Noisy Optimization Research Articles

Differentially private inference via noisy optimization

Differentially private inference via noisy optimization

Revisiting Implicit and Explicit Averaging for Noisy Optimization

Explicit and implicit averaging are two well-known strategies for noisy optimization. Both strategies can counteract the disruptive effect of noise; however, a critical question remains: which one is more efficient? This question has been raised in many studies, with conflicting preferences and, in some cases, findings. Nevertheless, theoretical findings on the noisy sphere problem with additive Gaussian noise supports the superiority of implicit averaging, which may have had a strong impact on the preference of implicit averaging in more recent evolutionary methods for noisy optimization. This study speculates that the analytically supported superiority of implicit averaging relies on specific features of the noisy sphere problem with additive noise, which cannot be generalized to other problems. It enumerates these features and designs controlled numerical experiments to investigate this potential reliance. Each experiment gradually suppresses one specific feature, and the progress rate is numerically calculated for different values of the sample size given a fixed evaluation budget. Our empirical results indicate that for a wide range of noise strength and evaluation budget per iteration, the more these specific features are suppressed, the more the optimal averaging strategy deviates from implicit towards explicit averaging, which confirms our speculations. Consequently, the optimal sample size, which is regarded as the trade-off between implicit and explicit averaging, depends on the problem characteristics and should be learned during optimization for maximum efficiency.

Differential Evolution With Domain Transform

Although a significant advancement of differential evolution (DE) for global optimization has been witnessed in the past two decades, the problems of premature convergence and stagnation are still open questions that hinder the performance. Both are likely to occur on complicated multi-modal functions, but the phenomena differ. Premature convergence refers to a rapid loss of population diversity when attracted to a local minimum while stagnation happens even though the population is diverse. To deal with these problems, this paper proposes a domain transform (DT) methodology. Different from existing fitness analysis which mainly utilizes the original fitness landscape information, DT yields a transformed fitness landscape with transform operation to the frequency domain and inverse transform operation back to the solution domain, between which the first few highest frequencies are removed. With the deletion operation, the transformed fitness landscape becomes smoother and facilitates the escape from local minima and stagnation on complicated multi-modal functions. Simulation results show that DT significantly improves the population successful update rate and population convergence. The constructed DTDE algorithm consequently exhibits remarkable improvements on the baseline algorithm and outperforms several state-of-the-art DE variants. DT has also been extended for noisy optimization and it performs better than the baseline, the classic resampling method, state-ofthe-art DE variants as well as several popular noisy evolutionary optimization algorithms.

Ability of Black-Box Optimisation to Efficiently Perform Simulation Studies in Power Engineering

Abstract In this study, the potential of the so-called black-box optimisation (BBO) to increase the efficiency of simulation studies in power engineering is evaluated. Three algorithms (“Multilevel Coordinate Search” (MCS) and “Stable Noisy Optimization by Branch and Fit” (SNOBFIT) by Huyer and Neumaier and “blackbox: A Procedure for Parallel Optimization of Expensive Black-box Functions” (blackbox) by Knysh and Korkolis) are implemented in MATLAB and compared for solving two use cases: the analysis of the maximum rotational speed of a gas turbine after a load rejection and the identification of transfer function parameters by measurements. The first use case has a high computational cost, whereas the second use case is computationally cheap. For each run of the algorithms, the accuracy of the found solution and the number of simulations or function evaluations needed to determine the optimum and the overall runtime are used to identify the potential of the algorithms in comparison to currently used methods. All methods provide solutions for potential optima that are at least 99.8% accurate compared to the reference methods. The number of evaluations of the objective functions differs significantly but cannot be directly compared as only the SNOBFIT algorithm does stop when the found solution does not improve further, whereas the other algorithms use a predefined number of function evaluations. Therefore, SNOBFIT has the shortest runtime for both examples. For computationally expensive simulations, it is shown that parallelisation of the function evaluations (SNOBFIT and blackbox) and quantisation of the input variables (SNOBFIT) are essential for the algorithmic performance. For the gas turbine overspeed analysis, only SNOBFIT can compete with the reference procedure concerning the runtime. Further studies will have to investigate whether the quantisation of input variables can be applied to other algorithms and whether the BBO algorithms can outperform the reference methods for problems with a higher dimensionality.

Quantum variational algorithms are swamped with traps

One of the most important properties of classical neural networks is how surprisingly trainable they are, though their training algorithms typically rely on optimizing complicated, nonconvex loss functions. Previous results have shown that unlike the case in classical neural networks, variational quantum models are often not trainable. The most studied phenomenon is the onset of barren plateaus in the training landscape of these quantum models, typically when the models are very deep. This focus on barren plateaus has made the phenomenon almost synonymous with the trainability of quantum models. Here, we show that barren plateaus are only a part of the story. We prove that a wide class of variational quantum models—which are shallow, and exhibit no barren plateaus—have only a superpolynomially small fraction of local minima within any constant energy from the global minimum, rendering these models untrainable if no good initial guess of the optimal parameters is known. We also study the trainability of variational quantum algorithms from a statistical query framework, and show that noisy optimization of a wide variety of quantum models is impossible with a sub-exponential number of queries. Finally, we numerically confirm our results on a variety of problem instances. Though we exclude a wide variety of quantum algorithms here, we give reason for optimism for certain classes of variational algorithms and discuss potential ways forward in showing the practical utility of such algorithms.

Noisy Optimization by Evolution Strategies With Online Population Size Learning

Optimization modeling of real-world application problems usually involves noise from various sources. Noisy optimization imposes challenges to optimization methods since the objective values can be different for multiple evaluations. In this article, we propose a novel online population size learning (OPL) technique of evolution strategies for handling noisy optimization problems. By re-evaluating a fraction of the candidates, we measure the strength of noise level of the re-evaluated candidate solutions and adapt the population size according to the noise level. The proposed OPL combines the advantages of both explicit averaging by re-evaluations and the implicit averaging by large population size and overcomes their limitations. We incorporate it with the covariance matrix adaptation evolution strategy (CMA-ES) and obtain OPL-CMA-ES. Compared with the existing noise handling technique, the proposed OPL is much simpler in both concepts and computation. We conduct comprehensive experiments to evaluate the algorithm’s performance on standard problems with Gaussian noise. We further evaluate the performance of OPL-CMA-ES on the black-box optimization benchmarks (BBOBs) noisy testbed, which is a standard platform for comparing black-box optimization algorithms, compared with the state-of-the-art noise-handling algorithms. The experimental results show that OPL-CMA-ES achieves remarkable performance and outperforms the compared variants.

Radial basis network simulation for noisy multiobjective optimization considering evolution control

Although some studies have achieved good optimization results when solving noisy multiobjective optimization problems (NMOPs), these studies require considerable computational resources to reduce the impact of noise on evolutionary algorithms. Inspired by the universal approximation theorem, we use a radial basis network (RBN) to estimate the noise-free fitness and embed it into the classical multiobjective algorithm NSGA-II to solve NMOPs, called RBN-NSGA-II. Specifically, the averaged fitness after individual resampling is used as the training sample of the RBN, and then, the trained RBN is used to simulate the fitness functions and estimate the noise-free fitness. Also, to better simulate the shape of the fitness functions, the RBN is adaptively updated based on its estimated error, and its training samples are gradually increased based on the population distribution in the search space. Experimental results demonstrate that RBN-NSGA-II markedly improves the optimization speed while maintaining good search ability compared with six state-of-the-art algorithms, which is important in practical noisy optimization scenarios.

Inter-nucleosomal potentials from nucleosomal positioning data

No systematic method exists to derive inter-nucleosomal potentials between nucleosomes along a chromosome consistently across a given genome. Such potentials can yield information on nucleosomal ordering, thermal as well as mechanical properties of chromosomes. Thus, indirectly, they shed light on a possible mechanical genomic code along a chromosome. To develop a method yielding effective inter-nucleosomal potentials between nucleosomes, a generalized Lennard-Jones potential for the parameterization is developed based on nucleosomal positioning data. This approach eliminates some of the problems that the underlying nucleosomal positioning data have, rendering the extraction difficult on the individual nucleosomal level. Furthermore, patterns on which to base a classification along a chromosome appear on larger domains, such as hetero- and euchromatin. An intuitive selection strategy for the noisy optimization problem is employed to derive effective exponents for the generalized potential. The method is tested on the Candida albicans genome. Applying k-means clustering based on potential parameters and thermodynamic compressibilities, a genome-wide clustering of nucleosome sequences is obtained for C. albicans. This clustering shows that a chromosome beyond the classical dichotomic categories of hetero- and euchromatin is more feature-rich.

Machine learning with knowledge constraints for process optimization of open-air perovskite solar cell manufacturing

Machine learning with knowledge constraints for process optimization of open-air perovskite solar cell manufacturing

Handling Sampling-induced Uncertainties in Tolerance-Cost Optimization

Embedding sampling-based tolerance analysis in tolerance-cost optimization is seen as the key to its industrial establishment as a powerful tool for cost-optimal tolerance allocation. However, the sampling-induced variance of the analysis results in each individual optimization step leads to a noisy optimization and thus significantly affects the stability, reproducibility, and reliability of the optimization process and its results. So far, the interrelations between sampling and optimization as well as specific countermeasures are only insufficiently discussed in literature. With the aim to overcome this shortcoming, the given article introduces several measures to enhance the reliability of sampling-based tolerance-cost optimization by reducing and managing the sampling-induced uncertainties in the definition and solution of the optimization problem as well as in the interpretation of the results. The findings serve as a guideline for both researchers and practitioners ensuring reliable tolerance-cost optimization results.

Self-Optimization of Continuous Flow Electrochemical Synthesis Using Fourier Transform Infrared Spectroscopy and Gas Chromatography.

A continuous-flow electrochemical synthesis platform has been developed to enable self-optimization of reaction conditions of organic electrochemical reactions using attenuated total reflection Fourier transform infrared spectroscopy (ATR FT-IR) and gas chromatography (GC) as online real-time monitoring techniques. We have overcome the challenges in using ATR FT-IR as the downstream analytical methods imposed when a large amount of hydrogen gas is produced from the counter electrode by designing two types of gas-liquid separators (GLS) for analysis of the product mixture flowing from the electrochemical reactor. In particular, we report an integrated GLS with an ATR FT-IR probe at the reactor outlet to give a facile and low-cost solution to determining the concentrations of products in gas-liquid two-phase flow. This approach provides a reliable method for quantifying low-volatile analytes, which can be problematic to be monitored by GC. Two electrochemical reactions the methoxylation of 1-formylpyrrolidine and the oxidation of 3-bromobenzyl alcohol were investigated to demonstrate that the optimal conditions can be located within the pre-defined multi-dimensional reaction parameter spaces without intervention of the operator by using the stable noisy optimization by branch and FIT (SNOBFIT) algorithm.

Noisy Optimization of Dispatching Policy for the Cranes at the Storage Yard in an Automated Container Terminal

In this paper, we claim that the operation schedule of automated stacking cranes (ASC) in the storage yard of automated container terminals can be built effectively and efficiently by using a crane dispatching policy, and propose a noisy optimization algorithm named N-RTS that can derive such a policy efficiently. To select a job for an ASC, our dispatching policy uses a multi-criteria scoring function to calculate the score of each candidate job using a weighted summation of the evaluations in those criteria. As the calculated score depends on the respective weights of these criteria, and thus a different weight vector gives rise to a different best candidate, a weight vector can be deemed as a policy. A good weight vector, or policy, can be found by a simulation-based search where a candidate policy is evaluated through a computationally expensive simulation of applying the policy to some operation scenarios. We may simplify the simulation to save time but at the cost of sacrificing the evaluation accuracy. N-RTS copes with this dilemma by maintaining a good balance between exploration and exploitation. Experimental results show that the policy derived by N-RTS outperforms other ASC scheduling methods. We also conducted additional experiments using some benchmark functions to validate the performance of N-RTS.

Olympus: a benchmarking framework for noisy optimization and experiment planning

Research challenges encountered across science, engineering, and economics can frequently be formulated as optimization tasks. In chemistry and materials science, recent growth in laboratory digitization and automation has sparked interest in optimization-guided autonomous discovery and closed-loop experimentation. Experiment planning strategies based on off-the-shelf optimization algorithms can be employed in fully autonomous research platforms to achieve desired experimentation goals with the minimum number of trials. However, the experiment planning strategy that is most suitable to a scientific discovery task is a priori unknown while rigorous comparisons of different strategies are highly time and resource demanding. As optimization algorithms are typically benchmarked on low-dimensional synthetic functions, it is unclear how their performance would translate to noisy, higher-dimensional experimental tasks encountered in chemistry and materials science. We introduce Olympus, a software package that provides a consistent and easy-to-use framework for benchmarking optimization algorithms against realistic experiments emulated via probabilistic deep-learning models. Olympus includes a collection of experimentally derived benchmark sets from chemistry and materials science and a suite of experiment planning strategies that can be easily accessed via a user-friendly Python interface. Furthermore, Olympus facilitates the integration, testing, and sharing of custom algorithms and user-defined datasets. In brief, Olympus mitigates the barriers associated with benchmarking optimization algorithms on realistic experimental scenarios, promoting data sharing and the creation of a standard framework for evaluating the performance of experiment planning strategies.

Expected improvement for expensive optimization: a review

The expected improvement (EI) algorithm is a very popular method for expensive optimization problems. In the past twenty years, the EI criterion has been extended to deal with a wide range of expensive optimization problems. This paper gives a comprehensive review of the EI extensions designed for parallel optimization, multiobjective optimization, constrained optimization, noisy optimization, multi-fidelity optimization and high-dimensional optimization. The main challenges of extending the EI approach to solve these complex optimization problems are pointed out, and the ideas proposed in literature to tackle these challenges are highlighted. For each reviewed algorithm, the surrogate modeling method, the computation of the infill criterion and the internal optimization of the infill criterion are carefully studied and compared. In addition, the monotonicity properties of the multiobjective EI criteria and constrained EI criteria are analyzed in detail. Through this review, we give an organized summary about the EI developments in the past twenty years and show a clear picture about how the EI approach has advanced. In the end of this paper, several interesting problems and future research topics about the EI developments are given.

How to catch a lion in the desert: on the solution of the coverage directed generation (CDG) problem

The testing and verification of a complex hardware or software system, such as modern integrated circuits (ICs) found in everything from smartphones to servers, can be a difficult process. One of the most difficult and time-consuming tasks a verification team faces is reaching coverage closure, or hitting all events in the coverage space. Coverage-directed-generation (CDG), or the automatic generation of tests that can hit hard-to-hit coverage events, and thus provide coverage closure, holds the potential to save verification teams significant simulation resources and time. In this paper, we propose a new approach to the CDG problem by formulating the CDG problem as a noisy derivative free optimization (DFO) problem. However, this formulation is complicated by the fact that derivatives of the objective function are unavailable, and the objective function evaluations are corrupted by noise. We solve this noisy optimization problem by utilizing techniques from direct optimization coupled with a robust noise estimator, and by leveraging techniques from inverse problems to estimate the gradient of the noisy objective function. We demonstrate the efficiency and reliability of this new approach through numerical experiments with an abstract model of part of IBM's NorthStar processor, a superscalar in-order processor designed for servers.

Improved differential evolution for noisy optimization

Abstract A novel approach is proposed in this paper to improve the optimization proficiency of the differential evolution (DE) algorithm in the presence of stochastic noise in the objective surface by utilizing the composite benefit of four strategies. The first strategy is devised with an aim to employ reinforcement learning scheme of stochastic learning automata for autonomous selection of the sample size of a trial solution (for its repeated fitness evaluation) based on the characteristics of the fitness landscape in its local neighborhood. The second stratagem is proposed to estimate the effective fitness measure from multiple fitness samples of a trial solution, resulting from sampling. The novelty of the second policy lies in considering the distribution of noisy samples during effective fitness evaluation, instead of their direct averaging. The third strategy deals with amelioration of the DE/current-to-best/1 mutation scheme to judiciously direct the search in promising region, even in prevailing existence of noise in the objective surface. Finally, the greedy selection policy of the traditional DE is modified by introducing the principle of probabilistic crowding induced niching to ensure both the population quality and the population diversity. Comparative analysis performed on simulation results for diverse noisy benchmark functions reveal the statistically significant superiority of the proposed algorithm to its contenders with respect to function error value.

Surrogate Optimization of Computationally Expensive Black-Box Problems with Hidden Constraints

We introduce the algorithm SHEBO (surrogate optimization of problems with hidden constraints and expensive black-box objectives), an efficient optimization algorithm that employs surrogate models to solve computationally expensive black-box simulation optimization problems that have hidden constraints. Hidden constraints are encountered when the objective function evaluation does not return a value for a parameter vector. These constraints are often encountered in optimization problems in which the objective function is computed by a black-box simulation code. SHEBO uses a combination of local and global search strategies together with an evaluability prediction function and a dynamically adjusted evaluability threshold to iteratively select new sample points. We compare the performance of our algorithm with that of the mesh-based algorithms mesh adaptive direct search (MADS, NOMAD [nonlinear optimization by mesh adaptive direct search] implementation) and implicit filtering and SNOBFIT (stable noisy optimization by branch and fit), which assigns artificial function values to points that violate the hidden constraints. Our numerical experiments for a large set of test problems with 2–30 dimensions and a 31-dimensional real-world application problem arising in combustion simulation show that SHEBO is an efficient solver that outperforms the other methods for many test problems.

Optimization of processor allocation for domain decomposed Monte Carlo calculations

Load imbalance plagues domain decomposed Monte Carlo calculations when sources are not uniform. Parallel efficiency for domain decomposed Monte Carlo transport calculations improves through a nonuniform allocation of processors over subdomains. We optimize the allocation with runtime diagnostics collected during a calibration step, then complete the full calculation. The diagnostic-based approach is compared to implicit filtering, an optimization algorithm for bound constrained noisy optimization problems. We consider both forward and hybrid radiation transport calculations to measure performance.

Running Time Analysis of the ( $$1+1$$ 1 + 1 )-EA for OneMax and LeadingOnes Under Bit-Wise Noise

In many real-world optimization problems, the objective function evaluation is subject to noise, and we cannot obtain the exact objective value. Evolutionary algorithms (EAs), a type of general-purpose randomized optimization algorithm, have been shown to be able to solve noisy optimization problems well. However, previous theoretical analyses of EAs mainly focused on noise-free optimization, which makes the theoretical understanding largely insufficient for the noisy case. Meanwhile, the few existing theoretical studies under noise often considered the one-bit noise model, which flips a randomly chosen bit of a solution before evaluation; while in many realistic applications, several bits of a solution can be changed simultaneously. In this paper, we study a natural extension of one-bit noise, the bit-wise noise model, which independently flips each bit of a solution with some probability. We analyze the running time of the ( $$1+1$$ )-EA solving OneMax and LeadingOnes under bit-wise noise for the first time, and derive the ranges of the noise level for polynomial and super-polynomial running time bounds. The analysis on LeadingOnes under bit-wise noise can be easily transferred to one-bit noise, and improves the previously known results. Since our analysis discloses that the ( $$1+1$$ )-EA can be efficient only under low noise levels, we also study whether the sampling strategy can bring robustness to noise. We prove that using sampling can significantly increase the largest noise level allowing a polynomial running time, that is, sampling is robust to noise.

Realization of learning induced self-adaptive sampling in noisy optimization

A novel method is proposed in this paper to extend the traditional artificial bee colony algorithm to competently optimize noisy objective function. The first strategy is concerned with adaptive selection of sample size for periodic evaluation of noisy objective function of a trial solution to reduce the possible risk of promoting noisy solutions over generations. The sample size of a given solution here is adapted based on the level of noise contamination in its local neighborhood. In this paper, the noise distribution in the locality of a candidate solution is captured by its local neighborhood fitness variance. The paper also recommends a new stratagem to identify local neighborhoods of trial solutions of a noisy optimization problem. The policy to adapt sample size here is realized using stochastic learning automata. The second strategy deals with evaluating the effective fitness of a trial solution based on a measure of relative distance between its noisy fitness samples. The adopted policy helps to capture the uncertainty in measure of fitness samples of a given solution due to noise contamination. Unlike conventional averaging, the proposed strategy aims at giving primacy to the true samples than the noisy samples of a trial solution during estimating its effective fitness. Last, a probabilistic selection strategy is introduced to diminish the jeopardizing effect of noise during deterministic selection of trial solutions to form next generation population. Computer simulations undertaken on the noisy versions of 38 benchmark functions reveal that the proposed algorithm surpasses its competitors with respect to function error value.