亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

Studying the properties of stochastic noise to optimize complex non-convex functions has been an active area of research in the field of machine learning. Prior work has shown that the noise of stochastic gradient descent improves optimization by overcoming undesirable obstacles in the landscape. Moreover, injecting artificial Gaussian noise has become a popular idea to quickly escape saddle points. Indeed, in the absence of reliable gradient information, the noise is used to explore the landscape, but it is unclear what type of noise is optimal in terms of exploration ability. In order to narrow this gap in our knowledge, we study a general type of continuous-time non-Markovian process, based on fractional Brownian motion, that allows for the increments of the process to be correlated. This generalizes processes based on Brownian motion, such as the Ornstein-Uhlenbeck process. We demonstrate how to discretize such processes which gives rise to the new algorithm fPGD. This method is a generalization of the known algorithms PGD and Anti-PGD. We study the properties of fPGD both theoretically and empirically, demonstrating that it possesses exploration abilities that, in some cases, are favorable over PGD and Anti-PGD. These results open the field to novel ways to exploit noise for training machine learning models.

Consider words of length $n$. The set of all periods of a word of length $n$ is a subset of $\{0,1,2,\ldots,n-1\}$. However, any subset of $\{0,1,2,\ldots,n-1\}$ is not necessarily a valid set of periods. In a seminal paper in 1981, Guibas and Odlyzko have proposed to encode the set of periods of a word into an $n$ long binary string, called an autocorrelation, where a one at position $i$ denotes a period of $i$. They considered the question of recognizing a valid period set, and also studied the number of valid period sets for length $n$, denoted $\kappa_n$. They conjectured that $\ln(\kappa_n)$ asymptotically converges to a constant times $\ln^2(n)$. If improved lower bounds for $\ln(\kappa_n)/\ln^2(n)$ were proposed in 2001, the question of a tight upper bound has remained opened since Guibas and Odlyzko's paper. Here, we exhibit an upper bound for this fraction, which implies its convergence and closes this long standing conjecture. Moreover, we extend our result to find similar bounds for the number of correlations: a generalization of autocorrelations which encodes the overlaps between two strings.

Best subset of groups selection (BSGS) is the process of selecting a small part of non-overlapping groups to achieve the best interpretability on the response variable. It has attracted increasing attention and has far-reaching applications in practice. However, due to the computational intractability of BSGS in high-dimensional settings, developing efficient algorithms for solving BSGS remains a research hotspot. In this paper,we propose a group-splicing algorithm that iteratively detects the relevant groups and excludes the irrelevant ones. Moreover, coupled with a novel group information criterion, we develop an adaptive algorithm to determine the optimal model size. Under mild conditions, it is certifiable that our algorithm can identify the optimal subset of groups in polynomial time with high probability. Finally, we demonstrate the efficiency and accuracy of our methods by comparing them with several state-of-the-art algorithms on both synthetic and real-world datasets.

QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent. This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs. We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas. We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies.

Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen. Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation.

In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schr\"odinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. A second-order scheme is first proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence in the $H^1$ norm without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to overcome the difficulty posed by the unavailability of a priori $L^\infty$ estimates of numerical solutions. Based on the integrating factor Runge-Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linear and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.

We investigate $L_2$ boosting in the context of kernel regression. Kernel smoothers, in general, lack appealing traits like symmetry and positive definiteness, which are critical not only for understanding theoretical aspects but also for achieving good practical performance. We consider a projection-based smoother (Huang and Chen, 2008) that is symmetric, positive definite, and shrinking. Theoretical results based on the orthonormal decomposition of the smoother reveal additional insights into the boosting algorithm. In our asymptotic framework, we may replace the full-rank smoother with a low-rank approximation. We demonstrate that the smoother's low-rank ($d(n)$) is bounded above by $O(h^{-1})$, where $h$ is the bandwidth. Our numerical findings show that, in terms of prediction accuracy, low-rank smoothers may outperform full-rank smoothers. Furthermore, we show that the boosting estimator with low-rank smoother achieves the optimal convergence rate. Finally, to improve the performance of the boosting algorithm in the presence of outliers, we propose a novel robustified boosting algorithm which can be used with any smoother discussed in the study. We investigate the numerical performance of the proposed approaches using simulations and a real-world case.

In the well-known complexity class NP, many combinatorial problems can be found, whose optimization counterpart are important for many practical settings. Those problems usually consider full knowledge about the input and optimize on this specific input. In a practical setting, however, uncertainty in the input data is a usual phenomenon, whereby this is normally not covered in optimization versions of NP problems. One concept to model the uncertainty in the input data, is \textit{recoverable robustness}. In this setting, a solution on the input is calculated, whereby a possible recovery to a good solution should be guaranteed, whenever uncertainty manifests itself. That is, a solution $\texttt{s}_0$ for the base scenario $\textsf{S}_0$ as well as a solution \texttt{s} for every possible scenario of scenario set \textsf{S} has to be calculated. In other words, not only solution $\texttt{s}_0$ for instance $\textsf{S}_0$ is calculated but solutions \texttt{s} for all scenarios from \textsf{S} are prepared to correct possible errors through uncertainty. This paper introduces a specific concept of recoverable robust problems: Hamming Distance Recoverable Robust Problems. In this setting, solutions $\texttt{s}_0$ and \texttt{s} have to be calculated, such that $\texttt{s}_0$ and \texttt{s} may only differ in at most $\kappa$ elements. That is, one can recover from a harmful scenario by choosing a different solution, which is not too far away from the first solution. This paper surveys the complexity of Hamming distance recoverable robust version of optimization problems, typically found in NP for different types of scenarios. The complexity is primarily situated in the lower levels of the polynomial hierarchy. The main contribution of the paper is that recoverable robust problems with compression-encoded scenarios and $m \in \mathbb{N}$ recoveries are $\Sigma^P_{2m+1}$-complete.

When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant.

Deep Learning algorithms have achieved the state-of-the-art performance for Image Classification and have been used even in security-critical applications, such as biometric recognition systems and self-driving cars. However, recent works have shown those algorithms, which can even surpass the human capabilities, are vulnerable to adversarial examples. In Computer Vision, adversarial examples are images containing subtle perturbations generated by malicious optimization algorithms in order to fool classifiers. As an attempt to mitigate these vulnerabilities, numerous countermeasures have been constantly proposed in literature. Nevertheless, devising an efficient defense mechanism has proven to be a difficult task, since many approaches have already shown to be ineffective to adaptive attackers. Thus, this self-containing paper aims to provide all readerships with a review of the latest research progress on Adversarial Machine Learning in Image Classification, however with a defender's perspective. Here, novel taxonomies for categorizing adversarial attacks and defenses are introduced and discussions about the existence of adversarial examples are provided. Further, in contrast to exisiting surveys, it is also given relevant guidance that should be taken into consideration by researchers when devising and evaluating defenses. Finally, based on the reviewed literature, it is discussed some promising paths for future research.