Asymptotic Convergence Rate Research Articles

The convergence properties of infeasible inexact proximal alternating linearized minimization

The proximal alternating linearized minimization method (PALM) suits well for solving block-structured optimization problems, which are ubiquitous in real applications. In the cases where subproblems do not have closed-form solutions, e.g., due to complex constraints, infeasible subsolvers are indispensable, giving rise to an infeasible inexact PALM (PALM-I). Numerous efforts have been devoted to analyzing feasible PALM, while little attention has been paid to PALM-I. The usage of PALM-I thus lacks theoretical guarantee. The essential difficulty of analyses consists in the objective value nonmonotonicity induced by the infeasibility. We study in the present work the convergence properties of PALM-I. In particular, we construct a surrogate sequence to surmount the nonmonotonicity issue and devise an implementable inexact criterion. Based upon these, we manage to establish the stationarity of any accumulation point and, moreover, show the iterate convergence and the asymptotic convergence rates under the assumption of the Lojasiewicz property. The prominent advantages of PALM-I on CPU time are illustrated via numerical experiments on problems arising from quantum physics and 3D anisotropic frictional contact.

Online parameter estimation for the McKean–Vlasov stochastic differential equation

We analyse the problem of online parameter estimation for a stochastic McKean–Vlasov equation, and the associated system of weakly interacting particles. We propose an online estimator for the parameters of the McKean–Vlasov SDE, or the interacting particle system, which is based on a continuous-time stochastic gradient ascent scheme with respect to the asymptotic log-likelihood of the interacting particle system. We characterise the asymptotic behaviour of this estimator in the limit as t→∞, and also in the joint limit as t→∞ and N→∞. In these two cases, we obtain almost sure or L1 convergence to the stationary points of a limiting contrast function, under suitable conditions which guarantee ergodicity and uniform-in-time propagation of chaos. We also establish, under the additional condition of global strong concavity, L2 convergence to the unique maximiser of the asymptotic log-likelihood of the McKean–Vlasov SDE, with an asymptotic convergence rate which depends on the learning rate, the number of observations, and the dimension of the non-linear process. Our theoretical results are supported by two numerical examples, a linear mean field model and a stochastic opinion dynamics model.

Optimized Schwarz Methods for the Cahn–Hilliard Equation

The Cahn–Hilliard equation was originally proposed to describe the phase separation phenomenon for a binary alloy in the quenching process and now has been widely applied in many other scientific fields. To investigate its solution behavior, one mainly depends on the numerical simulations because of the nonlinearity. However, it would require very small time steps to describe the phase separation procedure since it evolves very quickly in time, which implies that an efficient solver for the spatial problem at each time level is very important. In this paper, we investigate a coupled second order elliptic system of constant coefficients, which occurs in many different unconditionally energy stable time discretization schemes for the Cahn–Hilliard equation. For a min-max problem of convergence factor that is obtained via Fourier analysis in the case of two-subdomain domain decomposition, by using an asymptotic analysis we obtain the optimized transmission parameters in explicit form applied in the Robin and the two-sided Robin transmission conditions for both the overlapping and the nonoverlapping Schwarz domain decomposition algorithms, and we obtain as well the corresponding asymptotic convergence rates. Particularly, we also explore and analyze the many-subdomain cases directly and find that the algorithms are scalable until a certain number of subdomains is reached for small time steps, even if there is no coarse grid correction. We finally use numerical examples to illustrate our theoretical findings.

Improving reinforcement learning algorithms: Towards optimal learning rate policies

AbstractThis paper shows how to use results of statistical learning theory and stochastic algorithms to have a better understanding of the convergence of Reinforcement Learning (RL) once it is formulated as a fixed point problem. This can be used to propose improvement of RL learning rates. First, our analysis shows that the classical asymptotic convergence rate is pessimistic and can be replaced by with , and the number of iterations. Second, we propose a dynamic optimal policy for the choice of the learning rate used in RL. We decompose our policy into two interacting levels: the inner and outer levels. In the inner level, we present the PASS algorithm (for “PAst Sign Search”) which, based on a predefined sequence of learning rates, constructs a new sequence for which the error decreases faster. The convergence of PASS is proved and error bounds are established. In the outer level, we propose an optimal methodology for the selection of the predefined sequence. Third, we show empirically that our selection methodology of the learning rate outperforms significantly standard algorithms used in RL for the three following applications: the estimation of a drift, the optimal placement of limit orders, and the optimal execution of a large number of shares.

Secure Distributed Optimization Under Gradient Attacks

In this paper, we study secure distributed optimization against arbitrary gradient attacks in multi-agent networks. In distributed optimization, there is no central server to coordinate local updates, and each agent can only communicate with its neighbors on a predefined network. We consider the scenario where out of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> networked agents, a fixed but unknown fraction <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula> of the agents are under arbitrary gradient attacks in that their stochastic gradient oracles return arbitrary information to derail the optimization process, and the goal is to minimize the sum of local objective functions on unattacked agents. We propose a distributed stochastic gradient method that combines local variance reduction and clipping ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLIP-VRG</small> ). We show that, in a connected network, when the unattacked local objective functions are convex and smooth, share a common minimizer, and their sum is strongly convex, <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLIP-VRG</small> leads to almost sure convergence of the iterates to the exact sum cost minimizer at all agents. We quantify a tight upper bound on the fraction <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula> of attacked agents in terms of problem parameters such as the condition number of the associated sum cost that guarantee exact convergence of <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">CLIP-VRG</small> , and characterize its asymptotic convergence rate. Finally, we empirically demonstrate the effectiveness of the proposed method under gradient attacks on both synthetic and real-world image classification datasets.

Novel method of particular solutions for second-order variable coefficient differential equations on a square

In this work, a novel method of particular solutions (MPS, for short) is proposed with Legendre polynomials to solve second-order partial differential equations (PDEs) with a variable coefficient on a square. Specially, we prove the uniqueness of the solution for the given model, which is equipped with a sufficiently smooth variable coefficient excluding elliptic properties. Based on the characteristics of Legendre polynomials, particular solutions, which satisfy the natural boundary condition, are constructed for the variable coefficient within the PDEs. Meanwhile we investigate the a-priori error estimates in $ H^1 $-norm and $ L^2 $-norm of the MPS approximations, which depict the asymptotic super-exponential convergence rates of the numerical approximations for sufficiently smooth solutions. Some numerical examples and convergence rates are listed to depict the highlights of this proposed meshless method.

On a new higher order kernel for density estimation

Higher order kernels have been widely implemented for nonparametric function estimation, mainly due to their faster asymptotic rates of convergence and interpretability. This article constructs a novel higher order kernel of any pre‐specified even order by using Shannon's formula. The proposed kernels have a closed form expression and are easy to implement. We compare the constructed kernel with several popular higher order kernels in the context of density estimation. Further, we observe that the mean integrated squared error (MISE) of the density estimator corresponding to the proposed kernel is comparable with that of the popular kernels. The constructed kernel performs somewhat better for the Fejér‐de la Vallée Poussin and the ‐symmetric densities. The MISE of the corresponding kernel density estimator and a few related theoretical results are also presented.

Estimation of A Class of Nonparameteric Probit Models

In this paper, we consider a class of nonparametric probit models for binary dependent variables. We extend the standard parametric probit models to nonparametric models in which the parametric index function in the standard probit models is nonparametrically specified. Such a nonparametric specification for the index function in probit models allows us to consider a flexible functional form of the structural parameters. We propose to use a sieve maximum likelihood estimation approach to estimate the parameter and develop the asymptotic theory, including consistency, convergence rates, and asymptotic normality. &#x0D; Our asymptotic normality results are applicable to a wide class of functionals of the parameter, regardless of whether the functional of interest is regular or irregular. The Monte Carlo simulation result shows that the proposed sieve maximum likelihood estimator performs well in finite samples in the sense that the proposed sieve maximum likelihood estimator has a small variance and negligible bias in finite samples.

On the asymptotic rate of convergence of Stochastic Newton algorithms and their Weighted Averaged versions

Most machine learning methods can be regarded as the minimization of an unavailable risk function. To optimize the latter, with samples provided in a streaming fashion, we define general (weighted averaged) stochastic Newton algorithms, for which a theoretical analysis of their asymptotic efficiency is conducted. The corresponding implementations are shown not to require the inversion of a Hessian estimate at each iteration under a quite flexible framework that covers the case of linear, logistic or softmax regressions to name a few. Numerical experiments on simulated and real data give the empirical evidence of the pertinence of the proposed methods, which outperform popular competitors particularly in case of bad initializations.

Two-step Noda iteration for irreducible nonnegative matrices

ABSTRACT In this paper, we present a two-step Noda iteration for computing the Perron root and Perron vector of an irreducible nonnegative matrix by successively solving two linear systems with the same coefficient matrix. For every positive initial vector, the two-step Noda iteration always converges and has a cubic asymptotic convergence rate. As an application, the two-step Noda iteration is applied to compute the smallest eigenpair of irreducible nonsingular M-matrices. Numerical examples are provided for testing the proposed method.

Penalized time-varying model averaging

This paper proposes a new penalized time-varying model averaging method to determine optimal time-varying combination weights for candidate models, which avoids over-fitting and yields sparseness from various potential predictive variables. The asymptotic optimality and convergence rate of the selected weights are derived even when all candidate models are misspecified, and the consistency and normality of the proposed time-varying model averaging estimator are obtained when the true model is included in the candidate models. Simulation studies and empirical applications to inflation forecasting highlight the merits of the proposed method.

Bayesian fixed-domain asymptotics for covariance parameters in a Gaussian process model

Gaussian process models typically contain finite-dimensional parameters in the covariance function that need to be estimated from the data. We study the Bayesian fixed-domain asymptotics for the covariance parameters in a universal kriging model with an isotropic Matérn covariance function, which has many applications in spatial statistics. We show that when the dimension of domain is less than or equal to three, the joint posterior distribution of the microergodic parameter and the range parameter can be factored independently into the product of their marginal posteriors under fixed-domain asymptotics. The posterior of the microergodic parameter is asymptotically close in total variation distance to a normal distribution with shrinking variance, while the posterior distribution of the range parameter does not converge to any point mass distribution in general. Our theory allows an unbounded prior support for the range parameter and flexible designs of sampling points. We further study the asymptotic efficiency and convergence rates in posterior prediction for the Bayesian kriging predictor with covariance parameters randomly drawn from their posterior distribution. In the special case of one-dimensional Ornstein–Uhlenbeck process, we derive explicitly the limiting posterior of the range parameter and the posterior convergence rate for asymptotic efficiency in posterior prediction. We verify these asymptotic results in numerical experiments.

Brownian bridge expansions for Lévy area approximations and particular values of the Riemann zeta function

AbstractWe study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

A Sharp Estimate on the Transient Time of Distributed Stochastic Gradient Descent.

This paper is concerned with minimizing the average of n cost functions over a network in which agents may communicate and exchange information with each other. We consider the setting where only noisy gradient information is available. To solve the problem, we study the distributed stochastic gradient descent (DSGD) method and perform a non-asymptotic convergence analysis. For strongly convex and smooth objective functions, in expectation, DSGD asymptotically achieves the optimal network independent convergence rate compared to centralized stochastic gradient descent (SGD). Our main contribution is to characterize the transient time needed for DSGD to approach the asymptotic convergence rate. Moreover, we construct a "hard" optimization problem that proves the sharpness of the obtained result. Numerical experiments demonstrate the tightness of the theoretical results.

Scaling up stochastic gradient descent for non-convex optimisation

Stochastic gradient descent (SGD) is a widely adopted iterative method for optimizing differentiable objective functions. In this paper, we propose and discuss a novel approach to scale up SGD in applications involving non-convex functions and large datasets. We address the bottleneck problem arising when using both shared and distributed memory. Typically, the former is bounded by limited computation resources and bandwidth whereas the latter suffers from communication overheads. We propose a unified distributed and parallel implementation of SGD (named DPSGD) that relies on both asynchronous distribution and lock-free parallelism. By combining two strategies into a unified framework, DPSGD is able to strike a better trade-off between local computation and communication. The convergence properties of DPSGD are studied for non-convex problems such as those arising in statistical modelling and machine learning. Our theoretical analysis shows that DPSGD leads to speed-up with respect to the number of cores and number of workers while guaranteeing an asymptotic convergence rate of O(1/sqrt{T}) given that the number of cores is bounded by T^{1/4} and the number of workers is bounded by T^{1/2} where T is the number of iterations. The potential gains that can be achieved by DPSGD are demonstrated empirically on a stochastic variational inference problem (Latent Dirichlet Allocation) and on a deep reinforcement learning (DRL) problem (advantage actor critic - A2C) resulting in two algorithms: DPSVI and HSA2C. Empirical results validate our theoretical findings. Comparative studies are conducted to show the performance of the proposed DPSGD against the state-of-the-art DRL algorithms.

Distributed Newton Optimization With Maximized Convergence Rate

The distributed optimization problem is set up in a collection of nodes interconnected via a communication network. The goal is to find the minimizer of a global objective function formed by the sum of local functions known at individual nodes. A number of methods, having different advantages, are available for addressing this problem. The goal of this article is to achieve the maximum possible convergence rate. As the first step toward this end, we propose a new method, which we show converges faster than other available options. As the second step toward our goal, we complement the proposed method with a fully distributed method for estimating the optimal step size that maximizes the convergence rate. We provide theoretical guarantees for the convergence of the resulting method in a neighborhood of the solution. We present numerical experiments showing that, when using the same step size, our method converges significantly faster than its rivals. Experiments also show that the distributed step-size estimation method achieves an asymptotic convergence rate very close to the theoretical maximum.

Fast and scalable learning of sparse changes in high-dimensional graphical model structure

We focus on the problem of estimating the change in the dependency structures of two p-dimensional Gaussian Graphical models (GGMs). Previous studies for sparse change estimation in GGMs involve expensive and difficult non-smooth optimization. We propose a novel method, DIFFEE for estimating DIFFerential networks via an Elementary Estimator under a high-dimensional situation. DIFFEE is solved through a faster and closed-form solution that enables it to work in large-scale settings. Notice that GGM assumes data are generated from a Gaussian distribution. However, the Gaussian assumption is too strict and can not be satisfied with all the real-world data generated from a complex process. Therefore, we further extend DIFFEE to NPN-DIFFEE by assuming that data are drawn from the nonparanormal distribution (a large family of distributions) instead of a multivariate Gaussian distribution. Thus, NPN-DIFFEE is applicable to more general conditions. We conduct a rigorous statistical analysis showing that surprisingly DIFFEE achieves the same asymptotic convergence rates as the state-of-the-art estimators that are much more difficult to compute. Our experimental results on multiple synthetic datasets and one real-world data about brain connectivity show strong performance improvements over baselines, as well as significant computational benefits.

Additive models for symmetric positive-definite matrices and Lie groups

Summary We propose and investigate an additive regression model for symmetric positive-definite matrix-valued responses and multiple scalar predictors. The model exploits the Abelian group structure inherited from either of the log-Cholesky and log-Euclidean frameworks for symmetric positive-definite matrices and naturally extends to general Abelian Lie groups. The proposed additive model is shown to connect to an additive model on a tangent space. This connection not only entails an efficient algorithm to estimate the component functions, but also allows one to generalize the proposed additive model to general Riemannian manifolds. Optimal asymptotic convergence rates and normality of the estimated component functions are established, and numerical studies show that the proposed model enjoys good numerical performance, and is not subject to the curse of dimensionality when there are multiple predictors. The practical merits of the proposed model are demonstrated through an analysis of brain diffusion tensor imaging data.

Non-asymptotic superlinear convergence of standard quasi-Newton methods

In this paper, we study and prove the non-asymptotic superlinear convergence rate of the Broyden class of quasi-Newton algorithms which includes the Davidon–Fletcher–Powell (DFP) method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. The asymptotic superlinear convergence rate of these quasi-Newton methods has been extensively studied in the literature, but their explicit finite–time local convergence rate is not fully investigated. In this paper, we provide a finite–time (non-asymptotic) convergence analysis for Broyden quasi-Newton algorithms under the assumptions that the objective function is strongly convex, its gradient is Lipschitz continuous, and its Hessian is Lipschitz continuous at the optimal solution. We show that in a local neighborhood of the optimal solution, the iterates generated by both DFP and BFGS converge to the optimal solution at a superlinear rate of (1/k)^{k/2}, where k is the number of iterations. We also prove a similar local superlinear convergence result holds for the case that the objective function is self-concordant. Numerical experiments on several datasets confirm our explicit convergence rate bounds. Our theoretical guarantee is one of the first results that provide a non-asymptotic superlinear convergence rate for quasi-Newton methods.

Large-Scale Affine Matrix Rank Minimization With a Novel Nonconvex Regularizer.

Low-rank minimization aims to recover a matrix of minimum rank subject to linear system constraint. It can be found in various data analysis and machine learning areas, such as recommender systems, video denoising, and signal processing. Nuclear norm minimization is a dominating approach to handle it. However, such a method ignores the difference among singular values of target matrix. To address this issue, nonconvex low-rank regularizers have been widely used. Unfortunately, existing methods suffer from different drawbacks, such as inefficiency and inaccuracy. To alleviate such problems, this article proposes a flexible model with a novel nonconvex regularizer. Such a model not only promotes low rankness but also can be solved much faster and more accurate. With it, the original low-rank problem can be equivalently transformed into the resulting optimization problem under the rank restricted isometry property (rank-RIP) condition. Subsequently, Nesterov's rule and inexact proximal strategies are adopted to achieve a novel algorithm highly efficient in solving this problem at a convergence rate of O(1/K) , with K being the iterate count. Besides, the asymptotic convergence rate is also analyzed rigorously by adopting the Kurdyka- ojasiewicz (KL) inequality. Furthermore, we apply the proposed optimization model to typical low-rank problems, including matrix completion, robust principal component analysis (RPCA), and tensor completion. Exhaustively empirical studies regarding data analysis tasks, i.e., synthetic data analysis, image recovery, personalized recommendation, and background subtraction, indicate that the proposed model outperforms state-of-the-art models in both accuracy and efficiency.