Reserve price is an effective tool for revenue maximization in ad auctions. The optimal reserve price depends on bidders' value distributions, which, however, are generally unknown to auctioneers. A common practice for auctioneers is to first collect information about the value distributions by a sampling procedure and then apply the reserve price estimated with the sampled bids to the following auctions. In order to maximize the total revenue over finite auctions, it is important for the auctioneer to find a proper sample size to trade off between the cost of the sampling procedure and the optimality of the estimated reserve price. We investigate the sample size optimization problem for Generalized Second Price auctions, which is the most widely-used mechanism in ad auctions, and make three main contributions along this line. First, we bound the revenue losses in the form of competitive ratio during and after sampling. Second, we formulate the problem of finding the optimal sample size as a non-convex mixed integer optimization problem. Then we characterize the properties of the problem and prove the uniqueness of the optimal sample size. Third, we relax the integer optimization problem to a continuous form and develop an efficient algorithm based on the properties to solve it. Experimental results show that our approach can significantly improve the revenue for the auctioneer in finitely repeated ad auctions.

Most existing models of Stackelberg security games ignore the underlying topology of the space in which targets and defence resources are located. As a result, allocation of resources is restricted to a discrete collection of exogenously defined targets. However, in many practical security settings, defense resources can be located on a continuous plane. Better defense solutions could therefore be potentially achieved by placing resources in a space outside of actual targets (e.g., between targets). To address this limitation, we propose a model called Security Game on a Plane (SGP) in which targets are distributed on a 2-dimensional plane, and security resources, to be allocated on the same plane, protect targets within a certain effective distance. We investigate the algorithmic aspects of SGP. We find that computing a strong Stackelberg equilibrium of an SGP is NP-hard even for zero-sum games, and these are inapproximable in general. On the positive side, we find an exact solution technique for general SGPs based on an existing approach, and develop a PTAS (polynomial-time approximation scheme) for zero-sum SGP to more fundamentally overcome the computational obstacle. Our experiments demonstrate the value of considering SGP and effectiveness of our algorithms.

Obvious strategyproofness (OSP) is an appealing concept as it allows to maintain incentive compatibility even in the presence of agents that are not fully rational, e.g., those who struggle with contingent reasoning (Li 2015). However, it has been shown to impose some limitations, e.g., no OSP mechanism can return a stable matching (Ashlagi and Gonczarowski 2015). We here deepen the study of the limitations of OSP mechanisms by looking at their approximation guarantees for basic optimization problems paradigmatic of the area, i.e., machine scheduling and facility location. We prove a number of bounds on the approximation guarantee of OSP mechanisms, which show that OSP can come at a significant cost. However, rather surprisingly, we prove that OSP mechanisms can return optimal solutions when they use monitoring — a novel mechanism design paradigm that introduces a mild level of scrutiny on agents’ declarations (Kovacs, Meyer, and Ventre 2015).

In the late 19th century, Lars Edvard Phragmén proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants one minimizing the maximal load and one minimizing the variance of loads —and a sequential variant. We study Phragmén's methods from an axiomatic point of view, focussing on justified representation and related properties that have recently been introduced by Aziz et al. (2015a) and Sánchez-Fernández et al. (2017). We show that the sequential variant satisfies proportional justified representation, making it the first known polynomial-time computable method with this property. Moreover, we show that the optimization variants satisfy perfect representation. We also analyze the com- putational complexity of Phragmén's methods and provide mixed-integer programming based algorithms for computing them.

The Team-maxmin equilibrium prescribes the optimal strategies for a team of rational players sharing the same goal and without the capability of correlating their strategies in strategic games against an adversary. This solution concept can capture situations in which an agent controls multiple resources - corresponding to the team members - that cannot communicate. It is known that such equilibrium always exists and it is unique (except degenerate cases) and these properties make it a credible solution concept to be used in real-world applications, especially in security scenarios. Nevertheless, to the best of our knowledge, the Team-maxmin equilibrium is almost completely unexplored in the literature. In this paper, we investigate bounds of (in)efficiency of the Team-maxmin equilibrium w.r.t. the Nash equilibria and w.r.t. the Maxmin equilibrium when the team members can play correlated strategies. Furthermore, we study a number of algorithms to find and/or approximate an equilibrium, discussing their theoretical guarantees and evaluating their performance by using a standard testbed of game instances.

Matrix factorization is a popular approach to solving matrix estimation problems based on partial observations. Existing matrix factorization is based on least squares and aims to yield a low-rank matrix to interpret the conditional sample means given the observations. However, in many real applications with skewed and extreme data, least squares cannot explain their central tendency or tail distributions, yielding undesired estimates. In this paper, we propose expectile matrix factorization by introducing asymmetric least squares, a key concept in expectile regression analysis, into the matrix factorization framework. We propose an efficient algorithm to solve the new problem based on alternating minimization and quadratic programming. We prove that our algorithm converges to a global optimum and exactly recovers the true underlying low-rank matrices when noise is zero. For synthetic data with skewed noise and a real-world dataset containing web service response times, the proposed scheme achieves lower recovery errors than the existing matrix factorization method based on least squares in a wide range of settings.

Our learning task: Given a training set S {(x1 , y1), (x2 , y2), . . . The simplest case: y { 1, 1} called binary classification problem If y is a real number it becomes a regression problem More general case, y can be a vector and each element is drawn from a finite set.

The use of convex regularizers allows for easy optimization, though they often produce biased estimation and inferior prediction performance. Recently, nonconvex regularizers have attracted a lot of attention and outperformed convex ones. However, the resultant optimization problem is much harder. In this paper, for a large class of nonconvex regularizers, we propose to move the nonconvexity from the regularizer to the loss. The nonconvex regularizer is then transformed to a familiar convex regularizer, while the resultant loss function can still be guaranteed to be smooth. Learning with the convexified regularizer can be performed by existing efficient algorithms originally designed for convex regularizers (such as the proximal algorithm, Frank-Wolfe algorithm, alternating direction method of multipliers and stochastic gradient descent). Extensions are made when the convexified regularizer does not have closed-form proximal step, and when the loss function is nonconvex, nonsmooth. Extensive experiments on a variety of machine learning application scenarios show that optimizing the transformed problem is much faster than running the state-of-the-art on the original problem.

The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it strong rule for coordinate preselection}. Such a complex algorithmic structure grants superior empirical performance, but also poses significant challenge to theoretical analysis. To tackle this long lasting problem, we develop a new theory showing that these three features play pivotal roles in guaranteeing the outstanding statistical and computational performance of the pathwise coordinate optimization framework. Particularly, we analyze the existing pathwise coordinate optimization algorithms and provide new theoretical insights into them. The obtained insights further motivate the development of several modifications to improve the pathwise coordinate optimization framework, which guarantees linear convergence to a unique sparse local optimum with optimal statistical properties in parameter estimation and support recovery. This is the first result on the computational and statistical guarantees of the pathwise coordinate optimization framework in high dimensions. Thorough numerical experiments are provided to support our theory.

We extend chance-constrained path planning with direct method into continuous time. Chance-constrained path planning is a method to obtain the optimal path satisfying a specified risk (or probability of failure) value. Previous work expects trajectories' states as discrete information with respect to time. This discretized encoding makes the conversion from probabilistic path planning to deterministic path planning easy. However, risk guarantees are only produced for the discrete time model. The probability of constraints violation in continuous time could be larger than the discretized risk values. To address this problem, we modified the constraint encoding and risk assessment method. First, we introduce a computationally efficient mean path securing method, which uses fewer binary variables as compared with prior work. Second, we note that the deviation of the actual trajectory from the mean trajectory can be considered as a Brownian motion, for which the reflection principle holds in general. Therefore, we take advantage of the reflection principle to bound the probability of the constraint violation in continuous time. In numerical simulations, we confirmed faster solution generation, and the probability guarantees of the path in the continuous time model, with deterioration in the objective function.