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

In this article, we consider convergence of stochastic gradient descent schemes (SGD), including momentum stochastic gradient descent (MSGD), under weak assumptions on the underlying landscape. More explicitly, we show that on the event that the SGD stays bounded we have convergence of the SGD if there is only a countable number of critical points or if the objective function satisfies Lojasiewicz-inequalities around all critical levels as all analytic functions do. In particular, we show that for neural networks with analytic activation function such as softplus, sigmoid and the hyperbolic tangent, SGD converges on the event of staying bounded, if the random variables modelling the signal and response in the training are compactly supported.

In this paper, we define and study variants of several complexity classes of decision problems that are defined via some criteria on the number of accepting paths of an NPTM. In these variants, we modify the acceptance criteria so that they concern the total number of computation paths instead of the number of accepting ones. This direction reflects the relationship between the counting classes #P and TotP, which are the classes of functions that count the number of accepting paths and the total number of paths of NPTMs, respectively. The former is the well-studied class of counting versions of NP problems introduced by Valiant (1979). The latter contains all self-reducible counting problems in #P whose decision version is in P, among them prominent #P-complete problems such as Non-negative Permanent, #PerfMatch, and #DNF-Sat, thus playing a significant role in the study of approximable counting problems. We show that almost all classes introduced in this work coincide with their `#accepting paths'-definable counterparts, thus providing an alternative model of computation for them. Moreover, for each of these classes, we present a novel family of complete problems, which are defined via TotP-complete problems. This way, we show that all the aforementioned classes have complete problems that are defined via counting problems whose existence version is in P, in contrast to the standard way of obtaining completeness results via counting versions of NP-complete problems. To the best of our knowledge, prior to this work, such results were known only for parity-P and C=P.

We consider the following problem that we call the Shortest Two Disjoint Paths problem: given an undirected graph $G=(V,E)$ with edge weights $w:E \rightarrow \mathbb{R}$, two terminals $s$ and $t$ in $G$, find two internally vertex-disjoint paths between $s$ and $t$ with minimum total weight. As shown recently by Schlotter and Seb\H{o} (2022), this problem becomes NP-hard if edges can have negative weights, even if the weight function is conservative, there are no cycles in $G$ with negative total weight. We propose a polynomial-time algorithm that solves the Shortest Two Disjoint Paths problem for conservative weights in the case when the negative-weight edges form a constant number of trees in $G$.

In this work we present deep learning implementations of two popular theoretical constrained optimization algorithms in infinite dimensional Hilbert spaces, namely, the penalty and the augmented Lagrangian methods. We test these algorithms on some toy problems originating in either calculus of variations or physics. We demonstrate that both methods are able to produce decent approximations for the test problems and are comparable in terms of different errors produced. Leveraging the common occurrence of the Lagrange multiplier update rule being computationally less expensive than solving subproblems in the penalty method, we achieve significant speedups in cases when the output of the constraint function is itself a function.

Inverse Reinforcement Learning (IRL) techniques deal with the problem of deducing a reward function that explains the behavior of an expert agent who is assumed to act optimally in an underlying unknown task. In several problems of interest, however, it is possible to observe the behavior of multiple experts with different degree of optimality (e.g., racing drivers whose skills ranges from amateurs to professionals). For this reason, in this work, we extend the IRL formulation to problems where, in addition to demonstrations from the optimal agent, we can observe the behavior of multiple sub-optimal experts. Given this problem, we first study the theoretical properties of the class of reward functions that are compatible with a given set of experts, i.e., the feasible reward set. Our results show that the presence of multiple sub-optimal experts can significantly shrink the set of compatible rewards. Furthermore, we study the statistical complexity of estimating the feasible reward set with a generative model. To this end, we analyze a uniform sampling algorithm that results in being minimax optimal whenever the sub-optimal experts' performance level is sufficiently close to the one of the optimal agent.

Machine learning methods based on AdaBoost have been widely applied to various classification problems across many mission-critical applications including healthcare, law and finance. However, there is a growing concern about the unfairness and discrimination of data-driven classification models, which is inevitable for classical algorithms including AdaBoost. In order to achieve fair classification, a novel fair AdaBoost (FAB) approach is proposed that is an interpretable fairness-improving variant of AdaBoost. We mainly investigate binary classification problems and focus on the fairness of three different indicators (i.e., accuracy, false positive rate and false negative rate). By utilizing a fairness-aware reweighting technique for base classifiers, the proposed FAB approach can achieve fair classification while maintaining the advantage of AdaBoost with negligible sacrifice of predictive performance. In addition, a hyperparameter is introduced in FAB to show preferences for the fairness-accuracy trade-off. An upper bound for the target loss function that quantifies error rate and unfairness is theoretically derived for FAB, which provides a strict theoretical support for the fairness-improving methods designed for AdaBoost. The effectiveness of the proposed method is demonstrated on three real-world datasets (i.e., Adult, COMPAS and HSLS) with respect to the three fairness indicators. The results are accordant with theoretic analyses, and show that (i) FAB significantly improves classification fairness at a small cost of accuracy compared with AdaBoost; and (ii) FAB outperforms state-of-the-art fair classification methods including equalized odds method, exponentiated gradient method, and disparate mistreatment method in terms of the fairness-accuracy trade-off.

The generalized optimised Schwarz method proposed in [Claeys & Parolin, 2022] is a variant of the Despr\'es algorithm for solving harmonic wave problems where transmission conditions are enforced by means of a non-local exchange operator. We introduce and analyse an acceleration technique that significantly reduces the cost of applying this exchange operator without deteriorating the precision and convergence speed of the overall domain decomposition algorithm.

We study the problem of exact community recovery in the Geometric Stochastic Block Model (GSBM), where each vertex has an unknown community label as well as a known position, generated according to a Poisson point process in $\mathbb{R}^d$. Edges are formed independently conditioned on the community labels and positions, where vertices may only be connected by an edge if they are within a prescribed distance of each other. The GSBM thus favors the formation of dense local subgraphs, which commonly occur in real-world networks, a property that makes the GSBM qualitatively very different from the standard Stochastic Block Model (SBM). We propose a linear-time algorithm for exact community recovery, which succeeds down to the information-theoretic threshold, confirming a conjecture of Abbe, Baccelli, and Sankararaman. The algorithm involves two phases. The first phase exploits the density of local subgraphs to propagate estimated community labels among sufficiently occupied subregions, and produces an almost-exact vertex labeling. The second phase then refines the initial labels using a Poisson testing procedure. Thus, the GSBM enjoys local to global amplification just as the SBM, with the advantage of admitting an information-theoretically optimal, linear-time algorithm.

The Levin method is a well-known technique for evaluating oscillatory integrals, which operates by solving a certain ordinary differential equation in order to construct an antiderivative of the integrand. It was long believed that this approach suffers from "low-frequency breakdown," meaning that the accuracy of the calculated value of the integral deteriorates when the integrand is only slowly oscillating. Recently presented experimental evidence, however, suggests that if a Chebyshev spectral method is used to discretize the differential equation and the resulting linear system is solved via a truncated singular value decomposition, then no low-frequency breakdown occurs. Here, we provide a proof that this is the case, and our proof applies not only when the integrand is slowly oscillating, but even in the case of stationary points. Our result puts adaptive schemes based on the Levin method on a firm theoretical foundation and accounts for their behavior in the presence of stationary points. We go on to point out that by combining an adaptive Levin scheme with phase function methods for ordinary differential equations, a large class of oscillatory integrals involving special functions, including products of such functions and the compositions of such functions with slowly-varying functions, can be easily evaluated without the need for symbolic computations. Finally, we present the results of numerical experiments which illustrate the consequences of our analysis and demonstrate the properties of the adaptive Levin method.

Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.