Pietro Barbiero
Publications
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.
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 match the predictive performance of state-of-the-art CBMs while substantially improving transparency, interpretability, and intervenability.
Mixture of Concept Bottleneck Experts
Concept Bottleneck Models (CBMs) promote interpretability by grounding predictions in human-understandable concepts. However, existing CBMs typically fix their task predictor to a single linear or Boolean expression, limiting both predictive accuracy and adaptability to diverse user needs. We propose Mixture of Concept Bottleneck Experts (M-CBEs), a framework that generalizes existing CBMs along two dimensions: the number of experts and the functional form of each expert, exposing an underexplored region of the 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 under user-specified operator vocabularies. Empirical evaluation demonstrates that varying the mixture size and functional form provides a robust framework for navigating the accuracy-interpretability trade-off, adapting to different user and task needs.
Actionable Interpretability Must Be Defined in Terms of Symmetries
This paper argues that interpretability research in Artificial Intelligence is fundamentally ill-posed as existing definitions of interpretability are not *actionable*: they fail to provide formal principles from which concrete modelling and inferential rules can be derived. We posit that for a definition of interpretability to be actionable, it must be given in terms of *symmetries*. We hypothesise that four symmetries suffice to (i) motivate core interpretability properties, (ii) characterize the class of interpretable models, and (iii) derive a unified formulation of interpretable inference (e.g., alignment, interventions, and counterfactuals) as a form of Bayesian inversion.
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.