Neighbourhood Evaluation Criteria is a heuristical approximate algorithm that attempts to solve the Minimum Vertex Cover. degree count is kept in check for each vertex and the highest count based vertex is included in our cover set. In the case of multiple equivalent vertices, the one with the lowest neighbourhood influence is selected. In the case of still existing multiple equivalent vertices, the one with the lowest remaining active vertex count (the highest Independent Set enabling count) is selected as a tie-breaker.

We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.

We consider the problem of maximizing the $\ell_1$ norm of a linear map over the sphere, which arises in various machine learning applications such as orthogonal dictionary learning (ODL) and robust subspace recovery (RSR). The problem is numerically challenging due to its nonsmooth objective and nonconvex constraint, and its algorithmic aspects have not been well explored. In this paper, we show how the manifold structure of the sphere can be exploited to design fast algorithms for tackling this problem. Specifically, our contribution is threefold. First, we present a manifold proximal point algorithm (ManPPA) for the problem and show that it converges at a sublinear rate. Furthermore, we show that ManPPA can achieve a quadratic convergence rate when applied to the ODL and RSR problems. Second, we propose a stochastic variant of ManPPA called StManPPA, which is well suited for large-scale computation, and establish its sublinear convergence rate. Both ManPPA and StManPPA have provably faster convergence rates than existing subgradient-type methods. Third, using ManPPA as a building block, we propose a new approach to solving a matrix analog of the problem, in which the sphere is replaced by the Stiefel manifold. The results from our extensive numerical experiments on the ODL and RSR problems demonstrate the efficiency and efficacy of our proposed methods.

A new robust algorithm based of the explanation method SurvLIME called SurvLIME-KS is proposed for explaining machine learning survival models. The algorithm is developed to ensure robustness to cases of a small amount of training data or outliers of survival data. The first idea behind SurvLIME-KS is to apply the Cox proportional hazards model to approximate the black-box survival model at the local area around a test example due to the linear relationship of covariates in the model. The second idea is to incorporate the well-known Kolmogorov-Smirnov bounds for constructing sets of predicted cumulative hazard functions. As a result, the robust maximin strategy is used, which aims to minimize the average distance between cumulative hazard functions of the explained black-box model and of the approximating Cox model, and to maximize the distance over all cumulative hazard functions in the interval produced by the Kolmogorov-Smirnov bounds. The maximin optimization problem is reduced to the quadratic program. Various numerical experiments with synthetic and real datasets demonstrate the SurvLIME-KS efficiency.

Many applications require a learner to make sequential decisions given uncertainty regarding both the system's payoff function and safety constraints. In safety-critical systems, it is paramount that the learner's actions do not violate the safety constraints at any stage of the learning process. In this paper, we study a stochastic bandit optimization problem where the unknown payoff and constraint functions are sampled from Gaussian Processes (GPs) first considered in [Srinivas et al., 2010]. We develop a safe variant of GP-UCB called SGP-UCB, with necessary modifications to respect safety constraints at every round. The algorithm has two distinct phases. The first phase seeks to estimate the set of safe actions in the decision set, while the second phase follows the GP-UCB decision rule. Our main contribution is to derive the first sub-linear regret bounds for this problem. We numerically compare SGP-UCB against existing safe Bayesian GP optimization algorithms.

State-of-the-art subspace clustering methods are based on self-expressive model, which represents each data point as a linear combination of other data points. By enforcing such representation to be sparse, sparse subspace clustering is guaranteed to produce a subspace-preserving data affinity where two points are connected only if they are from the same subspace. On the other hand, however, data points from the same subspace may not be well-connected, leading to the issue of over-segmentation. We introduce dropout to address the issue of over-segmentation, which is based on randomly dropping out data points in self-expressive model. In particular, we show that dropout is equivalent to adding a squared $\ell_2$ norm regularization on the representation coefficients, therefore induces denser solutions. Then, we reformulate the optimization problem as a consensus problem over a set of small-scale subproblems. This leads to a scalable and flexible sparse subspace clustering approach, termed Stochastic Sparse Subspace Clustering, which can effectively handle large scale datasets. Extensive experiments on synthetic data and real world datasets validate the efficiency and effectiveness of our proposal.

A central concern in an interactive intelligent system is optimization of its actions, to be maximally helpful to its human user. In recommender systems for instance, the action is to choose what to recommend, and the optimization task is to recommend items the user prefers. The optimization is done based on earlier user's feedback (e.g. "likes" and "dislikes"), and the algorithms assume the feedback to be faithful. That is, when the user clicks "like," they actually prefer the item. We argue that this fundamental assumption can be extensively violated by human users, who are not passive feedback sources. Instead, they are in control, actively steering the system towards their goal. To verify this hypothesis, that humans steer and are able to improve performance by steering, we designed a function optimization task where a human and an optimization algorithm collaborate to find the maximum of a 1-dimensional function. At each iteration, the optimization algorithm queries the user for the value of a hidden function $f$ at a point $x$, and the user, who sees the hidden function, provides an answer about $f(x)$. Our study on 21 participants shows that users who understand how the optimization works, strategically provide biased answers (answers not equal to $f(x)$), which results in the algorithm finding the optimum significantly faster. Our work highlights that next-generation intelligent systems will need user models capable of helping users who steer systems to pursue their goals.

We consider infinite horizon dynamic programming problems, where the control at each stage consists of several distinct decisions, each one made by one of several agents. In an earlier work we introduced a policy iteration algorithm, where the policy improvement is done one-agent-at-a-time in a given order, with knowledge of the choices of the preceding agents in the order. As a result, the amount of computation for each policy improvement grows linearly with the number of agents, as opposed to exponentially for the standard all-agents-at-once method. For the case of a finite-state discounted problem, we showed convergence to an agent-by-agent optimal policy. In this paper, this result is extended to value iteration and optimistic versions of policy iteration, as well as to more general DP problems where the Bellman operator is a contraction mapping, such as stochastic shortest path problems with all policies being proper.

Not all generate-and-test search algorithms are created equal. Bayesian Optimization (BO) invests a lot of computation time to generate the candidate solution that best balances the predicted value and the uncertainty given all previous data, taking increasingly more time as the number of evaluations performed grows. Evolutionary Algorithms (EA) on the other hand rely on search heuristics that typically do not depend on all previous data and can be done in constant time. Both the BO and EA community typically assess their performance as a function of the number of evaluations. However, this is unfair once we start to compare the efficiency of these classes of algorithms, as the overhead times to generate candidate solutions are significantly different. We suggest to measure the efficiency of generate-and-test search algorithms as the expected gain in the objective value per unit of computation time spent. We observe that the preference of an algorithm to be used can change after a number of function evaluations. We therefore propose a new algorithm, a combination of Bayesian optimization and an Evolutionary Algorithm, BEA for short, that starts with BO, then transfers knowledge to an EA, and subsequently runs the EA. We compare the BEA with BO and the EA. The results show that BEA outperforms both BO and the EA in terms of time efficiency, and ultimately leads to better performance on well-known benchmark objective functions with many local optima. Moreover, we test the three algorithms on nine test cases of robot learning problems and here again we find that BEA outperforms the other algorithms.

Riemannian optimization has drawn a lot of attention due to its wide applications in practice. Riemannian stochastic first-order algorithms have been studied in the literature to solve large-scale machine learning problems over Riemannian manifolds. However, most of the existing Riemannian stochastic algorithms require the objective function to be differentiable, and they do not apply to the case where the objective function is nonsmooth. In this paper, we present two Riemannian stochastic proximal gradient methods for minimizing nonsmooth function over the Stiefel manifold. The two methods, named R-ProxSGD and R-ProxSPB, are generalizations of proximal SGD and proximal SpiderBoost in Euclidean setting to the Riemannian setting. Analysis on the incremental first-order oracle (IFO) complexity of the proposed algorithms is provided. Specifically, the R-ProxSPB algorithm finds an $\epsilon$-stationary point with $\mathcal{O}(\epsilon^{-3})$ IFOs in the online case, and $\mathcal{O}(n+\sqrt{n}\epsilon^{-3})$ IFOs in the finite-sum case with $n$ being the number of summands in the objective. Experimental results on online sparse PCA and robust low-rank matrix completion show that our proposed methods significantly outperform the existing methods that uses Riemannian subgradient information.