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Quickly and reliably finding accurate inverse kinematics (IK) solutions remains a challenging problem for robotic manipulation. Existing numerical solvers are broadly applicable, but rely on local search techniques to manage highly nonconvex objective functions. Recently, learning-based approaches have shown promise as a means to generate fast and accurate IK results; learned solvers can easily be integrated with other learning algorithms in end-to-end systems. However, learning-based methods have an Achilles' heel: each robot of interest requires a specialized model which must be trained from scratch. To address this key shortcoming, we investigate a novel distance-geometric robot representation coupled with a graph structure that allows us to leverage the flexibility of graph neural networks (GNNs). We use this approach to train the first learned generative graphical inverse kinematics (GGIK) solver that is, crucially, "robot-agnostic"-a single model is able to provide IK solutions for a variety of different robots. Additionally, the generative nature of GGIK allows the solver to produce a large number of diverse solutions in parallel with minimal additional computation time, making it appropriate for applications such as sampling-based motion planning. Finally, GGIK can complement local IK solvers by providing reliable initializations. These advantages, as well as the ability to use task-relevant priors and to continuously improve with new data, suggest that GGIK has the potential to be a key component of flexible, learning-based robotic manipulation systems.

We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual $L^2(\Omega)$ norm regularization term with a constant regularization parameter $\varrho$ is replaced by a suitable representation of the energy norm in $H^{-1}(\Omega)$ involving a variable, mesh-dependent regularization parameter $\varrho(x)$. It turns out that the error between the computed finite element state $\widetilde{u}_{\varrho h}$ and the desired state $\bar{u}$ (target) is optimal in the $L^2(\Omega)$ norm provided that $\varrho(x)$ behaves like the local mesh size squared. This is especially important when adaptive meshes are used in order to approximate discontinuous target functions. The adaptive scheme can be driven by the computable and localizable error norm $\| \widetilde{u}_{\varrho h} - \bar{u}\|_{L^2(\Omega)}$ between the finite element state $\widetilde{u}_{\varrho h}$ and the target $\bar{u}$. The numerical results not only illustrate our theoretical findings, but also show that the iterative solvers for the discretized reduced optimality system are very efficient and robust.

Node clustering is a powerful tool in the analysis of networks. We introduce a graph neural network framework to obtain node embeddings for directed networks in a self-supervised manner, including a novel probabilistic imbalance loss, which can be used for network clustering. Here, we propose directed flow imbalance measures, which are tightly related to directionality, to reveal clusters in the network even when there is no density difference between clusters. In contrast to standard approaches in the literature, in this paper, directionality is not treated as a nuisance, but rather contains the main signal. DIGRAC optimizes directed flow imbalance for clustering without requiring label supervision, unlike existing graph neural network methods, and can naturally incorporate node features, unlike existing spectral methods. Extensive experimental results on synthetic data, in the form of directed stochastic block models, and real-world data at different scales, demonstrate that our method, based on flow imbalance, attains state-of-the-art results on directed graph clustering when compared against 10 state-of-the-art methods from the literature, for a wide range of noise and sparsity levels, graph structures and topologies, and even outperforms supervised methods.

We study differentially private (DP) stochastic optimization (SO) with data containing outliers and loss functions that are not Lipschitz continuous. To date, the vast majority of work on DP SO assumes that the loss is Lipschitz (i.e. stochastic gradients are uniformly bounded), and their error bounds scale with the Lipschitz parameter of the loss. While this assumption is convenient, it is often unrealistic: in many practical problems where privacy is required, data may contain outliers or be unbounded, causing some stochastic gradients to have large norm. In such cases, the Lipschitz parameter may be prohibitively large, leading to vacuous excess risk bounds. Thus, building on a recent line of work [WXDX20, KLZ22], we make the weaker assumption that stochastic gradients have bounded $k$-th moments for some $k \geq 2$. Compared with works on DP Lipschitz SO, our excess risk scales with the $k$-th moment bound instead of the Lipschitz parameter of the loss, allowing for significantly faster rates in the presence of outliers. For convex and strongly convex loss functions, we provide the first asymptotically optimal excess risk bounds (up to a logarithmic factor). Moreover, in contrast to the prior works [WXDX20, KLZ22], our bounds do not require the loss function to be differentiable/smooth. We also devise an accelerated algorithm that runs in linear time and yields improved (compared to prior works) and nearly optimal excess risk for smooth losses. Additionally, our work is the first to address non-convex non-Lipschitz loss functions satisfying the Proximal-PL inequality; this covers some classes of neural nets, among other practical models. Our Proximal-PL algorithm has nearly optimal excess risk that almost matches the strongly convex lower bound. Lastly, we provide shuffle DP variations of our algorithms, which do not require a trusted curator (e.g. for distributed learning).

Classical results in general equilibrium theory assume divisible goods and convex preferences of market participants. In many real-world markets, participants have non-convex preferences and the allocation problem needs to consider complex constraints. Electricity markets are a prime example. In such markets, Walrasian prices are impossible, and heuristic pricing rules based on the dual of the relaxed allocation problem are used in practice. However, these rules have been criticized for high side-payments and inadequate congestion signals. We show that existing pricing heuristics optimize specific design goals that can be conflicting. The trade-offs can be substantial, and we establish that the design of pricing rules is fundamentally a multi-objective optimization problem addressing different incentives. In addition to traditional multi-objective optimization techniques using weighing of individual objectives, we introduce a novel parameter-free pricing rule that minimizes incentives for market participants to deviate locally. Our findings show how the new pricing rule capitalizes on the upsides of existing pricing rules under scrutiny today. It leads to prices that incur low make-whole payments while providing adequate congestion signals and low lost opportunity costs. Our suggested pricing rule does not require weighing of objectives, it is computationally scalable, and balances trade-offs in a principled manner, addressing an important policy issue in electricity markets.

We introduce two new tools to assess the validity of statistical distributions. These tools are based on components derived from a new statistical quantity, the $comparison$ $curve$. The first tool is a graphical representation of these components on a $bar$ $plot$ (B plot), which can provide a detailed appraisal of the validity of the statistical model, in particular when supplemented by acceptance regions related to the model. The knowledge gained from this representation can sometimes suggest an existing $goodness$-$of$-$fit$ test to supplement this visual assessment with a control of the type I error. Otherwise, an adaptive test may be preferable and the second tool is the combination of these components to produce a powerful $\chi^2$-type goodness-of-fit test. Because the number of these components can be large, we introduce a new selection rule to decide, in a data driven fashion, on their proper number to take into consideration. In a simulation, our goodness-of-fit tests are seen to be powerwise competitive with the best solutions that have been recommended in the context of a fully specified model as well as when some parameters must be estimated. Practical examples show how to use these tools to derive principled information about where the model departs from the data.

Variational Bayesian posterior inference often requires simplifying approximations such as mean-field parametrisation to ensure tractability. However, prior work has associated the variational mean-field approximation for Bayesian neural networks with underfitting in the case of small datasets or large model sizes. In this work, we show that invariances in the likelihood function of over-parametrised models contribute to this phenomenon because these invariances complicate the structure of the posterior by introducing discrete and/or continuous modes which cannot be well approximated by Gaussian mean-field distributions. In particular, we show that the mean-field approximation has an additional gap in the evidence lower bound compared to a purpose-built posterior that takes into account the known invariances. Importantly, this invariance gap is not constant; it vanishes as the approximation reverts to the prior. We proceed by first considering translation invariances in a linear model with a single data point in detail. We show that, while the true posterior can be constructed from a mean-field parametrisation, this is achieved only if the objective function takes into account the invariance gap. Then, we transfer our analysis of the linear model to neural networks. Our analysis provides a framework for future work to explore solutions to the invariance problem.

Background: Instrumental variables (IVs) can be used to provide evidence as to whether a treatment X has a causal effect on an outcome Y. Even if the instrument Z satisfies the three core IV assumptions of relevance, independence and the exclusion restriction, further assumptions are required to identify the average causal effect (ACE) of X on Y. Sufficient assumptions for this include: homogeneity in the causal effect of X on Y; homogeneity in the association of Z with X; and no effect modification (NEM). Methods: We describe the NO Simultaneous Heterogeneity (NOSH) assumption, which requires the heterogeneity in the X-Y causal effect to be mean independent of (i.e., uncorrelated with) both Z and heterogeneity in the Z-X association. This happens, for example, if there are no common modifiers of the X-Y effect and the Z-X association, and the X-Y effect is additive linear. We illustrate NOSH using simulations and by re-examining selected published studies. Results: When NOSH holds, the Wald estimand equals the ACE even if both homogeneity assumptions and NEM (which we demonstrate to be special cases of - and therefore stronger than - NOSH) are violated. Conclusions: NOSH is sufficient for identifying the ACE using IVs. Since NOSH is weaker than existing assumptions for ACE identification, doing so may be more plausible than previously anticipated.

Graph Convolutional Networks (GCNs) have been widely applied in various fields due to their significant power on processing graph-structured data. Typical GCN and its variants work under a homophily assumption (i.e., nodes with same class are prone to connect to each other), while ignoring the heterophily which exists in many real-world networks (i.e., nodes with different classes tend to form edges). Existing methods deal with heterophily by mainly aggregating higher-order neighborhoods or combing the immediate representations, which leads to noise and irrelevant information in the result. But these methods did not change the propagation mechanism which works under homophily assumption (that is a fundamental part of GCNs). This makes it difficult to distinguish the representation of nodes from different classes. To address this problem, in this paper we design a novel propagation mechanism, which can automatically change the propagation and aggregation process according to homophily or heterophily between node pairs. To adaptively learn the propagation process, we introduce two measurements of homophily degree between node pairs, which is learned based on topological and attribute information, respectively. Then we incorporate the learnable homophily degree into the graph convolution framework, which is trained in an end-to-end schema, enabling it to go beyond the assumption of homophily. More importantly, we theoretically prove that our model can constrain the similarity of representations between nodes according to their homophily degree. Experiments on seven real-world datasets demonstrate that this new approach outperforms the state-of-the-art methods under heterophily or low homophily, and gains competitive performance under homophily.

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.