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Programs involving discontinuities introduced by control flow constructs such as conditional branches pose challenges to mathematical optimization methods that assume a degree of smoothness in the objective function's response surface. Smooth interpretation (SI) is a form of abstract interpretation that approximates the convolution of a program's output with a Gaussian kernel, thus smoothing its output in a principled manner. Here, we combine SI with automatic differentiation (AD) to efficiently compute gradients of smoothed programs. In contrast to AD across a regular program execution, these gradients also capture the effects of alternative control flow paths. The combination of SI with AD enables the direct gradient-based parameter synthesis for branching programs, allowing for instance the calibration of simulation models or their combination with neural network models in machine learning pipelines. We detail the effects of the approximations made for tractability in SI and propose a novel Monte Carlo estimator that avoids the underlying assumptions by estimating the smoothed programs' gradients through a combination of AD and sampling. Using DiscoGrad, our tool for automatically translating simple C++ programs to a smooth differentiable form, we perform an extensive evaluation. We compare the combination of SI with AD and our Monte Carlo estimator to existing gradient-free and stochastic methods on four non-trivial and originally discontinuous problems ranging from classical simulation-based optimization to neural network-driven control. While the optimization progress with the SI-based estimator depends on the complexity of the program's control flow, our Monte Carlo estimator is competitive in all problems, exhibiting the fastest convergence by a substantial margin in our highest-dimensional problem.

Nonnegative tensor factorization (NTF) has become an important tool for feature extraction and part-based representation with preserved intrinsic structure information from nonnegative high-order data. However, the original NTF methods utilize Euclidean or Kullback-Leibler divergence as the loss function which treats each feature equally leading to the neglect of the side-information of features. To utilize correlation information of features and manifold information of samples, we introduce Wasserstein manifold nonnegative tensor factorization (WMNTF), which minimizes the Wasserstein distance between the distribution of input tensorial data and the distribution of reconstruction. Although some researches about Wasserstein distance have been proposed in nonnegative matrix factorization (NMF), they ignore the spatial structure information of higher-order data. We use Wasserstein distance (a.k.a Earth Mover's distance or Optimal Transport distance) as a metric and add a graph regularizer to a latent factor. Experimental results demonstrate the effectiveness of the proposed method compared with other NMF and NTF methods.

Molecular property optimization (MPO) problems are inherently challenging since they are formulated over discrete, unstructured spaces and the labeling process involves expensive simulations or experiments, which fundamentally limits the amount of available data. Bayesian optimization (BO) is a powerful and popular framework for efficient optimization of noisy, black-box objective functions (e.g., measured property values), thus is a potentially attractive framework for MPO. To apply BO to MPO problems, one must select a structured molecular representation that enables construction of a probabilistic surrogate model. Many molecular representations have been developed, however, they are all high-dimensional, which introduces important challenges in the BO process -- mainly because the curse of dimensionality makes it difficult to define and perform inference over a suitable class of surrogate models. This challenge has been recently addressed by learning a lower-dimensional encoding of a SMILE or graph representation of a molecule in an unsupervised manner and then performing BO in the encoded space. In this work, we show that such methods have a tendency to "get stuck," which we hypothesize occurs since the mapping from the encoded space to property values is not necessarily well-modeled by a Gaussian process. We argue for an alternative approach that combines numerical molecular descriptors with a sparse axis-aligned Gaussian process model, which is capable of rapidly identifying sparse subspaces that are most relevant to modeling the unknown property function. We demonstrate that our proposed method substantially outperforms existing MPO methods on a variety of benchmark and real-world problems. Specifically, we show that our method can routinely find near-optimal molecules out of a set of more than $>100$k alternatives within 100 or fewer expensive queries.

In robust optimization problems, the magnitude of perturbations is relatively small. Consequently, solutions within certain regions are less likely to represent the robust optima when perturbations are introduced. Hence, a more efficient search process would benefit from increased opportunities to explore promising regions where global optima or good local optima are situated. In this paper, we introduce a novel robust evolutionary algorithm named the dual-stage robust evolutionary algorithm (DREA) aimed at discovering robust solutions. DREA operates in two stages: the peak-detection stage and the robust solution-searching stage. The primary objective of the peak-detection stage is to identify peaks in the fitness landscape of the original optimization problem. Conversely, the robust solution-searching stage focuses on swiftly identifying the robust optimal solution using information obtained from the peaks discovered in the initial stage. These two stages collectively enable the proposed DREA to efficiently obtain the robust optimal solution for the optimization problem. This approach achieves a balance between solution optimality and robustness by separating the search processes for optimal and robust optimal solutions. Experimental results demonstrate that DREA significantly outperforms five state-of-the-art algorithms across 18 test problems characterized by diverse complexities. Moreover, when evaluated on higher-dimensional robust optimization problems (100-$D$ and 200-$D$), DREA also demonstrates superior performance compared to all five counterpart algorithms.

Numerous statistical methods have been developed to explore genomic imprinting and maternal effects, which are causes of parent-of-origin patterns in complex human diseases. However, most of them either only model one of these two confounded epigenetic effects, or make strong yet unrealistic assumptions about the population to avoid over-parameterization. A recent partial likelihood method (LIME) can identify both epigenetic effects based on case-control family data without those assumptions. Theoretical and empirical studies have shown its validity and robustness. However, because LIME obtains parameter estimation by maximizing partial likelihood, it is interesting to compare its efficiency with full likelihood maximizer. To overcome the difficulty in over-parameterization when using full likelihood, in this study we propose a Monte Carlo Expectation Maximization (MCEM) method to detect imprinting and maternal effects jointly. Those unknown mating type probabilities, the nuisance parameters, can be considered as latent variables in EM algorithm. Monte Carlo samples are used to numerically approximate the expectation function that cannot be solved algebraically. Our simulation results show that though this MCEM algorithm takes longer computational time, and can give higher bias in some simulations compared to LIME, it can generally detect both epigenetic effects with higher power and smaller standard error which demonstrates that it can be a good complement of LIME method.

Importance sampling is a powerful tool for correcting the distributional mismatch in many statistical and machine learning problems, but in practice its performance is limited by the usage of simple proposals whose importance weights can be computed analytically. To address this limitation, Liu and Lee (2017) proposed a Black-Box Importance Sampling (BBIS) algorithm that computes the importance weights for arbitrary simulated samples by minimizing the kernelized Stein discrepancy. However, this requires knowing the score function of the target distribution, which is not easy to compute for many Bayesian problems. Hence, in this paper we propose another novel BBIS algorithm using minimum energy design, BBIS-MED, that requires only the unnormalized density function, which can be utilized as a post-processing step to improve the quality of Markov Chain Monte Carlo samples. We demonstrate the effectiveness and wide applicability of our proposed BBIS-MED algorithm on extensive simulations and a real-world Bayesian model calibration problem where the score function cannot be derived analytically.

We develop a new efficient sequential approximate leverage score algorithm, SALSA, using methods from randomized numerical linear algebra (RandNLA) for large matrices. We demonstrate that, with high probability, the accuracy of SALSA's approximations is within $(1 + O({\varepsilon}))$ of the true leverage scores. In addition, we show that the theoretical computational complexity and numerical accuracy of SALSA surpass existing approximations. These theoretical results are subsequently utilized to develop an efficient algorithm, named LSARMA, for fitting an appropriate ARMA model to large-scale time series data. Our proposed algorithm is, with high probability, guaranteed to find the maximum likelihood estimates of the parameters for the true underlying ARMA model. Furthermore, it has a worst-case running time that significantly improves those of the state-of-the-art alternatives in big data regimes. Empirical results on large-scale data strongly support these theoretical results and underscore the efficacy of our new approach.

We propose a contour integral-based algorithm for computing a few singular values of a matrix or a few generalized singular values of a matrix pencil. Mathematically, the generalized singular values of a matrix pencil are the eigenvalues of an equivalent Hermitian-definite matrix pencil, known as the Jordan-Wielandt matrix pencil. However, direct application of the FEAST solver does not fully exploit the structure of this problem. We analyze several projection strategies on the Jordan-Wielandt matrix pencil, and propose an effective and robust scheme tailored to GSVD. Both theoretical analysis and numerical experiments demonstrate that our algorithm achieves rapid convergence and satisfactory accuracy.

We study the problem of estimating the distribution of the return of a policy using an offline dataset that is not generated from the policy, i.e., distributional offline policy evaluation (OPE). We propose an algorithm called Fitted Likelihood Estimation (FLE), which conducts a sequence of Maximum Likelihood Estimation (MLE) and has the flexibility of integrating any state-of-the-art probabilistic generative models as long as it can be trained via MLE. FLE can be used for both finite-horizon and infinite-horizon discounted settings where rewards can be multi-dimensional vectors. Our theoretical results show that for both finite-horizon and infinite-horizon discounted settings, FLE can learn distributions that are close to the ground truth under total variation distance and Wasserstein distance, respectively. Our theoretical results hold under the conditions that the offline data covers the test policy's traces and that the supervised learning MLE procedures succeed. Experimentally, we demonstrate the performance of FLE with two generative models, Gaussian mixture models and diffusion models. For the multi-dimensional reward setting, FLE with diffusion models is capable of estimating the complicated distribution of the return of a test policy.

We consider the problem of evaluating dynamic consistency in discrete time probabilistic filters that approximate stochastic system state densities with Gaussian mixtures. Dynamic consistency means that the estimated probability distributions correctly describe the actual uncertainties. As such, the problem of consistency testing naturally arises in applications with regards to estimator tuning and validation. However, due to the general complexity of the density functions involved, straightforward approaches for consistency testing of mixture-based estimators have remained challenging to define and implement. This paper derives a new exact result for Gaussian mixture consistency testing within the framework of normalized deviation squared (NDS) statistics. It is shown that NDS test statistics for generic multivariate Gaussian mixture models exactly follow mixtures of generalized chi-square distributions, for which efficient computational tools are available. The accuracy and utility of the resulting consistency tests are numerically demonstrated on static and dynamic mixture estimation examples.