the notion of factor width introduced by Bova and Szeider.

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 expressive efficient 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 Inception PCs, 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 Inception PCs can outperform both monotone and squared circuits on a range of tabular and image datasets.

Current LLM pipelines account for only one possible tokenization for a given string, ignoring exponentially many alternative tokenizations during training and inference.For example, the standard Llama3 tokenization of penguin is [p,enguin], yet [peng,uin] is another perfectly valid alternative.In this paper, we show that despite LLMs being trained solely on one tokenization, they still retain semantic understanding of other tokenizations, raising questions about their implications in LLM safety.Put succinctly, we answer the following question: can we adversarially tokenize an obviously malicious string to evade safety and alignment restrictions?We show that not only is adversarial tokenization an effective yet previously neglected axis of attack, but it is also competitive against existing state-of-the-art adversarial approaches without changing the text of the harmful request.We empirically validate this exploit across three state-of-the-art LLMs and adversarial datasets, revealing a previously unknown vulnerability in subword models.github.com/RenatoGeh/advtok

Probabilistic circuits (PCs) are 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.

During inference for transformer-based large language models (LLM), prefilling is the computation of the key-value (KV) cache for input tokens in the prompt prior to autoregressive generation. For longer input prompt lengths, prefilling will incur a significant overhead on decoding time. In this work, we highlight the following pitfall of prefilling: for batches containing high-varying prompt lengths, significant computation is wasted by the standard practice of padding sequences to the maximum length. As LLMs increasingly support longer context lengths, potentially up to 10 million tokens, variations in prompt lengths within a batch become more pronounced. To address this, we propose prepacking, a simple yet effective method to optimize prefilling computation. To avoid redundant computation on pad tokens, prepacking combines prompts of varying lengths into a sequence and packs multiple sequences into a compact batch using a bin-packing algorithm. It then modifies the attention mask and positional encoding to compute multiple prefilled KV-caches for multiple prompts within a single sequence. On standard curated dataset containing prompts with varying lengths, we obtain a significant speed and memory efficiency improvements as compared to the default padding-based prefilling computation within Huggingface across a range of base model configurations and inference serving scenarios.

We propose a set of modifications that improve training time and likelihood estimation of hierarchical mixture models implemented via Probabilistic Circuits (PCs). Our proposal reduces the complexity of mutual information estimation in the structure learning step of PCs from quadratic to linear in the number of inputs, without sacrificing likelihood estimation performance on image datasets. We repurpose invertible transformations from the lossless compression community to improve likelihood estimation by a factor of up to 25% on benchmark image datasets, making PCs competitive with current standard codecs on low-resolution datasets. Despite our improvements, experiments with low- and high-resolution image datasets indicate that the advantage of lossless neural compression and PCs over standard codecs, such as WebP, disappears as the image size increases, motivating future work on practical lossless neural compression.

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. By factoring in information about the entire sequence, our approach offers better contextual awareness during constrained generation compared to current greedy approaches. 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 sample weights, 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 also show that our approach achieves a perfect accuracy on Sudoku, compared to less than $50\%$ for GPT4-o and Gemini 1.5.

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.

Diffusion Probabilistic Models (DPMs) are generative models showing competitive performance in various domains, including image synthesis and 3D point cloud generation. Sampling from pre-trained DPMs involves multiple neural function evaluations (NFEs) to transform Gaussian noise samples into images, resulting in higher computational costs compared to single-step generative models such as GANs or VAEs. Therefore, reducing the number of NFEs while preserving generation quality is crucial. To address this, we propose LD3, a lightweight framework designed to learn the optimal time discretization for sampling. LD3 can be combined with various samplers and consistently improves generation quality without having to retrain resource-intensive neural networks. We demonstrate analytically and empirically that LD3 improves sampling efficiency with much less computational overhead. We evaluate our method with extensive experiments on 7 pre-trained models, covering unconditional and conditional sampling in both pixel-space and latent-space DPMs. We achieve FIDs of 2.38 (10 NFE), and 2.27 (10 NFE) on unconditional CIFAR10 and AFHQv2 in 5-10 minutes of training. LD3 offers an efficient approach to sampling from pre-trained diffusion models. Code is available at https://github.com/vinhsuhi/LD3.

Marginalization – summing a function over all assignments to a subset of its inputs – is a fundamental computational problem with applications from probabilistic inference to formal verification. Despite its computational hardness in general, there exist many classes of functions (e.g., probabilistic models) for which marginalization remains tractable, and they can all be commonly expressed by arithmetic circuits computing multilinear polynomials. This raises the question, can all functions with polynomial time marginalization algorithms be succinctly expressed by such circuits? We give a negative answer, exhibiting simple functions with tractable marginalization yet no efficient representation by known models, assuming $\\mathsf{FP} \\neq \#\\mathsf{P}$ (an assumption implied by $\\mathsf{P} \\neq \\mathsf{NP}$). To this end, we identify a hierarchy of complexity classes corresponding to stronger forms of marginalization, all of which are efficiently computable on the known circuit models. We conclude with a completeness result, showing that whenever there is an efficient real RAM performing virtual evidence marginalization for a function, then there are small arithmetic circuits for that function’s multilinear representation.

Non-autoregressive (NAR) generative models are valuable because they can handle diverse conditional generation tasks in a more principled way than their autoregressive (AR) counterparts, which are constrained by sequential dependency requirements. Recent advancements in NAR models, such as diffusion language models, have demonstrated superior performance in unconditional generation compared to AR models (e.g., GPTs) of similar sizes. However, such improvements do not always lead to improved conditional generation performance. We show that a key reason for this gap is the difficulty in generalizing to conditional probability queries unseen during training. As a result, strong unconditional generation performance does not guarantee high-quality conditional generation. This paper proposes Tractable Transformers (Tracformer), a Transformer-based generative model that is more robust to different conditional generation tasks. Unlike existing models that rely solely on global contextual features derived from full inputs, Tracformers incorporate a sparse Transformer encoder to capture both local and global contextual information. This information is routed through a decoder for conditional generation. Empirical results demonstrate that Tracformers achieve state-of-the-art conditional generation performance on text modeling compared to recent diffusion and AR model baselines.

As large language models (LMs) advance, there is an increasing need to control their outputs to align with human values (e.g., detoxification) or desired attributes (e.g., personalization, topic). However, autoregressive models focus on next-token predictions and struggle with global properties that require looking ahead. Existing solutions either post-train LMs for each new attribute—expensive and inflexible—or approximate the Expected Attribute Probability (EAP) of future sequences by sampling or training, which is slow and unreliable for rare attributes. We introduce **TRACE** (Tractable Probabilistic Reasoning for Adaptable Controllable gEneration), a novel framework that efficiently computes EAP and adapts to new attributes through tractable *probabilistic* reasoning and lightweight *control*. TRACE distills a Hidden Markov Model (HMM) from an LM and pairs it with a small classifier to estimate attribute probabilities, enabling exact EAP computation over the HMM’s predicted futures. This EAP is then used to reweigh the LM’s next-token probabilities for globally compliant continuations. Empirically, TRACE achieves state-of-the-art detoxification results with only 20% decoding overhead, yields 76 low-resource personalized LMs within seconds, and seamlessly extends to composite attributes.

Probabilistic Circuits (PCs) are tractable representations of probability distributions allowing for exact and efficient computation of likelihoods and marginals. Recent advancements have improved the scalability of PCs either by leveraging their sparse properties or through the use of tensorized operations for better hardware utilization. However, no existing method fully exploits both aspects simultaneously. In this paper, we propose a novel sparse and structured parameterization for the sum blocks in PCs. By replacing dense matrices with sparse Monarch matrices, we significantly reduce the memory and computation costs, enabling unprecedented scaling of PCs. From a theory perspective, our construction arises naturally from circuit multiplication; from a practical perspective, compared to previous efforts on scaling up tractable probabilistic models, our approach not only achieves state-of-the-art generative modeling performance on challenging benchmarks like Text8, LM1B and ImageNet, but also demonstrates superior scaling behavior, achieving the same performance with substantially less compute as measured by the number of floating-point operations (FLOPs) during training.

The generation speed of LLMs are bottlenecked by autoregressive decoding, where tokens are predicted sequentially one by one. Alternatively, diffusion large language models (dLLMs) theoretically allow for parallel token generation, but in practice struggle to achieve the speed of autoregressive models without significantly sacrificing quality. We therefore introduce adaptive parallel decoding (APD), a novel method that dynamically adjusts the number of tokens sampled in parallel. We achieve this by defining a multiplicative mixture between the dLLM marginal probabilities and the joint probability of sequences under a small auxiliary autoregressive model. This inverts the standard setup of speculative decoding, where the goal is to sample from a large autoregressive verifier by drafting from a smaller model. We further optimize APD by enabling KV caching and limiting the size of the masked input. Altogether, our method puts forward three tunable parameters to flexibly tradeoff throughput and quality. We show that APD provides markedly higher throughput with minimal quality degradations on downstream benchmarks.

Masked generative models (MGMs) have emerged as a powerful framework for image synthesis, combining parallel decoding with strong bidirectional context modeling. However, generating high-quality samples typically requires many iterative decoding steps, resulting in high inference costs. A straightforward way to speed up generation is by decoding more tokens in each step, thereby reducing the total number of steps. However, when many tokens are decoded simultaneously, the model can only estimate the univariate marginal distributions independently, failing to capture the dependency among them. As a result, reducing the number of steps significantly compromises generation fidelity. In this work, we introduce ReCAP (Reused Context-Aware Prediction), a plug-and-play module that accelerates inference in MGMs by constructing low-cost steps via reusing feature embeddings from previously decoded context tokens. ReCAP interleaves standard full evaluations with lightweight steps that cache and reuse context features, substantially reducing computation while preserving the benefits of fine-grained, iterative generation. We demonstrate its effectiveness on top of three representative MGMs (MaskGIT, MAGE, and MAR), including both discrete and continuous token spaces and covering diverse architectural designs. In particular, on ImageNet256 class-conditional generation, ReCAP achieves up to 2.4$\times$ faster inference than the base model with minimal performance drop, and consistently delivers better efficiency–fidelity trade-offs under various generation settings.

In domains such as molecular and protein generation, physical systems exhibit inherent symmetries that are critical to model. Two main strategies have emerged for learning invariant distributions: designing equivariant network architectures and using data augmentation to approximate equivariance. While equivariant architectures preserve symmetry by design, they often involve greater complexity and pose optimization challenges. Data augmentation, on the other hand, offers flexibility but may fall short in fully capturing symmetries. Our framework enhances both approaches by reducing training variance and providing a provably lower-variance gradient estimator. We achieve this by interpreting data augmentation as a Monte Carlo estimator of the training gradient and applying Rao–Blackwellization. This leads to more stable optimization, faster convergence, and reduced variance, all while requiring only a single forward and backward pass per sample. We also present a practical implementation of this estimator—incorporating the loss and sampling procedure—through a method we call Orbit Diffusion. Theoretically, we guarantee that our loss admits equivariant minimizers. Empirically, Orbit Diffusion achieves state-of-the-art results on GEOM-QM9 for molecular conformation generation, improves crystal structure prediction, and advances text-guided crystal generation on the Perov-5 and MP-20 benchmarks. Additionally, it enhances protein designability in protein structure generation. Code is available at https://github.com/vinhsuhi/Orbit-Diffusion.git.

Property-based testing validates software against an executable specification by evaluating it on randomly generated inputs. The standard way that PBT users generate test inputs is via generators that describe how to sample test inputs through random choices. To achieve a good distribution over test inputs, users must tune their generators, i.e., decide on the weights of these individual random choices. Unfortunately, it is very difficult to understand how to choose individual generator weights in order to achieve a desired distribution, so today this process is tedious and limits the distributions that can be practically achieved. In this paper, we develop techniques for the automatic and offline tuning of generators. Given a generator with undetermined symbolic weights and an objective function, our approach automatically learns values for these weights that optimize for the objective. We describe useful objective functions that allow users to (1) target desired distributions and (2) improve the diversity and validity of their test cases. We have implemented our approach in a novel discrete probabilistic programming system, Loaded Dice , that supports differentiation and parameter learning, and use it as a language for generators. We empirically demonstrate that our approach is effective at optimizing generator distributions according to the specified objective functions. We also perform a thorough evaluation on PBT benchmarks, demonstrating that, when automatically tuned for diversity and validity, the generators exhibit a 3.1–7.4× speedup in bug finding.

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 noncanonical 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 noncanonical tokenizations, we achieve improvements across a range of LLM evaluation benchmarks for a variety of architectures, including transformers and state space models.

Diffusion models are the current state of the art for generating photorealistic images. Controlling the sampling process for constrained image generation tasks such as inpainting, however, remains challenging since exact conditioning on such constraints is intractable. While existing methods use various techniques to approximate the constrained posterior, this paper proposes to exploit the ability of Tractable Probabilistic Models (TPMs) to exactly and efficiently compute the constrained posterior, and to leverage this signal to steer the denoising process of diffusion models. Specifically, this paper adopts a class of expressive TPMs termed Probabilistic Circuits (PCs). Building upon prior advances, we further scale up PCs and make them capable of guiding the image generation process of diffusion models. Empirical results suggest that our approach can consistently improve the overall quality and semantic coherence of inpainted images across three natural image datasets (i.e., CelebA-HQ, ImageNet, and LSUN) with only ~10% additional computational overhead brought by the TPM. Further, with the help of an image encoder and decoder, our method can readily accept semantic constraints on specific regions of the image, which opens up the potential for more controlled image generation tasks. In addition to proposing a new framework for constrained image generation, this paper highlights the benefit of more tractable models and motivates the development of expressive TPMs.

Message-passing graph neural networks (MPNNs) emerged as powerful tools for processing graph-structured input. However, they operate on a fixed input graph structure, ignoring potential noise and missing information. Furthermore, their local aggregation mechanism can lead to problems such as over-squashing and limited expressive power in capturing relevant graph structures. Existing solutions to these challenges have primarily relied on heuristic methods, often disregarding the underlying data distribution. Hence, devising principled approaches for learning to infer graph structures relevant to the given prediction task remains an open challenge. In this work, leveraging recent progress in exact and differentiable k-subset sampling, we devise probabilistically rewired MPNNs (PR-MPNNs), which learn to add relevant edges while omitting less beneficial ones. For the first time, our theoretical analysis explores how PR-MPNNs enhance expressive power, and we identify precise conditions under which they outperform purely randomized approaches. Empirically, we demonstrate that our approach effectively mitigates issues like over-squashing and under-reaching. In addition, on established real-world datasets, our method exhibits competitive or superior predictive performance compared to traditional MPNN models and recent graph transformer architectures.

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.

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.

A popular paradigm for offline Reinforcement Learning (RL) tasks is to first fit the offline trajectories to a sequence model, and then prompt the model for actions that lead to high expected return. In addition to obtaining accurate sequence models, this paper highlights that tractability, the ability to exactly and efficiently answer various probabilistic queries, plays an important role in offline RL. Specifically, due to the fundamental stochasticity from the offline data-collection policies and the environment dynamics, highly non-trivial conditional/constrained generation is required to elicit rewarding actions. While it is still possible to approximate such queries, we observe that such crude estimates undermine the benefits brought by expressive sequence models. To overcome this problem, this paper proposes Trifle (Tractable Inference for Offline RL), which leverages modern tractable generative models to bridge the gap between good sequence models and high expected returns at evaluation time. Empirically, Trifle achieves $7$ state-of-the-art scores and the highest average scores in $9$ Gym-MuJoCo benchmarks against strong baselines. Further, Trifle significantly outperforms prior approaches in stochastic environments and safe RL tasks with minimum algorithmic modifications.

Despite the success of Large Language Models (LLMs) on various tasks following human instructions, controlling model generation to follow strict constraints at inference time poses a persistent challenge. In this paper, we introduce Ctrl-G, a neuro-symbolic framework that enables tractable and adaptable control of LLM generation to follow logical constraints reliably. Ctrl-G combines any production-ready LLM with a Hidden Markov Model (HMM), guiding LLM outputs to adhere to logical constraints represented as deterministic finite automata. We show that Ctrl-G, when a TULU2-7B model is coupled with a 2B-parameter HMM, outperforms GPT4 in text editing: on the task of generating text insertions/continuations following logical constraints, our approach achieves over 30% higher satisfaction rate in human evaluation. When applied to medium-size language models (e.g., GPT2-large), Ctrl-G also beats its counterparts on standard benchmarks by large margins. Additionally, as a proof-of-concept study, we use Ctrl-G to assist LLM reasoning on the GSM benchmark, foreshadowing the application of Ctrl-G, as well as other constrained generation approaches, beyond traditional language generation tasks.

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.

Probabilistic programming languages (PPLs) are an expressive means for creating and reasoning about probabilistic models. Unfortunately hybrid probabilistic programs that involve both continuous and discrete structures are not well supported by today’s PPLs. In this paper we develop a new approximate inference algorithm for hybrid probabilistic programs that first discretizes the continuous distributions and then performs discrete inference on the resulting program. The key novelty is a form of discretization that we call bit blasting , which uses a binary representation of numbers such that a domain of 2 b discretized points can be succinctly represented as a discrete probabilistic program over poly b Boolean random variables. Surprisingly, we prove that many common continuous distributions can be bit blasted in a manner that incurs no loss of accuracy over an explicit discretization and supports efficient probabilistic inference. We have built a probabilistic programming system for hybrid programs called HyBit , which employs bit blasting followed by discrete probabilistic inference. We empirically demonstrate the benefits of our approach over existing sampling-based and symbolic inference approaches

This paper develops a novel methodology to simultaneously learn a neural network and extract generalized logic rules. Different from prior neural-symbolic methods that require background knowledge and candidate logical rules to be provided, we aim to induce task semantics with minimal priors. This is achieved by a two-step learning framework that iterates between optimizing neural predictions of task labels and searching for a more accurate representation of the hidden task semantics. Notably, supervision works in both directions: (partially) induced task semantics guide the learning of the neural network and induced neural predictions admit an improved semantic representation. We demonstrate that our proposed framework is capable of achieving superior out-of-distribution generalization performance on two tasks: (i) learning multi-digit addition, where it is trained on short sequences of digits and tested on long sequences of digits; (ii) predicting the optimal action in the Tower of Hanoi, where the model is challenged to discover a policy independent of the number of disks in the puzzle.

With the increased use of machine learning systems for decision making, questions about the fairness properties of such systems start to take center stage. Most existing work on algorithmic fairness assume complete observation of features at prediction time, as is the case for popular notions like statistical parity and equal opportunity. However, this is not sufficient for models that can make predictions with partial observation as we could miss patterns of bias and incorrectly certify a model to be fair. To address this, a recently introduced notion of fairness asks whether the model exhibits any discrimination pattern, in which an individual—characterized by (partial) feature observations—receives vastly different decisions merely by disclosing one or more sensitive attributes such as gender and race. By explicitly accounting for partial observations, this provides a much more fine-grained notion of fairness. In this paper, we propose an algorithm to search for discrimination patterns in a general class of probabilistic models, namely probabilistic circuits. Previously, such algorithms were limited to naive Bayes classifiers which make strong independence assumptions; by contrast, probabilistic circuits provide a unifying framework for a wide range of tractable probabilistic models and can even be compiled from certain classes of Bayesian networks and probabilistic programs, making our method much more broadly applicable. Furthermore, for an unfair model, it may be useful to quickly find discrimination patterns and distill them for better interpretability. As such, we also propose a sampling-based approach to more efficiently mine discrimination patterns, and introduce new classes of patterns such as minimal, maximal, and Pareto optimal patterns that can effectively summarize exponentially many discrimination patterns.

Numerous neuro-symbolic approaches have recently been proposed typically with the goal of adding symbolic knowledge to the output layer of a neural network. Ideally, such losses maximize the probability that the neural network’s predictions satisfy the underlying domain. Unfortunately, this type of probabilistic inference is often computationally infeasible. Neuro-symbolic approaches therefore commonly resort to fuzzy approximations of this probabilistic objective, sacrificing sound probabilistic semantics, or to sampling which is very seldom feasible. We approach the problem by first assuming the constraint decomposes conditioned on the features learned by the network. We iteratively strengthen our approximation, restoring the dependence between the constraints most responsible for degrading the quality of the approximation. This corresponds to computing the mutual information between pairs of constraints conditioned on the network’s learned features, and may be construed as a measure of how well aligned the gradients of two distributions are. We show how to compute this efficiently for tractable circuits. We test our approach on three tasks: predicting a minimum-cost path in Warcraft, predicting a minimum-cost perfect matching, and solving Sudoku puzzles, observing that it improves upon the baselines while sidestepping intractability.

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.

Probabilistic Circuits (PCs) are a general and unified computational framework for tractable probabilistic models that support efficient computation of various inference tasks (e.g., computing marginal probabilities). Towards enabling such reasoning capabilities in complex real-world tasks, Liu et al. (2022) propose to distill knowledge (through latent variable assignments) from less tractable but more expressive deep generative models. However, it is still unclear what factors make this distillation work well. In this paper, we theoretically and empirically discover that the performance of a PC can exceed that of its teacher model. Therefore, instead of performing distillation from the most expressive deep generative model, we study what properties the teacher model and the PC should have in order to achieve good distillation performance. This leads to a generic algorithmic improvement as well as other data-type-specific ones over the existing latent variable distillation pipeline. Empirically, we outperform SoTA TPMs by a large margin on challenging image modeling benchmarks. In particular, on ImageNet32, PCs achieve 4.06 bits-per-dimension, which is only 0.34 behind variational diffusion models (Kingma et al., 2021).

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 (TPMs) to impose lexical constraints in autoregressive text generation models, which we refer to as GeLaTo (Generating Language with Tractable Constraints). To demonstrate the effectiveness of this framework, we use distilled hidden Markov models, where we can efficiently compute ${\Pr}(\text{text} | \alpha)$, to guide autoregressive generation from GPT2. GeLaTo achieves state-of-the-art performance on challenging benchmarks for constrained text generation (e.g., CommonGen), beating various 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 TPMs.

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 (e.g. T5) and unveils the fundamental difference between learning to reason and learning to achieve high performance on NLP benchmarks using statistical features.

Neuro-symbolic AI bridges the gap between purely symbolic and neural approaches to learning. This often requires maximizing the likelihood of a symbolic constraint w.r.t the neural network's output distribution. Such output distributions are typically assumed to be fully-factorized. This limits the applicability of neuro-symbolic learning to the more expressive auto-regressive distributions, e.g., transformers. Under such distributions, computing the likelihood of even simple constraints is #P-hard. Instead of attempting to enforce the constraint on the entire likelihood distribution, we propose to do so on a random, local approximation thereof. More precisely, we approximate the likelihood of the constraint with the pseudolikelihood of the constraint centered around a model sample. Our approach is factorizable, allowing us to reuse solutions to sub-problems---a main tenet for the efficient computation of neuro-symbolic losses. It also provides a local, high fidelity approximation of the likelihood: it exhibits low entropy and KL-divergence around the model sample. We tested our approach on Sudoku and shortest-path prediction cast as auto-regressive generation, and observe that we greatly improve upon the base model's ability to predict logically-consistent outputs. We also tested our approach on the task of detoxifying large language models. We observe that using a simple constraint disallowing a list of toxic words, we are able to steer the model's outputs away from toxic generations, achieving SoTA compared to previous approaches.

High-quality labels are often very scarce, whereas unlabeled data with inferred weak labels occurs more naturally. In many cases, these weak labels dictate the frequency of each respective class over a set of instances. In this paper, we develop a unified approach to learning from such weakly-labeled data, which we call *count-based weakly-supervised learning*. At the heart of our approach is the ability to compute the probability of exactly $k$ out of $n$ outputs being set to true. This computation is differentiable, exact, and efficient. Building upon the previous computation, we derive a *count loss* penalizing the model for deviations in its distribution from an arithmetic constraint defined over label counts.

Bayesian neural networks (BNNs) provide a formalism to quantify and calibrate uncertainty in deep learning. Current inference approaches for BNNs often resort to few-sample estimation for scalability, which can harm predictive performance, while its alternatives tend to be computationally prohibitively expensive. We tackle this challenge by revealing a previously unseen connection between inference on BNNs and volume computation problems. With this observation, we introduce a novel collapsed inference scheme that performs Bayesian model averaging using collapsed samples. It improves over a Monte-Carlo sample by limiting sampling to a subset of the network weights while pairing it with some closed-form conditional distribution over the rest. A collapsed sample represents uncountably many models drawn from the approximate posterior and thus yields higher sample efficiency. Further, we show that the marginalization of a collapsed sample can be solved analytically and efficiently despite the non-linearity of neural networks by leveraging existing volume computation solvers. Our proposed use of collapsed samples achieves a balance between scalability and accuracy. On various regression and classification tasks, our collapsed Bayesian deep learning approach demonstrates significant improvements over existing methods and sets a new state of the art in terms of uncertainty estimation as well as predictive performance.

Distributions on integers are ubiquitous in probabilistic modeling but remain challenging for many of today’s probabilistic programming languages (PPLs). The core challenge comes from discrete structure: many of today’s PPL inference strategies rely on enumeration, sampling, or differentiation in order to scale, which fail for high-dimensional complex discrete distributions involving integers. Our insight is that there is structure in arithmetic that these approaches are not using. We present a binary encoding strategy for discrete distributions that exploits the rich logical structure of integer operations like summation and comparison. We leverage this structured encoding with knowledge compilation to perform exact probabilistic inference, and show that this approach scales to much larger integer distributions with arithmetic.

Deep learning excels at learning task information from large amounts of data, but struggles with learning from declarative high-level knowledge that can be more succinctly expressed directly. In this work, we introduce PYLON, a neuro-symbolic training framework that builds on PyTorch to augment procedurally trained models with declaratively specified knowledge. PYLON lets users programmatically specify constraints as Python functions and compiles them into a differentiable loss, thus training predictive models that fit the data whilst satisfying the specified constraints. PYLON includes both exact as well as approximate compilers to efficiently compute the loss, employing fuzzy logic, sampling methods, and circuits, ensuring scalability even to complex models and constraints. Crucially, a guiding principle in designing PYLON is the ease with which any existing deep learning codebase can be extended to learn from constraints in a few lines code: a function that expresses the constraint, and a single line to compile it into a loss. Our demo comprises of models in NLP, computer vision, logical games, and knowledge graphs that can be interactively trained using constraints as supervision.

Probabilistic circuits (PCs) are a class of tractable probabilistic models that allow efficient, often linear-time, inference of queries such as marginals and most probable explanations (MPE). However, marginal MAP, which is central to many decision-making problems, remains a hard query for PCs unless they satisfy highly restrictive structural constraints. In this paper, we develop a pruning algorithm that removes parts of the PC that are irrelevant to a marginal MAP query, shrinking the PC while maintaining the correct solution. This pruning technique is so effective that we are able to build a marginal MAP solver based solely on iteratively transforming the circuit—no search is required. We empirically demonstrate the efficacy of our approach on real-world datasets.

We design a predictive layer for structured-output prediction (SOP) that can be plugged into any neural network guaranteeing its predictions are consistent with a set of predefined symbolic constraints. Our Semantic Probabilistic Layer (SPL) can model intricate correlations, and hard constraints, over a structured output space all while being amenable to end-to-end learning via maximum likelihood.SPLs combine exact probabilistic inference with logical reasoning in a clean and modular way, learning complex distributions and restricting their support to solutions of the constraint. As such, they can faithfully, and efficiently, model complex SOP tasks beyond the reach of alternative neuro-symbolic approaches. We empirically demonstrate that SPLs outperform these competitors in terms of accuracy on challenging SOP tasks such as hierarchical multi-label classification, pathfinding and preference learning, while retaining perfect constraint satisfaction.

Probabilistic circuits (PCs) are a tractable representation of probability distributions allowing for exact and efficient computation of likelihoods and marginals. There has been significant recent progress on improving the scale and expressiveness of PCs. However, PC training performance plateaus as model size increases. We discover that most capacity in existing large PC structures is wasted: fully-connected parameter layers are only sparsely used. We propose two operations: pruning and growing, that exploit the sparsity of PC structures. Specifically, the pruning operation removes unimportant sub-networks of the PC for model compression and comes with theoretical guarantees. The growing operation increases model capacity by increasing the dimensions of latent states. By alternatingly applying pruning and growing, we increase the capacity that is meaningfully used, allowing us to significantly scale up PC learning. Empirically, our learner achieves state-of-the-art likelihoods on MNIST-family image datasets and an Penn Tree Bank language data compared to other PC learners and less tractable deep generative models such as flow-based models and variational autoencoders (VAEs).

In structured output prediction, the goal is to jointly predict several output variables that together encode a structured object – a path in a graph, an entity-relation triple, or an ordering of objects. Such a large output space makes learning hard and requires vast amounts of labeled data. Different approaches leverage alternate sources of supervision. One approach – entropy regularization – posits that decision boundaries should lie in low-probability regions. It extracts supervision from unlabeled examples, but remains agnostic to the structure of the output space. Conversely, neuro-symbolic approaches exploit the knowledge that not every prediction corresponds to a valid structure in the output space. Yet, they do not further restrict the learned output distribution.This paper introduces a framework that unifies both approaches. We propose a loss, neuro-symbolic entropy regularization, that encourages the model to confidently predict a valid object. It is obtained by restricting entropy regularization to the distribution over only the valid structures. This loss can be computed efficiently when the output constraint is expressed as a tractable logic circuit. Moreover, it seamlessly integrates with other neuro-symbolic losses that eliminate invalid predictions. We demonstrate the efficacy of our approach on a series of semi-supervised and fully-supervised structured-prediction experiments, where it leads to models whose predictions are more accurate as well as more likely to be valid.

SHAP explanations are a popular feature-attribution mechanism for explainable AI. They use game-theoretic notions to measure the influence of individual features on the prediction of a machine learning model. Despite a lot of recent interest from both academia and industry, it is not known whether SHAP explanations of common machine learning models can be computed efficiently. In this paper, we establish the complexity of computing the SHAP explanation in three important settings. First, we consider fully-factorized data distributions, and show that the complexity of computing the SHAP explanation is the same as the complexity of computing the expected value of the model. This fully-factorized setting is often used to simplify the SHAP computation, yet our results show that the computation can be intractable for commonly used models such as logistic regression. Going beyond fully-factorized distributions, we show that computing SHAP explanations is already intractable for a very simple setting: computing SHAP explanations of trivial classifiers over naive Bayes distributions. Finally, we show that even computing SHAP over the empirical distribution is #P-hard.

SHAP explanations are a popular feature-attribution mechanism for explainable AI. They use game-theoretic notions to measure the influence of individual features on the prediction of a machine learning model. Despite a lot of recent interest from both academia and industry, it is not known whether SHAP explanations of common machine learning models can be computed efficiently. In this paper, we establish the complexity of computing the SHAP explanation in three important settings. First, we consider fully-factorized data distributions, and show that the complexity of computing the SHAP explanation is the same as the complexity of computing the expected value of the model. This fully-factorized setting is often used to simplify the SHAP computation, yet our results show that the computation can be intractable for commonly used models such as logistic regression. Going beyond fully-factorized distributions, we show that computing SHAP explanations is already intractable for a very simple setting: computing SHAP explanations of trivial classifiers over naive Bayes distributions. Finally, we show that even computing SHAP over the empirical distribution is #P-hard.

Machine learning systems are increasingly being used to make impactful decisions such as loan applications and criminal justice risk assessments, and as such, ensuring fairness of these systems is critical. This is often challenging as the labels in the data are biased. This paper studies learning fair probability distributions from biased data by explicitly modeling a latent variable that represents a hidden, unbiased label. In particular, we aim to achieve demographic parity by enforcing certain independencies in the learned model. We also show that group fairness guarantees are meaningful only if the distribution used to provide those guarantees indeed captures the real-world data. In order to closely model the data distribution, we employ probabilistic circuits, an expressive and tractable probabilistic model, and propose an algorithm to learn them from incomplete data. We show on real-world datasets that our approach not only is a better model of how the data was generated than existing methods but also achieves competitive accuracy. Moreover, we also evaluate our approach on a synthetic dataset in which observed labels indeed come from fair labels but with added bias, and demonstrate that the fair labels are successfully retrieved.

Juice is an open-source Julia package providing tools for logic and probabilistic reasoning and learning based on logic circuits (LCs) and probabilistic circuits (PCs). It provides a range of efficient algorithms for probabilistic inference queries, such as computing marginal probabilities (MAR), as well as many more advanced queries. Certain structural circuit properties are needed to achieve this tractability, which Juice helps validate. Additionally, it supports several parameter and structure learning algorithms proposed in the recent literature. By leveraging parallelism (on both CPU and GPU), Juice provides a fast implementation of circuit-based algorithms, which makes it suitable for tackling large-scale datasets and models.

Due to the unreliability and limited capacity of existing quantum computer prototypes, quantum circuit simulation continues to be a vital tool for validating next generation quantum computers and for studying variational quantum algorithms, which are among the leading candidates for useful quantum computation. Existing quantum circuit simulators do not address the common traits of variational algorithms, namely: 1) their ability to work with noisy qubits and operations, 2) their repeated execution of the same circuits but with different parameters, and 3) the fact that they sample from circuit final wavefunctions to drive a classical optimization routine. We present a quantum circuit simulation toolchain based on logical abstractions targeted for simulating variational algorithms. Our proposed toolchain encodes quantum amplitudes and noise probabilities in a probabilistic graphical model, and it compiles the circuits to logical formulas that support efficient repeated simulation of and sampling from quantum circuits for different parameters. Compared to state-of-the-art state vector and density matrix quantum circuit simulators, our simulation approach offers greater performance when sampling from noisy circuits with at least eight to 20 qubits and with around 12 operations on each qubit, making the approach ideal for simulating near-term variational quantum algorithms. And for simulating noise-free shallow quantum circuits with 32 qubits, our simulation approach offers a 66× reduction in sampling cost versus quantum circuit simulation techniques based on tensor network contraction.

Abstract We revisit the symbolic verification of Markov chains with respect to finite horizon reachability properties. The prevalent approach iteratively computes step-bounded state reachability probabilities. By contrast, recent advances in probabilistic inference suggest symbolically representing all horizon-length paths through the Markov chain. We ask whether this perspective advances the state-of-the-art in probabilistic model checking. First, we formally describe both approaches in order to highlight their key differences. Then, using these insights we developRubicon, a tool that transpilesPrismmodels to the probabilistic inference tool . Finally, we demonstrate better scalability compared to probabilistic model checkers on selected benchmarks. All together, our results suggest that probabilistic inference is a valuable addition to the probabilistic model checking portfolio, withRubiconas a first step towards integrating both perspectives.

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 are strictly more expressive efficient than 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 great 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.

Understanding the behavior of learned classifiers is an important task, and various black-box explanations, logical reasoning approaches, and model-specific methods have been proposed. In this paper, we introduce probabilistic sufficient explanations, which formulate explaining an instance of classification as choosing the "simplest" subset of features such that only observing those features is "sufficient" to explain the classification. That is, sufficient to give us strong probabilistic guarantees that the model will behave similarly when all features are observed under the data distribution. In addition, we leverage tractable probabilistic reasoning tools such as probabilistic circuits and expected predictions to design a scalable algorithm for finding the desired explanations while keeping the guarantees intact. Our experiments demonstrate the effectiveness of our algorithm in finding sufficient explanations, and showcase its advantages compared to Anchors and logical explanations.

Probabilistic Circuits (PCs) are a promising avenue for probabilistic modeling. They combine advantages of probabilistic graphical models (PGMs) with those of neural networks (NNs). Crucially, however, they are tractable probabilistic models, supporting efficient and exact computation of many probabilistic inference queries, such as marginals and MAP. Further, since PCs are structured computation graphs, they can take advantage of deep-learning-style parameter updates, which greatly improves their scalability. However, this innovation also makes PCs prone to overfitting, which has been observed in many standard benchmarks. Despite the existence of abundant regularization techniques for both PGMs and NNs, they are not effective enough when applied to PCs. Instead, we re-think regularization for PCs and propose two intuitive techniques, data softening and entropy regularization, that both take advantage of PCs' tractability and still have an efficient implementation as a computation graph. Specifically, data softening provides a principled way to add uncertainty in datasets in closed form, which implicitly regularizes PC parameters. To learn parameters from a softened dataset, PCs only need linear time by virtue of their tractability. In entropy regularization, the exact entropy of the distribution encoded by a PC can be regularized directly, which is again infeasible for most other density estimation models. We show that both methods consistently improve the generalization performance of a wide variety of PCs. Moreover, when paired with a simple PC structure, we achieved state-of-the-art results on 10 out of 20 standard discrete density estimation benchmarks. Open-source code and experiments are available at https://github.com/UCLA-StarAI/Tractable-PC-Regularization.

Circuit representations are becoming the lingua franca to express and reason about tractable generative and discriminative models. In this paper, we show how complex inference scenarios for these models that commonly arise in machine learning---from computing the expectations of decision tree ensembles to information-theoretic divergences of sum-product networks---can be represented in terms of tractable modular operations over circuits. Specifically, we characterize the tractability of simple transformations---sums, products, quotients, powers, logarithms, and exponentials---in terms of sufficient structural constraints of the circuits they operate on, and present novel hardness results for the cases in which these properties are not satisfied. Building on these operations, we derive a unified framework for reasoning about tractable models that generalizes several results in the literature and opens up novel tractable inference scenarios.

Computing the expectation of kernel functions is a ubiquitous task in machine learning, with applications from classical support vector machines to exploiting kernel embeddings of distributions in probabilistic modeling, statistical inference, causal discovery, and deep learning. In all these scenarios, we tend to resort to Monte Carlo estimates as expectations of kernels are intractable in general. In this work, we characterize the conditions under which we can compute expected kernels exactly and efficiently, by leveraging recent advances in probabilistic circuit representations. We first construct a circuit representation for kernels and propose an approach to such tractable computation. We then demonstrate possible advancements for kernel embedding frameworks by exploiting tractable expected kernels to derive new algorithms for two challenging scenarios: 1) reasoning under missing data with kernel support vector regressors; 2) devising a collapsed black-box importance sampling scheme. Finally, we empirically evaluate both algorithms and show that they outperform standard baselines on a variety of datasets.

As machine learning is increasingly used to make real-world decisions, recent research efforts aim to define and ensure fairness in algorithmic decision making. Existing methods often assume a fixed set of observable features to define individuals, but lack a discussion of certain features not being observed at test time. In this paper, we study fairness of naive Bayes classifiers, which allow partial observations. In particular, we introduce the notion of a discrimination pattern, which refers to an individual receiving different classifications depending on whether some sensitive attributes were observed. Then a model is considered fair if it has no such pattern. We propose an algorithm to discover and mine for discrimination patterns in a naive Bayes classifier, and show how to learn maximum-likelihood parameters subject to these fairness constraints. Our approach iteratively discovers and eliminates discrimination patterns until a fair model is learned. An empirical evaluation on three real-world datasets demonstrates that we can remove exponentially many discrimination patterns by only adding a small fraction of them as constraints.

Probabilistic circuits (PCs) are a promising avenue for probabilistic modeling, as they permit a wide range of exact and efficient inference routines. Recent “deep-learning-style” implementations of PCs strive for a better scalability, but are still difficult to train on real-world data, due to their sparsely connected computational graphs. In this paper, we propose Einsum Networks (EiNets), a novel implementation design for PCs, improving prior art in several regards. At their core, EiNets combine a large number of arithmetic operations in a single monolithic einsum-operation, leading to speedups and memory savings of up to two orders of magnitude, in comparison to previous implementations. As an algorithmic contribution, we show that the implementation of Expectation-Maximization (EM) can be simplified for PCs, by leveraging automatic differentiation. Furthermore, we demonstrate that EiNets scale well to datasets which were previously out of reach, such as SVHN and CelebA, and that they can be used as faithful generative image models.

Weighted model integration (WMI) is an appealing framework for probabilistic inference: it allows for expressing the complex dependencies in real-world problems, where variables are both continuous and discrete, via the language of Satisfiability Modulo Theories (SMT), as well as to compute probabilistic queries with complex logical and arithmetic constraints. Yet, existing WMI solvers are not ready to scale to these problems. They either ignore the intrinsic dependency structure of the problem entirely, or they are limited to overly restrictive structures. To narrow this gap, we derive a factorized WMI computation enabling us to devise a scalable WMI solver based on message passing, called MP-WMI. Namely, MP-WMI is the first WMI solver that can (i) perform exact inference on the full class of tree-structured WMI problems, and (ii) perform inter-query amortization, e.g., to compute all marginal densities simultaneously. Experimental results show that our solver dramatically outperforms the existingWMI solvers on a large set of benchmarks.

Methods that learn the structure of Probabilistic Sentential Decision Diagrams (PSDD) from data have achieved state-of-the-art performance in tractable learning tasks. These methods learn PSDDs incrementally by optimizing the likelihood of the induced probability distribution given available data and are thus robust against missing values, a relevant trait to address the challenges of embedded applications, such as failing sensors and resource constraints. However PSDDs are outperformed by discriminatively trained models in classification tasks. In this work, we introduce D-LearnPSDD , a learner that improves the classification performance of the LearnPSDD algorithm by introducing a discriminative bias that encodes the conditional relation between the class and feature variables.

The widespread adoption of deep learning is often attributed to its automatic feature construction with minimal inductive bias. However, in many real-world tasks, the learned function is intended to satisfy domain-specific constraints. We focus on monotonicity constraints, which are common and require that the function's output increases with increasing values of specific input features. We develop a counterexample-guided technique to provably enforce monotonicity constraints at prediction time. Additionally, we propose a technique to use monotonicity as an inductive bias for deep learning. It works by iteratively incorporating monotonicity counterexamples in the learning process. Contrary to prior work in monotonic learning, we target general ReLU neural networks and do not further restrict the hypothesis space. We have implemented these techniques in a tool called COMET. Experiments on real-world datasets demonstrate that our approach achieves state-of-the-art results compared to existing monotonic learners, and can improve the model quality compared to those that were trained without taking monotonicity constraints into account.

Weighted model integration (WMI) is a framework to perform advanced probabilistic inference on hybrid domains, i.e., on distributions over mixed continuous-discrete random variables and in presence of complex logical and arithmetic constraints. In this work, we advance the WMI framework on both the theoretical and algorithmic side. First, we exactly trace the boundaries of tractability for WMI inference by proving that to be amenable to exact and efficient inference a WMI problem has to posses a tree-shaped structure with logarithmic diameter. While this result deepens our theoretical understanding of WMI it hinders the practical applicability of exact WMI solvers to real-world problems. To overcome this, we propose the first approximate WMI solver that does not resort to sampling, but performs exact inference on one approximate models. Our solution performs message passing in a relaxed problem structure iteratively to recover certain lost dependencies and, as our experiments suggest, is competitive with other SOTA WMI solvers.

We propose unifying techniques from probabilistic databases and relational embedding models with the goal of performing complex queries on incomplete and uncertain data. We formalize a probabilistic database model with respect to which all queries are done. This allows us to leverage the rich literature of theory and algorithms from probabilistic databases for solving problems. While this formalization can be used with any relational embedding model, the lack of a well-defined joint probability distribution causes simple query problems to become provably hard. With this in mind, we introduce TractOR, a relational embedding model designed to be a tractable probabilistic database, by exploiting typical embedding assumptions within the probabilistic framework. Using a principled, efficient inference algorithm that can be derived from its definition, we empirically demonstrate that TractOR is an effective and general model for these querying tasks.

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.

Probabilistic programming languages (PPLs) are an expressive means of representing and reasoning about probabilistic models. The computational challenge of probabilistic inference remains the primary roadblock for applying PPLs in practice. Inference is fundamentally hard, so there is no one-size-fits all solution. In this work, we target scalable inference for an important class of probabilistic programs: those whose probability distributions are discrete . Discrete distributions are common in many fields, including text analysis, network verification, artificial intelligence, and graph analysis, but they prove to be challenging for existing PPLs. We develop a domain-specific probabilistic programming language called Dice that features a new approach to exact discrete probabilistic program inference. Dice exploits program structure in order to factorize inference, enabling us to perform exact inference on probabilistic programs with hundreds of thousands of random variables. Our key technical contribution is a new reduction from discrete probabilistic programs to weighted model counting (WMC). This reduction separates the structure of the distribution from its parameters, enabling logical reasoning tools to exploit that structure for probabilistic inference. We (1) show how to compositionally reduce Dice inference to WMC, (2) prove this compilation correct with respect to a denotational semantics, (3) empirically demonstrate the performance benefits over prior approaches, and (4) analyze the types of structure that allow Dice to scale to large probabilistic programs.

This paper proposes a new classification model called logistic circuits. On MNIST and Fashion datasets, our learning algorithm outperforms neural networks that have an order of magnitude more parameters. Yet, logistic circuits have a distinct origin in symbolic AI, forming a discriminative counterpart to probabilistic-logical circuits such as ACs, SPNs, and PSDDs. We show that parameter learning for logistic circuits is convex optimization, and that a simple local search algorithm can induce strong model structures from data.

Modern search techniques either cannot efficiently incorporate human feedback to refine search results or cannot express structural or semantic properties of desired code. The key insight of our interactive code search technique ALICE is that user feedback can be actively incorporated to allow users to easily express and refine search queries. We design a query language to model the structure and semantics of code as logic facts. Given a code example with user annotations, ALICE automatically extracts a logic query from code features that are tagged as important. Users can refine the search query by labeling one or more examples as desired (positive) or irrelevant (negative). ALICE then infers a new logic query that separates positive examples from negative examples via active inductive logic programming. Our comprehensive simulation experiment shows that ALICE removes a large number of false positives quickly by actively incorporating user feedback. Its search algorithm is also robust to user labeling mistakes. Our choice of leveraging both positive and negative examples and using nested program structure as an inductive bias is effective in refining search queries. Compared with an existing interactive code search technique, ALICE does not require a user to manually construct a search pattern and yet achieves comparable precision and recall with much fewer search iterations. A case study with real developers shows that ALICE is easy to use and helps express complex code patterns.

Increasing amounts of available data have led to a heightened need for representing large-scale probabilistic knowledge bases. One approach is to use a probabilistic database, a model with strong assumptions that allow for efficiently answering many interesting queries. Recent work on open-world probabilistic databases strengthens the semantics of these probabilistic databases by discarding the assumption that any information not present in the data must be false. While intuitive, these semantics are not sufficiently precise to give reasonable answers to queries. We propose overcoming these issues by using constraints to restrict this open world. We provide an algorithm for one class of queries, and establish a basic hardness result for another. Finally, we propose an efficient and tight approximation for a large class of queries.

While discriminative classifiers often yield strong predictive performance, missing feature values at prediction time can still be a challenge. Classifiers may not behave as expected under certain ways of substituting the missing values, since they inherently make assumptions about the data distribution they were trained on. In this paper, we propose a novel framework that classifies examples with missing features by computing the expected prediction with respect to a feature distribution. Moreover, we use geometric programming to learn a naive Bayes distribution that embeds a given logistic regression classifier and can efficiently take its expected predictions. Empirical evaluations show that our model achieves the same performance as the logistic regression with all features observed, and outperforms standard imputation techniques when features go missing during prediction time. Furthermore, we demonstrate that our method can be used to generate ``sufficient explanations'' of logistic regression classifications, by removing features that do not affect the classification.

Computing expected predictions of discriminative models is a fundamental task in machine learning that appears in many interesting applications such as fairness, handling missing values, and data analysis. Unfortunately, computing expectations of a discriminative model with respect to a probability distribution defined by an arbitrary generative model has been proven to be hard in general. In fact, the task is intractable even for simple models such as logistic regression and a naive Bayes distribution. In this paper, we identify a pair of generative and discriminative models that enables tractable computation of expectations, as well as moments of any order, of the latter with respect to the former in case of regression. Specifically, we consider expressive probabilistic circuits with certain structural constraints that support tractable probabilistic inference. Moreover, we exploit the tractable computation of high-order moments to derive an algorithm to approximate the expectations for classification scenarios in which exact computations are intractable. Our framework to compute expected predictions allows for handling of missing data during prediction time in a principled and accurate way and enables reasoning about the behavior of discriminative models. We empirically show our algorithm to consistently outperform standard imputation techniques on a variety of datasets. Finally, we illustrate how our framework can be used for exploratory data analysis.

Smart portable applications increasingly rely on edge computing due to privacy and latency concerns. But guaranteeing always-on functionality comes with two major challenges: heavily resource-constrained hardware; and dynamic application conditions. Probabilistic models present an ideal solution to these challenges: they are robust to missing data, allow for joint predictions and have small data needs. In addition, ongoing efforts in field of tractable learning have resulted in probabilistic models with strict inference efficiency guarantees. However, the current notions of tractability are often limited to model complexity, disregarding the hardware's specifications and constraints. We propose a novel resource-aware cost metric that takes into consideration the hardware's properties in determining whether the inference task can be efficiently deployed. We use this metric to evaluate the performance versus resource trade-off relevant to the application of interest, and we propose a strategy that selects the device-settings that can optimally meet users' requirements. We showcase our framework on a mobile activity recognition scenario, and on a variety of benchmark datasets representative of the field of tractable learning and of the applications of interest.

We study the task of smoothing a circuit, i.e., ensuring that all children of a plus-gate mention the same variables. Circuits serve as the building blocks of state-of-the-art inference algorithms on discrete probabilistic graphical models and probabilistic programs. They are also important for discrete density estimation algorithms. Many of these tasks require the input circuit to be smooth. However, smoothing has not been studied in its own right yet, and only a trivial quadratic algorithm is known. This paper studies efficient smoothing for structured decomposable circuits. We propose a near-linear time algorithm for this task and explore lower bounds for smoothing decomposable circuits, using existing results on range-sum queries. Further, for the important case of All-Marginals, we show a more efficient linear-time algorithm. We validate experimentally the performance of our methods.

Probabilistic programming languages are a flexible tool for specifying statistical models, but this flexibility comes at the cost of efficient analysis. It is currently difficult to compactly represent the subtle independence properties of a probabilistic program, and exploit independence properties to decompose inference. Classical graphical model abstractions do capture some properties of the underlying distribution, enabling inference algorithms to operate at the level of the graph topology. However, we observe that graph-based abstractions are often too coarse to capture interesting properties of programs. We propose a form of sound abstraction for probabilistic programs wherein the abstractions are themselves simplified programs. We provide a theoretical foundation for these abstractions, as well as an algorithm to generate them. Experimentally, we also illustrate the practical benefits of our framework as a tool to decompose probabilistic program inference.

This paper develops a novel methodology for using symbolic knowledge in deep learning. From first principles, we derive a semantic loss function that bridges between neural output vectors and logical constraints. This loss function captures how close the neural network is to satisfying the constraints on its output. An experimental evaluation shows that it effectively guides the learner to achieve (near-)state-of-the-art results on semi-supervised multi-class classification. Moreover, it significantly increases the ability of the neural network to predict structured objects, such as rankings and paths. These discrete concepts are tremendously difficult to learn, and benefit from a tight integration of deep learning and symbolic reasoning methods.

This paper considers the problem of removing costly features from a Bayesian network classifier. We want the classifier to be robust to these changes, and maintain its classification behavior. To this end, we propose a closeness metric between Bayesian classifiers, called the expected classification agreement (ECA). Our corresponding trimming algorithm finds an optimal subset of features and a new classification threshold that maximize the expected agreement, subject to a budgetary constraint. It utilizes new theoretical insights to perform branch-and-bound search in the space of feature sets, while computing bounds on the ECA. Our experiments investigate both the runtime cost of trimming and its effect on the robustness and accuracy of the final classifier.

We introduce collapsed compilation, a novel approximate inference algorithm for discrete probabilistic graphical models. It is a collapsed sampling algorithm that incrementally selects which variable to sample next based on the partial compila- tion obtained so far. This online collapsing, together with knowledge compilation inference on the remaining variables, naturally exploits local structure and context- specific independence in the distribution. These properties are used implicitly in exact inference, but are difficult to harness for approximate inference. More- over, by having a partially compiled circuit available during sampling, collapsed compilation has access to a highly effective proposal distribution for importance sampling. Our experimental evaluation shows that collapsed compilation performs well on standard benchmarks. In particular, when the amount of exact inference is equally limited, collapsed compilation is competitive with the state of the art, and outperforms it on several benchmarks.

Large-scale probabilistic knowledge bases are becoming increasingly important in academia and industry alike. They are constantly extended with new data, powered by modern information extraction tools that associate probabilities with database tuples. In this paper, we revisit the semantics underlying such systems. In particular, the closed-world assumption of probabilistic databases, that facts not in the database have probability zero, clearly conflicts with their everyday use. To address this discrepancy, we propose an open-world probabilistic database semantics, which relaxes the probabilities of open facts to default intervals. For this open-world setting, we lift the existing data complexity dichotomy of probabilistic databases, and propose an efficient evaluation algorithm for unions of conjunctive queries. We also show that query evaluation can become harder for non-monotone queries.

In many applications, one can define a large set of features to support the classification task at hand. At test time, however, these become prohibitively expensive to evaluate, and only a small subset of features is used, often selected for their information-theoretic value. For threshold-based, Naive Bayes classifiers, recent work has suggested selecting features that maximize the expected robustness of the classifier, that is, the expected probability it maintains its decision after seeing more features. We propose the first algorithm to compute this expected same-decision probability for general Bayesian network classifiers, based on compiling the network into a tractable circuit representation. Moreover, we develop a search algorithm for optimal feature selection that utilizes efficient incremental circuit modifications. Experiments on Naive Bayes, as well as more general networks, show the efficacy and distinct behavior of this decision-making approach.

Counting the models of a propositional formula is an important problem: for example, it serves as the backbone of probabilistic inference by weighted model counting. A key algorithmic insight is component caching (CC), in which disjoint components of a formula, generated dynamically during a DPLL search, are cached so that they only have to be solved once. In the recent years, driven by SMT technology and probabilistic inference in hybrid domains, there is an increasing interest in counting the models of linear arithmetic sentences. To date, however, solvers for these are block-clause implementations, which are nonviable on large problem instances. In this paper, as a first step in extending CC to hybrid domains, we show how propositional CC systems can be leveraged when limited to piecewise polynomial densities. Our experiments demonstrate a large gap in performance when compared to existing approaches based on a variety of block-clause strategies.

Statistical relational models provide compact encodings of probabilistic dependencies in relational domains, but result in highly intractable graphical models. The goal of lifted inference is to carry out probabilistic inference without needing to reason about each individual separately, by instead treating exchangeable, undistinguished objects as a whole. In this paper, we study the domain recursion inference rule, which, despite its central role in early theoretical results on domain-lifted inference, has later been believed redundant. We show that this rule is more powerful than expected, and in fact significantly extends the range of models for which lifted inference runs in time polynomial in the number of individuals in the domain. This includes an open problem called S4, the symmetric transitivity model, and a first-order logic encoding of the birthday paradox. We further identify new classes S2FO2 and S2RU of domain-liftable theories, which respectively subsume FO2 and recursively unary theories, the largest classes of domain-liftable theories known so far, and show that using domain recursion can achieve exponential speedup even in theories that cannot fully be lifted with the existing set of inference rules.

Knowledge compilation is a powerful reasoning paradigm with many applications across AI and computer science more broadly. We consider the problem of bottom-up compilation of knowledge bases, which is usually predicated on the existence of a polytime function for combining compilations using Boolean operators (usually called an Apply function). While such a polytime Apply function is known to exist for certain languages (e.g., OBDDs) and not exist for others (e.g., DNNFs), its existence for certain languages remains unknown. Among the latter is the recently introduced language of Sentential Decision Diagrams (SDDs): while a polytime Apply function exists for SDDs, it was unknown whether such a function exists for the important subset of compressed SDDs which are canonical. We resolve this open question in this paper and consider some of its theoretical and practical implications. Some of the findings we report question the common wisdom on the relationship between bottom-up compilation, language canonicity and the complexity of the Apply function.

Lifted probabilistic inference algorithms have been successfully applied to a large number of symmetric graphical models. Unfortunately, the majority of real-world graphical models is asymmetric. This is even the case for relational representations when evidence is given. Therefore, more recent work in the community moved to making the models symmetric and then applying existing lifted inference algorithms. However, this approach has two shortcomings. First, all existing over-symmetric approximations require a relational representation such as Markov logic networks. Second, the induced symmetries often change the distribution significantly, making the computed probabilities highly biased. We present a framework for probabilistic sampling-based inference that only uses the induced approximate symmetries to propose steps in a Metropolis-Hastings style Markov chain. The framework, therefore, leads to improved probability estimates while remaining unbiased. Experiments demonstrate that the approach outperforms existing MCMC algorithms.

Tractable learning aims to learn probabilistic models where inference is guaranteed to be efficient. However, the particular class of queries that is tractable depends on the model and underlying representation. Usually this class is MPE or conditional probabilities $\Pr(\xs|\ys)$ for joint assignments~$\xs,\ys$. We propose a tractable learner that guarantees efficient inference for a broader class of queries. It simultaneously learns a Markov network and its tractable circuit representation, in order to guarantee and measure tractability. Our approach differs from earlier work by using Sentential Decision Diagrams (SDD) as the tractable language instead of Arithmetic Circuits (AC). SDDs have desirable properties, which more general representations such as ACs lack, that enable basic primitives for Boolean circuit compilation. This allows us to support a broader class of complex probability queries, including counting, threshold, and parity, in polytime.

The FO Model Counting problem (FOMC) is the following: given a sentence Φ in FO and a number n, compute the number of models of Φ over a domain of size n; the Weighted variant (WFOMC) generalizes the problem by associating a weight to each tuple and defining the weight of a model to be the product of weights of its tuples. In this paper we study the complexity of the symmetric WFOMC, where all tuples of a given relation have the same weight. Our motivation comes from an important application, inference in Knowledge Bases with soft constraints, like Markov Logic Networks, but the problem is also of independent theoretical interest. We study both the data complexity, and the combined complexity of FOMC and WFOMC. For the data complexity we prove the existence of an FO3 formula for which FOMC is #P1-complete, and the existence of a Conjunctive Query for which WFOMC is #P1-complete. We also prove that all γ-acyclic queries have polynomial time data complexity. For the combined complexity, we prove that, for every fragment FOk, k ≥ 2, the combined complexity of FOMC (or WFOMC) is #P-complete.

Abstract Recent advances in knowledge compilation introduced techniques to compilepositivelogic programs into propositional logic, essentially exploiting the constructive nature of the least fixpoint computation. This approach has several advantages over existing approaches: it maintains logical equivalence, does not require (expensive) loop-breaking preprocessing or the introduction of auxiliary variables, and significantly outperforms existing algorithms. Unfortunately, this technique is limited tonegation-freeprograms. In this paper, we show how to extend it to general logic programs under the well-founded semantics. We develop our work in approximation fixpoint theory, an algebraical framework that unifies semantics of different logics. As such, our algebraical results are also applicable to autoepistemic logic, default logic and abstract dialectical frameworks.

Abstract Probabilistic logic programs are logic programs in which some of the facts are annotated with probabilities. This paper investigates how classical inference and learning tasks known from the graphical model community can be tackled for probabilistic logic programs. Several such tasks, such as computing the marginals, given evidence and learning from (partial) interpretations, have not really been addressed for probabilistic logic programs before. The first contribution of this paper is a suite of efficient algorithms for various inference tasks. It is based on the conversion of the program and the queries and evidence to a weighted Boolean formula. This allows us to reduce inference tasks to well-studied tasks, such as weighted model counting, which can be solved using state-of-the-art methods known from the graphical model and knowledge compilation literature. The second contribution is an algorithm for parameter estimation in the learning from interpretations setting. The algorithm employs expectation-maximization, and is built on top of the developed inference algorithms. The proposed approach is experimentally evaluated. The results show that the inference algorithms improve upon the state of the art in probabilistic logic programming, and that it is indeed possible to learn the parameters of a probabilistic logic program from interpretations.

Exchangeability is a central notion in statistics and probability theory. The assumption that an infinite sequence of data points is exchangeable is at the core of Bayesian statistics. However, finite exchangeability as a statistical property that renders probabilistic inference tractable is less well-understood. We develop a theory of finite exchangeability and its relation to tractable probabilistic inference. The theory is complementary to that of independence and conditional independence. We show that tractable inference in probabilistic models with high treewidth and millions of variables can be explained with the notion of finite (partial) exchangeability. We also show that existing lifted inference algorithms implicitly utilize a combination of conditional independence and partial exchangeability.

Probabilistic inference can be realized using weighted model counting. Despite a lot of progress, computing weighted model counts exactly is still infeasible for many problems of interest, and one typically has to resort to approximation methods. We contribute a new bounded approximation method for weighted model counting based on probabilistic logic programming principles. Our bounded approximation algorithm is an anytime algorithm that provides lower and upper bounds on the weighted model count. An empirical evaluation on probabilistic logic programs shows that our approach is effective in many cases that are currently beyond the reach of exact methods.

Lifting aims at improving the efficiency of probabilistic inference by exploiting symmetries in the model. Various methods for lifted probabilistic inference have been proposed, but our understanding of these methods and the relationships between them is still limited, compared to their propositional counterparts. The only existing theoretical characterization of lifting is a completeness result for weighted first-order model counting. This paper addresses the question whether the same completeness result holds for other lifted inference algorithms. We answer this question positively for lifted variable elimination (LVE). Our proof relies on introducing a novel inference operator for LVE.

Lifted inference algorithms exploit symmetries in probabilistic models to speed up inference. They show impressive performance when calculating unconditional probabilities in relational models, but often resort to non-lifted inference when computing conditional probabilities. The reason is that conditioning on evidence breaks many of the model's symmetries, which preempts standard lifting techniques. Recent theoretical results show, for example, that conditioning on evidence which corresponds to binary relations is #P-hard, suggesting that no lifting is to be expected in the worst case. In this paper, we balance this grim result by identifying the Boolean rank of the evidence as a key parameter for characterizing the complexity of conditioning in lifted inference. In particular, we show that conditioning on binary evidence with bounded Boolean rank is efficient. This opens up the possibility of approximating evidence by a low-rank Boolean matrix factorization, which we investigate both theoretically and empirically.

Knowledge compilation is a powerful technique for compactly representing and efficiently reasoning about logical knowledge bases. It has been successfully applied to numerous problems in artificial intelligence, such as probabilistic inference and conformant planning. Conditioning, which updates a knowledge base with observed truth values for some propositions, is one of the fundamental operations employed for reasoning. In the propositional setting, conditioning can be efficiently applied in all cases. Recently, people have explored compilation for first-order knowledge bases. The majority of this work has centered around using first-order d-DNNF circuits as the target compilation language. However, conditioning has not been studied in this setting. This paper explores how to condition a first-order d-DNNF circuit. We show that it is possible to efficiently condition these circuits on unary relations. However, we prove that conditioning on higher arity relations is #P-hard. We study the implications of these findings on the application of performing lifted inference for first-order probabilistic models.This leads to a better understanding of which types of queries lifted inference can address.

We introduce aProbLog, a generalization of the probabilistic logic programming language ProbLog. An aProbLog program consists of a set of definite clauses and a set of algebraic facts; each such fact is labeled with an element of a semiring. A wide variety of labels is possible, ranging from probability values to reals (representing costs or utilities), polynomials, Boolean functions or data structures. The semiring is then used to calculate labels of possible worlds and of queries. We formally define the semantics of aProbLog and study the aProbLog inference problem, which is concerned with computing the label of a query. Two conditions are introduced that allow one to simplify the inference problem, resulting in four different algorithms and settings. Representative basic problems for each of these four settings are: is there a possible world where a query is true (SAT), how many such possible worlds are there (#SAT), what is the probability of a query being true (PROB), and what is the most likely world where the query is true (MPE). We further illustrate these settings with a number of tasks requiring more complex semirings.

Probabilistic logical languages provide powerful formalisms for knowledge representation and learning. Yet performing inference in these languages is extremely costly, especially if it is done at the propositional level. Lifted inference algorithms, which avoid repeated computation by treating indistinguishable groups of objects as one, help mitigate this cost. Seeking inspiration from logical inference, where lifted inference (e.g., resolution) is commonly performed, we develop a model theoretic approach to probabilistic lifted inference. Our algorithm compiles a first-order probabilistic theory into a first-order deterministic decomposable negation normal form (d-DNNF) circuit. Compilation offers the advantage that inference is polynomial in the size of the circuit. Furthermore, by borrowing techniques from the knowledge compilation literature our algorithm effectively exploits the logical structure (e.g., context-specific independencies) within the first-order model, which allows more computation to be done at the lifted level. An empirical comparison demonstrates the utility of the proposed approach.

Probabilistic logics are receiving a lot of attention today because of their expressive power for knowledge representation and learning. However, this expressivity is detrimental to the tractability of inference, when done at the propositional level. To solve this problem, various lifted inference algorithms have been proposed that reason at the first-order level, about groups of objects as a whole. Despite the existence of various lifted inference approaches, there are currently no completeness results about these algorithms. The key contribution of this paper is that we introduce a formal definition of lifted inference that allows us to reason about the completeness of lifted inference algorithms relative to a particular class of probabilistic models. We then show how to obtain a completeness result using a first-order knowledge compilation approach for theories of formulae containing up to two logical variables.

We introduce DTProbLog, a decision-theoretic extension of Prolog and its probabilistic variant ProbLog. DTProbLog is a simple but expressive probabilistic programming language that allows the modeling of a wide variety of domains, such as viral marketing. In DTProbLog, the utility of a strategy (a particular choice of actions) is defined as the expected reward for its execution in the presence of probabilistic effects. The key contribution of this paper is the introduction of exact, as well as approximate, solvers to compute the optimal strategy for a DTProbLog program and the decision problem it represents, by making use of binary and algebraic decision diagrams. We also report on experimental results that show the effectiveness and the practical usefulness of the approach.