Locally-Minimal Probabilistic Explanations

Formal abductive explanations offer crucial guarantees of rigor and so are of interest in high-stakes uses of machine learning (ML). One drawback of abductive explanations is explanation size, justified by the cognitive limits of human decision-makers. Probabilistic abductive explanations (PAXps) address this limitation, but their theoretical and practical complexity makes their exact computation most often unrealistic. This paper proposes novel efficient algorithms for the computation of locally-minimal PXAps, which offer high-quality approximations of PXAps in practice. The experimental results demonstrate the practical efficiency of the proposed algorithms.

Auditable Algorithms for Approximate Model Counting

Model counting, or counting the satisfying assignments of a Boolean formula, is a fundamental problem with diverse applications. Given #P-hardness of the problem, developing algorithms for approximate counting is an important research area. Building on the practical success of SAT-solvers, the focus has recently shifted from theory to practical implementations of approximate counting algorithms. This has brought to focus new challenges, such as the design of auditable approximate counters that not only provide an approximation of the model count, but also a certificate that a verifier with limited computational power can use to check if the count is indeed within the promised bounds of approximation. Towards generating certificates, we start by examining the best-known deterministic approximate counting algorithm that uses polynomially many calls to a $\Sigma_2^P$ oracle. We show that this can be audited via a $\Sigma_2^P$ oracle with the query constructed over $n^2 \log^2 n$ variables, where the original formula has $n$ variables. Since $n$ is often large, we ask if the count of variables in the certificate can be reduced -- a crucial question for potential implementation. We show that this is indeed possible with a tradeoff in the counting algorithm's complexity. Specifically, we develop new deterministic approximate counting algorithms that invoke a $\Sigma_3^P$ oracle, but can be certified using a $\Sigma_2^P$ oracle using certificates on far fewer variables: our final algorithm uses only $n \log n$ variables. Our study demonstrates that one can simplify auditing significantly if we allow the counting algorithm to access a slightly more powerful oracle. This shows for the first time how audit complexity can be traded for complexity of approximate counting.

Engineering an Exact Pseudo-Boolean Model Counter

Model counting, a fundamental task in computer science, involves determining the number of satisfying assignments to a Boolean formula, typically represented in conjunctive normal form (CNF). While model counting for CNF formulas has received extensive attention with a broad range of applications, the study of model counting for Pseudo-Boolean (PB) formulas has been relatively overlooked. Pseudo-Boolean formulas, being more succinct than propositional Boolean formulas, offer greater flexibility in representing real-world problems. Consequently, there is a crucial need to investigate efficient techniques for model counting for PB formulas. In this work, we propose the first exact Pseudo-Boolean model counter, PBCount, that relies on knowledge compilation approach via algebraic decision diagrams. Our extensive empirical evaluation shows that PBCount can compute counts for 1513 instances while the current state-of-the-art approach could only handle 1013 instances. Our work opens up several avenues for future work in the context of model counting for PB formulas, such as the development of preprocessing techniques and exploration of approaches other than knowledge compilation.

Total Variation Distance Estimation Is as Easy as Probabilistic Inference

In this paper, we establish a novel connection between total variation (TV) distance estimation and probabilistic inference. In particular, we present an efficient, structure-preserving reduction from relative approximation of TV distance to probabilistic inference over directed graphical models. This reduction leads to a fully polynomial randomized approximation scheme (FPRAS) for estimating TV distances between distributions over any class of Bayes nets for which there is an efficient probabilistic inference algorithm. In particular, it leads to an FPRAS for estimating TV distances between distributions that are defined by Bayes nets of bounded treewidth. Prior to this work, such approximation schemes only existed for estimating TV distances between product distributions. Our approach employs a new notion of $partial$ couplings of high-dimensional distributions, which might be of independent interest.

Scalable Probabilistic Routes

Inference and prediction of routes have become of interest over the past decade owing to a dramatic increase in package delivery and ride-sharing services. Given the underlying combinatorial structure and the incorporation of probabilities, route prediction involves techniques from both formal methods and machine learning. One promising approach for predicting routes uses decision diagrams that are augmented with probability values. However, the effectiveness of this approach depends on the size of the compiled decision diagrams. The scalability of the approach is limited owing to its empirical runtime and space complexity. In this work, our contributions are two-fold: first, we introduce a relaxed encoding that uses a linear number of variables with respect to the number of vertices in a road network graph to significantly reduce the size of resultant decision diagrams. Secondly, instead of a stepwise sampling procedure, we propose a single pass sampling-based route prediction. In our evaluations arising from a real-world road network, we demonstrate that the resulting system achieves around twice the quality of suggested routes while being an order of magnitude faster compared to state-of-the-art.

Explaining SAT Solving Using Causal Reasoning

The past three decades have witnessed notable success in designing efficient SAT solvers, with modern solvers capable of solving industrial benchmarks containing millions of variables in just a few seconds. The success of modern SAT solvers owes to the widely-used CDCL algorithm, which lacks comprehensive theoretical investigation. Furthermore, it has been observed that CDCL solvers still struggle to deal with specific classes of benchmarks comprising only hundreds of variables, which contrasts with their widespread use in real-world applications. Consequently, there is an urgent need to uncover the inner workings of these seemingly weak yet powerful black boxes. In this paper, we present a first step towards this goal by introducing an approach called CausalSAT, which employs causal reasoning to gain insights into the functioning of modern SAT solvers. CausalSAT initially generates observational data from the execution of SAT solvers and learns a structured graph representing the causal relationships between the components of a SAT solver. Subsequently, given a query such as whether a clause with low literals blocks distance (LBD) has a higher clause utility, CausalSAT calculates the causal effect of LBD on clause utility and provides an answer to the question. We use CausalSAT to quantitatively verify hypotheses previously regarded as "rules of thumb" or empirical findings such as the query above. Moreover, CausalSAT can address previously unexplored questions, like which branching heuristic leads to greater clause utility in order to study the relationship between branching and clause management. Experimental evaluations using practical benchmarks demonstrate that CausalSAT effectively fits the data, verifies four "rules of thumb", and provides answers to three questions closely related to implementing modern solvers.

Rounding Meets Approximate Model Counting

The problem of model counting, also known as #SAT, is to compute the number of models or satisfying assignments of a given Boolean formula $F$. Model counting is a fundamental problem in computer science with a wide range of applications. In recent years, there has been a growing interest in using hashing-based techniques for approximate model counting that provide $(\varepsilon, \delta)$-guarantees: i.e., the count returned is within a $(1+\varepsilon)$-factor of the exact count with confidence at least $1-\delta$. While hashing-based techniques attain reasonable scalability for large enough values of $\delta$, their scalability is severely impacted for smaller values of $\delta$, thereby preventing their adoption in application domains that require estimates with high confidence. The primary contribution of this paper is to address the Achilles heel of hashing-based techniques: we propose a novel approach based on rounding that allows us to achieve a significant reduction in runtime for smaller values of $\delta$. The resulting counter, called RoundMC, achieves a substantial runtime performance improvement over the current state-of-the-art counter, ApproxMC. In particular, our extensive evaluation over a benchmark suite consisting of 1890 instances shows that RoundMC solves 204 more instances than ApproxMC, and achieves a $4\times$ speedup over ApproxMC.

Synthesis with Explicit Dependencies

Quantified Boolean Formulas (QBF) extend propositional logic with quantification $\forall, \exists$. In QBF, an existentially quantified variable is allowed to depend on all universally quantified variables in its scope. Dependency Quantified Boolean Formulas (DQBF) restrict the dependencies of existentially quantified variables. In DQBF, existentially quantified variables have explicit dependencies on a subset of universally quantified variables called Henkin dependencies. Given a Boolean specification between the set of inputs and outputs, the problem of Henkin synthesis is to synthesize each output variable as a function of its Henkin dependencies such that the specification is met. Henkin synthesis has wide-ranging applications, including verification of partial circuits, controller synthesis, and circuit realizability. This work proposes a data-driven approach for Henkin synthesis called Manthan3. On an extensive evaluation of over 563 instances arising from past DQBF solving competitions, we demonstrate that Manthan3 is competitive with state-of-the-art tools. Furthermore, Manthan3 could synthesize Henkin functions for 26 benchmarks for which none of the state-of-the-art techniques could synthesize.

Fast Converging Anytime Model Counting

Model counting is a fundamental problem which has been influential in many applications, from artificial intelligence to formal verification. Due to the intrinsic hardness of model counting, approximate techniques have been developed to solve real-world instances of model counting. This paper designs a new anytime approach called PartialKC for approximate model counting. The idea is a form of partial knowledge compilation to provide an unbiased estimate of the model count which can converge to the exact count. Our empirical analysis demonstrates that PartialKC achieves significant scalability and accuracy over prior state-of-the-art approximate counters, including satss and STS. Interestingly, the empirical results show that PartialKC reaches convergence for many instances and therefore provides exact model counting performance comparable to state-of-the-art exact counters.

How Biased is Your Feature?: Computing Fairness Influence Functions with Global Sensitivity Analysis

Fairness in machine learning has attained significant focus due to the widespread application of machine learning in high-stake decision-making tasks. Unless regulated with a fairness objective, machine learning classifiers might demonstrate unfairness/bias towards certain demographic populations in the data. Thus, the quantification and mitigation of the bias induced by classifiers have become a central concern. In this paper, we aim to quantify the influence of different features on the bias of a classifier. To this end, we propose a framework of Fairness Influence Function (FIF), and compute it as a scaled difference of conditional variances in the prediction of the classifier. We also instantiate an algorithm, FairXplainer, that uses variance decomposition among the subset of features and a local regressor to compute FIFs accurately, while also capturing the intersectional effects of the features. Our experimental analysis validates that FairXplainer captures the influences of both individual features and higher-order feature interactions, estimates the bias more accurately than existing local explanation methods, and detects the increase/decrease in bias due to affirmative/punitive actions in the classifier.