arXiv subjects

Pietro Barbiero

Publications and source records attributed to Pietro Barbiero.

At least 19 recordsLinked to original sources

arXiv2026

Graphical Design of Interpretable Architectures

Designing, implementing, and comparing interpretable architectures requires a formal language to represent them. The most common representations fall short in one of two ways. Symbolic equations give no global view of an architecture at a glance. Probabilistic graphical models and flowcharts do not describe actual tensor manipulations, thus hiding key insights and limiting reproducibility. To close this gap, we introduce a graphical notation for designing interpretable AI architectures, adapted from Penrose tensor notation. This graphical notation gives a global view of an architecture and maps one to one onto PyTorch einsum code. We first use this notation to describe architectures that are interpretable by construction, including concept bottlenecks, sparse probes, prototype networks, neural additive models, and mixtures of linear models. We then diagram the key architectural components of Steerling-8B, a frontier interpretable language model. The diagram yields global insights into the architecture (e.g., showing that Steerling is a residual model), a geometric interpretation of each individual operation, and a direct translation into 33 lines of PyTorch code.

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arXiv2026

The Standard Interpretable Model: A general theory of interpretable machine learning to deductively design interpretable methods using Lagrangian mechanics

As Artificial Intelligence models grow in complexity, interpretability has become an indispensable tool for understanding, debugging, and controlling their computations. However, interpretability lacks general theories to deductively design interpretable methods. This gap between theories and methods results in a fragmented literature and inconsistent evaluation protocols. To fill this gap, we introduce the Standard Interpretable Model (SIM), a general theory grounded in Lagrangian mechanics that enables the deductive design of interpretable methods. Specifically, the SIM summarises, in a set of premises, what interpretability is for a target user. From these premises, the SIM systematically derives interpretability symmetries and corresponding constraints, which shape the landscape of a Lagrangian whose minima correspond to optimal interpretable models. To reach the minima, one can either update the parameter values of an opaque model to make it more interpretable or compile constraints into an interpretable architecture. We empirically show that the SIM identifies and solves limitations of existing methods (including traditional, concept-based, and mechanistic interpretability), highlights underexplored research directions, and informs the design of core programming interfaces. Beyond being a research method, the deductive nature of the SIM offers pedagogical grounding for interpretability curricula and may shift the scientific community's perspective of a discipline that has long been fragmented.

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arXiv2026

Prototype-Grounded Concept Models for Verifiable Concept Alignment

Concept Bottleneck Models (CBMs) aim to improve interpretability in Deep Learning by structuring predictions through human-understandable concepts, but they provide no way to verify whether learned concepts align with the human's intended meaning, hurting interpretability. We introduce Prototype-Grounded Concept Models (PGCMs), which ground concepts in learned visual prototypes: image parts that serve as explicit evidence for the concepts. This grounding enables direct inspection of concept semantics and supports targeted human intervention at the prototype level to correct misalignments. Empirically, PGCMs achieve similar predictive performance as state-of-the-art CBMs while substantially improving transparency, interpretability, and intervenability.

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arXiv2026

Federated Concept-Based Models: Interpretable models with distributed supervision

Concept-based Models (CMs) enhance interpretability in deep learning by grounding predictions in human-understandable concepts. However, concept annotations are costly and rarely available at scale within a single data source. Federated Learning (FL) could alleviate this limitation by enabling cross-institutional training over concept annotations distributed across multiple data owners. Yet, FL lacks interpretable modeling paradigms. Integrating CMs with FL is non-trivial: although FL supports heterogeneous and non-stationary client participation, it typically assumes a fixed shared architecture, whereas CMs may require architectural adaptation as the available concept set evolves. We propose Federated Concept-based Models (F-CMs), a new methodology for deploying CMs in evolving FL settings. F-CMs aggregate concept-level information across institutions and efficiently adapt the model architecture to changes in concept supervision while preserving privacy. Empirically, F-CMs maintain accuracy and intervention effectiveness comparable to training settings with full concept supervision, while outperforming on average non-adaptive federated baselines. Notably, F-CMs enable interpretable inference on concepts unavailable to a given institution, a key novelty over existing approaches.

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arXiv2026

Interpretability in Deep Time Series Models Demands Semantic Alignment

Deep time series models continue to improve predictive performance, yet their deployment remains limited by their black-box nature. In response, existing interpretability approaches in the field keep focusing on explaining the internal model computations, without addressing whether they align or not with how a human would reason about the studied phenomenon. Instead, we state interpretability in deep time series models should pursue semantic alignment: predictions should be expressed in terms of variables that are meaningful to the end user, mediated by spatial and temporal mechanisms that admit user-dependent constraints. In this paper, we formalize this requirement and state that, once established, semantic alignment must be preserved under temporal evolution: a constraint with no analog in static settings. Provided with this definition, we outline a blueprint for semantically aligned deep time series models, identify properties that support trust, and discuss implications for model design.

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arXiv2026

Mixture of Concept Bottleneck Experts

Concept Bottleneck Models (CBMs) promote interpretability by grounding predictions in human-understandable concepts. However, existing CBMs typically constrain their task predictor to a single expression whose functional form is set a priori, limiting both predictive accuracy and adaptability to diverse user needs. We propose Mixture of Concept Bottleneck Experts (M-CBE), a framework that generalizes existing CBMs along two dimensions: the number of expressions, referred to as experts, employed by the task predictor to map concepts to the task, and the functional form each expression takes, thus exposing an underexplored region of this design space. We investigate this region by instantiating two novel models: Linear M-CBE, which learns a finite set of linear expressions, and Symbolic M-CBE, which leverages symbolic regression to discover expert functions from data subject to user-specified operator vocabularies. Empirical evaluation demonstrates that varying the number of expressions and their functional form provides a robust framework for navigating the accuracy-interpretability trade-off.

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arXiv2026

Actionable Interpretability Must Be Defined in Terms of Symmetries

This paper argues that interpretability research in Artificial Intelligence (AI) is fundamentally ill-posed as existing definitions of interpretability fail to describe how interpretability can be formally tested or designed for. We posit that actionable definitions of interpretability must be formulated in terms of *symmetries* that inform model design and lead to testable conditions. Under a probabilistic view, we hypothesise that four symmetries (inference equivariance, information invariance, concept-closure invariance, and structural invariance) suffice to (i) formalise interpretable models as a subclass of probabilistic models, (ii) yield a unified formulation of interpretable inference (e.g., alignment, interventions, and counterfactuals) as a form of Bayesian inversion, and (iii) provide a formal framework to verify compliance with safety standards and regulations.

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arXiv2025

FLUX: Efficient Descriptor-Driven Clustered Federated Learning under Arbitrary Distribution Shifts

Federated Learning (FL) enables collaborative model training across multiple clients while preserving data privacy. Traditional FL methods often use a global model to fit all clients, assuming that clients' data are independent and identically distributed (IID). However, when this assumption does not hold, the global model accuracy may drop significantly, limiting FL applicability in real-world scenarios. To address this gap, we propose FLUX, a novel clustering-based FL (CFL) framework that addresses the four most common types of distribution shifts during both training and test time. To this end, FLUX leverages privacy-preserving client-side descriptor extraction and unsupervised clustering to ensure robust performance and scalability across varying levels and types of distribution shifts. Unlike existing CFL methods addressing non-IID client distribution shifts, FLUX i) does not require any prior knowledge of the types of distribution shifts or the number of client clusters, and ii) supports test-time adaptation, enabling unseen and unlabeled clients to benefit from the most suitable cluster-specific models. Extensive experiments across four standard benchmarks, two real-world datasets and ten state-of-the-art baselines show that FLUX improves performance and stability under diverse distribution shifts, achieving an average accuracy gain of up to 23 percentage points over the best-performing baselines, while maintaining computational and communication overhead comparable to FedAvg.

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arXiv2025

Scalable Evaluation and Neural Models for Compositional Generalization

Compositional generalization-a key open challenge in modern machine learning-requires models to predict unknown combinations of known concepts. However, assessing compositional generalization remains a fundamental challenge due to the lack of standardized evaluation protocols and the limitations of current benchmarks, which often favor efficiency over rigor. At the same time, general-purpose vision architectures lack the necessary inductive biases, and existing approaches to endow them compromise scalability. As a remedy, this paper introduces: 1) a rigorous evaluation framework that unifies and extends previous approaches while reducing computational requirements from combinatorial to constant; 2) an extensive and modern evaluation on the status of compositional generalization in supervised vision backbones, training more than 5000 models; 3) Attribute Invariant Networks, a class of models establishing a new Pareto frontier in compositional generalization, achieving a 23.43% accuracy improvement over baselines while reducing parameter overhead from 600% to 16% compared to fully disentangled counterparts. Our code is available at https://github.com/IBM/scalable-compositional-generalization.

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arXiv2025

Foundations of Interpretable Models

We argue that existing definitions of interpretability are not actionable in that they fail to inform users about general, sound, and robust interpretable model design. This makes current interpretability research fundamentally ill-posed. To address this issue, we propose a definition of interpretability that is general, simple, and subsumes existing informal notions within the interpretable AI community. We show that our definition is actionable, as it directly reveals the foundational properties, underlying assumptions, principles, data structures, and architectural features necessary for designing interpretable models. Building on this, we propose a general blueprint for designing interpretable models and introduce the first open-sourced library with native support for interpretable data structures and processes.

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arXiv2025

Interpretable Hierarchical Concept Reasoning through Attention-Guided Graph Learning

Concept-Based Models (CBMs) are a class of deep learning models that provide interpretability by explaining predictions through high-level concepts. These models first predict concepts and then use them to perform a downstream task. However, current CBMs offer interpretability only for the final task prediction, while the concept predictions themselves are typically made via black-box neural networks. To address this limitation, we propose Hierarchical Concept Memory Reasoner (H-CMR), a new CBM that provides interpretability for both concept and task predictions. H-CMR models relationships between concepts using a learned directed acyclic graph, where edges represent logic rules that define concepts in terms of other concepts. During inference, H-CMR employs a neural attention mechanism to select a subset of these rules, which are then applied hierarchically to predict all concepts and the final task. Experimental results demonstrate that H-CMR matches state-of-the-art performance while enabling strong human interaction through concept and model interventions. The former can significantly improve accuracy at inference time, while the latter can enhance data efficiency during training when background knowledge is available.

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arXiv2025

Towards Better Generalization and Interpretability in Unsupervised Concept-Based Models

To increase the trustworthiness of deep neural networks, it is critical to improve the understanding of how they make decisions. This paper introduces a novel unsupervised concept-based model for image classification, named Learnable Concept-Based Model (LCBM) which models concepts as random variables within a Bernoulli latent space. Unlike traditional methods that either require extensive human supervision or suffer from limited scalability, our approach employs a reduced number of concepts without sacrificing performance. We demonstrate that LCBM surpasses existing unsupervised concept-based models in generalization capability and nearly matches the performance of black-box models. The proposed concept representation enhances information retention and aligns more closely with human understanding. A user study demonstrates the discovered concepts are also more intuitive for humans to interpret. Finally, despite the use of concept embeddings, we maintain model interpretability by means of a local linear combination of concepts.

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arXiv2025

If Concept Bottlenecks are the Question, are Foundation Models the Answer?

Concept Bottleneck Models (CBMs) are neural networks designed to conjoin high performance with ante-hoc interpretability. CBMs work by first mapping inputs (e.g., images) to high-level concepts (e.g., visible objects and their properties) and then use these to solve a downstream task (e.g., tagging or scoring an image) in an interpretable manner. Their performance and interpretability, however, hinge on the quality of the concepts they learn. The go-to strategy for ensuring good quality concepts is to leverage expert annotations, which are expensive to collect and seldom available in applications. Researchers have recently addressed this issue by introducing "VLM-CBM" architectures that replace manual annotations with weak supervision from foundation models. It is however unclear what is the impact of doing so on the quality of the learned concepts. To answer this question, we put state-of-the-art VLM-CBMs to the test, analyzing their learned concepts empirically using a selection of significant metrics. Our results show that, depending on the task, VLM supervision can sensibly differ from expert annotations, and that concept accuracy and quality are not strongly correlated. Our code is available at https://github.com/debryu/CQA.

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arXiv2025

Avoiding Leakage Poisoning: Concept Interventions Under Distribution Shifts

In this paper, we investigate how concept-based models (CMs) respond to out-of-distribution (OOD) inputs. CMs are interpretable neural architectures that first predict a set of high-level concepts (e.g., stripes, black) and then predict a task label from those concepts. In particular, we study the impact of concept interventions (i.e., operations where a human expert corrects a CM's mispredicted concepts at test time) on CMs' task predictions when inputs are OOD. Our analysis reveals a weakness in current state-of-the-art CMs, which we term leakage poisoning, that prevents them from properly improving their accuracy when intervened on for OOD inputs. To address this, we introduce MixCEM, a new CM that learns to dynamically exploit leaked information missing from its concepts only when this information is in-distribution. Our results across tasks with and without complete sets of concept annotations demonstrate that MixCEMs outperform strong baselines by significantly improving their accuracy for both in-distribution and OOD samples in the presence and absence of concept interventions.

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arXiv2025

Causal Concept Graph Models: Beyond Causal Opacity in Deep Learning

Causal opacity denotes the difficulty in understanding the "hidden" causal structure underlying the decisions of deep neural network (DNN) models. This leads to the inability to rely on and verify state-of-the-art DNN-based systems, especially in high-stakes scenarios. For this reason, circumventing causal opacity in DNNs represents a key open challenge at the intersection of deep learning, interpretability, and causality. This work addresses this gap by introducing Causal Concept Graph Models (Causal CGMs), a class of interpretable models whose decision-making process is causally transparent by design. Our experiments show that Causal CGMs can: (i) match the generalisation performance of causally opaque models, (ii) enable human-in-the-loop corrections to mispredicted intermediate reasoning steps, boosting not just downstream accuracy after corrections but also the reliability of the explanations provided for specific instances, and (iii) support the analysis of interventional and counterfactual scenarios, thereby improving the model's causal interpretability and supporting the effective verification of its reliability and fairness.

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arXiv2025

Deferring Concept Bottleneck Models: Learning to Defer Interventions to Inaccurate Experts

Concept Bottleneck Models (CBMs) are machine learning models that improve interpretability by grounding their predictions on human-understandable concepts, allowing for targeted interventions in their decision-making process. However, when intervened on, CBMs assume the availability of humans that can identify the need to intervene and always provide correct interventions. Both assumptions are unrealistic and impractical, considering labor costs and human error-proneness. In contrast, Learning to Defer (L2D) extends supervised learning by allowing machine learning models to identify cases where a human is more likely to be correct than the model, thus leading to deferring systems with improved performance. In this work, we gain inspiration from L2D and propose Deferring CBMs (DCBMs), a novel framework that allows CBMs to learn when an intervention is needed. To this end, we model DCBMs as a composition of deferring systems and derive a consistent L2D loss to train them. Moreover, by relying on a CBM architecture, DCBMs can explain why defer occurs on the final task. Our results show that DCBMs achieve high predictive performance and interpretability at the cost of deferring more to humans.

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arXiv2025

Logic Explanation of AI Classifiers by Categorical Explaining Functors

The most common methods in explainable artificial intelligence are post-hoc techniques which identify the most relevant features used by pretrained opaque models. Some of the most advanced post hoc methods can generate explanations that account for the mutual interactions of input features in the form of logic rules. However, these methods frequently fail to guarantee the consistency of the extracted explanations with the model's underlying reasoning. To bridge this gap, we propose a theoretically grounded approach to ensure coherence and fidelity of the extracted explanations, moving beyond the limitations of current heuristic-based approaches. To this end, drawing from category theory, we introduce an explaining functor which structurally preserves logical entailment between the explanation and the opaque model's reasoning. As a proof of concept, we validate the proposed theoretical constructions on a synthetic benchmark verifying how the proposed approach significantly mitigates the generation of contradictory or unfaithful explanations.

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arXiv2025

Causally Reliable Concept Bottleneck Models

Concept-based models are an emerging paradigm in deep learning that constrains the inference process to operate through human-interpretable variables, facilitating explainability and human interaction. However, these architectures, on par with popular opaque neural models, fail to account for the true causal mechanisms underlying the target phenomena represented in the data. This hampers their ability to support causal reasoning tasks, limits out-of-distribution generalization, and hinders the implementation of fairness constraints. To overcome these issues, we propose Causally reliable Concept Bottleneck Models (C$^2$BMs), a class of concept-based architectures that enforce reasoning through a bottleneck of concepts structured according to a model of the real-world causal mechanisms. We also introduce a pipeline to automatically learn this structure from observational data and unstructured background knowledge (e.g., scientific literature). Experimental evidence suggests that C$^2$BMs are more interpretable, causally reliable, and improve responsiveness to interventions w.r.t. standard opaque and concept-based models, while maintaining their accuracy.

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