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We exploit the complementary strengths of vision and proprioception to achieve point goal navigation in a legged robot. Legged systems are capable of traversing more complex terrain than wheeled robots, but to fully exploit this capability, we need the high-level path planner in the navigation system to be aware of the walking capabilities of the low-level locomotion policy on varying terrains. We achieve this by using proprioceptive feedback to estimate the safe operating limits of the walking policy, and to sense unexpected obstacles and terrain properties like smoothness or softness of the ground that may be missed by vision. The navigation system uses onboard cameras to generate an occupancy map and a corresponding cost map to reach the goal. The FMM (Fast Marching Method) planner then generates a target path. The velocity command generator takes this as input to generate the desired velocity for the locomotion policy using as input additional constraints, from the safety advisor, of unexpected obstacles and terrain determined speed limits. We show superior performance compared to wheeled robot (LoCoBot) baselines, and other baselines which have disjoint high-level planning and low-level control. We also show the real-world deployment of our system on a quadruped robot with onboard sensors and compute. Videos at //navigation-locomotion.github.io/camera-ready

In this work, we present a new high order Discontinuous Galerkin time integration scheme for second-order (in time) differential systems that typically arise from the space discretization of the elastodynamics equation. By rewriting the original equation as a system of first order differential equations we introduce the method and show that the resulting discrete formulation is well-posed, stable and retains super-optimal rate of convergence with respect to the discretization parameters, namely the time step and the polynomial approximation degree. A set of two- and three-dimensional numerical experiments confirm the theoretical bounds. Finally, the method is applied to real geophysical applications.

In this paper, we formulate and study substructuring type algorithm for the Cahn-Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature and since then it has appeared in many fields ranging from tumour growth simulation, image processing, thin liquid films, population dynamics etc. Being a non-linear equation, it is important to develop robust numerical techniques to solve the CH equation. Here we present the formulation of Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to CH equation and study their convergence behaviour. We consider the domain-decomposition based DN and NN methods in one and two space dimension for two subdomains and extend the study for multi-subdomain setting for NN method. We verify our findings with numerical results.

Recent research on graph neural network (GNN) models successfully applied GNNs to classical graph algorithms and combinatorial optimisation problems. This has numerous benefits, such as allowing applications of algorithms when preconditions are not satisfied, or reusing learned models when sufficient training data is not available or can't be generated. Unfortunately, a key hindrance of these approaches is their lack of explainability, since GNNs are black-box models that cannot be interpreted directly. In this work, we address this limitation by applying existing work on concept-based explanations to GNN models. We introduce concept-bottleneck GNNs, which rely on a modification to the GNN readout mechanism. Using three case studies we demonstrate that: (i) our proposed model is capable of accurately learning concepts and extracting propositional formulas based on the learned concepts for each target class; (ii) our concept-based GNN models achieve comparative performance with state-of-the-art models; (iii) we can derive global graph concepts, without explicitly providing any supervision on graph-level concepts.

Deep learning based recommender systems (DLRSs) often have embedding layers, which are utilized to lessen the dimensionality of categorical variables (e.g. user/item identifiers) and meaningfully transform them in the low-dimensional space. The majority of existing DLRSs empirically pre-define a fixed and unified dimension for all user/item embeddings. It is evident from recent researches that different embedding sizes are highly desired for different users/items according to their popularity. However, manually selecting embedding sizes in recommender systems can be very challenging due to the large number of users/items and the dynamic nature of their popularity. Thus, in this paper, we propose an AutoML based end-to-end framework (AutoEmb), which can enable various embedding dimensions according to the popularity in an automated and dynamic manner. To be specific, we first enhance a typical DLRS to allow various embedding dimensions; then we propose an end-to-end differentiable framework that can automatically select different embedding dimensions according to user/item popularity; finally we propose an AutoML based optimization algorithm in a streaming recommendation setting. The experimental results based on widely used benchmark datasets demonstrate the effectiveness of the AutoEmb framework.

Alternating Direction Method of Multipliers (ADMM) is a widely used tool for machine learning in distributed settings, where a machine learning model is trained over distributed data sources through an interactive process of local computation and message passing. Such an iterative process could cause privacy concerns of data owners. The goal of this paper is to provide differential privacy for ADMM-based distributed machine learning. Prior approaches on differentially private ADMM exhibit low utility under high privacy guarantee and often assume the objective functions of the learning problems to be smooth and strongly convex. To address these concerns, we propose a novel differentially private ADMM-based distributed learning algorithm called DP-ADMM, which combines an approximate augmented Lagrangian function with time-varying Gaussian noise addition in the iterative process to achieve higher utility for general objective functions under the same differential privacy guarantee. We also apply the moments accountant method to bound the end-to-end privacy loss. The theoretical analysis shows that DP-ADMM can be applied to a wider class of distributed learning problems, is provably convergent, and offers an explicit utility-privacy tradeoff. To our knowledge, this is the first paper to provide explicit convergence and utility properties for differentially private ADMM-based distributed learning algorithms. The evaluation results demonstrate that our approach can achieve good convergence and model accuracy under high end-to-end differential privacy guarantee.

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.

This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.

Querying graph structured data is a fundamental operation that enables important applications including knowledge graph search, social network analysis, and cyber-network security. However, the growing size of real-world data graphs poses severe challenges for graph databases to meet the response-time requirements of the applications. Planning the computational steps of query processing - Query Planning - is central to address these challenges. In this paper, we study the problem of learning to speedup query planning in graph databases towards the goal of improving the computational-efficiency of query processing via training queries.We present a Learning to Plan (L2P) framework that is applicable to a large class of query reasoners that follow the Threshold Algorithm (TA) approach. First, we define a generic search space over candidate query plans, and identify target search trajectories (query plans) corresponding to the training queries by performing an expensive search. Subsequently, we learn greedy search control knowledge to imitate the search behavior of the target query plans. We provide a concrete instantiation of our L2P framework for STAR, a state-of-the-art graph query reasoner. Our experiments on benchmark knowledge graphs including DBpedia, YAGO, and Freebase show that using the query plans generated by the learned search control knowledge, we can significantly improve the speed of STAR with negligible loss in accuracy.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.