Variance Reduction Algorithms Research Articles

Variance reduction via simultaneous importance sampling and control variates techniques using vegas

Monte Carlo (MC) integration is an important calculational technique in the physical sciences. Practical considerations require that the calculations are performed as accurately as possible for a given set of computational resources. To improve the accuracy of MC integration, a number of useful variance reduction algorithms have been developed, including importance sampling and control variates. In this work, we demonstrate how these two methods can be applied simultaneously, thus combining their benefits. We provide a python wrapper, named CoVVVR, which implements our approach in the VEGAS program. The improvements are quantified with several benchmark examples from the literature.

Monte Carlo global variance reduction method combining source bias and weight window

The global tally problem has a wide range of applications in major research fields such as Monte Carlo simulations of pin-by-pin reactor models and time-dependent particle transport problems in multi-physics coupling calculations. Due to the uneven power distribution of the simulated system, the statistical errors of all tallies are unevenly distributed, resulting in some low global efficiency. For this kind of problem with global characteristics, it is necessary to develop global variance reduction techniques to obtain the accurate distribution of target tallies in the entire space. A large number of global variance reduction algorithms have been studied based on the consideration of flattening global tally error distribution, so as to improve global efficiency. This work focuses on the combination of two efficient global variance reduction algorithms, namely, the uniform tally density algorithm and the weight window algorithm, which belong to source bias and transport process bias, respectively. In tally, a method is proposed to adjust the weight window parameters by using the bias factor of the uniform tally density algorithm. Then, the weight window method will be used to reduce the weight fluctuation caused by the uniform tally density method. In this way, an organic combination of these two algorithms can be realized. A series of comparative tests are carried out based on the Hoogenboom-Martin pressurized water reactor benchmark, and it is verified that the hybrid global variance reduction algorithm proposed in this work is better than the single weight window algorithm or the uniform tally density algorithm. In terms of reducing the maximum error, the global efficiency of the hybrid algorithm is 2.6 times and 3 times that of the weight window algorithm and the uniform tally density algorithm, respectively. In addition, through the comparative analysis of computational asymmetry degree and computational efficiency, it is verified that the uniform tally density algorithm has better performance than the classical uniform fission site algorithm, and the performance advantages of the uniform tally density algorithm are quantitatively evaluated based on some new indicators. The results show that the hybrid global variance reduction algorithm proposed in this work can solve the global tally problem efficiently, thereby further promoting research in related fields.

An Investigation of Stochastic Variance Reduction Algorithms for Relative Difference Penalized 3D PET Image Reconstruction.

Penalised PET image reconstruction algorithms are often accelerated during early iterations with the use of subsets. However, these methods may exhibit limit cycle behaviour at later iterations due to variations between subsets. Desirable converged images can be achieved for a subclass of these algorithms via the implementation of a relaxed step size sequence, but the heuristic selection of parameters will impact the quality of the image sequence and algorithm convergence rates. In this work, we demonstrate the adaption and application of a class of stochastic variance reduction gradient algorithms for PET image reconstruction using the relative difference penalty and numerically compare convergence performance to BSREM. The two investigated algorithms are: SAGA and SVRG. These algorithms require the retention in memory of recently computed subset gradients, which are utilised in subsequent updates. We present several numerical studies based on Monte Carlo simulated data and a patient data set for fully 3D PET acquisitions. The impact of the number of subsets, different preconditioners and step size methods on the convergence of regions of interest values within the reconstructed images is explored. We observe that when using constant preconditioning, SAGA and SVRG demonstrate reduced variations in voxel values between subsequent updates and are less reliant on step size hyper-parameter selection than BSREM reconstructions. Furthermore, SAGA and SVRG can converge significantly faster to the penalised maximum likelihood solution than BSREM, particularly in low count data.

A kernel-free Laplacian quadratic surface optimal margin distribution machine with application to credit risk assessment

This paper proposes a novel binary classification approach named Laplacian quadratic surface optimal margin distribution machine (LapQSODM), for semisupervised learning. This new model exploits the geometric information embedded in unlabeled samples through a manifold regularizer to overcome the problem of insufficient labeled samples. Different from the traditional support vector machines (SVMs) based on the largest minimum margin idea and using the kernel technique to address the nonlinearity in the datasets, our proposed model optimizes the margin distribution and directly generates a quadratic surface to perform the classification. This new model not only improves the generalization performance but also avoids the difficulty of searching for an appropriate kernel function and tuning its parameters. For the regular-scale datasets, we extend the classical conjugate gradient algorithm for the proposed method and design an easy iterative method to calculate its exact step size; for the large-scale datasets, we develop an unbiased estimation for the LapQSODM gradient by utilizing one or several samples and design a stochastic gradient descent with variance reduction (SVRG) algorithm that is efficient and effective. Then, we conduct a comprehensive numerical experiment on both artificial and public benchmark datasets. The experimental results show LapQSODM has a better generalization performance than most of the well-known benchmark methods and is more robust due to the optimization of the margin distribution. Finally, we apply LapQSODM to three realistic credit risk assessment problems. The promising numerical results demonstrate the great potential and ability of our model in the real practice.

A stochastic variance reduction algorithm with Bregman distances for structured composite problems

We develop a novel stochastic primal–dual splitting method with Bregman distances for solving structured composite problems involving infimal convolutions in non-Euclidean spaces. The sublinear convergence in expectation of the primal–dual gap is proved under mild conditions on stepsize for the general case. The linear convergence rate is obtained under additional condition like the strong convexity relative to Bregman functions.

Laplacian large margin distribution machine for semi-supervised classification

Semi-Supervised Learning (SSL) has attracted much attention in the field of machine learning and data mining. As an extension of Support Vector Machine (SVM), the Semi-Supervised Support Vector Machine (S3VM) was proposed for SSL. Recent studies have disclosed that optimising the margin distribution is more crucial than maximising the minimum margin in generating a better classification. However, the existing S3VM models still follow the idea of maximising the minimum margin. Therefore, this paper proposes a novel Laplacian Large margin Distribution Machine (LapLDM) for SSL to enhance the classification performance. This method can optimise the margin distribution by maximising the first-order (margin mean) and minimising the second-order (margin variance) statistics of margins, and exploit the geometry information of marginal distribution embedded in the unlabelled data through the Laplacian regularizer. Then this paper develops a Preconditioned Conjugate Gradient (PCG) algorithm to solve the nonlinear LapLDM model on those regular-scaled data sets and a Stochastic Gradient Descent with Variance Reduction (SVRG) algorithm to solve the linear LapLDM model on those large-scaled data sets. These algorithms can accelerate the implementing efficiencies of proposed models and make them available for those large-scaled problems. Finally, the numerical results on four artificial and fourteen public benchmark data sets demonstrate that the LapLDM is superior to some well-known S3VM models.

Efficient Asynchronous Semi-Stochastic Block Coordinate Descent Methods for Large-Scale SVD

Eigenvector computation such as Singular Value Decomposition (SVD) is one of the most fundamental problems in machine learning, optimization and numerical linear algebra. In recent years, many stochastic variance reduction algorithms and randomized coordinate descent algorithms have been developed to efficiently solve the leading eigenvalue problem. By taking full advantage of both variance reduction and randomized coordinate descent techniques, this paper proposes a novel Semi-stochastic Block Coordinate Descent algorithm (SBCD-SVD), which is more suitable than existing algorithms for large-scale leading eigenvalue problems of SVD, and can obtain linear convergence. Unlike existing stochastic variance reduction and randomized coordinate descent methods, our algorithm inherits their advantages. Moreover, we propose a new Asynchronous parallel Semi-stochastic Block Coordinate Descent algorithm (ASBCD-SVD) and one new Asynchronous parallel Sparse approximated Variance Reduction algorithm (ASVR-SVD) for large-scale dense and sparse datasets, respectively. Finally, we prove that both dense and sparse asynchronous parallel variants can converge linearly. Extensive experimental results show that our algorithms attain high parallel speedup and achieve almost the same performance with significantly shorter time, and thus they can be widely used in various practice applications.

Large-Scale Phase Retrieval via Stochastic Reweighted Amplitude Flow

Phase retrieval, recovering a signal from phaseless measurements, is generally considered to be an NP-hard problem. This paper adopts an amplitude-based nonconvex optimization cost function to develop a new stochastic gradient algorithm, named stochastic reweighted phase retrieval (SRPR). SRPR is a stochastic gradient iteration algorithm, which runs in two stages: First, we use a truncated sample stochastic variance reduction algorithm to initialize the objective function. The second stage is the gradient refinement stage, which uses continuous updating of the amplitude-based stochastic weighted gradient algorithm to improve the initial estimate. Because of the stochastic method, each iteration of the two stages of SRPR involves only one equation. Therefore, SRPR is simple, scalable, and fast. Compared with the state-of-the-art phase retrieval algorithm, simulation results show that SRPR has a faster convergence speed and fewer magnitude-only measurements required to reconstruct the signal, under the real- or complex- cases.

Improved adaptive variance reduction algorithm based on RMC code for deep penetration problems

An adaptive variance reduction algorithm based on RMC code was developed for deep penetration problems. The adaptive variance reduction algorithm uses the conservation of penetration rate and a predictor–corrector algorithm to quickly determine exponential importance parameters or equal-gradient importance parameters with fast iterative convergence. The adaptive variance reduction algorithm greatly accelerates one-dimensional deep penetration problems. However, the previous algorithm does not have capacity of energy bias nor does it support calculations on three-dimensional models. The adaptive variance reduction algorithm is improved to allow for energy bias and for three-dimensional models. The improved algorithm was applied to the HBR2 benchmark with the average Figure of Merit (FOM) of RMC code improved 243 fold. Therefore, this improved adaptive variance reduction algorithm can efficiently deal with complex engineering shielding problems.

A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec

The purely numerical evaluation of multi-loop integrals and amplitudes can be a viable alternative to analytic approaches, in particular in the presence of several mass scales, provided sufficient accuracy can be achieved in an acceptable amount of time. For many multi-loop integrals, the fraction of time required to perform the numerical integration is significant and it is therefore beneficial to have efficient and well-implemented numerical integration methods. With this goal in mind, we present a new stand-alone integrator based on the use of (quasi-Monte Carlo) rank-1 shifted lattice rules. For integrals with high variance we also implement a variance reduction algorithm based on fitting a smooth function to the inverse cumulative distribution function of the integrand dimension-by-dimension. Additionally, the new integrator is interfaced to pySecDec to allow the straightforward evaluation of multi-loop integrals and dimensionally regulated parameter integrals. In order to make use of recent advances in parallel computing hardware, our integrator can be used both on CPUs and CUDA compatible GPUs where available. Program summaryProgram Title: pySecDec, qmcProgram Files doi:http://dx.doi.org/10.17632/dnrkf5jxzh.2Licensing provisions: GNU General Public License v3Programming language: python, FORM, C++, CUDAExternal routines/libraries: catch [1], gsl [2], numpy [3], sympy [4], Nauty [5], Cuba [6], FORM [7], Normaliz [8]. The program can also be used in a mode which does not require Normaliz.Journal reference of previous version: Comput. Phys. Commun. 222 (2018) 313–326.Does the new version supersede the previous version?: YesNature of problem: Extraction of ultraviolet and infrared singularities from parametric integrals appearing in higher order perturbative calculations in quantum field theory. Numerical integration in the presence of integrable singularities (e.g. kinematic thresholds).Solution method: Algebraic extraction of singularities within dimensional regularization using iterated sector decomposition. This leads to a Laurent series in the dimensional regularization parameter ϵ (and optionally other regulators), where the coefficients are finite integrals over the unit-hypercube. Those integrals are evaluated numerically by Monte Carlo integration. The integrable singularities are handled by choosing a suitable integration contour in the complex plane, in an automated way. The parameter integrals forming the coefficients of the Laurent series in the regulator(s) are provided in the form of libraries which can be linked to the calculation of (multi-) loop amplitudes.Restrictions: Depending on the complexity of the problem, limited by memory and CPU/GPU time.

An adaptive variance reduction algorithm based on RMC code for solving deep penetration problems

The deep penetration problems will be encountered when the Monte Carlo code is used to perform shielding simulation. By analyzing the transport process of particles in the shielding layers, the conservation of the penetration rate is proved, and an adaptive variance reduction method guided by the conservation of penetration rate is developed. This method can realize the exponential or equal-gradient distribution of the importance functions, and quickly achieve the iterative convergence. This new algorithm obtains good results in deep penetration problems, such as concrete and water shielding layers which are commonly used in engineering design. At the same time, this new algorithm was implemented by RMC code, and it can fully improve the Figure of Merit of RMC code for solving the deep penetration problems.

Riemannian Stochastic Variance Reduced Gradient Algorithm with Retraction and Vector Transport

In recent years, stochastic variance reduction algorithms have attracted considerable attention for minimizing the average of a large but finite number of loss functions. This paper proposes a novel Riemannian extension of the Euclidean stochastic variance reduced gradient (R-SVRG) algorithm to a manifold search space. The key challenges of averaging, adding, and subtracting multiple gradients are addressed with retraction and vector transport. For the proposed algorithm, we present a global convergence analysis with a decaying step size as well as a local convergence rate analysis with a fixed step size under some natural assumptions. In addition, the proposed algorithm is applied to the computation problem of the Riemannian centroid on the symmetric positive definite (SPD) manifold as well as the principal component analysis and low-rank matrix completion problems on the Grassmann manifold. The results show that the proposed algorithm outperforms the standard Riemannian stochastic gradient descent algorithm in each case.

Stochastic Conjugate Gradient Algorithm With Variance Reduction.

Conjugate gradient (CG) methods are a class of important methods for solving linear equations and nonlinear optimization problems. In this paper, we propose a new stochastic CG algorithm with variance reduction1 and we prove its linear convergence with the Fletcher and Reeves method for strongly convex and smooth functions. We experimentally demonstrate that the CG with variance reduction algorithm converges faster than its counterparts for four learning models, which may be convex, nonconvex or nonsmooth. In addition, its area under the curve performance on six large-scale data sets is comparable to that of the LIBLINEAR solver for the L2 -regularized L2 -loss but with a significant improvement in computational efficiency.1CGVR algorithm is available on github: https://github.com/xbjin/cgvr.

On the Efficient Simulation of Outage Probability in a Log-Normal Fading Environment

The outage probability (OP) of the signal-to-interference-plus-noise ratio (SINR) is an important metric that is used to evaluate the performance of wireless systems. One difficulty toward assessing the OP is that, in realistic scenarios, closed-form expressions cannot be derived. This is, for instance, the case of the Log-normal environment, in which evaluating the OP of the SINR amounts to computing the probability that a sum of correlated Log-normal variates exceeds a given threshold. Since such a probability does not admit a closed-form expression, it has thus far been evaluated by several approximation techniques, the accuracies of which are not guaranteed in the region of small OPs. For these regions, simulation techniques based on variance reduction algorithms are a good alternative, being quick and highly accurate for estimating rare event probabilities. This constitutes the major motivation behind this paper. More specifically, we propose a generalized hybrid importance sampling scheme, based on a combination of a mean shifting and a covariance matrix scaling, to evaluate the OP of the SINR in a Log-normal environment. We further our analysis by providing a detailed study of two particular cases. Finally, the performance of these techniques is performed both theoretically and through various simulation results.

On the generalization of the hazard rate twisting-based simulation approach

Estimating the probability that a sum of random variables (RVs) exceeds a given threshold is a well-known challenging problem. A naive Monte Carlo simulation is the standard technique for the estimation of this type of probability. However, this approach is computationally expensive, especially when dealing with rare events. An alternative approach is represented by the use of variance reduction techniques, known for their efficiency in requiring less computations for achieving the same accuracy requirement. Most of these methods have thus far been proposed to deal with specific settings under which the RVs belong to particular classes of distributions. In this paper, we propose a generalization of the well-known hazard rate twisting Importance Sampling-based approach that presents the advantage of being logarithmic efficient for arbitrary sums of RVs. The wide scope of applicability of the proposed method is mainly due to our particular way of selecting the twisting parameter. It is worth observing that this interesting feature is rarely satisfied by variance reduction algorithms whose performances were only proven under some restrictive assumptions. It comes along with a good efficiency, illustrated by some selected simulation results comparing the performance of the proposed method with some existing techniques.

Fast variance reduction for steady-state simulation and sensitivity analysis of stochastic chemical systems using shadow function estimators.

We address the problem of estimating steady-state quantities associated to systems of stochastic chemical kinetics. In most cases of interest, these systems are analytically intractable, and one has to resort to computational methods to estimate stationary values of cost functions. In this work, we introduce a novel variance reduction algorithm for stochastic chemical kinetics, inspired by related methods in queueing theory, in particular the use of shadow functions. Using two numerical examples, we demonstrate the efficiency of the method for the calculation of steady-state parametric sensitivities and evaluate its performance in comparison to other estimation methods.

Bandpass Second-Degree Digital-Integrator-Based Power System Frequency Estimation Under Nonsinusoidal Conditions

A novel digital signal processing algorithm for online estimation of the fundamental frequency of the distorted power system signals is presented. The basic algorithm relies on the development of an efficient variance reduction algorithm; and design of a new stable bandpass infinite impulse response (IIR) second-degree digital integrator (SDDI) with reduced approximation error. Compared with the well-established technique such as the enhanced-phase-locked-loop (EPLL) system, the proposed algorithm provides the following: 1) higher degree of immunity and insensitivity to harmonics and noise and 2) faster response during step frequency change. Structural simplicity, wide range of application, controls over speed and accuracy, and parameter robustness are other salient features of the method. The only limitation as compared with the EPLL system is its slower transient response during step change in signal magnitude. Based on simulation studies, performances of the proposed algorithm at different operating conditions have been presented, and its accuracy and response time have been compared with the EPLL systems.

Second-degree digital differentiator-based power system frequency estimation under non-sinusoidal conditions

This study presents development and implementation of a novel digital signal processing algorithm for on-line estimation of the fundamental frequency of non-sinusoidal power system signals. The basic algorithm relies on the development of a new variance reduction algorithm and design of an optimised infinite impulse response (IIR) second-degree digital differentiator (SDDD). The design of SDDD consists of obtaining a second-degree integrator from Tick integrator, and then modifying and optimising its transfer function appropriately to attain a stable second-degree differentiator. Compared with the well-established technique such as the enhanced phase-locked-loop (EPLL) system, the proposed algorithm provides faster transient response, higher degree of immunity and insensitivity to harmonics and noise. Structural simplicity, wide range of application and robustness against sampling frequency variation are other salient features of the method. The only limitation as compared with the EPLL system is its slight reduced accuracy (around 3 mHz) under static sinusoidal conditions. Based on simulation studies, performances of the proposed algorithm at different operating conditions have been presented and its accuracy and response time have been compared with the EPLL systems.

QMC techniques for CAT bond pricing *

Pricing of catastrophe bonds leads to integrals with discontinuous and formally infinite-dimensional integrands. We investigate the suitability of Quasi-Monte Carlo methods for the numerical evaluation of these integrals and develop several variance-reduction algorithms. Furthermore, the performance of Quasi-Monte Carlo sequences for asymptotically efficient rare event simulation is examined. Various numerical illustrations are given.

GEANT4 code for simulation of a germanium gamma-ray detector and its application to efficiency calibration

The GEANT4 software was developed by RD44, a world-wide collaboration of national institutes, laboratories and large High-Energy Physics experiments. GEANT4 is a public software package composed of tools which can be used to accurately simulate the passage of particles through matter. In this article, the first attempt to use GEANT4 to model a reverse electrode germanium detector (REGe), and to improve also its efficiency calibration procedure, is presented. A variance reduction algorithm based on a directional bias scheme is implemented into GEANT4 in order to accelerate the efficiency computations. A fast optimisation method to model the detector geometry using standard point sources is also presented and validated for point, Marinelli and air filter sources. The simulated full-energy peak efficiencies agreed with the measured values to within 1% between 36 and 1460 keV for these three counting geometries.