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

We present Path Integral Sampler~(PIS), a novel algorithm to draw samples from unnormalized probability density functions. The PIS is built on the Schr\"odinger bridge problem which aims to recover the most likely evolution of a diffusion process given its initial distribution and terminal distribution. The PIS draws samples from the initial distribution and then propagates the samples through the Schr\"odinger bridge to reach the terminal distribution. Applying the Girsanov theorem, with a simple prior diffusion, we formulate the PIS as a stochastic optimal control problem whose running cost is the control energy and terminal cost is chosen according to the target distribution. By modeling the control as a neural network, we establish a sampling algorithm that can be trained end-to-end. We provide theoretical justification of the sampling quality of PIS in terms of Wasserstein distance when sub-optimal control is used. Moreover, the path integrals theory is used to compute importance weights of the samples to compensate for the bias induced by the sub-optimality of the controller and time-discretization. We experimentally demonstrate the advantages of PIS compared with other start-of-the-art sampling methods on a variety of tasks.

We consider pessimistic bilevel stochastic programs in which the follower maximizes over a fixed compact convex set a strictly convex quadratic function, whose Hessian depends on the leader's decision. The resulting random variable is evaluated by a convex risk measure. Under assumptions including real analyticity of the lower-level goal function, we prove existence of optimal solutions. We discuss an alternate model where the leader hedges against optimal lower-level solutions, and show that in this case solvability can be guaranteed under weaker conditions both in a deterministic and in a stochastic setting. The approach is applied to a mechanical shape optimization problem in which the leader decides on an optimal material distribution to minimize a tracking-type cost functional, whereas the follower chooses forces from an admissible set to maximize a compliance objective. The material distribution is considered to be stochastically perturbed in the actual construction phase. Computational results illustrate the bilevel optimization concept and demonstrate the interplay of follower and leader in shape design and testing.

We study the Maximum Independent Set (MIS) problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of MIS is $\gamma$-stable if it has a unique optimal solution that remains the unique optimum under multiplicative perturbations of the weights by a factor of at most $\gamma\geq 1$. The goal then is to efficiently recover the unique optimal solution. In this work, we solve stable instances of MIS on several graphs classes: we solve $\widetilde{O}(\Delta/\sqrt{\log \Delta})$-stable instances on graphs of maximum degree $\Delta$, $(k - 1)$-stable instances on $k$-colorable graphs and $(1 + \varepsilon)$-stable instances on planar graphs. For general graphs, we present a strong lower bound showing that there are no efficient algorithms for $O(n^{\frac{1}{2} - \varepsilon})$-stable instances of MIS, assuming the planted clique conjecture. We also give an algorithm for $(\varepsilon n)$-stable instances. As a by-product of our techniques, we give algorithms and lower bounds for stable instances of Node Multiway Cut. Furthermore, we prove a general result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. Moreover, we initiate the study of certified algorithms, a notion recently introduced by Makarychev and Makarychev (2018), which is a class of $\gamma$-approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance. We obtain $\Delta$-certified algorithms for MIS on graphs of maximum degree $\Delta$, and $(1+\varepsilon)$-certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Furer (1994) and prove that it is a $\left(\frac{\Delta + 1}{3} + \varepsilon\right)$-certified algorithm for MIS on graphs of maximum degree $\Delta$ where all weights are equal to 1.

Integrity constraints such as functional dependencies (FD) and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been investigated in both the database and the AI literature, under the assumption that all constraints hold {\em exactly}. However, many applications today consider constraints that hold only {\em approximately}. In this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the {\em relaxation problem}: when does an exact implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second, we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Then, we show that the implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Finally, we show how some of the results in the paper can be derived using the {\em I-measure} theory, which relates between information theoretic measures and set theory. Our results recover, and sometimes extend, previously known results about the implication problem: the implication of MVDs and FDs can be checked by considering only 2-tuple relations.

The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence $2$-approximate) matching in $O(n)$ worst-case update time in $n$-node graphs. We present the first deterministic algorithm which outperforms the folklore algorithm in terms of {\em both} approximation ratio and worst-case update time. Specifically, we give a $(2-\Omega(1))$-approximate algorithm with $O(m^{3/8})=O(n^{3/4})$ worst-case update time in $n$-node, $m$-edge graphs. For sufficiently small constant $\epsilon>0$, no deterministic $(2+\epsilon)$-approximate algorithm with worst-case update time $O(n^{0.99})$ was known. Our second result is the first deterministic $(2+\epsilon)$-approximate weighted matching algorithm with $O_\epsilon(1)\cdot O(\sqrt[4]{m}) = O_\epsilon(1)\cdot O(\sqrt{n})$ worst-case update time. Our main technical contributions are threefold: first, we characterize the tight cases for \emph{kernels}, which are the well-studied matching sparsifiers underlying much of the $(2+\epsilon)$-approximate dynamic matching literature. This characterization, together with multiple ideas -- old and new -- underlies our result for breaking the approximation barrier of $2$. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the \emph{recourse} of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural $\frac{3}{2}$ factor in the approximation ratio which such approaches naturally incur.

Given a simple graph $G$ and an integer $k$, the goal of $k$-Clique problem is to decide if $G$ contains a complete subgraph of size $k$. We say an algorithm approximates $k$-Clique within a factor $g(k)$ if it can find a clique of size at least $k / g(k)$ when $G$ is guaranteed to have a $k$-clique. Recently, it was shown that approximating $k$-Clique within a constant factor is W[1]-hard [Lin21]. We study the approximation of $k$-Clique under the Exponential Time Hypothesis (ETH). The reduction of [Lin21] already implies an $n^{\Omega(\sqrt[6]{\log k})}$-time lower bound under ETH. We improve this lower bound to $n^{\Omega(\log k)}$. Using the gap-amplification technique by expander graphs, we also prove that there is no $k^{o(1)}$ factor FPT-approximation algorithm for $k$-Clique under ETH. We also suggest a new way to prove the Parameterized Inapproximability Hypothesis (PIH) under ETH. We show that if there is no $n^{O(\frac{k}{\log k})}$ algorithm to approximate $k$-Clique within a constant factor, then PIH is true.

We consider the twin problems of estimating the effective rank and the Schatten norms $\|{\bf A}\|_{s}$ of a rectangular $p\times q$ matrix ${\bf A}$ from noisy observations. When $s$ is an even integer, we introduce a polynomial-time estimator of $\|{\bf A}\|_s$ that achieves the minimax rate $(pq)^{1/4}$. Interestingly, this optimal rate does not depend on the underlying rank of the matrix. When $s$ is not an even integer, the optimal rate is much slower. A simple thresholding estimator of the singular values achieves the rate $(q\wedge p)(pq)^{1/4}$, which turns out to be optimal up to a logarithmic multiplicative term. The tight minimax rate is achieved by a more involved polynomial approximation method. This allows us to build estimators for a class of effective rank indices. As a byproduct, we also characterize the minimax rate for estimating the sequence of singular values of a matrix.

In this work we consider the well-known Secretary Problem -- a number $n$ of elements, each having an adversarial value, are arriving one-by-one according to some random order, and the goal is to choose the highest value element. The decisions are made online and are irrevocable -- if the algorithm decides to choose or not to choose the currently seen element, based on the previously observed values, it cannot change its decision later regarding this element. The measure of success is the probability of selecting the highest value element, minimized over all adversarial assignments of values. We show existential and constructive upper bounds on approximation of the success probability in this problem, depending on the entropy of the randomly chosen arrival order, including the lowest possible entropy $O(\log\log (n))$ for which the probability of success could be constant. We show that below entropy level $\mathcal{H}<0.5\log\log n$, all algorithms succeed with probability $0$ if random order is selected uniformly at random from some subset of permutations, while we are able to construct in polynomial time a non-uniform distribution with entropy $\mathcal{H}$ resulting in success probability of at least $\Omega\left(\frac{1}{(\log\log n +3\log\log\log n -\mathcal{H})^{2+\epsilon}}\right)$, for any constant $\epsilon>0$. We also prove that no algorithm using entropy $\mathcal{H}=O((\log\log n)^a)$ can improve our result by more than polynomially, for any constant $0<a<1$. For entropy $\log\log (n)$ and larger, our analysis precisely quantifies both multiplicative and additive approximation of the success probability. In particular, we improve more than doubly exponentially on the best previously known additive approximation guarantee for the secretary problem.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.