Noisy Functions Research Articles

On the Choice of Random Directions for Stochastic Approximation Algorithms

Stochastic approximation provides a simple and effective approach for finding roots and minima of functions whose evaluations are contaminated with noise. We investigate variants of the random direction stochastic approximation (RDSA) algorithm for optimizing noisy loss functions in high-dimensional spaces. The most popular variant is random selection from a Bernoulli distribution, which also goes by the name simultaneous perturbation stochastic approximation (SPSA). Viable alternatives include an axis-aligned distribution, normal distribution, and uniform distribution on a spherical shell. Although there are special cases where the Bernoulli distribution is optimal, we identify other cases where it performs worse than the alternatives. We show that performance depends on the orientation of the loss function with respect to its coordinate axes, and consider averages over all orientations. We find that the average asymptotic performance depends only on the radial fourth moment of the distribution of random directions, and is identical for the Bernoulli, axis-aligned, and spherical shell distributions. Of these variants, the spherical shell is optimal in the sense of minimum variance over random orientations of the loss function with respect to the coordinate axes. We also show that for unaligned loss functions, the performance of the Kiefer-Wolfowitz-Blum finite difference stochastic approximation (FDSA) is asymptotically equivalent to the RDSA algorithms, and we observe numerically that the pre-asymptotic performance of FDSA is often superior.

Gradient-based approximation methods applied to the optimal design of vehicle suspension systems using computational models with severe inherent noise

The principal aim of this paper is to evaluate the feasibility of using gradient-based approximation methods for the optimisation of the spring and damper characteristics of an off-road vehicle, for both ride comfort and handling. The Sequential Quadratic Programming algorithm and the relatively new Dynamic-Q method are the two successive approximation methods used for the optimisation. The determination of the objective function value is performed using computationally expensive numerical simulations that exhibit severe inherent numerical noise. The use of forward finite differences and central finite differences for the determination of the gradients of the objective function within Dynamic-Q is also investigated. This is done in investigating methods for overcoming the difficulties associated with the optimisation of noisy objective functions. A recreational off-road vehicle is modelled in ADAMS, and coupled to MATLAB for the execution of the optimisation process. The full vehicle ADAMS model includes suspension kinematics, a load-dependent tyre model, as well as non-linear springs and dampers. Up to four design variables are considered in modelling the suspension characteristics. It is found that both algorithms perform well in optimising handling. However, difficulties are encountered in obtaining improvements in the design process when ride comfort is considered. Nevertheless, meaningful design configurations are still achievable through the proposed optimisation process, at a relatively low cost in terms of the number of simulations that have to be performed.

Constrained optimization of a preparative ion-exchange step for antibody purification

Today, the optimization of chromatographic separation is usually based on experimental work and rule of thumb. The process and analytical technology (PAT) initiative, of the US Food and Drug Administration, has provided the opportunity of using model-based approach when designing downstream processing of pharmaceutical substances. A nonlinear chromatography model was used in this study to optimize a preparative ion-exchange separation step involving two components. Separation was simulated with the general rate model employing Langmuir kinetics. Optimization was performed with an indirect method allowing constraints on the purity, thus avoiding sub-optimization, which can lead to noisy objective functions. The six decision variables used in the optimizations were flow rate, loading volume, initial salt concentration in the elution, final salt concentration in the linear elution gradient and the two cut points. A graphical representation of the effect of the decision variables on the objective function was used to verify that the optimization had converged to the true optimum. The optimal operating points, using productivity and yield separately as objective functions, were found and compared with the product of productivity and yield as objective function. The optimum obtained with this objective function had a lower productivity, than the productivity function, but much higher yield, which makes it a good substitute for a cost function.

Bias and pathology in minimax search

This article presents the results of experiments designed to gain insight into the effect of the minimax algorithm on the error of a heuristic evaluation function. Two types of effect of minimax are considered: (a) evaluation accuracy (Are the minimax backed-up values more accurate than the heuristic values themselves?), and (b) decision accuracy (Are moves played by deeper minimax search better than those by shallower search?). The experiments were performed in the King–Rook–King (KRK) chess endgame and in randomly generated game trees. The results show that, counter-intuitively, evaluation accuracy may decline with search depth, whereas at the same time decision accuracy improves with depth. In the article, this is explained by the fact that minimax in combination with a noisy evaluation function introduces a bias into the backed-up evaluations, which masks the evaluation effectiveness of minimax, but this bias still permits decision accuracy to improve with depth. This observed behaviour of minimax in the KRK endgame is discussed in the light of previous studies of pathology in minimax. It is shown that explaining the behaviour of minimax in an actual chess endgame in terms of previously known results requires special care.

Gradient Estimation Schemes for Noisy Functions

In this paper, we analyze different schemes for obtaining gradient estimates when the underlying functions are noisy. Good gradient estimation is important e.g. for nonlinear programming solvers. As error criterion, we take the norm of the difference between the real and estimated gradients. The total error can be split into a deterministic error and a stochastic error. For three finite-difference schemes and two design of experiments (DoE) schemes, we analyze both the deterministic errors and stochastic errors. We derive also optimal stepsizes for each scheme, such that the total error is minimized. Some of the schemes have the nice property that this stepsize minimizes also the variance of the error. Based on these results, we show that, to obtain good gradient estimates for noisy functions, it is worthwhile to use DoE schemes. We recommend to implement such schemes in NLP solvers.

노이즈 필터링을 적용한 반응표면 기반 순차적 근사 최적화

In this paper, a new method for constrained optimization of noisy functions is proposed. In approximate optimization using response surface methods, if constraints have severe noise, the approximate feasible region defined by approximate constraints is apt to include some of the infeasible region defined by actual constraints. This can cause the approximate optimum to converge into the infeasible region. In the proposed method, the approximate optimization is performed with the approximate constraints shifted by their deviations, which are calculated using a diagonal quadratic response surface method. This can prevent the approximate optimum from converging into the infeasible region. To fit the objective and constraints into diagonal quadratic models, we select the center and 4 additional points along each axis of design variables as experimental points. The deviation of each function is calculated using the differences between the real and approximate function values at the experimental points. A sequential approximate optimization technique based on the trust region algorithm is adopted to manage approximate models. The proposed approach is validated by solving some design problems. The results of the problems show the effectiveness of the proposed method.

不確実な関数を最適化する多目的遺伝的アルゴリズムの提案

In this paper, genetic algorithms for multi-objective optimization problems with uncertainty, which attract attention for applications to simulation-based and experiment-based optimization of real systems, are discussed. First, difficulties faced by conventional multi-objective GAs in their application to multi-objective optimization of noisy fitness functions are described. Second, to cope with these problems, a multi-objective GA that has a fitness estimation method and a new selection operator is proposed. The effectiveness of the proposed method is demonstrated by numerical simulations and real-world experiments.

Optimal sampling in a noisy genetic algorithm for risk-based remediation design

A groundwater management model has been developed that predicts human health risks and uses a noisy genetic algorithm to identify promising risk-based corrective action (RBCA) designs. Noisy genetic algorithms are simple genetic algorithms that operate in noisy environments. The noisy genetic algorithm uses a type of noisy fitness function (objective function) called the sampling fitness function, which utilises Monte-Carlo-type sampling to find robust designs. Unlike Monte Carlo simulation modelling, however, the noisy genetic algorithm is highly efficient and can identify robust designs with only a few samples per design. For hydroinformatic problems with complex fitness functions, however, it is important that the sampling be as efficient as possible. In this paper, methods for identifying efficient sampling strategies are investigated and their performance evaluated using a case study of a RBCA design problem. Guidelines for setting the parameter values used in these methods are also developed. Applying these guidelines to the case study resulted in highly efficient sampling strategies that found RBCA designs with 98% reliability using as few as 4 samples per design. Moreover, these designs were identified with fewer simulation runs than would likely be required to identify designs using trial-and-error Monte Carlo simulation. These findings show considerable promise for applying these methods to complex hydroinformatic problems where substantial uncertainty exists but extensive sampling cannot feasibly be done.

A Class of Trust-Region Methods for Parallel Optimization

We present a new class of optimization methods that incorporates a parallel direct search (PDS) method within a trust-region Newton framework. This approach combines the inherent parallelism of PDS with the rapid and robust convergence properties of Newton methods. Numerical tests have yielded favorable results for both standard test problems and engineering applications. In addition, the new method appears to be more robust in the presence of noisy functions, which are inherent in many engineering simulations.

A Derivative-Free Algorithm for Bound Constrained Optimization

In this work, we propose a new globally convergent derivative-free algorithm for the minimization of a continuously differentiable function in the case that some of (or all) the variables are bounded. This algorithm investigates the local behaviour of the objective function on the feasible set by sampling it along the coordinate directions. Whenever a “suitable” descent feasible coordinate direction is detected a new point is produced by performing a linesearch along this direction. The information progressively obtained during the iterates of the algorithm can be used to build an approximation model of the objective function. The minimum of such a model is accepted if it produces an improvement of the objective function value. We also derive a bound for the limit accuracy of the algorithm in the minimization of noisy functions. Finally, we report the results of a preliminary numerical experience.

CURVATURE IN DIGITAL 2D IMAGES

The paper presents an analysis of sources of errors when estimating derivatives of numerical or noisy functions. A method of minimizing the errors is suggested. When being applied to the estimation of the curvature of digital curves, the analysis shows that under the conditions typical for digital image processing the curvature can rarely be estimated with a precision higher than 50%. Ways of overcoming the difficulties are discussed and a new method for estimating the curvature is suggested and investigated as to its precision. The method is based on specifying boundaries of regions in gray value images with subpixel precision. The method has an essentially higher precision than the known methods.

On stable numerical differentiation

Based on a regularized Volterra equation, two different approaches for numeri- cal differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete fa- vorably with the variational regularization method for stable calculating the derivatives of noisy functions.

A fully‐distributed parallel global search algorithm

The n‐dimensional direct search algorithm, DIRECT, developed by Jones, Perttunen, and Stuckman has attracted recent attention from the multidisciplinary design optimization community. Since DIRECT only requires function values (or ranking) and balances global exploration with local refinement better than n‐dimensional bisection, it is well suited to the noisy function values typical of realistic simulations. While not efficient for high accuracy optimization, DIRECT is appropriate for the sort of global design space exploration done in large scale engineering design. Direct and pattern search schemes have the potential to exploit massive parallelism, but efficient use of massively parallel machines is non‐trivial to achieve. A fully‐distributed control version of DIRECT that is designed for massively parallel (distributed memory) architectures is presented. Parallel results are presented for a multidisciplinary design optimization problem – configuration design of a high speed civil transport.

Polynomial Response Surface Approximations for the Multidisciplinary Design Optimization of a High Speed Civil Transport

Surrogate functions have become an important tool in multidisciplinary design optimization to deal with noisy functions, high computational cost, and the practical difficulty of integrating legacy disciplinary computer codes. A combination of mathematical, statistical, and engineering techniques, well known in other contexts, have made polynomial surrogate functions viable for MDO. Despite the obvious limitations imposed by sparse high fidelity data in high dimensions and the locality of low order polynomial approximations, the success of the panoply of techniques based on polynomial response surface approximations for MDO shows that the implementation details are more important than the underlying approximation method (polynomial, spline, DACE, kernel regression, etc.). This paper selectively surveys some of the ancillary techniques—statistics, global search, parallel computing, variable complexity modeling—that augment the construction and use of polynomial surrogates.

A Direct Search Algorithm for Optimization with Noisy Function Evaluations

We consider the unconstrained optimization of a function when each function evaluation is subject to a random noise. We assume that there is some control over the variance of the noise term, in the sense that additional computational effort will reduce the amount of noise. This situation may occur when function evaluations involve simulation or the approximate solution of a numerical problem. It also occurs in an experimental setting when averaging repeated observations at the same point can lead to a better estimate of the underlying function value. We describe a new direct search algorithm for this type of problem. We prove convergence of the new algorithm when the noise is controlled so that the standard deviation of the noise approaches zero faster than the step size. We also report some numerical results on the performance of the new algorithm.

Detection of central positions of noisy rounded square function using modified Haar wavelet transform and Haar-Gaussian wavelet transform

We propose and demonstrate new wavelet transforms, the modified Haar wavelet transform and Haar-Gaussian wavelet transform, for the detection of central positions of a noisy rounded square function. The advantage is the immunity of the wavelet transform to random noise. The alternative technique based on first- and second-order differentiation cannot be applied when the signal function is corrupted by noise. The proposed technique can be used for lead inspection of surface mount devices in the electronic industry. © 2000 Society of Photo-Optical Instrumenta- tion Engineers. (S0091-3286(00)01705-0)

Smoothed Sensitivity Equation Method for Fluid Dynamic Design Problems

We consider shape optimization problems involving compressible fluid flows, which are characterized by non-smooth and/or noisy objective functions. Such functions are difficult to optimize using derivative-based techniques. To overcome such a difficulty, we suggest an approach for estimating the sensitivity derivatives, based on a suitable smoothing of the sensitivity equations. The smoothing affects only the sensitivity derivatives and not the accuracy of the analysis. The basic mechanism by which the smoothing process achieves this result is illustrated with the help of an inverse design problem involving an inviscid quasi-one-dimensional flow having a closed-form solution. The convergence properties and the computational efficiency of the approach are demonstrated on two inverse design problems involving two-dimensional inviscid, compressible flows

Nonparametric estimation of smoothed principal components analysis of sampled noisy functions

This study deals with the simultaneous nonparametric estimations of n curves or observations of a random process corrupted by noise in which sample paths belong to a finite dimension functional subspace. The estimation, by means of B-splines, leads to a new kind of functional principal components analysis. Asymptotic rates of convergence are given for the mean and the eigenelements of the empirical covariance operator. Heuristic arguments show that a well chosen smoothing parameter may improve the estimation of the subspace which contains the sample path of the process. Finally, simulations suggest that the estimation method studied here is advantageous when there are a small number of design points.

Supporting decision making in combinatorially explosive multicriteria situations

Several real-world problems, including distributed system design and product design among others, are characterized by combinatorially explosive solution spaces as well as multiple, conflicting criteria. Strategies for finding near-optimal solutions, developed for combinatorial problems, are not applicable in such situations, which require a balance between extensive computation and continual interaction. This makes support or automation of these decisions a difficult task. Current approaches to solve these problems fall in three categories: analytical, genetic algorithm-based approaches and local generators. They frequently assume well-behaved functions and clear understanding of interdependencies among criteria. Many such problems, however, present noisy and discontinuous evaluation functions and contain holistic interdependencies among these functions, rendering these solution approaches inadequate for these problems. We propose a theoretically grounded approach for decision support for this class of problems. The approach combines broad and deep searches with decision-maker feedback that allows the decision-maker to guide and/or stop the search. Specifically, it provides the decision-maker information about (a) the search spaces explored/probed so far, and (b) the search space not yet explored (or may never be explored). We operationalize the approach in a two-phase solution procedure. The first phase — broad characterization of search spaces — requires choices about randomization, sampling and decision space estimation techniques. The second phase — iterative local probes — requires choices about local search heuristics, and fuzzy interpretations based on which the decision-maker can evaluate alternatives and/or stop the search. We demonstrate a specific instantiation of the approach for a multicriteria object assignment problem to verify the feasibility of our approach.

Numerical inversion of the Laplace transform

A generalization of Doetsch’s formula [Math. Z. 42, 263 (1937)] is derived to develop a stable numerical inversion of the one-sided Laplace transform Ĉ(β). The necessary input is only the values of Ĉ(β) on the positive real axis. The method is applicable provided that the functions Ĉ(β) belong to the function space Lα2 defined by the condition that G(x)=exαĈ(ex), α>0 has to be square integrable. The inversion algorithm consists of two sequential Fourier transforms where the second Fourier integration requires a cutoff, whose magnitude depends on the accuracy of the data. For high accuracy data, the cutoff tends to infinity and the inversion is very accurate. The presence of noise in the signal causes a lowering of the cutoff and a lowering of the accuracy of the inverted data. The optimal cutoff value is shown to be one which leads to an inversion which remains consistent with the original data and its noise level. The method is demonstrated for some model problems: a harmonic partition function, resonant transmission through a barrier, noisy correlation functions, and noisy Monte Carlo generated data for tunneling coefficients obtained via the recently introduced quantum transition state theory (QTST).