Marcelo Arenas’s research while affiliated with University of Santiago Chile and other places

Formal XAI is an emerging field that focuses on providing explanations with mathematical guarantees for the decisions made by machine learning models. A significant amount of work in this area is centered on the computation of ``sufficient reasons''. Given a model M and an input instance x, a sufficient reason for the decision x is a subset S of the features of x such that for any instance z that has the same values as x for every feature in S, it holds that M(x) = M(z). Intuitively, this means that the features in S are sufficient to fully justify the classification of x by M. For sufficient reasons to be useful in practice, they should be as small as possible, and a natural way to reduce the size of sufficient reasons is to consider a probabilistic relaxation; the probability of M(x) = M(z) must be at least some value delta in (0,1], where z is a random instance that coincides with x on the features in S. Computing small delta-sufficient reasons (delta-SRs) is known to be a theoretically hard problem; even over decision trees — traditionally deemed simple and interpretable models — strong inapproximability results make the efficient computation of small delta-SRs unlikely. We propose the notion of (delta, epsilon)-SR, a simple relaxation of delta-SRs, and show that this kind of explanations can be computed efficiently over linear models.

Formal XAI is an emerging field that focuses on providing explanations with mathematical guarantees for the decisions made by machine learning models. A significant amount of work in this area is centered on the computation of "sufficient reasons". Given a model M and an input instance $\vec{x}$, a sufficient reason for the decision $M(\vec{x})$ is a subset S of the features of $\vec{x}$ such that for any instance $\vec{z}$ that has the same values as $\vec{x}$ for every feature in S, it holds that $M(\vec{x}) = M(\vec{z})$. Intuitively, this means that the features in S are sufficient to fully justify the classification of $\vec{x}$ by M. For sufficient reasons to be useful in practice, they should be as small as possible, and a natural way to reduce the size of sufficient reasons is to consider a probabilistic relaxation; the probability of $M(\vec{x}) = M(\vec{z})$ must be at least some value $\delta \in (0,1]$, for a random instance $\vec{z}$ that coincides with $\vec{x}$ on the features in S. Computing small $\delta$-sufficient reasons ($\delta$-SRs) is known to be a theoretically hard problem; even over decision trees--traditionally deemed simple and interpretable models--strong inapproximability results make the efficient computation of small $\delta$-SRs unlikely. We propose the notion of $(\delta, \epsilon)$-SR, a simple relaxation of $\delta$-SRs, and show that this kind of explanation can be computed efficiently over linear models.

The set of answers to a query may be very large, potentially overwhelming users when presented with the entire set. In such cases, presenting only a small subset of the answers to the user may be preferable. A natural requirement for this subset is that it should be as diverse as possible to reflect the variety of the entire population. To achieve this, the diversity of a subset is measured using a metric that determines how different two solutions are and a diversity function that extends this metric from pairs to sets. In the past, several studies have shown that finding a diverse subset from an explicitly given set is intractable even for simple metrics (like Hamming distance) and simple diversity functions (like summing all pairwise distances). This complexity barrier becomes even more challenging when trying to output a diverse subset from a set that is only implicitly given (such as the query answers for a given query and a database). Until now, tractable cases have been found only for restricted problems and particular diversity functions. To overcome these limitations, we focus in this work on the notion of ultrametrics, which have been widely studied and used in many applications. Starting from any ultrametric d and a diversity function δ extending d , we provide sufficient conditions over δ for having polynomial-time algorithms to construct diverse answers. To the best of our knowledge, these conditions are satisfied by all the diversity functions considered in the literature. Moreover, we complement these results with lower bounds that show specific cases when these conditions are not satisfied and finding diverse subsets becomes intractable. We conclude by applying these results to the evaluation of conjunctive queries, demonstrating efficient algorithms for finding a diverse subset of solutions for acyclic conjunctive queries when the attribute order is used to measure diversity.

The formal XAI community has studied a plethora of interpretability queries aiming to understand the classifications made by decision trees. However, a more uniform understanding of what questions we can hope to answer about these models, traditionally deemed to be easily interpretable, has remained elusive. In an initial attempt to understand uniform languages for interpretability, Arenas et al. proposed FOIL, a logic for explaining black-box ML models, and showed that it can express a variety of interpretability queries. However, we show that FOIL is limited in two important senses: (i) it is not expressive enough to capture some crucial queries, and (ii) its model agnostic nature results in a high computational complexity for decision trees. In this paper, we carefully craft two fragments of first-order logic that allow for efficiently interpreting decision trees: Q-DT-FOIL and its optimization variant OPT-DT-FOIL. We show that our proposed logics can express not only a variety of interpretability queries considered by previous literature, but also elegantly allows users to specify different objectives the sought explanations should optimize for. Using finite model-theoretic techniques, we show that the different ingredients of Q-DT-FOIL are necessary for its expressiveness, and yet that queries in Q-DT-FOIL can be evaluated with a polynomial number of queries to a SAT solver, as well as their optimization versions in OPT-DT-FOIL. Besides our theoretical results, we provide a SAT-based implementation of the evaluation for OPT-DT-FOIL that is performant on industry-size decision trees.

The set of answers to a query may be very large, potentially overwhelming users when presented with the entire set. In such cases, presenting only a small subset of the answers to the user may be preferable. A natural requirement for this subset is that it should be as diverse as possible to reflect the variety of the entire population. To achieve this, the diversity of a subset is measured using a metric that determines how different two solutions are and a diversity function that extends this metric from pairs to sets. In the past, several studies have shown that finding a diverse subset from an explicitly given set is intractable even for simple metrics (like Hamming distance) and simple diversity functions (like summing all pairwise distances). This complexity barrier becomes even more challenging when trying to output a diverse subset from a set that is only implicitly given such as the query answers of a query and a database. Until now, tractable cases have been found only for restricted problems and particular diversity functions. To overcome these limitations, we focus on the notion of ultrametrics, which have been widely studied and used in many applications. Starting from any ultrametric d and a diversity function $\delta$ extending d, we provide sufficient conditions over $\delta$ for having polynomial-time algorithms to construct diverse answers. To the best of our knowledge, these conditions are satisfied by all diversity functions considered in the literature. Moreover, we complement these results with lower bounds that show specific cases when these conditions are not satisfied and finding diverse subsets becomes intractable. We conclude by applying these results to the evaluation of conjunctive queries, demonstrating efficient algorithms for finding a diverse subset of solutions for acyclic conjunctive queries when the attribute order is used to measure diversity.

In this systems paper, we present MillenniumDB: a novel graph database engine that is modular, persistent, and open source. MillenniumDB is based on a graph data model, which we call domain graphs, that provides a simple abstraction upon which a variety of popular graph models can be supported, thus providing a flexible data management engine for diverse types of knowledge graph. The engine itself is founded on a combination of tried and tested techniques from relational data management, state-of-the-art algorithms for worst-case-optimal joins, as well as graph-specific algorithms for evaluating path queries. In this paper, we present the main design principles underlying MillenniumDB, describing the abstract graph model and query semantics supported, the concrete data model and query syntax implemented, as well as the storage, indexing, query planning and query evaluation techniques used. We evaluate MillenniumDB over real-world data and queries from the Wikidata knowledge graph, where we find that it outperforms other popular persistent graph database engines (including both enterprise and open source alternatives) that support similar query features.

In this systems paper, we present MillenniumDB: a novel graph database engine that is modular, persistent, and open source. MillenniumDB is based on a graph data model, which we call domain graphs, that provides a simple abstraction upon which a variety of popular graph models can be supported, thus providing a flexible data management engine for diverse types of knowledge graph. The engine itself is founded on a combination of tried and tested techniques from relational data management, state-of-the-art algorithms for worst-case-optimal joins, as well as graph-specific algorithms for evaluating path queries. In this paper, we present the main design principles underlying MillenniumDB, describing the abstract graph model and query semantics supported, the concrete data model and query syntax implemented, as well as the storage, indexing, query planning and query evaluation techniques used. We evaluate MillenniumDB over real-world data and queries from the Wikidata knowledge graph, where we find that it outperforms other popular persistent graph database engines (including both enterprise and open source alternatives) that support similar query features.

Counting the answers to a query is a fundamental problem in databases, with several applications in the evaluation, optimization, and visualization of queries. Unfortunately, counting query answers is a #P-hard problem in most cases, so it is unlikely to be solvable in polynomial time. Recently, new results on approximate counting have been developed, specifically by showing that some problems in automata theory admit fully polynomial-time randomized approximation schemes. These results have several implications for the problem of counting the answers to a query; in particular, for graph and conjunctive queries. In this work, we present the main ideas of these approximation results, by using labeled DAGs instead of automata to simplify the presentation. In addition, we review how to apply these results to count query answers in different areas of databases.

Formal XAI (explainable AI) is a growing area that focuses on computing explanations with mathematical guarantees for the decisions made by ML models. Inside formal XAI, one of the most studied cases is that of explaining the choices taken by decision trees, as they are traditionally deemed as one of the most interpretable classes of models. Recent work has focused on studying the computation of "sufficient reasons", a kind of explanation in which given a decision tree T and an instance x, one explains the decision T(x) by providing a subset y of the features of x such that for any other instance z compatible with y, it holds that $T(z) = T(x)$, intuitively meaning that the features in y are already enough to fully justify the classification of x by T. It has been argued, however, that sufficient reasons constitute a restrictive notion of explanation, and thus the community has started to study their probabilistic counterpart, in which one requires that the probability of $T(z) = T(x)$ must be at least some value $\delta \in (0, 1]$, where z is a random instance that is compatible with y. Our paper settles the computational complexity of $\delta$-sufficient-reasons over decision trees, showing that both (1) finding $\delta$-sufficient-reasons that are minimal in size, and (2) finding $\delta$-sufficient-reasons that are minimal inclusion-wise, do not admit polynomial-time algorithms (unless P=NP). This is in stark contrast with the deterministic case ($\delta = 1$) where inclusion-wise minimal sufficient-reasons are easy to compute. By doing this, we answer two open problems originally raised by Izza et al. On the positive side, we identify structural restrictions of decision trees that make the problem tractable, and show how SAT solvers might be able to tackle these problems in practical settings.