Variance Reduction Techniques Research Articles

Empirical Sample Complexity of Neural Network Mixed State Reconstruction

Quantum state reconstruction using Neural Quantum States has been proposed as a viable tool to reduce quantum shot complexity in practical applications, and its advantage over competing techniques has been shown in numerical experiments focusing mainly on the noiseless case. In this work, we numerically investigate the performance of different quantum state reconstruction techniques for mixed states: the finite-temperature Ising model. We show how to systematically reduce the quantum resource requirement of the algorithms by applying variance reduction techniques. Then, we compare the two leading neural quantum state encodings of the state, namely, the Neural Density Operator and the positive operator-valued measurement representation, and illustrate their different performance as the mixedness of the target state varies. We find that certain encodings are more efficient in different regimes of mixedness and point out the need for designing more efficient encodings in terms of both classical and quantum resources.

A simulation-based genetic algorithm for a semi-automated warehouse scheduling problem with processing time variability

For warehouse operations, efficiently scheduling the available resources is crucial to improve the productivity and customer satisfaction. This paper proposes a simulation-based evolutionary algorithm for order scheduling and multi-robot task assignment in a robotic mobile fulfillment system. The algorithm proactively deals with the effects of the processing time variability by evaluating schedules based on both its system performance as well as its robustness under uncertain conditions. The algorithm implements an efficient resource allocation method and a variance reduction technique to reduce the overall computational burden. The experimental results show that the techniques to reduce the computational time are effective and can significantly reduce the amount of simulations required for the fitness evaluation. If a candidate schedule is allocated insufficient simulation replications it can lead to an inaccurate estimate of its long-term average performance. This could lead to an average performance loss of 7.3 %. Furthermore, the proactive scheduler is able to generate schedules that are more robust compared to deterministically generated. A reduction in the average operational cost of about 5 % can be reached, compared to a deterministically generated schedule. The paper reveals the relevance of identifying and modeling uncertainty when designing schedules in an operational system, rather than looking for optimal schedules for ideal scenarios.

Reduced Variance Random Batch Methods for Nonlocal PDEs

Random Batch Methods (RBM) for mean-field interacting particle systems enable the reduction of the quadratic computational cost associated with particle interactions to a near-linear cost. The essence of these algorithms lies in the random partitioning of the particle ensemble into smaller batches at each time step. The interaction of each particle within these batches is then evolved until the subsequent time step. This approach effectively decreases the computational cost by an order of magnitude while increasing the amount of fluctuations due to the random partitioning. In this work, we propose a variance reduction technique for RBM applied to nonlocal PDEs of Fokker-Planck type based on a control variate strategy. The core idea is to construct a surrogate model that can be computed on the full set of particles at a linear cost while maintaining enough correlations with the original particle dynamics. Examples from models of collective behavior in opinion spreading and swarming dynamics demonstrate the great potential of the present approach.

Single-Loop Sampling for Estimating Failure-Probability Upper-Bound Function

Under random-interval mixed uncertainties of structures, failure-probability upper-bound function (FPUBF), which varies with the distribution parameters of random inputs, can not only provide the influence of distribution parameters on the failure-probability upper bound (FPUB), but also contribute to decoupling a reliability-based design optimization model. Although FPUBF can be estimated by repeatedly evaluating FPUBs at different distribution parameter realizations, it suffers from unaffordable computational cost resulting from this double-loop framework. To address this issue, this paper proposes a single-loop sampling strategy (SL) to estimate FPUBF at arbitrary realizations in the interested distribution parameter region. Instead of the huge computational cost of a double-loop framework, the SL estimates the entire FPUBF only by one simulation analysis. Moreover, importance sampling (IS) variance reduction technique is introduced, and a single-loop IS probability density function (PDF), or SL-IS-PDF, is constructed to more efficiently estimate FPUBF by reducing the required size of the candidate sample pool. For approximating the optimal SL-IS-PDF and identifying the states of candidate samples efficiently, the double-loop adaptive Kriging model of performance function is introduced to further reduce the number of performance function evaluations. A numerical example and two composite structure examples are employed to verify the accuracy, efficiency, and feasibility of the proposed methods.

Verification of shielding calculation capability of cosRMC with SINBAD fusion benchmarks

In the design of fusion reactors, extensive shielding calculations are necessary to verify the feasibility and safety of the design. The issue of deep penetration is a common and significant problem in fusion reactor shielding calculations. When the thickness of the shielding layer exceeds a certain value, the statistical error in Monte Carlo method calculations significantly increases. Therefore, it is imperative to adopt variance reduction techniques to enhance the computational accuracy for deep penetration problems. cosRMC is a reactor Monte Carlo code, and its accuracy in calculating neutron-deep-penetration shielding problems should be verified. For this purpose, three deep penetration benchmarks from Shielding Integral Benchmark Archive Database (SINBAD) were simulated by cosRMC and neutron cross section data library ENDF/B-VII.0. These benchmarks describe the penetration behaviors of neutrons in models with different geometric shapes and materials. This paper shows detailed cosRMC models based on benchmark experimental models, used to calculate neutron flux spectra at different penetration depths, which are recorded by detectors in the related benchmark experiments. In this way, a direct comparison between cosRMC results and experimental measurements is provided. In order to improve the accuracy of cosRMC in calculating deep penetration problems, the variance reduction technique of cell importance was used. The neutron flux spectrum calculated by cosRMC was compared with the neutron flux spectrum results calculated by MCNP5 and the experimental values of the benchmarks. The comparison results show that the calculations of cosRMC are in compliance with the MCNP5 reference values and experimental values, and the relative deviations between calculations of cosRMC and MCNP5 is within three time of standard errors. Therefore, this paper has verified that cosRMC can applies to neutron deep penetration problems of fusion reactors.

Construction of GVR weight windows maps from very low density transport simulations

Fusion-related facilities present relevant neutron radiation fields even after penetrating through a considerable thickness of shielding material. Neutronic analyses performed via Monte Carlo codes, then, need Global Variance Reduction (GVR) techniques so that low statistical uncertainty is reached efficiently throughout the geometry.Mesh-based Weight Windows is a flexible methodology used extensively for variance reduction purposes, both for Local and Global Variance Reduction. Purely stochastic GVR methodologies based on Weight Windows usually construct weight maps so that they are proportional to the forward particle flux, which is unknown a priori. Therefore, an iterative cycle is established. In each iteration, a weight map is obtained from the forward flux that allows the next iteration to reach further into the geometry, until all of it is populated.However, this iterative cycle may take a considerable amount of computer time, as many iterations are needed to fully populate the geometry. An alternative to achieve relevant penetration in a single iteration is to perform calculations at very low densities. However, a reconstruction method is needed to estimate the flux at the real density. This work studies a scheme to reconstruct the fluxes from low density calculations and compares it to already existing techniques.

Neural network-based source biasing to speed-up challenging MCNP simulations

Nuclear analysis of fusion facilities, especially in the context of ITER, is complex due to the need for precise modeling of complex geometry and radiation sources. Monte Carlo (MC) codes, such as MCNP, are used in this context due to their high precision and capability to deal with these cases. To speed up calculations, variance reduction (VR) techniques are crucial for carrying out the simulations in reasonable times. Particularly, the cases where only a small source phase space contributes significantly to the tally must be optimized by an exhaustive sampling of this phase space. This paper introduces a novel VR method for optimizing the calculation. This method obtains the histories that contribute most to the tally and uses this information for a neural network (NN) to sample the source´s phase space. Here we present a preliminary implementation of the method to study its viability in complex cases. The method's efficacy is demonstrated in a representative source-geometry configuration, significantly reducing computational time for the tally convergence. The promising results suggest applicability to more intricate ITER-like scenarios, prompting further development.

DISCount: Counting in Large Image Collections with Detector-Based Importance Sampling

Many applications use computer vision to detect and count objects in massive image collections. However, automated methods may fail to deliver accurate counts, especially when the task is very difficult or requires a fast response time. For example, during disaster response, aid organizations aim to quickly count damaged buildings in satellite images to plan relief missions, but pre-trained building and damage detectors often perform poorly due to domain shifts. In such cases, there is a need for human-in-the-loop approaches to accurately count with minimal human effort. We propose DISCount -- a detector-based importance sampling framework for counting in large image collections. DISCount uses an imperfect detector and human screening to estimate low-variance unbiased counts. We propose techniques for counting over multiple spatial or temporal regions using a small amount of screening and estimate confidence intervals. This enables end-users to stop screening when estimates are sufficiently accurate, which is often the goal in real-world applications. We demonstrate our method with two applications: counting birds in radar imagery to understand responses to climate change, and counting damaged buildings in satellite imagery for damage assessment in regions struck by a natural disaster. On the technical side we develop variance reduction techniques based on control variates and prove the (conditional) unbiasedness of the estimators. DISCount leads to a 9-12x reduction in the labeling costs to obtain the same error rates compared to naive screening for tasks we consider, and surpasses alternative covariate-based screening approaches.

UNEX-RL: Reinforcing Long-Term Rewards in Multi-Stage Recommender Systems with UNidirectional EXecution

In recent years, there has been a growing interest in utilizing reinforcement learning (RL) to optimize long-term rewards in recommender systems. Since industrial recommender systems are typically designed as multi-stage systems, RL methods with a single agent face challenges when optimizing multiple stages simultaneously. The reason is that different stages have different observation spaces, and thus cannot be modeled by a single agent. To address this issue, we propose a novel UNidirectional-EXecution-based multi-agent Reinforcement Learning (UNEX-RL) framework to reinforce the long-term rewards in multi-stage recommender systems. We show that the unidirectional execution is a key feature of multi-stage recommender systems, bringing new challenges to the applications of multi-agent reinforcement learning (MARL), namely the observation dependency and the cascading effect. To tackle these challenges, we provide a cascading information chain (CIC) method to separate the independent observations from action-dependent observations and use CIC to train UNEX-RL effectively. We also discuss practical variance reduction techniques for UNEX-RL. Finally, we show the effectiveness of UNEX-RL on both public datasets and an online recommender system with over 100 million users. Specifically, UNEX-RL reveals a 0.558% increase in users' usage time compared with single-agent RL algorithms in online A/B experiments, highlighting the effectiveness of UNEX-RL in industrial recommender systems.

Meta model-based and cross entropy-based importance sampling algorithms for efficiently solving system failure probability function

The multi-mode system failure probability function (SFPF) can quantify how the distribution parameters of the random input vector affect the system safety and decouple the system reliability-based design optimization model. However, for a problem with a time-consuming implicit performance function and rare failure domain, efficiently solving the SFPF remains significantly challenging. Therefore, in this study, two efficient algorithms are proposed, namely, the meta model-based importance sampling and cross entropy-based importance sampling. The contributions of this study are twofold. The first is constructing a single-loop optimal importance sampling density (SL-OISD) method to decouple the double-loop framework for analyzing the SFPF. The second is establishing two methods to efficiently approximate the SL-OISD and complete the SFPF estimation. The first method is based on the meta model of the system performance function, which is abbreviated as SL-Meta-IS. The second method is based on minimizing the cross entropy between the Gaussian mixture density model and SL-OISD, which is abbreviated as SL-CE-IS. To reduce the number of evaluating the system performance function when approximating the SL-OISD, sampling the SL-OISD, and identifying the state of the samples for completing the SFPF estimation, an adaptive Kriging model of the system performance function is introduced into SL-Meta-IS and SL-CE-IS. Owing to decoupling the double-loop framework into a single-loop framework, replacing the time-consuming system performance function with the economic Kriging model, and employing importance sampling variance reduction techniques to address issues related to the rare failure domain, the proposed SL-Meta-IS and SL-CE-IS methods greatly enhance the efficiency of SFPF estimations. The numerical and practical examples demonstrate that the two proposed methods are superior to the existing algorithms; moreover, the efficiency of SL-CE-IS is higher than that of SL-Meta-IS.

Development of multilevel mesh adaptivity for global variance reduction in Monte Carlo code cosRMC

The mesh-based weight window method is a widely used global variance reduction technique for Monte Carlo particle transport calculation. In order to meet the fine voxel division required for high-quality weight window distribution, the uniform structured mesh division will lead to large memory footprint pressure. This work focused on the research of the multilevel adaptive mesh algorithm, using a recursive algorithm to adjust the voxel resolution adaptively according to the material of geometry. Then the adaptive mesh algorithm was applied to the weight window generator based on the response matrix method by establishing the neighbor relationship of adaptive mesh. And an optimization method for memory allocation of the response matrix solver was proposed, effectively reducing the memory footprint of the response matrix solver in parallel computing. In the calculation of the CFETR model, the calculation efficiency under the weight window based on adaptive mesh was increased by 53.3 % compared with the weight window based on uniform mesh, and the proportion of voxels with a statistical error less than 10 % reached 93.33 %, which verified the effectiveness of the multilevel adaptive mesh algorithm.

The Effect of Branchless Collisions and Population Control on Correlations in Monte Carlo Power Iteration

The investigation of correlations in Monte Carlo power iteration has long been dominated by the question of generational correlations and their effects on the estimation of statistical uncertainties. More recently, there has been a growing interest in spatial correlations, prompted by the discovery of neutron clustering. Despite several attempts, a comprehensive framework concerning how Monte Carlo sampling strategies, population control, and variance reduction methods affect the strength of such correlations is still lacking. In this work, we propose a set of global and local (i.e., space-dependent) tallies that can be used to characterize the impact of correlations. These tallies encompass Shannon entropy, pair distance, normalized variance, and Feynman moment. In order to have a clean yet fully meaningful setting, we carry out our analysis in a few homogeneous and heterogeneous benchmark problems of varying dominance ratio. Several classes of collision sampling strategies, population control, and variance reduction techniques are tested, and their relative advantages and drawbacks are assessed with respect to the proposed tallies. The major finding of our study is that branchless collisions, which suppress the emergence of branches in neutron histories, also considerably reduce the effects of correlations in most of the explored configurations.

CPU-GPU-coupled acceleration method for point flux calculation in Monte Carlo particle transport.

In Monte Carlo particle transport simulations, point flux tallying is a variance reduction technique that performs well with small detectors and finds broad application in source-detector problems and local point dose calculations. However, its use in large-scale point flux tallying computation adds substantial computational time. To address this issue, we propose a CPU-GPU-coupled acceleration method, which separates the complex logic and computationally intensive parts of particle transport calculation and assigns them to the CPU and GPU, respectively. This proposed method greatly enhances the efficiency of large-scale point flux tallies, providing significant convenience for subsequent dose calculations and other related steps. We validated our method by comparing the performance of a pure CPU program with our CPU-GPU accelerated program using the NUREG/CR-6115 PWR benchmark problem. The results indicate identical outcomes for photon point flux estimation, with the accelerated program being ~50 times faster.

Randomly Shifted Lattice Rules with Importance Sampling and Applications

In financial and statistical computations, calculating expectations often requires evaluating integrals with respect to a Gaussian measure. Monte Carlo methods are widely used for this purpose due to their dimension-independent convergence rate. Quasi-Monte Carlo is the deterministic analogue of Monte Carlo and has the potential to substantially enhance the convergence rate. Importance sampling is a widely used variance reduction technique. However, research into the specific impact of importance sampling on the integrand, as well as the conditions for convergence, is relatively scarce. In this study, we combine the randomly shifted lattice rule with importance sampling. We prove that, for unbounded functions, randomly shifted lattice rules combined with a suitably chosen importance density can achieve convergence as quickly as O(N−1+ϵ), given N samples for arbitrary ϵ values under certain conditions. We also prove that the conditions of convergence for Laplace importance sampling are stricter than those for optimal drift importance sampling. Furthermore, using a generalized linear mixed model and Randleman–Bartter model, we provide the conditions under which functions utilizing Laplace importance sampling achieve convergence rates of nearly O(N−1+ϵ) for arbitrary ϵ values.

Painless Stochastic Conjugate Gradient for Large-Scale Machine Learning.

Conjugate gradient (CG), as an effective technique to speed up gradient descent algorithms, has shown great potential and has widely been used for large-scale machine-learning problems. However, CG and its variants have not been devised for the stochastic setting, which makes them extremely unstable, and even leads to divergence when using noisy gradients. This article develops a novel class of stable stochastic CG (SCG) algorithms with a faster convergence rate via the variance-reduced technique and an adaptive step size rule in the mini-batch setting. Actually, replacing the use of a line search in the CG-type approaches which is time-consuming, or even fails for SCG, this article considers using the random stabilized Barzilai-Borwein (RSBB) method to obtain an online step size. We rigorously analyze the convergence properties of the proposed algorithms and show that the proposed algorithms attain a linear convergence rate for both the strongly convex and nonconvex settings. Also, we show that the total complexity of the proposed algorithms matches that of modern stochastic optimization algorithms under different cases. Scores of numerical experiments on machine-learning problems demonstrate that the proposed algorithms outperform state-of-the-art stochastic optimization algorithms.

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.

Structured Output Feedback Control for Linear Quadratic Regulator Using Policy Gradient Method

We consider the static output feedback control for Linear Quadratic Regulator problems with structured constraints under the assumption that system parameters are unknown. To solve the problem in the model free setting, we propose the policy gradient algorithm based on the gradient projection method and show its global convergence to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> -stationary points. In addition, we introduce a variance reduction technique and show both theoretically and numerically that it significantly reduces the variance in the gradient estimation. We also show in the numerical experiments that the model free approach efficiently solves the problem.

A Machine Learning–Based Tire Life Prediction Framework for Increasing Life of Commercial Vehicle Tires

Abstract In the commercial freight industry, tire retreading decisions are often conservative due to limited knowledge of a tire’s remaining service life. This practice leads to increased costs and material waste. This paper proposes a machine learning–based approach for estimating tire casing life and retreadability, focusing on usage data rather than wear information. This approach could extend the tire’s lifespan and reduce landfill waste. Data integration from diverse tire casing measurement sources presents challenges, including imbalanced removal data. Our methodology addresses these challenges by using historical inspection, telematics, and finite element modeling (FEM) datasets. We introduce “Tire Casing Energy” as a comprehensive usage input and apply a Variance-Reduction Synthetic Minority Oversampling Technique (VR-SMOTE) for data imbalance rectification. A random forest model is used to estimate the state of the tire casing and the casing removal probability, with Bayesian optimization applied for hyperparameter tuning, enhancing model accuracy. The proposed prediction framework is able to differentiate different truck fleets and tire locations based on their usage parameters. With the aid of this machine learning model, the importance and sensitivity of different tire usage parameters can be obtained, which is beneficial to maximize tire life.

Effect of spatial variability of cone penetration resistance on probabilistic axial capacity of pile foundations

The Cone Penetration Test (CPT) is site-specific and it is one of the predominant tests in geotechnical investigations used for varied offshore and onshore facilities. The evaluation of axial load carrying capacity of pile is one such use. Due to their continual records, CPT-based methods are appropriate in the offshore environments. Uncertainties are prevailing in most of the engineered systems and implementation of spatial correlation in the probabilistic geotechnical studies possess a paramount importance to overcome the uncertainties. In the present study, the influence of spatial correlation in the CPT data towards the load carrying capacity of pile group in sandy soil is examined. The in-situ test approach involving the spatial correlation of the CPT data in the analytical formulations is established in the study. The inclusion of spatial correlation intervening the spatial averages of CPT data obtained along the soil profile for the pile length, qc,a and at its base, qc,s is studied. Variance reduction technique is utilized for developing the one-dimensional spatial correlation. The failure probability and its relationship with the factor of safety is illustrated based on the Rackwitz’s first-order reliability method (FORM). The results of the analyses have shown that, the inclusion of spatial correlation imparts a significant effect on failure probability and which when ignored will result in an unsafe design. A test pile group is analyzed to exemplify the reliability assessment of axial pile load carrying capacity by considering the spatial correlation of CPT data. This approach can be utilized for the safer pile design practices within the probabilistic framework.

Powered stochastic optimization with hypergradient descent for large-scale learning systems

Stochastic optimization (SO) algorithms based on the Powerball function, namely powered stochastic optimization (PoweredSO) algorithms, have been confirmed, effectively, and demonstrated great potential in the context of large-scale optimization and machine learning tasks. Nevertheless, the issue of how to determine the learning rate for PoweredSO is a challenge and still unsolved problem. In this paper, we propose a class of adaptive PoweredSO approaches that are efficient, scalable and robust. It takes advantage of the hypergradient descent (HD) technique to automatically acquire an online learning rate for PoweredSO-like methods. In the first part, we study the behavior of the canonical PoweredSO algorithm, the Powerball stochastic gradient descent (pbSGD) method, with HD. The existing PoweredSO algorithms also suffer from the high variance because they take the similar algorithmic framework to SO algorithms, arising from sampling tactics. Therefore, the second portion develops an adaptive powered variance-reduced optimization method via utilizing both variance-reduced technique and HD. Moreover, we present the convergence analysis of the proposed algorithms and explore their iteration complexity on non-convex cases. Numerical experiments are conducted on machine learning tasks, verifying the superior performance over modern SO algorithms.