Parking pressure has been steadily increasing in cities as well as in university and corporate campuses. To relieve this pressure, this paper studies a car-pooling platform that would match riders and drivers, while guaranteeing a ride back and exploiting spatial and temporal locality. In particular, the paper formalizes the Commute Trip Sharing Problem (CTSP) to find a routing plan that maximizes ride sharing for a set of commute trips. The CTSP is a generalization of the vehicle routing problem with routes that satisfy time window, capacity, pairing, precedence, ride duration, and driver constraints. The paper introduces two exact algorithms for the CTPS: A route-enumeration algorithm and a branch-and-price algorithm. Experimental results show that, on a high-fidelity, real-world dataset of commute trips from a mid-size city, both algorithms optimally solve small and medium-sized problems and produce high-quality solutions for larger problem instances. The results show that car pooling, if widely adopted, has the potential to reduce vehicle usage by up to 57% and decrease vehicle miles traveled by up to 46% while only incurring a 22% increase in average ride time per commuter for the trips considered.

Learning directed acyclic graphs (DAGs) from data is a challenging task both in theory and in practice, because the number of possible DAGs scales superexponentially with the number of nodes. In this paper, we study the problem of learning an optimal DAG from continuous observational data. We cast this problem in the form of a mathematical programming model which can naturally incorporate a super-structure in order to reduce the set of possible candidate DAGs. We use the penalized negative log-likelihood score function with both $\ell_0$ and $\ell_1$ regularizations and propose a new mixed-integer quadratic optimization (MIQO) model, referred to as a layered network (LN) formulation. The LN formulation is a compact model, which enjoys as tight an optimal continuous relaxation value as the stronger but larger formulations under a mild condition. Computational results indicate that the proposed formulation outperforms existing mathematical formulations and scales better than available algorithms that can solve the same problem with only $\ell_1$ regularization. In particular, the LN formulation clearly outperforms existing methods in terms of computational time needed to find an optimal DAG in the presence of a sparse super-structure.

Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying graphical models from data. Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. Useful structured graphs include the multi-component graph, bipartite graph, connected graph, sparse graph, and regular graph. In general, structured graph learning is an NP-hard combinatorial problem, therefore, designing a general tractable optimization method is extremely challenging. In this paper, we introduce a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory. To impose a particular structure on a graph, we first show how to formulate the combinatorial constraints as an analytical property of the graph matrix. Then we develop an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices. The proposed algorithms are provably convergent, computationally efficient, and practically amenable for numerous graph-based tasks. Extensive numerical experiments with both synthetic and real data sets illustrate the effectiveness of the proposed algorithms. The code for all the simulations is made available as an open source repository.

Most of real-world graphs are {\em dynamic}, i.e., they change over time. However, while the regression problem has been studied for {\em static} graphs, it has not been investigated for {\em dynamic} graphs, yet. In the current paper, first we present the notion of {\em update-efficient matrix embedding} that defines the conditions sufficient for a matrix embedding to be used for the dynamic graph regression problem, efficiently. We also show that some of the standard matrix embeddings, e.g., the (weighted) adjacency matrix, satisfy these conditions. Then, we prove that given an $n \times m$ update-efficient matrix embedding, after an update operation in the graph, the optimal solution of the graph regression problem for the revised graph can be computed in $O(nm)$ time. In particular, using the (weighted) adjacency matrix as the matrix embedding of $G$, it takes $O(n^2)$ time to update the optimal solution, where $n$ is the number of nodes of the revised graph. To the best of our knowledge, this is the first result on updating the solution of the graph regression problem, in a time considerably less than the time of computing the solution from the scratch. Finally, we study a generalization of the dynamic graph regression problem and show that it can be solved in $O(nm + mm')$ space.

Many applications of machine learning, such as human health research, involve processing private or sensitive information. Privacy concerns may impose significant hurdles to collaboration in scenarios where there are multiple sites holding data and the goal is to estimate properties jointly across all datasets. Differentially private decentralized algorithms can provide strong privacy guarantees. However, the accuracy of the joint estimates may be poor when the datasets at each site are small. This paper proposes a new framework, Correlation Assisted Private Estimation (CAPE), for designing privacy-preserving decentralized algorithms with better accuracy guarantees in an honest-but-curious model. CAPE can be used in conjunction with the functional mechanism for statistical and machine learning optimization problems. A tighter characterization of the functional mechanism is provided that allows CAPE to achieve the same performance as a centralized algorithm in the decentralized setting using all datasets. Empirical results on regression and neural network problems for both synthetic and real datasets show that differentially private methods can be competitive with non-private algorithms in many scenarios of interest.

Video-based vehicle detection and tracking is one of the most important components for Intelligent Transportation Systems (ITS). When it comes to road junctions, the problem becomes even more difficult due to the occlusions and complex interactions among vehicles. In order to get a precise detection and tracking result, in this work we propose a novel tracking-by-detection framework. In the detection stage, we present a sequential detection model to deal with serious occlusions. In the tracking stage, we model group behavior to treat complex interactions with overlaps and ambiguities. The main contributions of this paper are twofold: 1) Shape prior is exploited in the sequential detection model to tackle occlusions in crowded scene. 2) Traffic force is defined in the traffic scene to model group behavior, and it can assist to handle complex interactions among vehicles. We evaluate the proposed approach on real surveillance videos at road junctions and the performance has demonstrated the effectiveness of our method.

In recent years, Generative Adversarial Networks (GANs) have drawn a lot of attentions for learning the underlying distribution of data in various applications. Despite their wide applicability, training GANs is notoriously difficult. This difficulty is due to the min-max nature of the resulting optimization problem and the lack of proper tools of solving general (non-convex, non-concave) min-max optimization problems. In this paper, we try to alleviate this problem by proposing a new generative network that relies on the use of random discriminators instead of adversarial design. This design helps us to avoid the min-max formulation and leads to an optimization problem that is stable and could be solved efficiently. The performance of the proposed method is evaluated using handwritten digits (MNIST) and Fashion products (Fashion-MNIST) data sets. While the resulting images are not as sharp as adversarial training, the use of random discriminator leads to a much faster algorithm as compared to the adversarial counterpart. This observation, at the minimum, illustrates the potential of the random discriminator approach for warm-start in training GANs.

Decomposition methods have been proposed to approximate solutions to large sequential decision making problems. In contexts where an agent interacts with multiple entities, utility decomposition can be used to separate the global objective into local tasks considering each individual entity independently. An arbitrator is then responsible for combining the individual utilities and selecting an action in real time to solve the global problem. Although these techniques can perform well empirically, they rely on strong assumptions of independence between the local tasks and sacrifice the optimality of the global solution. This paper proposes an approach that improves upon such approximate solutions by learning a correction term represented by a neural network. We demonstrate this approach on a fisheries management problem where multiple boats must coordinate to maximize their catch over time as well as on a pedestrian avoidance problem for autonomous driving. In each problem, decomposition methods can scale to multiple boats or pedestrians by using strategies involving one entity. We verify empirically that the proposed correction method significantly improves the decomposition method and outperforms a policy trained on the full scale problem without utility decomposition.

The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothness condition, which is satisfied by any finite-order polynomial objective function. Further, this work develops new Bregman-divergence based algorithms that are guaranteed to converge to a second-order stationary point for any relatively smooth problem. In addition, the proposed optimization methods cover both the proximal alternating minimization and the proximal alternating linearized minimization when we specialize the Bregman divergence to the Euclidian distance. Therefore, this work not only develops guaranteed optimization methods for non-Lipschitz smooth problems but also solves an open problem of showing the second-order convergence guarantees for these alternating minimization methods.

Gaussian processes (GP) for machine learning have been studied systematically over the past two decades and they are by now widely used in a number of diverse applications. However, GP kernel design and the associated hyper-parameter optimization are still hard and to a large extend open problems. In this paper, we consider the task of GP regression for time series modeling and analysis. The underlying stationary kernel can be approximated arbitrarily close by a new proposed grid spectral mixture (GSM) kernel, which turns out to be a linear combination of low-rank sub-kernels. In the case where a large number of the sub-kernels are used, either the Nystr\"{o}m or the random Fourier feature approximations can be adopted to deal efficiently with the computational demands. The unknown GP hyper-parameters consist of the non-negative weights of all sub-kernels as well as the noise variance; their estimation is performed via the maximum-likelihood (ML) estimation framework. Two efficient numerical optimization methods for solving the unknown hyper-parameters are derived, including a sequential majorization-minimization (MM) method and a non-linearly constrained alternating direction of multiplier method (ADMM). The MM matches perfectly with the proven low-rank property of the proposed GSM sub-kernels and turns out to be a part of efficiency, stable, and efficient solver, while the ADMM has the potential to generate better local minimum in terms of the test MSE. Experimental results, based on various classic time series data sets, corroborate that the proposed GSM kernel-based GP regression model outperforms several salient competitors of similar kind in terms of prediction mean-squared-error and numerical stability.