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Most modern deep reinforcement learning (RL) algorithms are motivated by either the general policy improvement (GPI) or trust-region learning (TRL) frameworks. However, algorithms that strictly respect these theoretical frameworks have proven unscalable. Surprisingly, the only known scalable algorithms violate the GPI/TRL assumptions, e.g. due to required regularisation or other heuristics. The current explanation of their empirical success is essentially by "analogy": they are deemed approximate adaptations of theoretically sound methods. Unfortunately, studies have shown that in practice these algorithms differ greatly from their conceptual ancestors. In contrast, in this paper, we introduce a novel theoretical framework, named Mirror Learning, which provides theoretical guarantees to a large class of algorithms, including TRPO and PPO. While the latter two exploit the flexibility of our framework, GPI and TRL fit in merely as pathologically restrictive or impractical corner cases thereof. This suggests that the empirical performance of state-of-the-art methods is a direct consequence of their theoretical properties, rather than of aforementioned approximate analogies. Mirror learning sets us free to boldly explore novel, theoretically sound RL algorithms, a thus far uncharted wonderland.

In 2018 Bornmann and Haunschild (2018a) introduced a new indicator called the Mantel-Haenszel quotient (MHq) to measure alternative metrics (or altmetrics) of scientometric data. In this article we review the Mantel-Haenszel statistics, point out two errors in the literature, and introduce a new indicator. First, we correct the interpretation of MHq and mention that it is still a meaningful indicator. Second, we correct the variance formula for MHq, which leads to narrower confidence intervals. A simulation study shows the superior performance of our variance estimator and confidence intervals. Since MHq does not match its original description in the literature, we propose a new indicator, the Mantel-Haenszel row risk ratio (MHRR), to meet that need. Interpretation and statistical inference for MHRR are discussed. For both MHRR and MHq, a value greater (less) than one means performance is better (worse) than in the reference set called the world.

Reference priors are theoretically attractive for the analysis of geostatistical data since they enable automatic Bayesian analysis and have desirable Bayesian and frequentist properties. But their use is hindered by computational hurdles that make their application in practice challenging. In this work, we derive a new class of default priors that approximate reference priors for the parameters of some Gaussian random fields. It is based on an approximation to the integrated likelihood of the covariance parameters derived from the spectral approximation of stationary random fields. This prior depends on the structure of the mean function and the spectral density of the model evaluated at a set of spectral points associated with an auxiliary regular grid. In addition to preserving the desirable Bayesian and frequentist properties, these approximate reference priors are more stable, and their computations are much less onerous than those of exact reference priors. Unlike exact reference priors, the marginal approximate reference prior of correlation parameter is always proper, regardless of the mean function or the smoothness of the correlation function. This property has important consequences for covariance model selection. An illustration comparing default Bayesian analyses is provided with a data set of lead pollution in Galicia, Spain.

We study a social learning scheme where at every time instant, each agent chooses to receive information from one of its neighbors at random. We show that under this sparser communication scheme, the agents learn the truth eventually and the asymptotic convergence rate remains the same as the standard algorithms which use more communication resources. We also derive large deviation estimates of the log-belief ratios for a special case where each agent replaces its belief with that of the chosen neighbor.

Statistical divergences (SDs), which quantify the dissimilarity between probability distributions, are a basic constituent of statistical inference and machine learning. A modern method for estimating those divergences relies on parametrizing an empirical variational form by a neural network (NN) and optimizing over parameter space. Such neural estimators are abundantly used in practice, but corresponding performance guarantees are partial and call for further exploration. We establish non-asymptotic absolute error bounds for a neural estimator realized by a shallow NN, focusing on four popular $\mathsf{f}$-divergences -- Kullback-Leibler, chi-squared, squared Hellinger, and total variation. Our analysis relies on non-asymptotic function approximation theorems and tools from empirical process theory to bound the two sources of error involved: function approximation and empirical estimation. The bounds characterize the effective error in terms of NN size and the number of samples, and reveal scaling rates that ensure consistency. For compactly supported distributions, we further show that neural estimators of the first three divergences above with appropriate NN growth-rate are minimax rate-optimal, achieving the parametric convergence rate.

We design and analyze VA-LUCB, a parameter-free algorithm, for identifying the best arm under the fixed-confidence setup and under a stringent constraint that the variance of the chosen arm is strictly smaller than a given threshold. An upper bound on VA-LUCB's sample complexity is shown to be characterized by a fundamental variance-aware hardness quantity $H_{VA}$. By proving a lower bound, we show that sample complexity of VA-LUCB is optimal up to a factor logarithmic in $H_{VA}$. Extensive experiments corroborate the dependence of the sample complexity on the various terms in $H_{VA}$. By comparing VA-LUCB's empirical performance to a close competitor RiskAverse-UCB-BAI by David et al. (2018), our experiments suggest that VA-LUCB has the lowest sample complexity for this class of risk-constrained best arm identification problems, especially for the riskiest instances.

We develop a universal model based on the classical complex matter fields that allow the optimal mapping of many real-life NP-hard combinatorial optimisation problems into the problem of minimising a spin Hamiltonian. We explicitly formulate one-to-one mapping for three famous problems: graph colouring, the travelling salesman, and the modular N-queens problem. We show that such a formulation allows for several orders of magnitude improvement in the search for the global minimum compared to the standard Ising formulation. At the same time, the amplitude dynamics escape from the local minima.

We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.

We address the question of characterizing and finding optimal representations for supervised learning. Traditionally, this question has been tackled using the Information Bottleneck, which compresses the inputs while retaining information about the targets, in a decoder-agnostic fashion. In machine learning, however, our goal is not compression but rather generalization, which is intimately linked to the predictive family or decoder of interest (e.g. linear classifier). We propose the Decodable Information Bottleneck (DIB) that considers information retention and compression from the perspective of the desired predictive family. As a result, DIB gives rise to representations that are optimal in terms of expected test performance and can be estimated with guarantees. Empirically, we show that the framework can be used to enforce a small generalization gap on downstream classifiers and to predict the generalization ability of neural networks.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.