A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. This paper introduces several classes of distributed decision problems, proves separation among them and presents some complete problems. More specifically, we consider the standard LOCAL model of computation and define LD (for local decision) as the class of decision problems that can be solved in constant number of communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD, and ask whether LD=BPLD. We provide a partial answer to this question by showing that in many cases, randomization does not help for deciding hereditary languages. In addition, we define the notion of local many-one reductions, and introduce the (nondeterministic) class NLD of decision problems for which there exists a certificate that can be verified in constant number of communication rounds. We prove that there exists an NLD-complete problem. We also show that there exist problems not in NLD. On the other hand, we prove that the class NLD#n, which is NLD assuming that each processor can access an oracle that provides the number of nodes in the network, contains all (decidable) languages. For this class we provide a natural complete problem as well.
Finally, we note that our notion of NLD seems to be related to the theory of lifts , e.g., @cite_41 .
@cite_41
2953378433 2962722175 2769449302 2948130861
A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. This paper introduces several classes of distributed decision problems, proves separation among them and presents some complete problems. More specifically, we consider the standard LOCAL model of computation and define LD (for local decision) as the class of decision problems that can be solved in constant number of communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD, and ask whether LD=BPLD. We provide a partial answer to this question by showing that in many cases, randomization does not help for deciding hereditary languages. In addition, we define the notion of local many-one reductions, and introduce the (nondeterministic) class NLD of decision problems for which there exists a certificate that can be verified in constant number of communication rounds. We prove that there exists an NLD-complete problem. We also show that there exist problems not in NLD. On the other hand, we prove that the class NLD#n, which is NLD assuming that each processor can access an oracle that provides the number of nodes in the network, contains all (decidable) languages. For this class we provide a natural complete problem as well. This paper focuses on the problem of learning 6- DOF grasping with a parallel jaw gripper in simulation. Our key idea is constraining and regularizing grasping interaction learning through 3D geometry prediction. We introduce a deep geometry-aware grasping network (DGGN) that decomposes the learning into two steps. First, we learn to build mental geometry-aware representation by reconstructing the scene (i.e., 3D occupancy grid) from RGBD input via generative 3D shape modeling. Second, we learn to predict grasping outcome with its internal geometry-aware representation. The learned outcome prediction model is used to sequentially propose grasping solutions via analysis-by-synthesis optimization. Our contributions are fourfold: (1) To best of our knowledge, we are presenting for the first time a method to learn a 6-DOF grasping net from RGBD input; (2) We build a grasping dataset from demonstrations in virtual reality with rich sensory and interaction annotations. This dataset includes 101 everyday objects spread across 7 categories, additionally, we propose a data augmentation strategy for effective learning; (3) We demonstrate that the learned geometry-aware representation leads to about 10 relative performance improvement over the baseline CNN on grasping objects from our dataset. (4) We further demonstrate that the model generalizes to novel viewpoints and object instances. By lifting the ReLU function into a higher dimensional space, we develop a smooth multi-convex formulation for training feed-forward deep neural networks (DNNs). This allows us to develop a block coordinate descent (BCD) training algorithm consisting of a sequence of numerically well-behaved convex optimizations. Using ideas from proximal point methods in convex analysis, we prove that this BCD algorithm will converge globally to a stationary point with R-linear convergence rate of order one. In experiments with the MNIST database, DNNs trained with this BCD algorithm consistently yielded better test-set error rates than identical DNN architectures trained via all the stochastic gradient descent (SGD) variants in the Caffe toolbox. The lottery ticket hypothesis proposes that over-parameterization of deep neural networks (DNNs) aids training by increasing the probability of a "lucky" sub-network initialization being present rather than by helping the optimization process. This phenomenon is intriguing and suggests that initialization strategies for DNNs can be improved substantially, but the lottery ticket hypothesis has only previously been tested in the context of supervised learning for natural image tasks. Here, we evaluate whether "winning ticket" initializations exist in two different domains: reinforcement learning (RL) and in natural language processing (NLP). For RL, we analyzed a number of discrete-action space tasks, including both classic control and pixel control. For NLP, we examined both recurrent LSTM models and large-scale Transformer models. Consistent with work in supervised image classification, we confirm that winning ticket initializations generally outperform parameter-matched random initializations, even at extreme pruning rates. Together, these results suggest that the lottery ticket hypothesis is not restricted to supervised learning of natural images, but rather represents a broader phenomenon in DNNs.