Antonio Vergari

Kareem Ahmed, Stefano Teso, Kai-Wei Chang et al.

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 including hierarchical multi-label classification, pathfinding and preference learning, while retaining perfect constraint satisfaction.

Andreas Grivas, Antonio Vergari, Adam Lopez

Sigmoid output layers are widely used in multi-label classification (MLC) tasks, in which multiple labels can be assigned to any input. In many practical MLC tasks, the number of possible labels is in the thousands, often exceeding the number of input features and resulting in a low-rank output layer. In multi-class classification, it is known that such a low-rank output layer is a bottleneck that can result in unargmaxable classes: classes which cannot be predicted for any input. In this paper, we show that for MLC tasks, the analogous sigmoid bottleneck results in exponentially many unargmaxable label combinations. We explain how to detect these unargmaxable outputs and demonstrate their presence in three widely used MLC datasets. We then show that they can be prevented in practice by introducing a Discrete Fourier Transform (DFT) output layer, which guarantees that all sparse label combinations with up to $k$ active labels are argmaxable. Our DFT layer trains faster and is more parameter efficient, matching the F1@k score of a sigmoid layer while using up to 50% fewer trainable parameters. Our code is publicly available at https://github.com/andreasgrv/sigmoid-bottleneck.

Lena Zellinger, Nicola Branchini, Lennert De Smet et al.

Classical mixture models (MMs) are widely used tractable proposals for approximate inference settings such as variational inference (VI) and importance sampling (IS). Recently, mixture models with negative coefficients, called subtractive mixture models (SMMs), have been proposed as a potentially more expressive alternative. However, how to effectively use SMMs for VI and IS is still an open question as they do not provide latent variable semantics and therefore cannot use sampling schemes for classical MMs. In this work, we study how to circumvent this issue by designing several expectation estimators for IS and learning schemes for VI with SMMs, and we empirically evaluate them for distribution approximation. Finally, we discuss the additional challenges in estimation stability and learning efficiency that they carry and propose ways to overcome them. Code is available at: https://github.com/april-tools/delta-vi.

Davide Scassola, Dylan Ponsford, Adrián Javaloy et al.

Tabular data is more challenging to generate than text and images, due to its heterogeneous features and much lower sample sizes. On this task, diffusion-based models are the current state-of-the-art (SotA) model class, achieving almost perfect performance on commonly used benchmarks. In this paper, we question the perception of progress for tabular data generation. First, we highlight the limitations of current protocols to evaluate the fidelity of generated data, and advocate for alternative ones. Next, we revisit a simple baseline -- hierarchical mixture models in the form of deep probabilistic circuits (PCs) -- which delivers competitive or superior performance to SotA models for a fraction of the cost. PCs are the generative counterpart of decision forests, and as such can natively handle heterogeneous data as well as deliver tractable probabilistic generation and inference. Finally, in a rigorous empirical analysis we show that the apparent saturation of progress for SotA models is largely due to the use of inadequate metrics. As such, we highlight that there is still much to be done to generate realistic tabular data. Code available at https://github.com/april-tools/tabpc.

Adrián Javaloy, Antonio Vergari

Orthogonality constraints are ubiquitous in robust and probabilistic machine learning. Unfortunately, current optimizers are computationally expensive and do not scale to problems with hundreds or thousands of constraints. One notable exception is the Landing algorithm (Ablin et al., 2024) which, however comes at the expense of temporarily relaxing orthogonality. In this work, we revisit and improve on the ideas behind Landing, enabling the inclusion of modern adaptive optimizers while ensuring that orthogonal constraints are effectively met. Remarkably, these improvements come at little to no cost, and reduce the number of required hyperparemeters. Our algorithm POGO is fast and GPU-friendly, consisting of only 5 matrix products, and in practice maintains orthogonality at all times. On several challenging benchmarks, POGO greatly outperforms recent optimizers and shows it can optimize problems with thousands of orthogonal matrices in minutes while alternatives would take hours. As such, POGO sets a milestone to finally exploit orthogonality constraints in ML at scale. A PyTorch implementation of POGO is publicly available at https://github.com/adrianjav/pogo.

Kamil Faber, Dominik Zurek, Marcin Pietron et al.

Continual learning (CL) is one of the most promising trends in recent machine learning research. Its goal is to go beyond classical assumptions in machine learning and develop models and learning strategies that present high robustness in dynamic environments. The landscape of CL research is fragmented into several learning evaluation protocols, comprising different learning tasks, datasets, and evaluation metrics. Additionally, the benchmarks adopted so far are still distant from the complexity of real-world scenarios, and are usually tailored to highlight capabilities specific to certain strategies. In such a landscape, it is hard to objectively assess strategies. In this work, we fill this gap for CL on image data by introducing two novel CL benchmarks that involve multiple heterogeneous tasks from six image datasets, with varying levels of complexity and quality. Our aim is to fairly evaluate current state-of-the-art CL strategies on a common ground that is closer to complex real-world scenarios. We additionally structure our benchmarks so that tasks are presented in increasing and decreasing order of complexity -- according to a curriculum -- in order to evaluate if current CL models are able to exploit structure across tasks. We devote particular emphasis to providing the CL community with a rigorous and reproducible evaluation protocol for measuring the ability of a model to generalize and not to forget while learning. Furthermore, we provide an extensive experimental evaluation showing that popular CL strategies, when challenged with our benchmarks, yield sub-par performance, high levels of forgetting, and present a limited ability to effectively leverage curriculum task ordering. We believe that these results highlight the need for rigorous comparisons in future CL works as well as pave the way to design new CL strategies that are able to deal with more complex scenarios.

Lorenzo Loconte, Aleksanteri M. Sladek, Stefan Mengel et al.

Mixture models are traditionally represented and learned by adding several distributions as components. Allowing mixtures to subtract probability mass or density can drastically reduce the number of components needed to model complex distributions. However, learning such subtractive mixtures while ensuring they still encode a non-negative function is challenging. We investigate how to learn and perform inference on deep subtractive mixtures by squaring them. We do this in the framework of probabilistic circuits, which enable us to represent tensorized mixtures and generalize several other subtractive models. We theoretically prove that the class of squared circuits allowing subtractions can be exponentially more expressive than traditional additive mixtures; and, we empirically show this increased expressiveness on a series of real-world distribution estimation tasks.

Lorenzo Loconte, Antonio Mari, Gennaro Gala et al.

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

Diego Calanzone, Stefano Teso, Antonio Vergari

Large language models (LLMs) are a promising venue for natural language understanding and generation. However, current LLMs are far from reliable: they are prone to generating non-factual information and, more crucially, to contradicting themselves when prompted to reason about relations between entities of the world. These problems are currently addressed with large scale fine-tuning or by delegating reasoning to external tools. In this work, we strive for a middle ground and introduce a loss based on neuro-symbolic reasoning that teaches an LLM to be logically consistent with an external set of facts and rules and improves self-consistency even when the LLM is fine-tuned on a limited set of facts. Our approach also allows to easily combine multiple logical constraints at once in a principled way, delivering LLMs that are more consistent w.r.t. all constraints and improve over several baselines w.r.t. a given constraint. Moreover, our method allows LLMs to extrapolate to unseen but semantically similar factual knowledge, represented in unseen datasets, more systematically.

Lorenzo Loconte, Stefan Mengel, Antonio Vergari

Designing expressive generative models that support exact and efficient inference is a core question in probabilistic ML. Probabilistic circuits (PCs) offer a framework where this tractability-vs-expressiveness trade-off can be analyzed theoretically. Recently, squared PCs encoding subtractive mixtures via negative parameters have emerged as tractable models that can be exponentially more expressive than monotonic PCs, i.e., PCs with positive parameters only. In this paper, we provide a more precise theoretical characterization of the expressiveness relationships among these models. First, we prove that squared PCs can be less expressive than monotonic ones. Second, we formalize a novel class of PCs -- sum of squares PCs -- that can be exponentially more expressive than both squared and monotonic PCs. Around sum of squares PCs, we build an expressiveness hierarchy that allows us to precisely unify and separate different tractable model classes such as Born Machines and PSD models, and other recently introduced tractable probabilistic models by using complex parameters. Finally, we empirically show the effectiveness of sum of squares circuits in performing distribution estimation.

Filippo Corponi, Bryan M. Li, Gerard Anmella et al.

Personal sensing, leveraging data passively and near-continuously collected with wearables from patients in their ecological environment, is a promising paradigm to monitor mood disorders (MDs), a major determinant of worldwide disease burden. However, collecting and annotating wearable data is very resource-intensive. Studies of this kind can thus typically afford to recruit only a couple dozens of patients. This constitutes one of the major obstacles to applying modern supervised machine learning techniques to MDs detection. In this paper, we overcome this data bottleneck and advance the detection of MDs acute episode vs stable state from wearables data on the back of recent advances in self-supervised learning (SSL). This leverages unlabelled data to learn representations during pre-training, subsequently exploited for a supervised task. First, we collected open-access datasets recording with an Empatica E4 spanning different, unrelated to MD monitoring, personal sensing tasks -- from emotion recognition in Super Mario players to stress detection in undergraduates -- and devised a pre-processing pipeline performing on-/off-body detection, sleep-wake detection, segmentation, and (optionally) feature extraction. With 161 E4-recorded subjects, we introduce E4SelfLearning, the largest to date open access collection, and its pre-processing pipeline. Second, we show that SSL confidently outperforms fully-supervised pipelines using either our novel E4-tailored Transformer architecture (E4mer) or classical baseline XGBoost: 81.23% against 75.35% (E4mer) and 72.02% (XGBoost) correctly classified recording segments from 64 (half acute, half stable) patients. Lastly, we illustrate that SSL performance is strongly associated with the specific surrogate task employed for pre-training as well as with unlabelled data availability.

Nickil Maveli, Antonio Vergari, Shay B. Cohen

Code-LLMs, LLMs pre-trained on large code corpora, have shown great progress in learning rich representations of the structure and syntax of code, successfully using it to generate or classify code fragments. At the same time, understanding if they are able to do so because they capture code semantics, and how well, is still an open question. In this paper, we tackle this problem by introducing SeqCoBench, a benchmark for systematically assessing how Code-LLMs can capture code functional equivalence. SeqCoBench contains over 20 code transformations that either preserve or alter the semantics of Python programs. We conduct extensive evaluations in different settings, including zero-shot and parameter-efficient finetuning methods on state-of-the-art (Code)-LLMs to see if they can discern semantically equivalent or different pairs of programs in SeqCoBench. We find that the performance gap between these LLMs and classical match-based retrieval scores is minimal, with both approaches showing a concerning lack of depth in understanding code semantics.

Lena Zellinger, Antonio Vergari

When approximating an intractable density via variational inference (VI) the variational family is typically chosen as a simple parametric family that very likely does not contain the target. This raises the question: Under which conditions can we recover characteristics of the target despite misspecification? In this work, we extend previous results on robust VI with location-scale families under target symmetries. We derive sufficient conditions guaranteeing exact recovery of the mean when using the forward Kullback-Leibler divergence and $α$-divergences. We further show how and why optimization can fail to recover the target mean in the absence of our sufficient conditions, providing initial guidelines on the choice of the variational family and $α$-value.

Gennaro Gala, Cassio de Campos, Robert Peharz et al.

Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

Chen Hu, Yintao Tai, Antonio Vergari et al.

Pixel-based language models are gaining momentum as alternatives to traditional token-based approaches, promising to circumvent tokenization challenges. However, the inherent perceptual diversity across languages poses a significant hurdle for multilingual generalization in pixel space. This paper introduces MIXAR, the first generative pixel-based language model trained on eight different languages utilizing a range of different scripts. We empirically evaluate MIXAR against previous pixel-based models as well as comparable tokenizer-based models, demonstrating substantial performance improvement on discriminative and generative multilingual tasks. Additionally, we show how MIXAR is robust to languages never seen during the training. These results are further strengthened when scaling the model to 0.5B parameters which not only improves its capabilities in generative tasks like LAMBADA but also its robustness when challenged with input perturbations such as orthographic attacks.

Andreas Grivas, Lorenzo Loconte, Emile van Krieken et al.

Multi-token prediction (MTP) is a prominent strategy to significantly speed up generation in large language models (LLMs), including byte-level LLMs, which are tokeniser-free but prohibitively slow. However, existing MTP methods often sacrifice expressiveness by assuming independence between future tokens. In this work, we investigate the trade-off between expressiveness and latency in MTP within the framework of probabilistic circuits (PCs). Our framework, named MTPC, allows one to explore different ways to encode the joint distributions over future tokens by selecting different circuit architectures, generalising classical models such as (hierarchical) mixture models, hidden Markov models and tensor networks. We show the efficacy of MTPC by retrofitting existing byte-level LLMs, such as EvaByte. Our experiments show that, when combined with speculative decoding, MTPC significantly speeds up generation compared to MTP with independence assumptions, while guaranteeing to retain the performance of the original verifier LLM. We also rigorously study the optimal trade-off between expressiveness and latency when exploring the possible parameterisations of MTPC, such as PC architectures and partial layer sharing between the verifier and draft LLMs.

Nicolas Lazzari, Valentina Presutti, Antonio Vergari

Background: Neuro-symbolic methods enhance the reliability of neural network classifiers through logical constraints, but they lack native support for ontologies. Objectives: We aim to develop a neuro-symbolic method that reliably outputs predictions consistent with a Description Logic ontology that formalizes domain-specific knowledge. Methods: We encode a Description Logic ontology as a circuit, a feed-forward differentiable computational graph that supports tractable execution of queries and transformations. We show that the circuit can be used to (i) generate synthetic datasets that capture the semantics of the ontology; (ii) efficiently perform deductive reasoning on a GPU; (iii) implement neuro-symbolic models whose predictions are approximately or provably consistent with the knowledge defined in the ontology. Results We show that the synthetic dataset generated using the circuit qualitatively captures the semantics of the ontology while being challenging for Machine Learning classifiers, including neural networks. Moreover, we show that compiling the ontology into a circuit is a promising approach for scalable deductive reasoning, with runtimes up to three orders of magnitude faster than available reasoners. Finally, we show that our neuro-symbolic classifiers reliably produce consistent predictions when compared to neural network baselines, maintaining competitive performances or even outperforming them. Conclusions By compiling Description Logic ontologies into circuits, we obtain a tighter integration between the Deep Learning and Knowledge Representation fields. We show that a single circuit representation can be used to tackle different challenging tasks closely related to real-world applications.

Weixin Chen, Antonio Vergari, Han Zhao

Vision-Language Models (VLMs) have been applied to a wide range of reasoning tasks, yet it remains unclear whether they can reason robustly under distribution shifts. In this paper, we study covariate shifts in which the perceptual input distribution changes while the underlying prediction rules do not. To investigate this question, we consider visual deductive reasoning tasks, where a model is required to answer a query given an image and logical rules defined over the object concepts in the image. Empirically, we find that VLMs fine-tuned through gradient-based end-to-end training can achieve high in-distribution accuracy but fail to generalize under such shifts, suggesting that fine-tuning does not reliably induce the underlying reasoning function. This motivates a neuro-symbolic perspective that decouples perception from reasoning. However, we further observe that recent neuro-symbolic approaches that rely on black-box components for reasoning can still exhibit inconsistent robustness across tasks. To address this issue, we propose VLC, a neuro-symbolic method that combines VLM-based concept recognition with circuit-based symbolic reasoning. In particular, task rules are compiled into a symbolic program, specifically a circuit, which executes the rules exactly over the object concepts recognized by the VLM. Experiments on three visual deductive reasoning tasks with distinct rule sets show that VLC consistently achieves strong performance under covariate shifts, highlighting its ability to support robust reasoning.

Andrea Valenti, Davide Bacciu, Antonio Vergari

While showing impressive performance on various kinds of learning tasks, it is yet unclear whether deep learning models have the ability to robustly tackle reasoning tasks. than by learning the underlying reasoning process that is actually required to solve the tasks. Measuring the robustness of reasoning in machine learning models is challenging as one needs to provide a task that cannot be easily shortcut by exploiting spurious statistical correlations in the data, while operating on complex objects and constraints. reasoning task. To address this issue, we propose ChemAlgebra, a benchmark for measuring the reasoning capabilities of deep learning models through the prediction of stoichiometrically-balanced chemical reactions. ChemAlgebra requires manipulating sets of complex discrete objects -- molecules represented as formulas or graphs -- under algebraic constraints such as the mass preservation principle. We believe that ChemAlgebra can serve as a useful test bed for the next generation of machine reasoning models and as a promoter of their development.

Lorenzo Loconte, Adrián Javaloy, Antonio Vergari

Squared tensor networks (TNs) and their extension as computational graphs--squared circuits--have been used as expressive distribution estimators, yet supporting closed-form marginalization. However, the squaring operation introduces additional complexity when computing the partition function or marginalizing variables, which hinders their applicability in ML. To solve this issue, canonical forms of TNs are parameterized via unitary matrices to simplify the computation of marginals. However, these canonical forms do not apply to circuits, as they can represent factorizations that do not directly map to a known TN. Inspired by the ideas of orthogonality in canonical forms and determinism in circuits enabling tractable maximization, we show how to parameterize squared circuits to overcome their marginalization overhead. Our parameterizations unlock efficient marginalization even in factorizations different from TNs, but encoded as circuits, whose structure would otherwise make marginalization computationally hard. Finally, our experiments on distribution estimation show how our proposed conditions in squared circuits come with no expressiveness loss, while enabling more efficient learning.

Leander Kurscheidt, Gabriele Masina, Roberto Sebastiani et al.

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

Mayank Jobanputra, Nils Philipp Walter, Maitrey Mehta et al.

We present a systematic study of subtraction in large language models (LLMs). While prior benchmarks emphasize addition and multiplication, subtraction has received comparatively little attention despite being structurally distinct as a non-commutative operation. We evaluate eight pretrained LLMs spanning four families on addition and subtraction problems. Our experiments reveal that subtraction accuracy lags behind addition by a wide margin. We find that the errors for ($a-b$) are concentrated in cases where ($a<b$). In such cases, LLMs frequently produce the correct magnitude but omit the negative sign. Probing analyses show that LLMs internally encode whether results should be negative, yet this information is often not reflected in generated outputs. We further test well-known techniques such as few-shot learning and instruction-tuning to see if they can improve the LLMs' performance. Our results suggest that while few-shot prompting yields modest gains, the instruction-tuned models achieve near-perfect accuracies in generating the negative sign. Together, these findings provide a clearer characterization of the limitations and recoverability of LLMs' arithmetic capabilities in subtraction.

Aryo Pradipta Gema, Dominik Grabarczyk, Wolf De Wulf et al.

Knowledge graphs are powerful tools for representing and organising complex biomedical data. Several knowledge graph embedding algorithms have been proposed to learn from and complete knowledge graphs. However, a recent study demonstrates the limited efficacy of these embedding algorithms when applied to biomedical knowledge graphs, raising the question of whether knowledge graph embeddings have limitations in biomedical settings. This study aims to apply state-of-the-art knowledge graph embedding models in the context of a recent biomedical knowledge graph, BioKG, and evaluate their performance and potential downstream uses. We achieve a three-fold improvement in terms of performance based on the HITS@10 score over previous work on the same biomedical knowledge graph. Additionally, we provide interpretable predictions through a rule-based method. We demonstrate that knowledge graph embedding models are applicable in practice by evaluating the best-performing model on four tasks that represent real-life polypharmacy situations. Results suggest that knowledge learnt from large biomedical knowledge graphs can be transferred to such downstream use cases. Our code is available at https://github.com/aryopg/biokge.

Partha Ghosh, Mehdi S. M. Sajjadi, Antonio Vergari et al.

Variational Autoencoders (VAEs) provide a theoretically-backed and popular framework for deep generative models. However, learning a VAE from data poses still unanswered theoretical questions and considerable practical challenges. In this work, we propose an alternative framework for generative modeling that is simpler, easier to train, and deterministic, yet has many of the advantages of VAEs. We observe that sampling a stochastic encoder in a Gaussian VAE can be interpreted as simply injecting noise into the input of a deterministic decoder. We investigate how substituting this kind of stochasticity, with other explicit and implicit regularization schemes, can lead to an equally smooth and meaningful latent space without forcing it to conform to an arbitrarily chosen prior. To retrieve a generative mechanism to sample new data, we introduce an ex-post density estimation step that can be readily applied also to existing VAEs, improving their sample quality. We show, in a rigorous empirical study, that the proposed regularized deterministic autoencoders are able to generate samples that are comparable to, or better than, those of VAEs and more powerful alternatives when applied to images as well as to structured data such as molecules. \footnote{An implementation is available at: \url{https://github.com/ParthaEth/Regularized_autoencoders-RAE-}}

Alejandro Molina, Antonio Vergari, Karl Stelzner et al.

We introduce SPFlow, an open-source Python library providing a simple interface to inference, learning and manipulation routines for deep and tractable probabilistic models called Sum-Product Networks (SPNs). The library allows one to quickly create SPNs both from data and through a domain specific language (DSL). It efficiently implements several probabilistic inference routines like computing marginals, conditionals and (approximate) most probable explanations (MPEs) along with sampling as well as utilities for serializing, plotting and structure statistics on an SPN. Moreover, many of the algorithms proposed in the literature to learn the structure and parameters of SPNs are readily available in SPFlow. Furthermore, SPFlow is extremely extensible and customizable, allowing users to promptly distill new inference and learning routines by injecting custom code into a lightweight functional-oriented API framework. This is achieved in SPFlow by keeping an internal Python representation of the graph structure that also enables practical compilation of an SPN into a TensorFlow graph, C, CUDA or FPGA custom code, significantly speeding-up computations.

Emanuele Marconato, Samuele Bortolotti, Emile van Krieken et al.

Neuro-Symbolic (NeSy) predictors that conform to symbolic knowledge - encoding, e.g., safety constraints - can be affected by Reasoning Shortcuts (RSs): They learn concepts consistent with the symbolic knowledge by exploiting unintended semantics. RSs compromise reliability and generalization and, as we show in this paper, they are linked to NeSy models being overconfident about the predicted concepts. Unfortunately, the only trustworthy mitigation strategy requires collecting costly dense supervision over the concepts. Rather than attempting to avoid RSs altogether, we propose to ensure NeSy models are aware of the semantic ambiguity of the concepts they learn, thus enabling their users to identify and distrust low-quality concepts. Starting from three simple desiderata, we derive bears (BE Aware of Reasoning Shortcuts), an ensembling technique that calibrates the model's concept-level confidence without compromising prediction accuracy, thus encouraging NeSy architectures to be uncertain about concepts affected by RSs. We show empirically that bears improves RS-awareness of several state-of-the-art NeSy models, and also facilitates acquiring informative dense annotations for mitigation purposes.

Emile van Krieken, Pasquale Minervini, Edoardo M. Ponti et al.

State-of-the-art neurosymbolic learning systems use probabilistic reasoning to guide neural networks towards predictions that conform to logical constraints over symbols. Many such systems assume that the probabilities of the considered symbols are conditionally independent given the input to simplify learning and reasoning. We study and criticise this assumption, highlighting how it can hinder optimisation and prevent uncertainty quantification. We prove that loss functions bias conditionally independent neural networks to become overconfident in their predictions. As a result, they are unable to represent uncertainty over multiple valid options. Furthermore, we prove that these loss functions are difficult to optimise: they are non-convex, and their minima are usually highly disconnected. Our theoretical analysis gives the foundation for replacing the conditional independence assumption and designing more expressive neurosymbolic probabilistic models.

Yintao Tai, Xiyang Liao, Alessandro Suglia et al.

Recent work showed the possibility of building open-vocabulary large language models (LLMs) that directly operate on pixel representations. These models are implemented as autoencoders that reconstruct masked patches of rendered text. However, these pixel-based LLMs are limited to discriminative tasks (e.g., classification) and, similar to BERT, cannot be used to generate text. Therefore, they cannot be used for generative tasks such as free-form question answering. In this work, we introduce PIXAR, the first pixel-based autoregressive LLM that performs text generation. Consisting of only a decoder, PIXAR can perform free-form generative tasks while keeping the number of parameters on par with previous encoder-decoder models. Furthermore, we highlight the challenges of generating text as non-noisy images and show this is due to using a maximum likelihood objective. To overcome this problem, we propose an adversarial pretraining stage that improves the readability and accuracy of PIXAR by 8.1 on LAMBADA and 8.5 on bAbI -- making it comparable to GPT-2 on text generation tasks. This paves the way to build open-vocabulary LLMs that operate on perceptual input only and calls into question the necessity of the usual symbolic input representation, i.e., text as (sub)tokens.

Cosimo Gregucci, Bo Xiong, Daniel Hernandez et al.

Complex query answering (CQA) on knowledge graphs (KGs) is gaining momentum as a challenging reasoning task. In this paper, we show that the current benchmarks for CQA might not be as complex as we think, as the way they are built distorts our perception of progress in this field. For example, we find that in these benchmarks, most queries (up to 98% for some query types) can be reduced to simpler problems, e.g., link prediction, where only one link needs to be predicted. The performance of state-of-the-art CQA models decreases significantly when such models are evaluated on queries that cannot be reduced to easier types. Thus, we propose a set of more challenging benchmarks composed of queries that require models to reason over multiple hops and better reflect the construction of real-world KGs. In a systematic empirical investigation, the new benchmarks show that current methods leave much to be desired from current CQA methods.

Diego Calanzone, Stefano Teso, Antonio Vergari

Large language models (LLMs) are a promising venue for natural language understanding and generation tasks. However, current LLMs are far from reliable: they are prone to generate non-factual information and, more crucially, to contradict themselves when prompted to reason about beliefs of the world. These problems are currently addressed with large scale fine-tuning or by delegating consistent reasoning to external tools. In this work, we strive for a middle ground and introduce a training objective based on principled probabilistic reasoning that teaches a LLM to be consistent with external knowledge in the form of a set of facts and rules. Fine-tuning with our loss on a limited set of facts enables our LLMs to be more logically consistent than previous baselines and allows them to extrapolate to unseen but semantically similar factual knowledge more systematically.

Leander Kurscheidt, Paolo Morettin, Roberto Sebastiani et al.

In safety-critical applications, guaranteeing the satisfaction of constraints over continuous environments is crucial, e.g., an autonomous agent should never crash into obstacles or go off-road. Neural models struggle in the presence of these constraints, especially when they involve intricate algebraic relationships. To address this, we introduce a differentiable probabilistic layer that guarantees the satisfaction of non-convex algebraic constraints over continuous variables. This probabilistic algebraic layer (PAL) can be seamlessly plugged into any neural architecture and trained via maximum likelihood without requiring approximations. PAL defines a distribution over conjunctions and disjunctions of linear inequalities, parameterized by polynomials. This formulation enables efficient and exact renormalization via symbolic integration, which can be amortized across different data points and easily parallelized on a GPU. We showcase PAL and our integration scheme on a number of benchmarks for algebraic constraint integration and on real-world trajectory data.

Emile van Krieken, Pasquale Minervini, Edoardo Ponti et al.

Neurosymbolic (NeSy) predictors combine neural perception with symbolic reasoning to solve tasks like visual reasoning. However, standard NeSy predictors assume conditional independence between the symbols they extract, thus limiting their ability to model interactions and uncertainty - often leading to overconfident predictions and poor out-of-distribution generalisation. To overcome the limitations of the independence assumption, we introduce neurosymbolic diffusion models (NeSyDMs), a new class of NeSy predictors that use discrete diffusion to model dependencies between symbols. Our approach reuses the independence assumption from NeSy predictors at each step of the diffusion process, enabling scalable learning while capturing symbol dependencies and uncertainty quantification. Across both synthetic and real-world benchmarks - including high-dimensional visual path planning and rule-based autonomous driving - NeSyDMs achieve state-of-the-art accuracy among NeSy predictors and demonstrate strong calibration.

Lorenzo Loconte, Antonio Vergari

Squared tensor networks (TNs) and their generalization as parameterized computational graphs -- squared circuits -- have been recently used as expressive distribution estimators in high dimensions. However, the squaring operation introduces additional complexity when marginalizing variables or computing the partition function, which hinders their usage in machine learning applications. Canonical forms of popular TNs are parameterized via unitary matrices as to simplify the computation of particular marginals, but cannot be mapped to general circuits since these might not correspond to a known TN. Inspired by TN canonical forms, we show how to parameterize squared circuits to ensure they encode already normalized distributions. We then use this parameterization to devise an algorithm to compute any marginal of squared circuits that is more efficient than a previously known one. We conclude by formally showing the proposed parameterization comes with no expressiveness loss for many circuit classes.

Emile van Krieken, Pasquale Minervini, Edoardo Ponti et al.

The ubiquitous independence assumption among symbolic concepts in neurosymbolic (NeSy) predictors is a convenient simplification: NeSy predictors use it to speed up probabilistic reasoning. Recent works like van Krieken et al. (2024) and Marconato et al. (2024) argued that the independence assumption can hinder learning of NeSy predictors and, more crucially, prevent them from correctly modelling uncertainty. There is, however, scepticism in the NeSy community around the scenarios in which the independence assumption actually limits NeSy systems (Faronius and Dos Martires, 2025). In this work, we settle this question by formally showing that assuming independence among symbolic concepts entails that a model can never represent uncertainty over certain concept combinations. Thus, the model fails to be aware of reasoning shortcuts, i.e., the pathological behaviour of NeSy predictors that predict correct downstream tasks but for the wrong reasons.

Nickil Maveli, Antonio Vergari, Shay B. Cohen

LLMs demonstrate strong performance on code benchmarks, yet round-trip code execution reveals limitations in their ability to maintain consistent reasoning across forward and backward execution. We present RoundTripCodeEval (RTCE), a comprehensive benchmark consisting of four distinct code execution reasoning tasks designed to rigorously test round-trip consistency. RTCE provides an execution-free, exact-match evaluation of bijection fidelity, assessing whether models preserve a consistent one-to-one mapping between encoding and decoding operations across various algorithms and directions. We systematically evaluate state-of-the-art Code-LLMs using zero-shot prompting, supervised fine-tuning on execution traces, and self-reflection mechanisms. Each yields modest improvements, but none closes the gap, indicating that current LLMs struggle with true round-trip consistency, which demonstrates that they lack the internal coherence required for trustworthy code reasoning. RTCE surfaces several new and previously unmeasured insights that are not captured by existing I/O-prediction, execution-reasoning, or round-trip natural-language benchmarks. We will release the code and the dataset upon acceptance.

Emanuele Marconato, Samuele Bortolotti, Emile van Krieken et al.

Neuro-symbolic (NeSy) AI aims to develop deep neural networks whose predictions comply with prior knowledge encoding, e.g. safety or structural constraints. As such, it represents one of the most promising avenues for reliable and trustworthy AI. The core idea behind NeSy AI is to combine neural and symbolic steps: neural networks are typically responsible for mapping low-level inputs into high-level symbolic concepts, while symbolic reasoning infers predictions compatible with the extracted concepts and the prior knowledge. Despite their promise, it was recently shown that - whenever the concepts are not supervised directly - NeSy models can be affected by Reasoning Shortcuts (RSs). That is, they can achieve high label accuracy by grounding the concepts incorrectly. RSs can compromise the interpretability of the model's explanations, performance in out-of-distribution scenarios, and therefore reliability. At the same time, RSs are difficult to detect and prevent unless concept supervision is available, which is typically not the case. However, the literature on RSs is scattered, making it difficult for researchers and practitioners to understand and tackle this challenging problem. This overview addresses this issue by providing a gentle introduction to RSs, discussing their causes and consequences in intuitive terms. It also reviews and elucidates existing theoretical characterizations of this phenomenon. Finally, it details methods for dealing with RSs, including mitigation and awareness strategies, and maps their benefits and limitations. By reformulating advanced material in a digestible form, this overview aims to provide a unifying perspective on RSs to lower the bar to entry for tackling them. Ultimately, we hope this overview contributes to the development of reliable NeSy and trustworthy AI models.

Steven Braun, Sahil Sidheekh, Antonio Vergari et al.

Probabilistic circuits (PCs) are powerful probabilistic models that enable exact and tractable inference, making them highly suitable for probabilistic reasoning and inference tasks. While dominant in neural networks, representation learning with PCs remains underexplored, with prior approaches relying on external neural embeddings or activation-based encodings. To address this gap, we introduce autoencoding probabilistic circuits (APCs), a novel framework leveraging the tractability of PCs to model probabilistic embeddings explicitly. APCs extend PCs by jointly modeling data and embeddings, obtaining embedding representations through tractable probabilistic inference. The PC encoder allows the framework to natively handle arbitrary missing data and is seamlessly integrated with a neural decoder in a hybrid, end-to-end trainable architecture enabled by differentiable sampling. Our empirical evaluation demonstrates that APCs outperform existing PC-based autoencoding methods in reconstruction quality, generate embeddings competitive with, and exhibit superior robustness in handling missing data compared to neural autoencoders. These results highlight APCs as a powerful and flexible representation learning method that exploits the probabilistic inference capabilities of PCs, showing promising directions for robust inference, out-of-distribution detection, and knowledge distillation.

Lena Zellinger, Nicola Branchini, Víctor Elvira et al.

Many Monte Carlo (MC) and importance sampling (IS) methods use mixture models (MMs) for their simplicity and ability to capture multimodal distributions. Recently, subtractive mixture models (SMMs), i.e. MMs with negative coefficients, have shown greater expressiveness and success in generative modeling. However, their negative parameters complicate sampling, requiring costly auto-regressive techniques or accept-reject algorithms that do not scale in high dimensions. In this work, we use the difference representation of SMMs to construct an unbiased IS estimator ($Δ\text{Ex}$) that removes the need to sample from the SMM, enabling high-dimensional expectation estimation with SMMs. In our experiments, we show that $Δ\text{Ex}$ can achieve comparable estimation quality to auto-regressive sampling while being considerably faster in MC estimation. Moreover, we conduct initial experiments with $Δ\text{Ex}$ using hand-crafted proposals, gaining first insights into how to construct safe proposals for $Δ\text{Ex}$.

Adrián Javaloy, Antonio Vergari, Isabel Valera

In machine learning (ML), we often need to choose one among hundreds of trained ML models at hand, based on various objectives such as accuracy, robustness, fairness or scalability. However, it is often unclear how to compare, aggregate and, ultimately, trade-off these objectives, making it a time-consuming task that requires expert knowledge, as objectives may be measured in different units and scales. In this work, we investigate how objectives can be automatically normalized and aggregated to systematically help the user navigate their Pareto front. To this end, we make incomparable objectives comparable using their cumulative functions, approximated by their relative rankings. As a result, our proposed approach, COPA, can aggregate them while matching user-specific preferences, allowing practitioners to meaningfully navigate and search for models in the Pareto front. We demonstrate the potential impact of COPA in both model selection and benchmarking tasks across diverse ML areas such as fair ML, domain generalization, AutoML and foundation models, where classical ways to normalize and aggregate objectives fall short.

Samuele Bortolotti, Emanuele Marconato, Tommaso Carraro et al.

The advent of powerful neural classifiers has increased interest in problems that require both learning and reasoning. These problems are critical for understanding important properties of models, such as trustworthiness, generalization, interpretability, and compliance to safety and structural constraints. However, recent research observed that tasks requiring both learning and reasoning on background knowledge often suffer from reasoning shortcuts (RSs): predictors can solve the downstream reasoning task without associating the correct concepts to the high-dimensional data. To address this issue, we introduce rsbench, a comprehensive benchmark suite designed to systematically evaluate the impact of RSs on models by providing easy access to highly customizable tasks affected by RSs. Furthermore, rsbench implements common metrics for evaluating concept quality and introduces novel formal verification procedures for assessing the presence of RSs in learning tasks. Using rsbench, we highlight that obtaining high quality concepts in both purely neural and neuro-symbolic models is a far-from-solved problem. rsbench is available at: https://unitn-sml.github.io/rsbench.

Gennaro Gala, Cassio de Campos, Antonio Vergari et al.

Probabilistic integral circuits (PICs) have been recently introduced as probabilistic models enjoying the key ingredient behind expressive generative models: continuous latent variables (LVs). PICs are symbolic computational graphs defining continuous LV models as hierarchies of functions that are summed and multiplied together, or integrated over some LVs. They are tractable if LVs can be analytically integrated out, otherwise they can be approximated by tractable probabilistic circuits (PC) encoding a hierarchical numerical quadrature process, called QPCs. So far, only tree-shaped PICs have been explored, and training them via numerical quadrature requires memory-intensive processing at scale. In this paper, we address these issues, and present: (i) a pipeline for building DAG-shaped PICs out of arbitrary variable decompositions, (ii) a procedure for training PICs using tensorized circuit architectures, and (iii) neural functional sharing techniques to allow scalable training. In extensive experiments, we showcase the effectiveness of functional sharing and the superiority of QPCs over traditional PCs.

Emanuele Marconato, Stefano Teso, Antonio Vergari et al.

Neuro-Symbolic (NeSy) predictive models hold the promise of improved compliance with given constraints, systematic generalization, and interpretability, as they allow to infer labels that are consistent with some prior knowledge by reasoning over high-level concepts extracted from sub-symbolic inputs. It was recently shown that NeSy predictors are affected by reasoning shortcuts: they can attain high accuracy but by leveraging concepts with unintended semantics, thus coming short of their promised advantages. Yet, a systematic characterization of reasoning shortcuts and of potential mitigation strategies is missing. This work fills this gap by characterizing them as unintended optima of the learning objective and identifying four key conditions behind their occurrence. Based on this, we derive several natural mitigation strategies, and analyze their efficacy both theoretically and empirically. Our analysis shows reasoning shortcuts are difficult to deal with, casting doubts on the trustworthiness and interpretability of existing NeSy solutions.

Lorenzo Loconte, Nicola Di Mauro, Robert Peharz et al.

Some of the most successful knowledge graph embedding (KGE) models for link prediction -- CP, RESCAL, TuckER, ComplEx -- can be interpreted as energy-based models. Under this perspective they are not amenable for exact maximum-likelihood estimation (MLE), sampling and struggle to integrate logical constraints. This work re-interprets the score functions of these KGEs as circuits -- constrained computational graphs allowing efficient marginalisation. Then, we design two recipes to obtain efficient generative circuit models by either restricting their activations to be non-negative or squaring their outputs. Our interpretation comes with little or no loss of performance for link prediction, while the circuits framework unlocks exact learning by MLE, efficient sampling of new triples, and guarantee that logical constraints are satisfied by design. Furthermore, our models scale more gracefully than the original KGEs on graphs with millions of entities.

Stefano Teso, Antonio Vergari

In this position paper, we study interactive learning for structured output spaces, with a focus on active learning, in which labels are unknown and must be acquired, and on skeptical learning, in which the labels are noisy and may need relabeling. These scenarios require expressive models that guarantee reliable and efficient computation of probabilistic quantities to measure uncertainty. We identify conditions under which a class of probabilistic models -- which we denote CRISPs -- meet all of these conditions, thus delivering tractable computation of the above quantities while preserving expressiveness. Building on prior work on tractable probabilistic circuits, we illustrate how CRISPs enable robust and efficient active and skeptical learning in large structured output spaces.

Wenzhe Li, Zhe Zeng, Antonio Vergari et al.

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.

Antonio Vergari, YooJung Choi, Anji Liu et al.

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 deep mixture models -- can be represented in terms of tractable modular operations over circuits. Specifically, we characterize the tractability of a vocabulary 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.

Alessandro Suglia, Yonatan Bisk, Ioannis Konstas et al.

Guessing games are a prototypical instance of the "learning by interacting" paradigm. This work investigates how well an artificial agent can benefit from playing guessing games when later asked to perform on novel NLP downstream tasks such as Visual Question Answering (VQA). We propose two ways to exploit playing guessing games: 1) a supervised learning scenario in which the agent learns to mimic successful guessing games and 2) a novel way for an agent to play by itself, called Self-play via Iterated Experience Learning (SPIEL). We evaluate the ability of both procedures to generalize: an in-domain evaluation shows an increased accuracy (+7.79) compared with competitors on the evaluation suite CompGuessWhat?!; a transfer evaluation shows improved performance for VQA on the TDIUC dataset in terms of harmonic average accuracy (+5.31) thanks to more fine-grained object representations learned via SPIEL.

Alessandro Suglia, Antonio Vergari, Ioannis Konstas et al.

In visual guessing games, a Guesser has to identify a target object in a scene by asking questions to an Oracle. An effective strategy for the players is to learn conceptual representations of objects that are both discriminative and expressive enough to ask questions and guess correctly. However, as shown by Suglia et al. (2020), existing models fail to learn truly multi-modal representations, relying instead on gold category labels for objects in the scene both at training and inference time. This provides an unnatural performance advantage when categories at inference time match those at training time, and it causes models to fail in more realistic "zero-shot" scenarios where out-of-domain object categories are involved. To overcome this issue, we introduce a novel "imagination" module based on Regularized Auto-Encoders, that learns context-aware and category-aware latent embeddings without relying on category labels at inference time. Our imagination module outperforms state-of-the-art competitors by 8.26% gameplay accuracy in the CompGuessWhat?! zero-shot scenario (Suglia et al., 2020), and it improves the Oracle and Guesser accuracy by 2.08% and 12.86% in the GuessWhat?! benchmark, when no gold categories are available at inference time. The imagination module also boosts reasoning about object properties and attributes.

Meihua Dang, Antonio Vergari, Guy Van den Broeck

Probabilistic circuits (PCs) represent a probability distribution as a computational graph. Enforcing structural properties on these graphs guarantees that several inference scenarios become tractable. Among these properties, structured decomposability is a particularly appealing one: it enables the efficient and exact computations of the probability of complex logical formulas, and can be used to reason about the expected output of certain predictive models under missing data. This paper proposes Strudel, a simple, fast and accurate learning algorithm for structured-decomposable PCs. Compared to prior work for learning structured-decomposable PCs, Strudel delivers more accurate single PC models in fewer iterations, and dramatically scales learning when building ensembles of PCs. It achieves this scalability by exploiting another structural property of PCs, called determinism, and by sharing the same computational graph across mixture components. We show these advantages on standard density estimation benchmarks and challenging inference scenarios.

Pasha Khosravi, Antonio Vergari, YooJung Choi et al.

Decision trees are a popular family of models due to their attractive properties such as interpretability and ability to handle heterogeneous data. Concurrently, missing data is a prevalent occurrence that hinders performance of machine learning models. As such, handling missing data in decision trees is a well studied problem. In this paper, we tackle this problem by taking a probabilistic approach. At deployment time, we use tractable density estimators to compute the "expected prediction" of our models. At learning time, we fine-tune parameters of already learned trees by minimizing their "expected prediction loss" w.r.t.\ our density estimators. We provide brief experiments showcasing effectiveness of our methods compared to few baselines.