Given a convex and differentiable objective $Q(\M)$ for a real, symmetric matrix $\M$ in the positive definite (PD) cone---used to compute Mahalanobis distances---we propose a fast general metric learning framework that is entirely projection-free. We first assume that $\M$ resides in a space $\cS$ of generalized graph Laplacian matrices (graph metric matrices) corresponding to balanced signed graphs. Unlike low-rank metric matrices common in the literature, $\cS$ includes the important diagonal-only matrices as a special case. The key theorem to circumvent full eigen-decomposition and enable fast metric matrix optimization is Gershgorin disc alignment (GDA): given graph metric matrix $\M \in \cS$ and diagonal matrix $\S$, where $S_{ii} = 1/v_i$ and $\v$ is the first eigenvector of $\M$, we prove that Gershgorin disc left-ends of similar transform $\B = \S \M \S^{-1}$ are perfectly aligned at the smallest eigenvalue $\lambda_{\min}$. Using this theorem, we replace the PD cone constraint in the metric learning problem with tightest possible linear constraints per iteration, so that the alternating optimization of the diagonal / off-diagonal terms in $\M$ can be solved efficiently as linear programs via Frank-Wolfe iterations. We update $\v$ using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as matrix entries in $\M$ are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection methods, and produces competitive binary classification performance.

Mission schedule of satellites is an important part of space operation nowadays, since the number and types of satellites in orbit are increasing tremendously and their corresponding tasks are also becoming more and more complicated. In this paper, a mission schedule model combined with Proximal Policy Optimization Algorithm(PPO) is proposed. Different from the traditional heuristic planning method, this paper incorporate reinforcement learning algorithms into it and find a new way to describe the problem. Several constraints including data download are considered in this paper.

Deep learning based semi-supervised learning (SSL) algorithms have led to promising results in recent years. However, they tend to introduce multiple tunable hyper-parameters, making them less practical in real SSL scenarios where the labeled data is scarce for extensive hyper-parameter search. In this paper, we propose a novel meta-learning based SSL algorithm (Meta-Semi) that requires tuning only one additional hyper-parameter, compared with a standard supervised deep learning algorithm, to achieve competitive performance under various conditions of SSL. We start by defining a meta optimization problem that minimizes the loss on labeled data through dynamically reweighting the loss on unlabeled samples, which are associated with soft pseudo labels during training. As the meta problem is computationally intensive to solve directly, we propose an efficient algorithm to dynamically obtain the approximate solutions. We show theoretically that Meta-Semi converges to the stationary point of the loss function on labeled data under mild conditions. Empirically, Meta-Semi outperforms state-of-the-art SSL algorithms significantly on the challenging semi-supervised CIFAR-100 and STL-10 tasks, and achieves competitive performance on CIFAR-10 and SVHN.

Feature selection is a core area of data mining with a recent innovation of graph-driven unsupervised feature selection for linked data. In this setting we have a dataset $\mathbf{Y}$ consisting of $n$ instances each with $m$ features and a corresponding $n$ node graph (whose adjacency matrix is $\mathbf{A}$) with an edge indicating that the two instances are similar. Existing efforts for unsupervised feature selection on attributed networks have explored either directly regenerating the links by solving for $f$ such that $f(\mathbf{y}_i,\mathbf{y}_j) \approx \mathbf{A}_{i,j}$ or finding community structure in $\mathbf{A}$ and using the features in $\mathbf{Y}$ to predict these communities. However, graph-driven unsupervised feature selection remains an understudied area with respect to exploring more complex guidance. Here we take the novel approach of first building a block model on the graph and then using the block model for feature selection. That is, we discover $\mathbf{F}\mathbf{M}\mathbf{F}^T \approx \mathbf{A}$ and then find a subset of features $\mathcal{S}$ that induces another graph to preserve both $\mathbf{F}$ and $\mathbf{M}$. We call our approach Block Model Guided Unsupervised Feature Selection (BMGUFS). Experimental results show that our method outperforms the state of the art on several real-world public datasets in finding high-quality features for clustering.

The growing demand for edge computing resources, particularly due to increasing popularity of Internet of Things (IoT), and distributed machine/deep learning applications poses a significant challenge. On the one hand, certain edge service providers (ESPs) may not have sufficient resources to satisfy their applications according to the associated service-level agreements. On the other hand, some ESPs may have additional unused resources. In this paper, we propose a resource-sharing framework that allows different ESPs to optimally utilize their resources and improve the satisfaction level of applications subject to constraints such as communication cost for sharing resources across ESPs. Our framework considers that different ESPs have their own objectives for utilizing their resources, thus resulting in a multi-objective optimization problem. We present an $N$-person \emph{Nash Bargaining Solution} (NBS) for resource allocation and sharing among ESPs with \emph{Pareto} optimality guarantee. Furthermore, we propose a \emph{distributed}, primal-dual algorithm to obtain the NBS by proving that the strong-duality property holds for the resultant resource sharing optimization problem. Using synthetic and real-world data traces, we show numerically that the proposed NBS based framework not only enhances the ability to satisfy applications' resource demands, but also improves utilities of different ESPs.

Differentiable physics is a powerful approach to learning and control problems that involve physical objects and environments. While notable progress has been made, the capabilities of differentiable physics solvers remain limited. We develop a scalable framework for differentiable physics that can support a large number of objects and their interactions. To accommodate objects with arbitrary geometry and topology, we adopt meshes as our representation and leverage the sparsity of contacts for scalable differentiable collision handling. Collisions are resolved in localized regions to minimize the number of optimization variables even when the number of simulated objects is high. We further accelerate implicit differentiation of optimization with nonlinear constraints. Experiments demonstrate that the presented framework requires up to two orders of magnitude less memory and computation in comparison to recent particle-based methods. We further validate the approach on inverse problems and control scenarios, where it outperforms derivative-free and model-free baselines by at least an order of magnitude.

In recent years, reinforcement learning (RL) systems with general goals beyond a cumulative sum of rewards have gained traction, such as in constrained problems, exploration, and acting upon prior experiences. In this paper, we consider policy optimization in Markov Decision Problems, where the objective is a general concave utility function of the state-action occupancy measure, which subsumes several of the aforementioned examples as special cases. Such generality invalidates the Bellman equation. As this means that dynamic programming no longer works, we focus on direct policy search. Analogously to the Policy Gradient Theorem \cite{sutton2000policy} available for RL with cumulative rewards, we derive a new Variational Policy Gradient Theorem for RL with general utilities, which establishes that the parametrized policy gradient may be obtained as the solution of a stochastic saddle point problem involving the Fenchel dual of the utility function. We develop a variational Monte Carlo gradient estimation algorithm to compute the policy gradient based on sample paths. We prove that the variational policy gradient scheme converges globally to the optimal policy for the general objective, though the optimization problem is nonconvex. We also establish its rate of convergence of the order $O(1/t)$ by exploiting the hidden convexity of the problem, and proves that it converges exponentially when the problem admits hidden strong convexity. Our analysis applies to the standard RL problem with cumulative rewards as a special case, in which case our result improves the available convergence rate.

Algorithms for bandit convex optimization and online learning often rely on constructing noisy gradient estimates, which are then used in appropriately adjusted first-order algorithms, replacing actual gradients. Depending on the properties of the function to be optimized and the nature of ``noise'' in the bandit feedback, the bias and variance of gradient estimates exhibit various tradeoffs. In this paper we propose a novel framework that replaces the specific gradient estimation methods with an abstract oracle. With the help of the new framework we unify previous works, reproducing their results in a clean and concise fashion, while, perhaps more importantly, the framework also allows us to formally show that to achieve the optimal root-$n$ rate either the algorithms that use existing gradient estimators, or the proof techniques used to analyze them have to go beyond what exists today.

As we seek to deploy machine learning models beyond virtual and controlled domains, it is critical to analyze not only the accuracy or the fact that it works most of the time, but if such a model is truly robust and reliable. This paper studies strategies to implement adversary robustly trained algorithms towards guaranteeing safety in machine learning algorithms. We provide a taxonomy to classify adversarial attacks and defenses, formulate the Robust Optimization problem in a min-max setting and divide it into 3 subcategories, namely: Adversarial (re)Training, Regularization Approach, and Certified Defenses. We survey the most recent and important results in adversarial example generation, defense mechanisms with adversarial (re)Training as their main defense against perturbations. We also survey mothods that add regularization terms that change the behavior of the gradient, making it harder for attackers to achieve their objective. Alternatively, we've surveyed methods which formally derive certificates of robustness by exactly solving the optimization problem or by approximations using upper or lower bounds. In addition, we discuss the challenges faced by most of the recent algorithms presenting future research perspectives.

The problem of linking functional connectomics to behavior is extremely challenging due to the complex interactions between the two distinct, but related, data domains. We propose a coupled manifold optimization framework which projects fMRI data onto a low dimensional matrix manifold common to the cohort. The patient specific loadings simultaneously map onto a behavioral measure of interest via a second, non-linear, manifold. By leveraging the kernel trick, we can optimize over a potentially infinite dimensional space without explicitly computing the embeddings. As opposed to conventional manifold learning, which assumes a fixed input representation, our framework directly optimizes for embedding directions that predict behavior. Our optimization algorithm combines proximal gradient descent with the trust region method, which has good convergence guarantees. We validate our framework on resting state fMRI from fifty-eight patients with Autism Spectrum Disorder using three distinct measures of clinical severity. Our method outperforms traditional representation learning techniques in a cross validated setting, thus demonstrating the predictive power of our coupled objective.