Gradient-free Methods Research Articles

Causality-based counterfactual explanation for classification models

Counterfactual explanation is one branch of interpretable machine learning that produces a perturbation sample to change the model’s original decision. The generated samples can act as a recommendation for end-users to achieve their desired outputs. Most of the current counterfactual explanation approaches are the gradient-based method, which can only optimize the differentiable loss functions with continuous variables. Accordingly, the gradient-free methods are proposed to handle the categorical variables, which however have several major limitations: (1) causal relationships among features are typically ignored when generating the counterfactuals, possibly resulting in impractical guidelines for decision-makers; (2) the counterfactual explanation algorithm requires a great deal of effort into parameter tuning for determining the optimal weight for each loss functions which must be conducted repeatedly for different datasets and settings. In this work, to address the above limitations, we propose a prototype-based counterfactual explanation framework (ProCE). ProCE is capable of preserving the causal relationship underlying the features of the counterfactual data. In addition, we design a novel gradient-free optimization based on the multi-objective genetic algorithm that generates the counterfactual explanations for the mixed-type of continuous and categorical features. Numerical experiments demonstrate that our method compares favorably with state-of-the-art methods and therefore is applicable to existing prediction models. All the source codes and data are available at https://github.com/tridungduong16/multiobj-scm-cf.

Stabilization of SIMPLE-like RANS solvers for computing accurate gradients using the complex-step derivative method

The Reynolds-averaged Navier-Stokes (RANS) approach in Computational Fluid Dynamics (CFD) is increasingly vital for aerodynamic design across wind turbines, gas turbines, aircraft, and rotorcraft. Enhancing the process through CFD-based design optimization requires numerous design variables, favoring gradient-based methods for better scalability than gradient-free methods. This research uses numerical differentiation techniques, particularly the complex-step derivative method, to compute gradients. Challenges arise during aerodynamic shape optimization when unconventional shapes disrupt RANS solver assumptions, causing convergence failures. These failures undermine optimization, prompting the need to enhance solver convergence for robust optimization. The modified-Boostconv method is a residual recombination method bolstering unstable eigenvalues to stabilize convergence of iterative solvers for nonlinear systems of equations. This study extends the modified-Boostconv method to combine it with the complex-step derivative technique, creating robust optimizations via RANS-CFD with accurate gradients, even for these numerically unstable cases. The main issue encountered during the complexification of the modified-Boostconv method is how to correctly ensure that the model reduction leading to the least-squares problem satisfies the complex-step derivative method. We test whether the dot product operation involved in the model reduction should be a Hermitian or non-Hermitian inner product. The problem is first tested for a simple analytical case using the logistic equation in combination with the modified-Boostconv method. It shows that the non-Hermitian inner product should be used. This is also confirmed with a similar study using the RANS solver.

Convergence guarantees for forward gradient descent in the linear regression model

Renewed interest in the relationship between artificial and biological neural networks motivates the study of gradient-free methods. Considering the linear regression model with random design, we theoretically analyze in this work the biologically motivated (weight-perturbed) forward gradient scheme that is based on random linear combination of the gradient. If d denotes the number of parameters and k the number of samples, we prove that the mean squared error of this method converges for k≳d2log(d) with rate d2log(d)/k. Compared to the dimension dependence d for stochastic gradient descent, an additional factor dlog(d) occurs.

Decentralized Gradient-Free Methods for Stochastic Non-smooth Non-convex Optimization

We consider decentralized gradient-free optimization of minimizing Lipschitz continuous functions that satisfy neither smoothness nor convexity assumption. We propose two novel gradient-free algorithms, the Decentralized Gradient-Free Method (DGFM) and its variant, the Decentralized Gradient-Free Method+ (DGFM+). Based on the techniques of randomized smoothing and gradient tracking, DGFM requires the computation of the zeroth-order oracle of a single sample in each iteration, making it less demanding in terms of computational resources for individual computing nodes. Theoretically, DGFM achieves a complexity of O(d^(3/2)δ^(-1)ε^(-4)) for obtaining a (δ,ε)-Goldstein stationary point. DGFM+, an advanced version of DGFM, incorporates variance reduction to further improve the convergence behavior. It samples a mini-batch at each iteration and periodically draws a larger batch of data, which improves the complexity to O(d^(3/2)δ^(-1)ε^(-3)). Moreover, experimental results underscore the empirical advantages of our proposed algorithms when applied to real-world datasets.

Gradient-free Federated Learning Methods with l1 and l2-randomization for Non-smooth Convex Stochastic Optimization Problems

This paper studies non-smooth problems of convex stochastic optimization. Using the smoothing technique based on the replacement of the function value at the considered point by the averaged function value over a ball (in l1-norm or l2-norm) of a small radius centered at this point, and then the original problem is reduced to a smooth problem (whose Lipschitz constant of the gradient is inversely proportional to the radius of the ball). An essential property of the smoothing used is the possibility of calculating an unbiased estimation of the gradient of a smoothed function based only on realizations of the original function. The obtained smooth stochastic optimization problem is proposed to be solved in a distributed federated learning architecture (the problem is solved in parallel: nodes make local steps, e.g. stochastic gradient descent, then communicate—all with all, then all this is repeated). The goal of the article is to build on the basis of modern achievements in the field of gradient–free non-smooth optimization and in the field of federated learning gradient-free methods for solving problems of non-smooth stochastic optimization in the architecture of federated learning.

Gradient-free methods for non-smooth convex stochastic optimization with heavy-tailed noise on convex compact

Gradient-free methods for non-smooth convex stochastic optimization with heavy-tailed noise on convex compact

Benchmarking of different optimizers in the variational quantum algorithms for applications in quantum chemistry.

Classical optimizers play a crucial role in determining the accuracy and convergence of variational quantum algorithms; leading algorithms use a near-term quantum computer to solve the ground state properties of molecules, simulate dynamics of different quantum systems, and so on. In the literature, many optimizers, each having its own architecture, have been employed expediently for different applications. In this work, we consider a few popular and efficacious optimizers and assess their performance in variational quantum algorithms for applications in quantum chemistry in a realistic noisy setting. We benchmark the optimizers with critical analysis based on quantum simulations of simple molecules, such as hydrogen, lithium hydride, beryllium hydride, water, and hydrogen fluoride. The errors in the ground state energy, dissociation energy, and dipole moment are the parameters used as yardsticks. All the simulations were carried out with an ideal quantum circuit simulator, a noisy quantum circuit simulator, and finally a noisy simulator with noise embedded from the IBM Cairo quantum device to understand the performance of the classical optimizers in ideal and realistic quantum environments. We used the standard unitary coupled cluster ansatz for simulations, and the number of qubits varied from two starting from the hydrogen molecule to ten qubits in hydrogen fluoride. Based on the performance of these optimizers in the ideal quantum circuits, the conjugate gradient, limited-memory Broyden-Fletcher-Goldfarb-Shanno bound, and sequential least squares programming optimizers are found to be the best-performing gradient-based optimizers. While constrained optimization by linear approximation (COBYLA) and Powell's conjugate direction algorithm for unconstrained optimization (POWELL) perform most efficiently among the gradient-free methods, in noisy quantum circuit conditions, simultaneous perturbation stochastic approximation, POWELL, and COBYLA are among the best-performing optimizers.

Genetic algorithms as classical optimizer for the Quantum Approximate Optimization Algorithm

Optimization is one of the research areas where quantum computing could bring significant benefits. In this scenario, a hybrid quantum–classical variational algorithm, the Quantum Approximate Optimization Algorithm (QAOA), is receiving much attention for its potential to efficiently solve combinatorial optimization problems. This approach works by using a classical optimizer to identify appropriate parameters of a problem-dependent quantum circuit, which ultimately performs the optimization process. Unfortunately, learning the most appropriate QAOA circuit parameters is a complex task that is affected by several issues, such as search landscapes characterized by many local optima. Moreover, gradient-based optimizers, which have been pioneered in this context, tend to waste quantum computing resources. Therefore, gradient-free approaches are emerging as promising methods to address this parameter-setting task. Following this trend, this paper proposes, for the first time, the use of genetic algorithms as gradient-free methods for optimizing the QAOA circuit. The proposed evolutionary approach has been evaluated in solving the MaxCut problem for graphs with 5 to 9 nodes on a noisy quantum device. As the results show, the proposed genetic algorithm statistically outperforms the state-of-the-art gradient-free optimizers by achieving solutions with a better approximation ratio.

A natural Hessian approximation for ensemble based optimization

A key challenge in reservoir management and other fields of engineering involves optimizing a nonlinear function iteratively. Due to the lack of available gradients in commercial reservoir simulators the attention over the last decades has been on gradient free methods or gradient approximations. In particular, the ensemble-based optimization has gained popularity over the last decade due to its simplicity and efficient implementation when considering an ensemble of reservoir models. Typically, a regression type gradient approximation is used in a backtracking or line search setting. This paper introduces an approximation of the Hessian utilizing a Monte Carlo approximation of the natural gradient with respect to the covariance matrix. This Hessian approximation can further be implemented in a trust region approach in order to improve the efficiency of the algorithm. The advantages of using such approximations are demonstrated by testing the proposed algorithm on the Rosenbrock function and on a synthetic reservoir field.

Monte-Carlo estimation of geometric sensitivities in Solar Power Tower systems of flat mirrors

Optical optimizations for a Concentrated Solar Power (CSP) system are currently limited largely to gradient-free methods, since the gradient is hard to obtain by using the existing numerical optics tools available in the community. This article aims to build new algorithms of the Monte-Carlo type, which numerically estimate the gradient of power impacting the receiver with respect to the geometric parameters that characterize the geometric status of the heliostats in the heliostats field, for a Solar Power Tower (SPT) system.Physical models will be built for the specific intensity and also for its derivatives to the geometric parameters of the heliostats, also called geometric sensitivities of intensity. Similar to the intensity but with their own models, they are regarded as physical quantities emitted, absorbed and reflected in the system. They carry the perturbations of intensity as information, corresponding to the relations between the geometric parameters and the physical events in the SPT system: blocking, spillage, shadowing, etc. These relations will be distinguished and discussed.Finally, not only the gradient but also the contributions of physical events (blocking, spillage, shadowing, etc.) to the gradient can be estimated for further sensitivity analysis and optimization processes.

Towards Differentiable Agent-Based Simulation

Simulation-based optimization using agent-based models is typically carried out under the assumption that the gradient describing the sensitivity of the simulation output to the input cannot be evaluated directly. To still apply gradient-based optimization methods, which efficiently steer the optimization towards a local optimum, gradient estimation methods can be employed. However, many simulation runs are needed to obtain accurate estimates if the input dimension is large. Automatic differentiation (AD) is a family of techniques to compute gradients of general programs directly. Here, we explore the use of AD in the context of time-driven agent-based simulations. By substituting common discrete model elements such as conditional branching with smooth approximations, we obtain gradient information across discontinuities in the model logic. On the examples of a synthetic grid-based model, an epidemics model, and a microscopic traffic model, we study the fidelity and overhead of the differentiable simulations as well as the convergence speed and solution quality achieved by gradient-based optimization compared with gradient-free methods. In traffic signal timing optimization problems with high input dimension, the gradient-based methods exhibit substantially superior performance. A further increase in optimization progress is achieved by combining gradient-free and gradient-based methods. We demonstrate that the approach enables gradient-based training of neural network-controlled simulation entities embedded in the model logic. Finally, we show that the performance overhead of differentiable agent-based simulations can be reduced substantially by exploiting sparsity in the model logic.

Policy Gradient Approach to Compilation of Variational Quantum Circuits

We propose a method for finding approximate compilations of quantum unitary transformations, based on techniques from policy gradient reinforcement learning. The choice of a stochastic policy allows us to rephrase the optimization problem in terms of probability distributions, rather than variational gates. In this framework, the optimal configuration is found by optimizing over distribution parameters, rather than over free angles. We show numerically that this approach can be more competitive than gradient-free methods, for a comparable amount of resources, both for noiseless and noisy circuits. Another interesting feature of this approach to variational compilation is that it does not need a separate register and long-range interactions to estimate the end-point fidelity, which is an improvement over methods which rely on the Hilbert-Schmidt test. We expect these techniques to be relevant for training variational circuits in other contexts.

Evolutionary bagging for ensemble learning

Ensemble learning has gained success in machine learning with major advantages over other learning methods. Bagging is a prominent ensemble learning method that creates subgroups of data, known as bags, that are trained by individual machine learning methods such as decision trees. Random forest is a prominent example of bagging with additional features in the learning process. Evolutionary algorithms have been prominent for optimisation problems and also been used for machine learning. Evolutionary algorithms are gradient-free methods that work with a population of candidate solutions that maintain diversity for creating new solutions. In conventional bagged ensemble learning, the bags are created once and the content, in terms of the training examples, are fixed over the learning process. In our paper, we propose evolutionary bagged ensemble learning, where we utilise evolutionary algorithms to evolve the content of the bags in order to iteratively enhance the ensemble by providing diversity in the bags. The results show that our evolutionary ensemble bagging method outperforms conventional ensemble methods (bagging and random forests) for several benchmark datasets under certain constraints. We find that evolutionary bagging can inherently sustain a diverse set of bags without reduction in performance accuracy.

Topological dimensionality reduction-based machine learning for efficient gradient-free 3D topology optimization

Powerful gradient-free topology optimization methods are needed for structural design concerning complex responses. In this paper, a novel gradient-free optimization method is proposed by integrating the material-field series expansion topological parameterization and the deep neural networks, providing two-fold advances: firstly, it generally reduces the massive topological design variables to fewer than 200, while keeps the capability to represent relative complex 3D topologies and clear boundaries; secondly, by constructing a sequential neural network surrogate model, it sufficiently explores the reduced design space and is capable of handling multi-peak and discontinuous optimization problems. The effectiveness of this method is illustrated via several design problems, among which the optimized material effective bulk modulus achieves 98% of the H-S bound and the highly-nonlinear peak weld stress in a phone dropping process is decreased by 16.59%. This method reduces the computational time by 1–4 orders of magnitude compared with the coarse-mesh-based gradient-free methods, and it is the first time to successfully conduct gradient-free 3D topology optimization with thousands of finite elements. The method’s ease of implementation and compatibility with various simulation software, brings topology optimization into complex industrial applications and proves that gradient-free technology represents an effective optimization benchmark for improving structural performance.

Линейно сходящиеся безградиентные методы для минимизации параболической аппроксимации

Finding the global minimum of non-convex functions is one of the main and most difficult problems in modern optimization. In the first part of the paper, we consider a certain class of "good" non-convex functions that can be bounded above and below by a parabolic function. We show that using only the zeroth-order oracle, one can obtain the linear speed $\log \left(\frac{1}{\varepsilon}\right)$ of finding the global minimum on a cube. The second part of the paper looks at the nonconvex problem in a slightly different way. We assume that minimizing the quadratic function, but at the same time we have access to a zeroth-order oracle with noise and this noise is proportional to the distance to the solution. Dealing with such noise assumptions for gradient-free methods is new in the literature. We show that here it is also possible to achieve the linear rate of convergence.

Gradient-Free Optimization in Thermoacoustics: Application to a Low-Order Model

Abstract Machine learning and automatized routines for parameter optimization have experienced a surge in development in the past years, mostly caused by the increasing availability of computing capacity. Gradient-free optimization can avoid cumbersome theoretical studies as input parameters are purely adapted based on output data. As no knowledge about the objective function is provided to the algorithms, this approach might reveal unconventional solutions to complex problems that were out of scope of classical solution strategies. In this study, the potential of these optimization methods on thermoacoustic problems is examined. The optimization algorithms are applied to a generic low-order thermoacoustic can-combustor model with several fuel injectors at different locations. We use three optimization algorithms – the well established downhill simplex method, the recently proposed explorative gradient method, and an evolutionary algorithm – to find optimal fuel distributions across the fuel lines while maintaining the amount of consumed fuel constant. The objective is to have minimal pulsation amplitudes. We compare the results and efficiency of the gradient-free algorithms. Additionally, we employ model-based linear stability analysis to calculate the growth rates of the dominant thermoacoustic modes. This allows us to highlight general and thermoacoustic-specific features of the optimization methods and results. The findings of this study show the potential of gradient-free optimization methods on combustor design for tackling thermoacoustic problems, and motivate further research in this direction.

Evolutionary Machine Learning With Minions: A Case Study in Feature Selection

Many decisions in a machine learning (ML) pipeline involve nondifferentiable and discontinuous objectives and search spaces. Examples include feature selection, model selection, and hyperparameter tuning, where candidate solutions in an outer optimization loop must be evaluated via a learning subsystem. Evolutionary algorithms (EAs) are prominent gradient-free methods to handle such tasks. However, EAs are known to pose steep computational challenges, especially when dealing with large-instance datasets. As opposed to prior works that often fall back on parallel computing hardware to resolve this big data problem of EAs, in this article, we propose a novel algorithm-centric solution based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evolutionary multitasking</i> . Our approach involves the creation of a band of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">minions</i> , i.e., small data proxies to the main target task, that are constructed by subsampling a fraction of the large dataset. We then combine the minions with the main task in a single multitask optimization framework, boosting evolutionary search by using small data to quickly optimize for the large dataset. Our key algorithmic contribution in this setting is to allocate computational resources to each of the tasks in a principled manner. The article considers wrapper-based feature selection as an illustrative case study of the broader idea of using multitasking to speedup outer loop evolutionary configurations of any ML subsystem. The experiments reveal that multitasking can indeed speedup baseline EAs, by more than 40% on some datasets.

Topology Optimization for Electromagnetics: A Survey

The development of technologies for the additive manufacturing, in particular of metallic materials, is offering the possibility of producing parts with complex geometries. This opens up to the possibility of using topological optimization methods for the design of electromagnetic devices. Hence, a wide variety of approaches, originally developed for solid mechanics, have recently become attractive also in the field of electromagnetics. The general distinction between gradient-based and gradient-free methods drives the structure of the paper, with the latter becoming particularly attractive in the last years due to the concepts of artificial neural networks. The aim of this paper is twofold. On one hand, the paper aims at summarizing and describing the state-of-art on topology optimization techniques while on the other it aims at showing how the latter methodologies developed in non-electromagnetic framework (e.g., solid mechanics field) can be applied for the optimization of electromagnetic devices. Discussions and comparisons are both supported by theoretical aspects and numerical results.

Adjoint-Based Unsteady Optimization of Turbomachinery Operating with Nonideal Compressible Flows

The lack of established optimal design guidelines for turbomachinery operating in the nonideal flow regime (e.g., organic Rankine cycle turbines, compressors, compressors for refrigeration systems) demands for effective and efficient automated design methods. Past research work focused on gradient-free methods applied to computational fluid-dynamic simulations. The application of the adjoint method is a cost-effective alternative as it enables gradient-based optimization irrespective of the number of design variables. This paper presents the application of a fully turbulent unsteady adjoint method for the automated design of multirow turbomachinery partly operating in the nonideal flow regime. The method therefore allows for the solution of constrained unsteady fluid-dynamic optimization problems, in which the thermodynamic properties of the working fluid need to be modeled by means of complex equations of state. The optimal designs computed with unsteady simulations obtained with the harmonic balance method are then compared with optimal design resulting from mixing-plane simulations. The method is applied to the optimization of 1) a two-dimensional turbine cascade subject to time-varying inlet conditions, and 2) a two-dimensional turbine stage of an organic Rankine cycle power system. The results demonstrate the importance of computing fluid properties using accurate thermodynamic models and of using unsteady simulations for shape optimization of these machines.

A comparison of various classical optimizers for a variational quantum linear solver

Variational Hybrid Quantum Classical Algorithms (VHQCAs) are a class of quantum algorithms intended to run on noisy intermediate-scale quantum (NISQ) devices. These algorithms employ a parameterized quantum circuit (ansatz) and a quantum-classical feedback loop. A classical device is used to optimize the parameters in order to minimize a cost function that can be computed far more efficiently on a quantum device. The cost function is constructed such that finding the ansatz parameters that minimize its value, solves some problem of interest. We focus specifically on the Variational Quantum Linear Solver (VQLS), and examine the effect of several gradient-free and gradient-based classical optimizers on performance. We focus on both the average rate of convergence of the classical optimizers studied, as well as the distribution of their average termination cost values, and how these are affected by noise. Our work demonstrates that realistic noise levels on NISQ devices present a challenge to the optimization process. All classical optimizers appear to be very negatively affected by the presence of realistic noise. If noise levels are significantly improved, there may be a good reason for preferring gradient-based methods in the future, which performed better than the gradient-free methods with the only shot-noise present. The gradient-free optimizers, Simultaneous Perturbation Stochastic Approximation (SPSA) and Powell's method, and the gradient-based optimizers, AMSGrad and BFGS performed the best in the noisy simulation, and appear to be less affected by noise than the rest of the methods. SPSA appears to be the best performing method. COBYLA, Nelder-Mead and Conjugate-Gradient methods appear to be the most heavily affected by noise, with even slight noise levels significantly impacting their performance.