ABSTRACT

Alternatives to recurrent neural networks, in particular, architectures based on self-attention, are gaining momentum for processing input sequences. In spite of their relevance, the computational properties of such networks have not yet been fully explored. We study the computational power of the Transformer, one of the most paradigmatic architectures exemplifying self-attention. We show that the Transformer with hard-attention is Turing complete exclusively based on their capacity to compute and access internal dense representations of the data. Our study also reveals some minimal sets of elements needed to obtain this completeness result.

ABSTRACT

We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the domains of these unknown values are finite and, for a Boolean query q, we consider the following two problems: Given as input an incomplete database D, (a) return the number of completions of D that satisfy q; or (b) return the number of valuations of the nulls of D yielding a completion that satisfies q. We obtain dichotomies between #P-hardness and polynomial-time computability for these problems when q is a self-join–free conjunctive query and study the impact on the complexity of the following two restrictions: (1) every null occurs at most once in D (what is called Codd tables); and (2) the domain of each null is the same. Roughly speaking, we show that counting completions is much harder than counting valuations: For instance, while the latter is always in #P, we prove that the former is not in #P under some widely believed theoretical complexity assumption. Moreover, we find that both (1) and (2) can reduce the complexity of our problems. We also study the approximability of these problems and show that, while counting valuations always has a fully polynomial-time randomized approximation scheme (FPRAS), in most cases counting completions does not. Finally, we consider more expressive query languages and situate our problems with respect to known complexity classes.

ABSTRACT

Several queries and scores have recently been proposed to explain individual predictions over ML models. Examples include queries based on “anchors”, which are parts of an instance that are sufficient to justify its classification, and “feature-perturbation” scores such as SHAP. Given the need for flexible, reliable, and easy-to-apply interpretability methods for ML models, we foresee the need for developing declarative languages to naturally specify different explainability queries. We do this in a principled way by rooting such a language in a logic called FOIL, which allows for expressing many simple but important explainability queries, and might serve as a core for more expressive interpretability languages. We study the computational complexity of FOIL queries over two classes of ML models often deemed to be easily interpretable: decision trees and more general decision diagrams. Since the number of possible inputs for an ML model is exponential in its dimension, tractability of the FOIL evaluation problem is delicate but can be achieved by either restricting the structure of the models, or the fragment of FOIL being evaluated. We also present a prototype implementation of FOIL wrapped in a high-level declarative language and perform experiments showing that such a language can be used in practice.

Publisher: Advances in Neural Information Processing Systems, Link>

ABSTRACT

In Machine Learning, the SHAP-score is a version of the Shapley value that is used to explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is an intractable problem, we prove a strong positive result stating that the SHAP-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits are studied in the field of Knowledge Compilation and generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees and Ordered Binary Decision Diagrams (OBDDs). We also establish the computational limits of the SHAP-score by observing that computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider. It also implies that computing SHAP-scores is intractable as well over the class of propositional formulas in DNF. Based on this negative result, we look for the existence of fully-polynomial randomized approximation schemes (FPRAS) for computing SHAP-scores over such class. In contrast to the model counting problem for DNF formulas, which admits an FPRAS, we prove that no such FPRAS exists for the computation of SHAP-scores. Surprisingly, this negative result holds even for the class of monotone formulas in DNF. These techniques can be further extended to prove another strong negative result: Under widely believed complexity assumptions, there is no polynomial-time algorithm that checks, given a monotone DNF formula φ and features x,y, whether the SHAP-score of x in φ is smaller than the SHAP-score of y in φ.

ABSTRACT

This is a release of parts 1, 2, and 4 of the upcoming book “Principles of Databases”, which will be about the foundational and mathematical principles of databases in its various forms. The first two parts focus on an overview of the relational model, and on processing some of the most commonly occurring relational queries. Forthcoming parts will focus on additional aspects of the relational model and will cover tree-structured and graph-structured data as well.

Publisher: Journal of Computer and System Sciences, Link>

ABSTRACT

The sudden loss of smell is among the earliest and most prevalent symptoms of COVID-19 when measured with a clinical psychophysical test. Research has shown the potential impact of frequent screening for olfactory dysfunction, but existing tests are expensive and time consuming. We developed a low-cost ($0.50/test) rapid psychophysical olfactory test (KOR) for frequent testing and a model-based COVID-19 screening framework using a Bayes Network symptoms model. We trained and validated the model on two samples: suspected COVID-19 cases in five healthcare centers (n = 926; 33% prevalence, 309 RT-PCR confirmed) and healthy miners (n = 1,365; 1.1% prevalence, 15 RT-PCR confirmed). The model predicted COVID-19 status with 76% and 96% accuracy in the healthcare and miners samples, respectively (healthcare: AUC = 0.79 [0.75–0.82], sensitivity: 59%, specificity: 87%; miners: AUC = 0.71 [0.63–0.79], sensitivity: 40%, specificity: 97%, at 0.50 infection probability threshold). Our results highlight the potential for low-cost, frequent, accessible, routine COVID-19 testing to support society's reopening.

ABSTRACT

Scores based on Shapley values are widely used for providing explanations to classification results over machine learning models. A prime example of this is the influential SHAP-score, a version of the Shapley value that can help explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is a computationally intractable problem, it has recently been claimed that the SHAP-score can be computed in polynomial time over the class of decision trees. In this paper, we provide a proof of a stronger result over Boolean models: the SHAP-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits, also known as tractable Boolean circuits, generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees, Ordered Binary Decision Diagrams (OBDDs) and Free Binary Decision Diagrams (FBDDs). We also establish the computational limits of the notion of SHAP-score by observing that, under a mild condition, computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider, as removing one or the other renders the problem of computing the SHAP-score intractable (namely, #P-hard).

ABSTRACT

Scores based on Shapley values are widely used for providing explanations to classification results over machine learning models. A prime example of this is the influential SHAP-score, a version of the Shapley value that can help explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is a computationally intractable problem, it has recently been claimed that the SHAP-score can be computed in polynomial time over the class of decision trees. In this paper, we provide a proof of a stronger result over Boolean models: the SHAP-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits, also known as tractable Boolean circuits, generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees, Ordered Binary Decision Diagrams (OBDDs) and Free Binary Decision Diagrams (FBDDs). We also establish the computational limits of the notion of SHAP-score by observing that, under a mild condition, computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider, as removing one or the other renders the problem of computing the SHAP-score intractable (namely, #P-hard).