We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a Gaussian random field. In this setting, the posterior distribution on the objective function gives rise to a posterior distribution on excursion sets. Several approaches exist to summarize the distribution of such sets based on random closed set theory. While the recently proposed Vorob'ev approach exploits analytical formulae, further notions of variability require Monte Carlo estimators relying on Gaussian random field conditional simulations. In the present work we propose a method to choose Monte Carlo simulation points and obtain quasi-realizations of the conditional field at fine designs through affine predictors. The points are chosen optimally in the sense that they minimize the posterior expected distance in measure between the excursion set and its reconstruction. The proposed method reduces the computational costs due to Monte Carlo simulations and enables the computation of quasi-realizations on fine designs in large dimensions. We apply this reconstruction approach to obtain realizations of an excursion set on a fine grid which allow us to give a new measure of uncertainty based on the distance transform of the excursion set. Finally we present a safety engineering test case where the simulation method is employed to compute a Monte Carlo estimate of a contour line.

A BSTRACT Existing approaches to combine both additive and multiplicative neural units either use a fixed assignment of operations or require discrete optimization to determine what function a neuron should perform. However, this leads to an extensive increase in the computational complexity of the training procedure. We present a novel, parameterizable transfer function based on the mathematical concept of non-integer functional iteration that allows the operation each neuron performs to be smoothly and, most importantly, differentiablely adjusted between addition and multiplication. This allows the decision between addition and multiplication to be integrated into the standard backpropagation training procedure. The value of such a product unit is given byy i σ ( j x W ij j).

The paper focuses on finding the m best solutions to combinatorial optimization problems using best-first or depth-first branch and bound search. Specifically, we present a new algorithm m-A*, extending the well-known A* to the m-best task, and for the first time prove that all its desirable properties, including soundness, completeness and optimal efficiency, are maintained. Since best-first algorithms require extensive memory, we also extend the memory-efficient depth-first branch and bound to the m-best task. We adapt both algorithms to optimization tasks over graphical models (e.g., Weighted CSP and MPE in Bayesian networks), provide complexity analysis and an empirical evaluation. Our experiments confirm theory that the best-first approach is largely superior when memory is available, but depth-first branch and bound is more robust. We also show that our algorithms are competitive with related schemes recently developed for the m-best task.

While there exist several approaches in the constraint programming community to learn a constraint theory, few of them have considered the learning of constraint optimization problems.To alleviate this situation, we introduce an initial approach to learning first-order weighted MAX-SAT theories. It employs inductive logic programming techniques to learn a set of first-order clauses and then uses preference learning techniques to learn the weights of the clauses.In order to learn these weighted clauses, the clausal optimization system uses examples of possible worlds and a set of preferences that state which examples are preferred over other ones.The technique is also empirically evaluated on a number of examples.These experiments show that the system is capable of learning clauses and weights that accurately capture underlying models.

Many problems from the area of AI have been shown tractable for bounded treewidth. In order to put such results into practice, quite involved dynamic programming (DP) algorithms on tree decompositions have to be designed and implemented. These algorithms typically show recurring patterns that call for tasks like subset minimization. In this paper, we provide a new method for obtaining DP algorithms from simpler principles, where the necessary data structures and algorithms for subset minimization are automatically generated. Moreover, we discuss how this method can be implemented in systems that perform more space-efficiently than current approaches.

We present RELOOP, a domain-specific language for relational optimization embedded in Python. It allows the user to express relational optimization problems in a natural syntax that follows logic and linear algebra, rather than in the restrictive standard form required by solvers, and can automatically compile the model to a lower-order but equivalent model. Moreover, RELOOP makes it easy to combine relational optimization with high-level features of Python such as loops, parallelism and interfaces to relational databases.

Empirical risk minimization frequently employs convex surrogates to underlying discrete loss functions in order to achieve computational tractability during optimization. However, classical convex surrogates can only tightly bound modular loss functions, sub-modular functions or supermodular functions separately while maintaining polynomial time computation. In this work, a novel generic convex surrogate for general non-modular loss functions is introduced, which provides for the first time a tractable solution for loss functions that are neither super-modular nor submodular. This convex surro-gate is based on a submodular-supermodular decomposition for which the existence and uniqueness is proven in this paper. It takes the sum of two convex surrogates that separately bound the supermodular component and the submodular component using slack-rescaling and the Lov{\'a}sz hinge, respectively. It is further proven that this surrogate is convex , piecewise linear, an extension of the loss function, and for which subgradient computation is polynomial time. Empirical results are reported on a non-submodular loss based on the S{{\o}}rensen-Dice difference function, and a real-world face track dataset with tens of thousands of frames, demonstrating the improved performance, efficiency, and scalabil-ity of the novel convex surrogate.

We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range of applications in data science, where the objective is used for inducing sparsity in the solutions while the constraint set models the noise tolerance and incorporates other prior information for data fitting. To solve this class of constrained optimization problems, a common approach is the penalty method. However, there is little theory on exact penalization for problems with nonconvex and non-Lipschitz objective functions. In this paper, we study the existence of exact penalty parameters regarding local minimizers, stationary points and $\epsilon$-minimizers under suitable assumptions. Moreover, we discuss a penalty method whose subproblems are solved via a nonmonotone proximal gradient method with a suitable update scheme for the penalty parameters, and prove the convergence of the algorithm to a KKT point of the constrained problem. Preliminary numerical results demonstrate the efficiency of the penalty method for finding sparse solutions of underdetermined linear systems.

This paper presents a Bayesian optimization method with exponential convergence without the need of auxiliary optimization and without the delta-cover sampling. Most Bayesian optimization methods require auxiliary optimization: an additional non-convex global optimization problem, which can be time-consuming and hard to implement in practice. Also, the existing Bayesian optimization method with exponential convergence requires access to the delta-cover sampling, which was considered to be impractical. Our approach eliminates both requirements and achieves an exponential convergence rate.

This tutorial introduces the CMA Evolution Strategy (ES), where CMA stands for Covariance Matrix Adaptation. The CMA-ES is a stochastic, or randomized, method for real-parameter (continuous domain) optimization of non-linear, non-convex functions. We try to motivate and derive the algorithm from intuitive concepts and from requirements of non-linear, non-convex search in continuous domain.