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MEANS IN COMPLETE MANIFOLDS: UNIQUENESS AND APPROXIMATION

ESAIM-PROBABILITY AND STATISTICS 2014年第1期18卷 185-206页

作者： Arnaudon, Marc Miclo, Laurent Univ Poitiers CNRS UMR 7348 Lab Math & Applicat F-86962 Futuroscope France CNRS UMR 5219 Inst Math Toulouse F-31062 Toulouse 9 France

Let M be a complete Riemannian manifold, N is an element of N and p >= 1. We prove that almost everywhere on x = (x(1), ..., x(N)) is an element of M-N for Lebesgue measure in M-N, the measure mu(x) = 1/N Sigma(N)(k= 1) delta(xk) has a unique p-mean e(p)(x). As a consequence, if X = (X-1, ..., X-N) is a M-N-valued random variable with absolutely continuous law, then almost surely mu(X(omega)) has a unique p-mean. In particular if (X-n)(n >= 1) is an independent sample of an absolutely continuous law in M, then the process e(p, n)(omega) = e(p)(X-1(omega), ..., X-n(omega)) is well-defined. Assume M is compact and consider a probability measure nu in M. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to nu. When the set is a singleton, it converges to the p-mean.

关键词： stochastic algorithms diffusion processes simulated annealing homogenization probability measures on compact Riemannian manifolds intrinsic p-means instantaneous invariant measures Gibbs measures spectral gap at small temperature

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On the convergence of inhomogeneous Markov chains approximating equilibrium placements of flexible objects

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2001年第2期19卷 179-208页

作者： Albrecht, A Cheung, SK Leung, KS Wong, CK Chinese Univ Hong Kong Dept Comp Sci & Engn Sha Tin Hong Kong Peoples R China

This paper deals with the minimization of local forces in two-dimensional placements of flexible objects within rigid boundaries. The objects are disks of the same size but, in general, of different materials. Potential applications include the design of new amorphous polymeric and related granular materials as well as the design of package cushioning systems. The problem is considered on a grid structure with a fixed step size w and for a fixed diameter of the discs, i.e., the number of placed disks may increase as the size of the placement region increases. The near-equilibrium configurations have to be calculated from uniformly distributed random initial placements. The final arrangements of disks must ensure that any particular object is deformed only within the limits of elasticity of the material. The main result concerns epsilon -approximations of the probability distribution on the set of equilibrium placements. Under a natural assumption about the configuration space, we prove that a run-time of n(gamma) + log(O(1)) (1/epsilon) is sufficient to approach with probability 1 - epsilon the minimum value of the objective function, where gamma depends on the maximum Gamma of the escape depth of local minima within the underlying energy landscape. The result is derived from a careful analysis of the interaction among probabilities assigned to configurations from adjacent distance levels to minimum placements. The overall approach for estimating the convergence rate is relatively independent of the particular placement problem and can be applied to various optimization problems with similar properties of the associated landscape of the objective function.

关键词： Markov chains placement problems flexible objects stochastic algorithms simulated annealing composite materials

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NECESSARY AND SUFFICIENT CONDITIONS FOR STABILITY OF A BIN-PACKING SYSTEM

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JOURNAL OF APPLIED PROBABILITY 1986年第4期23卷 989-999页

作者： COURCOUBETIS, C WEBER, RR UNIV CAMBRIDGE CAMBRIDGE ENGLAND

Objects of various integer sizes, o1, · ··, on, are to be packed together into bins of size N as they arrive at a service facility. The number of objects of size oi which arrive by time t is , where the components of are independent renewal processes, with At/t → λ as t → ∞. The empty space in those bins which are neither empty nor full at time t is called the wasted space and the system is declared stabilizable if for some finite B there exists a bin-packing algorithm whose use guarantees the expected wasted space is less than B for all t. We show that the system is stabilizable if the arrival processes are Poisson and λ lies in the interior of a certain convex polyhedral cone Λ. In this case there exists a bin-packing algorithm which stabilizes the system without needing to know λ. However, if λ lies on the boundary of Λ the wasted space grows as and if λ is exterior to Λ it grows as O(t); these conclusions hold even if objects may be repacked as often as desired.

关键词： ONLINE algorithms RANDOM WALKS stochastic algorithms

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Nonsmooth Optimization over the Stiefel Manifold and Beyond: Proximal Gradient Method and Recent Variants

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SIAM REVIEW 2024年第2期66卷 319-352页

作者： Chen, Shixiang Ma, Shiqian So, Anthony Man-Cho Zhang, Tong Univ Sci & Technol China Sch Math Sci Hefei Anhui Peoples R China Rice Univ Dept Computat Appl Math & Operat Res Houston TX 77005 USA Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China Hong Kong Univ Sci & Technol Clear Water Bay Hong Kong Peoples R China

We consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this class of problems converge slowly in practice, involve subproblems that can be as difficult as the original problem, or lack rigorous convergence guarantees. In this paper, we propose a manifold proximal gradient method (ManPG) for solving this class of problems. We prove that the proposed method converges globally to a stationary point and establish its iteration complexity for obtaining an \epsilon -stationary point. Furthermore, we present numerical results on the sparse PCA and compressed modes problems to demonstrate the advantages of the proposed method. We also discuss some recent advances related to ManPG for Riemannian optimization with nonsmooth objective functions.

关键词： manifold optimization Stiefel manifold nonsmooth proximal gradient method iteration complexity semismooth Newton method stochastic algorithms zeroth-order algorithms

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Simulated annealing without a cooling schedule

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 1998年第1-2期66卷 9-20页

作者： Muller, D Univ Hannover Inst Informat Hannover Germany

An optimization algorithm is presented which may be considered as a variant of Simulated Annealing with an ensemble of K interacting Systems. This algorithm avoids the explicit specification of a temperature parameter and a cooling schedule. The Boltzmann distribution of energies or cost values is accomplished by enforcing an approximately constant total energy of the ensemble during a certain sequence of steps. The implicit temperature is lowered by a cooling step which is controlled solely by the state of the ensemble as a whole. To compare this algorithm with classical Simulated Annealing a problem with known optimal solution is studied with both algorithms. In particular the success probability P and the number of function evaluations required for reaching the optimal solution are compared.

关键词： optimization simulated annealing parallelization stochastic algorithms

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Accelerated First-Order Optimization algorithms for Machine Learning

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PROCEEDINGS OF THE IEEE 2020年第11期108卷 2067-2082页

作者： Li, Huan Fang, Cong Lin, Zhouchen Nankai Univ Inst Robot & Automat Informat Syst Coll Artificial Intelligence Tianjin 300071 Peoples R China Nanjing Univ Aeronaut & Astronaut Coll Comp Sci & Technol Nanjing 210016 Peoples R China Princeton Univ Dept Elect Engn Princeton NJ 08544 USA Peking Univ Sch Elect Engn & Comp Sci EECS Key Lab Machine Percept MOE Beijing 100871 Peoples R China

Numerical optimization serves as one of the pillars of machine learning. To meet the demands of big data applications, lots of efforts have been put on designing theoretically and practically fast algorithms. This article provides a comprehensive survey on accelerated first-order algorithms with a focus on stochastic algorithms. Specifically, this article starts with reviewing the basic accelerated algorithms on deterministic convex optimization, then concentrates on their extensions to stochastic convex optimization, and at last introduces some recent developments on acceleration for nonconvex optimization.

关键词： Numerical analysis Machine learning algorithms Optimization methods Machine learning Convergence Complexity theory Approximation algorithms Convex functions Acceleration convex optimization deterministic algorithms machine learning nonconvex optimization stochastic algorithms

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Density estimation and random variate generation using multilayer networks

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IEEE TRANSACTIONS ON NEURAL NETWORKS 2002年第3期13卷 497-520页

作者： Magdon-Ismail, M Atiya, A Rensselaer Polytech Inst Dept Comp Sci Troy NY 12180 USA Countrywide Capital Markets Calbasas CA 91302 USA

In this paper we consider two important topics: density estimation and random variate generation. We will present a framework that is easily implemented using the familiar multilayer neural network. First, we develop two new methods for density estimation, a stochastic method and a related deterministic method. Both methods are based on approximating the distribution function, the density being obtained by differentiation. In the second part of the paper, we develop new random number generation methods. Our methods do not suffer from some of the restrictions of existing methods in that they can be used to generate numbers from any density provided that certain smoothness conditions are satisfied. One of our methods is based on an observed inverse relationship between the density estimation process and random number generation. We present two variants of this method, a stochastic, and a deterministic version. We propose a second method that is based on a novel control formulation of the problem, where a "controller network" is trained to shape a given density into the desired density. We justify the use of all the methods that we propose by providing theoretical convergence results. In particular, we prove that the L-infinity convergence to the true density for both the density estimation and random variate generation techniques occurs at a rate 0 ((log log N/N)((1-epsilon))(/2)) where N is the number of data points and e can be made arbitrarily small for sufficiently smooth target densities. This bound is very close to the optimally achievable convergence rate under similar smoothness conditions. Also, for comparison, the L-2 root mean square (rms) convergence rate of a positive kernel density estimator is 0 (N-2/5) when the optimal kernel width is used. We present numerical simulations to illustrate the performance of the proposed density estimation and random variate generation methods. In addition, we present an extended introduction and bibliography that serves

关键词： convergence rate density estimation distribution function estimation error multilayer network neural network random number generation stochastic algorithms

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Asymptotic distribution theory for Hoare's selection algorithm

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ADVANCES IN APPLIED PROBABILITY 1996年第1期28卷 252-269页

作者： Grubel, R Rosler, U CHRISTIAN ALBRECHTS UNIV KIEL MATH SEMINARD-24098 KIELGERMANY

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ..., n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

关键词： stochastic algorithms asymptotic distribution order statistics random binary numbers random walk in a random environment

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On Error-Saturation Nonlinearities in NLMS Adaptation

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2009年第10期57卷 4105-4111页

作者： Bershad, Neil J. Univ Calif Irvine Henry Samueli Sch Engn Irvine CA 92660 USA

The effect of a saturation-type error nonlinearity in the weight update equation in normalized least mean-square (NLMS) adaptation is investigated for system identification for a white Gaussian data model. Nonlinear recursions are derived for the weight mean error and mean-square deviation (MSD) that include the effect of an error function (erf) saturation-type nonlinearity on the error sequence driving the algorithm. The nonlinear recursion for the MSD is solved numerically and shown in excellent agreement with Monte Carlo simulations, supporting the theoretical model assumptions. The theory is extended to tracking a Markov channel and accurately predicts the tracking behavior as well. The saturation behavior of the algorithm is easily studied by varying a single parameter in the error function, varying from a linear device to a hard limiter. For the white data case, the excess mean square-error (EMSE) is simply related to the MSD. The tradeoff between the extent of error saturation, steady-state EMSE, and algorithm convergence rate is studied using these results.

关键词： Adaptive filters analysis NLMS nonlinear systems stochastic algorithms

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Optimal shape design of a class of permanent magnet motors in a multiple-objectives context

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COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING 2022年第6期41卷 1994-2009页

作者： Di Barba, Paolo Mognaschi, Maria Evelina Petkovska, Lidija Cvetkovski, Goga Vladimir Univ Pavia Dept Elect Comp & Biomed Engn Pavia Italy Ss Cyril & Methodius Univ Fac Elect Engn & Informat Technol Skopje North Macedonia

Purpose This paper aims to deal with the optimal shape design of a class of permanent magnet motors by minimizing multiple objectives according to an original interpretation of Pareto optimality. The proposed method solves a many-objective problems characterized by five objective functions and five design variables with evolution strategy algorithms, classically used for single- and multi-objective (two objective functions) optimization problems. Design/methodology/approach Two approaches are proposed in the paper: the All-Objectives (AO) and the Many-Objectives (MO) optimization approach. The former is based on a single-objective optimization of a preference function, i.e. a normalized weighted sum. In contrast, in the MO a multi-objective optimization algorithm is applied to the minimization of a weight-free preference function and simultaneously to a maximization of the distance of the current solution from the prototype. The optimizations are based on an equivalent circuit model of the Permanent Magnet (PM) motor, but the results are assessed by means of finite element analyses (FEAs). Findings An extensive study of the solutions obtained by means of the different optimization approaches is provided by means of post-processing analyses. Both the approaches find non-dominated solutions with respect to the prototype that are substantially improving the initial solution. The points of strength along with the weakness points of each solution with respect to the prototype are analysed in depth. Practical implications The paper gives a good guide to the designers of electric motors, focussed on a shape design optimization. Originality/value Considering simultaneously five objective functions in an automated optimal design procedure is challenging. The proposed approach, based on a well-known and established optimization algorithm, but exploiting a new concept of degree of conflict, can lead to new results in the field of automated optimal design in a many-objective co

关键词： Shape design Permanent magnet motor stochastic algorithms All-objective optimization Many-objective optimization Permanent magnet machine Multiobjective optimization Shape optimization Design optimization methodology

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