Paolo Morettin

Giuseppe Spallitta, Gabriele Masina, Paolo Morettin et al.

The development of efficient exact and approximate algorithms for probabilistic inference is a long-standing goal of artificial intelligence research. Whereas substantial progress has been made in dealing with purely discrete or purely continuous domains, adapting the developed solutions to tackle hybrid domains, characterised by discrete and continuous variables and their relationships, is highly non-trivial. Weighted Model Integration (WMI) recently emerged as a unifying formalism for probabilistic inference in hybrid domains. Despite a considerable amount of recent work, allowing WMI algorithms to scale with the complexity of the hybrid problem is still a challenge. In this paper we highlight some substantial limitations of existing state-of-the-art solutions, and develop an algorithm that combines SMT-based enumeration, an efficient technique in formal verification, with an effective encoding of the problem structure. This allows our algorithm to avoid generating redundant models, resulting in drastic computational savings. Additionally, we show how SMT-based approaches can seamlessly deal with different integration techniques, both exact and approximate, significantly expanding the set of problems that can be tackled by WMI technology. An extensive experimental evaluation on both synthetic and real-world datasets confirms the substantial advantage of the proposed solution over existing alternatives. The application potential of this technology is further showcased on a prototypical task aimed at verifying the fairness of probabilistic programs.

Giuseppe Spallitta, Gabriele Masina, Paolo Morettin et al.

Weighted Model Integration (WMI) is a popular formalism aimed at unifying approaches for probabilistic inference in hybrid domains, involving logical and algebraic constraints. Despite a considerable amount of recent work, allowing WMI algorithms to scale with the complexity of the hybrid problem is still a challenge. In this paper we highlight some substantial limitations of existing state-of-the-art solutions, and develop an algorithm that combines SMT-based enumeration, an efficient technique in formal verification, with an effective encoding of the problem structure. This allows our algorithm to avoid generating redundant models, resulting in substantial computational savings. An extensive experimental evaluation on both synthetic and real-world datasets confirms the advantage of the proposed solution over existing alternatives.

Vincent Derkinderen, Pedro Zuidberg Dos Martires, Samuel Kolb et al.

Propositional model counting (#SAT) can be solved efficiently when the input formula is in deterministic decomposable negation normal form (d-DNNF). Translating an arbitrary formula into a representation that allows inference tasks, such as counting, to be performed efficiently, is called knowledge compilation. Top-down knowledge compilation is a state-of-the-art technique for solving #SAT problems that leverages the traces of exhaustive DPLL search to obtain d-DNNF representations. While knowledge compilation is well studied for propositional approaches, knowledge compilation for the (quantifier free) counting modulo theory setting (#SMT) has been studied to a much lesser degree. In this paper, we discuss compilation strategies for #SMT. We specifically advocate for a top-down compiler based on the traces of exhaustive DPLL(T) search.

Samuele Bortolotti, Emanuele Marconato, Paolo Morettin et al.

Concept-based Models are neural networks that learn a concept extractor to map inputs to high-level concepts and an inference layer to translate these into predictions. Ensuring these modules produce interpretable concepts and behave reliably in out-of-distribution is crucial, yet the conditions for achieving this remain unclear. We study this problem by establishing a novel connection between Concept-based Models and reasoning shortcuts (RSs), a common issue where models achieve high accuracy by learning low-quality concepts, even when the inference layer is fixed and provided upfront. Specifically, we extend RSs to the more complex setting of Concept-based Models and derive theoretical conditions for identifying both the concepts and the inference layer. Our empirical results highlight the impact of RSs and show that existing methods, even combined with multiple natural mitigation strategies, often fail to meet these conditions in practice.

Kareem Ahmed, Stefano Teso, Paolo Morettin et al. · amazon-science

Structured output prediction problems are ubiquitous in machine learning. The prominent approach leverages neural networks as powerful feature extractors, otherwise assuming the independence of the outputs. These outputs, however, jointly encode an object, e.g. a path in a graph, and are therefore related through the structure underlying the output space. We discuss the semantic loss, which injects knowledge about such structure, defined symbolically, into training by minimizing the network's violation of such dependencies, steering the network towards predicting distributions satisfying the underlying structure. At the same time, it is agnostic to the arrangement of the symbols, and depends only on the semantics expressed thereby, while also enabling efficient end-to-end training and inference. We also discuss key improvements and applications of the semantic loss. One limitations of the semantic loss is that it does not exploit the association of every data point with certain features certifying its membership in a target class. We should therefore prefer minimum-entropy distributions over valid structures, which we obtain by additionally minimizing the neuro-symbolic entropy. We empirically demonstrate the benefits of this more refined formulation. Moreover, the semantic loss is designed to be modular and can be combined with both discriminative and generative neural models. This is illustrated by integrating it into generative adversarial networks, yielding constrained adversarial networks, a novel class of deep generative models able to efficiently synthesize complex objects obeying the structure of the underlying domain.

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.

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.

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.

Paolo Morettin, Andrea Passerini, Roberto Sebastiani

In machine learning (ML) verification, the majority of procedures are non-quantitative and therefore cannot be used for verifying probabilistic models, or be applied in domains where hard guarantees are practically unachievable. The probabilistic formal verification (PFV) of ML models is in its infancy, with the existing approaches limited to specific ML models, properties, or both. This contrasts with standard formal methods techniques, whose successful adoption in real-world scenarios is also due to their support for a wide range of properties and diverse systems. We propose a unifying framework for the PFV of ML systems based on Weighted Model Integration (WMI), a relatively recent formalism for probabilistic inference with algebraic and logical constraints. Crucially, reducing the PFV of ML models to WMI enables the verification of many properties of interest over a wide range of systems, addressing multiple limitations of deterministic verification and ad-hoc algorithms. We substantiate the generality of the approach on prototypical tasks involving the verification of group fairness, monotonicity, robustness to noise, probabilistic local robustness and equivalence among predictors. We characterize the challenges related to the scalability of the approach and, through our WMI-based perspective, we show how successful scaling techniques in the ML verification literature can be generalized beyond their original scope.

Luca Di Liello, Pierfrancesco Ardino, Jacopo Gobbi et al.

Generative Adversarial Networks (GANs) struggle to generate structured objects like molecules and game maps. The issue is that structured objects must satisfy hard requirements (e.g., molecules must be chemically valid) that are difficult to acquire from examples alone. As a remedy, we propose Constrained Adversarial Networks (CANs), an extension of GANs in which the constraints are embedded into the model during training. This is achieved by penalizing the generator proportionally to the mass it allocates to invalid structures. In contrast to other generative models, CANs support efficient inference of valid structures (with high probability) and allows to turn on and off the learned constraints at inference time. CANs handle arbitrary logical constraints and leverage knowledge compilation techniques to efficiently evaluate the disagreement between the model and the constraints. Our setup is further extended to hybrid logical-neural constraints for capturing very complex constraints, like graph reachability. An extensive empirical analysis shows that CANs efficiently generate valid structures that are both high-quality and novel.

Zhe Zeng, Paolo Morettin, Fanqi Yan et al.

Weighted model integration (WMI) is a very appealing framework for probabilistic inference: it allows to express the complex dependencies of 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 at all, or they are limited to too restrictive structures. To narrow this gap, we derive a factorized formalism of WMI enabling us to devise a scalable WMI solver based on message passing, MP-WMI. Namely, MP-WMI is the first WMI solver which allows to: 1) perform exact inference on the full class of tree-structured WMI problems; 2) compute all marginal densities in linear time; 3) amortize inference inter query. Experimental results show that our solver dramatically outperforms the existing WMI solvers on a large set of benchmarks.

Zhe Zeng, Fanqi Yan, Paolo Morettin et al.

Weighted model integration (WMI) is a very appealing framework for probabilistic inference: it allows to express the complex dependencies of real-world hybrid scenarios where variables are heterogeneous in nature (both continuous and discrete) via the language of Satisfiability Modulo Theories (SMT); as well as computing probabilistic queries with arbitrarily complex logical constraints. Recent work has shown WMI inference to be reducible to a model integration (MI) problem, under some assumptions, thus effectively allowing hybrid probabilistic reasoning by volume computations. In this paper, we introduce a novel formulation of MI via a message passing scheme that allows to efficiently compute the marginal densities and statistical moments of all the variables in linear time. As such, we are able to amortize inference for arbitrarily rich MI queries when they conform to the problem structure, here represented as the primal graph associated to the SMT formula. Furthermore, we theoretically trace the tractability boundaries of exact MI. Indeed, we prove that in terms of the structural requirements on the primal graph that make our MI algorithm tractable - bounding its diameter and treewidth - the bounds are not only sufficient, but necessary for tractable inference via MI.