Sampling algorithms based on discretizations of Stochastic Differential Equations (SDEs) compose a rich and popular subset of MCMC methods. This work provides a general framework for the non-asymptotic analysis of sampling error in 2-Wasserstein distance, which also leads to a bound of mixing time. The method applies to any consistent discretization of contractive SDEs. When applied to Langevin Monte Carlo algorithm, it establishes $\tilde{\mathcal{O}}\left( \frac{\sqrt{d}}{\epsilon} \right)$ mixing time, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures at infinity. This bound improves the best previously known $\tilde{\mathcal{O}}\left( \frac{d}{\epsilon} \right)$ result and is optimal (in terms of order) in both dimension $d$ and accuracy tolerance $\epsilon$ for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.
相關內容
We revisit the divide-and-conquer sequential Monte Carlo (DaC-SMC) algorithm and firmly establish it as a well-founded method by showing that it possesses the same basic properties as conventional sequential Monte Carlo (SMC) algorithms do. In particular, we derive pertinent laws of large numbers, $L^p$ inequalities, and central limit theorems; and we characterize the bias in the normalized estimates produced by the algorithm and argue the absence thereof in the unnormalized ones. We further consider its practical implementation and several interesting variants; obtain expressions for its globally and locally optimal intermediate targets, auxiliary measures, and proposal kernels; and show that, in comparable conditions, DaC-SMC proves more statistically efficient than its direct SMC analogue. We close the paper with a discussion of our results, open questions, and future research directions.
The primary emphasis of this work is the development of a finite element based space-time discretization for solving the stochastic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of R^d , d $\in$ {2, 3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale $\alpha$, namely we first assume $\alpha$ to be controlled by the step size of the space discretization so that it vanishes when passing to the limit, then we provide an alternative study when $\alpha$ is fixed. A preparatory analysis of uniform estimates in both $\alpha$ and discretization parameters is carried out. Starting out from the stochastic LANS-$\alpha$ model, we achieve convergence toward the continuous strong solutions of the stochastic Navier-Stokes equations in 2D when $\alpha$ vanishes at the limit. Additionally, convergence toward the continuous strong solutions of the stochastic LANS-$\alpha$ model is accomplished if $\alpha$ is fixed.
In this paper, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. {\it A~priori} error estimates of optimal order are derived for velocity and pressure in the energy norm and the $L^2$-norm, respectively. Moreover, a reliable and efficient {\it a~posteriori} error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix-Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces. The theoretical results are illustrated by the numerical experiments.
In this paper we present a finite element analysis for a Dirichlet boundary control problem governed by the Stokes equation. The Dirichlet control is considered in a convex closed subset of the energy space $\mathbf{H}^1(\Omega).$ Most of the previous works on the Stokes Dirichlet boundary control problem deals with either tangential control or the case where the flux of the control is zero. This choice of the control is very particular and their choice of the formulation leads to the control with limited regularity. To overcome this difficulty, we introduce the Stokes problem with outflow condition and the control acts on the Dirichlet boundary only hence our control is more general and it has both the tangential and normal components. We prove well-posedness and discuss on the regularity of the control problem. The first-order optimality condition for the control leads to a Signorini problem. We develop a two-level finite element discretization by using $\mathbf{P}_1$ elements(on the fine mesh) for the velocity and the control variable and $P_0$ elements (on the coarse mesh) for the pressure variable. The standard energy error analysis gives $\frac{1}{2}+\frac{\delta}{2}$ order of convergence when the control is in $\mathbf{H}^{\frac{3}{2}+\delta}(\Omega)$ space. Here we have improved it to $\frac{1}{2}+\delta,$ which is optimal. Also, when the control lies in less regular space we derived optimal order of convergence up to the regularity. The theoretical results are corroborated by a variety of numerical tests.
We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove that our algorithm achieves optimal $\mathcal{O}(n/k)$ and $\mathcal{O}(n^2/k^2)$ convergence rates (up to a constant factor) in two cases: general convexity and strong convexity, respectively, where $k$ is the iteration counter and n is the number of block-coordinates. Our convergence rates are obtained through three criteria: primal objective residual and primal feasibility violation, dual objective residual, and primal-dual expected gap. Moreover, our rates for the primal problem are on the last iterate sequence. Our dual convergence guarantee requires additionally a Lipschitz continuity assumption. We specify our algorithm to handle two important special cases, where our rates are still applied. Finally, we verify our algorithm on two well-studied numerical examples and compare it with two existing methods. Our results show that the proposed method has encouraging performance on different experiments.
We consider the problem of testing for long-range dependence for time-varying coefficient regression models. The covariates and errors are assumed to be locally stationary, which allows complex temporal dynamics and heteroscedasticity. We develop KPSS, R/S, V/S, and K/S-type statistics based on the nonparametric residuals, and propose bootstrap approaches equipped with a difference-based long-run covariance matrix estimator for practical implementation. Under the null hypothesis, the local alternatives as well as the fixed alternatives, we derive the limiting distributions of the test statistics, establish the uniform consistency of the difference-based long-run covariance estimator, and justify the bootstrap algorithms theoretically. In particular, the exact local asymptotic power of our testing procedure enjoys the order $O( \log^{-1} n)$, the same as that of the classical KPSS test for long memory in strictly stationary series without covariates. We demonstrate the effectiveness of our tests by extensive simulation studies. The proposed tests are applied to a COVID-19 dataset in favor of long-range dependence in the cumulative confirmed series of COVID-19 in several countries, and to the Hong Kong circulatory and respiratory dataset, identifying a new type of 'spurious long memory'.
We consider robust variants of the standard optimal transport, named robust optimal transport, where marginal constraints are relaxed via Kullback-Leibler divergence. We show that Sinkhorn-based algorithms can approximate the optimal cost of robust optimal transport in $\widetilde{\mathcal{O}}(\frac{n^2}{\varepsilon})$ time, in which $n$ is the number of supports of the probability distributions and $\varepsilon$ is the desired error. Furthermore, we investigate a fixed-support robust barycenter problem between $m$ discrete probability distributions with at most $n$ number of supports and develop an approximating algorithm based on iterative Bregman projections (IBP). For the specific case $m = 2$, we show that this algorithm can approximate the optimal barycenter value in $\widetilde{\mathcal{O}}(\frac{mn^2}{\varepsilon})$ time, thus being better than the previous complexity $\widetilde{\mathcal{O}}(\frac{mn^2}{\varepsilon^2})$ of the IBP algorithm for approximating the Wasserstein barycenter.
We consider \emph{Gibbs distributions}, which are families of probability distributions over a discrete space $\Omega$ with probability mass function of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval $[\beta_{\min}, \beta_{\max}]$ and $H( \omega ) \in \{0 \} \cup [1, n]$. The \emph{partition function} is the normalization factor $Z(\beta)=\sum_{\omega \in\Omega}e^{\beta H(\omega)}$. Two important parameters of these distributions are the log partition ratio $q = \log \tfrac{Z(\beta_{\max})}{Z(\beta_{\min})}$ and the counts $c_x = |H^{-1}(x)|$. These are correlated with system parameters in a number of physical applications and sampling algorithms. Our first main result is to estimate the counts $c_x$ using roughly $\tilde O( \frac{q}{\varepsilon^2})$ samples for general Gibbs distributions and $\tilde O( \frac{n^2}{\varepsilon^2} )$ samples for integer-valued distributions (ignoring some second-order terms and parameters), and we show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs and perfect matchings in a graph. We develop a key subroutine to estimate the partition function $Z$. Specifically, it generates a data structure to estimate $Z(\beta)$ for \emph{all} values $\beta$, without further samples. Constructing the data structure requires $O(\frac{q \log n}{\varepsilon^2})$ samples for general Gibbs distributions and $O(\frac{n^2 \log n}{\varepsilon^2} + n \log q)$ samples for integer-valued distributions. This improves over a prior algorithm of Huber (2015) which computes a single point estimate $Z(\beta_\max)$ using $O( q \log n( \log q + \log \log n + \varepsilon^{-2}))$ samples. We show matching lower bounds, demonstrating that this complexity is optimal as a function of $n$ and $q$ up to logarithmic terms.
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.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191