Formal Verification of Input-Output Mappings of Tree Ensembles

Recent advances in machine learning and artificial intelligence are now being considered in safety-critical autonomous systems where software defects may cause severe harm to humans and the environment. Design organizations in these domains are currently unable to provide convincing arguments that their systems are safe to operate when machine learning algorithms are used to implement their software. In this paper, we present an efficient method to extract equivalence classes from decision trees and tree ensembles, and to formally verify that their input-output mappings comply with requirements. The idea is that, given that safety requirements can be traced to desirable properties on system input-output patterns, we can use positive verification outcomes in safety arguments. This paper presents the implementation of the method in the tool VoTE (Verifier of Tree Ensembles), and evaluates its scalability on two case studies presented in current literature. We demonstrate that our method is practical for tree ensembles trained on low-dimensional data with up to 25 decision trees and tree depths of up to 20. Our work also studies the limitations of the method with high-dimensional data and preliminarily investigates the trade-off between large number of trees and time taken for verification

Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

Boosting Answer Set Optimization with Weighted Comparator Networks

Answer set programming (ASP) is a paradigm for modeling knowledge intensive domains and solving challenging reasoning problems. In ASP solving, a typical strategy is to preprocess problem instances by rewriting complex rules into simpler ones. Normalization is a rewriting process that removes extended rule types altogether in favor of normal rules. Recently, such techniques led to optimization rewriting in ASP, where the goal is to boost answer set optimization by refactoring the optimization criteria of interest. In this paper, we present a novel, general, and effective technique for optimization rewriting based on comparator networks, which are specific kinds of circuits for reordering the elements of vectors. The idea is to connect an ASP encoding of a comparator network to the literals being optimized and to redistribute the weights of these literals over the structure of the network. The encoding captures information about the weight of an answer set in auxiliary atoms in a structured way that is proven to yield exponential improvements during branch-and-bound optimization on an infinite family of example programs. The used comparator network can be tuned freely, e.g., to find the best size for a given benchmark class. Experiments show accelerated optimization performance on several benchmark problems.Comment: 36 page

Boosting Haplotype Inference with Local Search

Abstract. A very challenging problem in the genetics domain is to infer haplotypes from genotypes. This process is expected to identify genes affecting health, disease and response to drugs. One of the approaches to haplotype inference aims to minimise the number of different haplotypes used, and is known as haplotype inference by pure parsimony (HIPP). The HIPP problem is computationally difficult, being NP-hard. Recently, a SAT-based method (SHIPs) has been proposed to solve the HIPP problem. This method iteratively considers an increasing number of haplotypes, starting from an initial lower bound. Hence, one important aspect of SHIPs is the lower bounding procedure, which reduces the number of iterations of the basic algorithm, and also indirectly simplifies the resulting SAT model. This paper describes the use of local search to improve existing lower bounding procedures. The new lower bounding procedure is guaranteed to be as tight as the existing procedures. In practice the new procedure is in most cases considerably tighter, allowing significant improvement of performance on challenging problem instances.

Enabling Incrementality in the Implicit Hitting Set Approach to MaxSAT Under Changing Weights

Recent advances in solvers for the Boolean satisfiability (SAT) based optimization paradigm of maximum satisfiability (MaxSAT) have turned MaxSAT into a viable approach to finding provably optimal solutions for various types of hard optimization problems. In various types of real-world problem settings, a sequence of related optimization problems need to solved. This calls for studying ways of enabling incremental computations in MaxSAT, with the hope of speeding up the overall computation times. However, current state-of-the-art MaxSAT solvers offer no or limited forms of incrementality. In this work, we study ways of enabling incremental computations in the context of the implicit hitting set (IHS) approach to MaxSAT solving, as both one of the key MaxSAT solving approaches today and a relatively well-suited candidate for extending to incremental computations. In particular, motivated by several recent applications of MaxSAT in the context of interpretability in machine learning calling for this type of incrementality, we focus on enabling incrementality in IHS under changes to the objective function coefficients (i.e., to the weights of soft clauses). To this end, we explain to what extent different search techniques applied in IHS-based MaxSAT solving can and cannot be adapted to this incremental setting. As practical result, we develop an incremental version of an IHS MaxSAT solver, and show it provides significant runtime improvements in recent application settings which can benefit from incrementality but in which MaxSAT solvers have so-far been applied only non-incrementally, i.e., by calling a MaxSAT solver from scratch after each change to the problem instance at hand

Deep Graph Laplacian Regularization for Robust Denoising of Real Images

Recent developments in deep learning have revolutionized the paradigm of image restoration. However, its applications on real image denoising are still limited, due to its sensitivity to training data and the complex nature of real image noise. In this work, we combine the robustness merit of model-based approaches and the learning power of data-driven approaches for real image denoising. Specifically, by integrating graph Laplacian regularization as a trainable module into a deep learning framework, we are less susceptible to overfitting than pure CNN-based approaches, achieving higher robustness to small datasets and cross-domain denoising. First, a sparse neighborhood graph is built from the output of a convolutional neural network (CNN). Then the image is restored by solving an unconstrained quadratic programming problem, using a corresponding graph Laplacian regularizer as a prior term. The proposed restoration pipeline is fully differentiable and hence can be end-to-end trained. Experimental results demonstrate that our work is less prone to overfitting given small training data. It is also endowed with strong cross-domain generalization power, outperforming the state-of-the-art approaches by a remarkable margin