Balancing common disease treatment and epidemic control is a key objective of medical supplies procurement in hospitals during a pandemic such as COVID-19. This problem can be formulated as a bi-objective optimization problem for simultaneously optimizing the effects of common disease treatment and epidemic control. However, due to the large number of supplies, difficulties in evaluating the effects, and the strict budget constraint, it is difficult for existing evolutionary multiobjective algorithms to efficiently approximate the Pareto front of the problem. In this paper, we present an approach that first transforms the original high-dimensional, constrained multiobjective optimization problem to a low-dimensional, unconstrained multiobjective optimization problem, and then evaluates each solution to the transformed problem by solving a set of simple single-objective optimization subproblems, such that the problem can be efficiently solved by existing evolutionary multiobjective algorithms. We applied the transform-and-divide evolutionary optimization approach to six hospitals in Zhejiang Province, China, during the peak of COVID-19. Results showed that the proposed approach exhibits significantly better performance than that of directly solving the original problem. Our study has also shown that transform-and-divide evolutionary optimization based on problem-specific knowledge can be an efficient solution approach to many other complex problems and, therefore, enlarge the application field of evolutionary algorithms.
Tasks in multi-task learning often correlate, conflict, or even compete with each other. As a result, a single solution that is optimal for all tasks rarely exists. Recent papers introduced the concept of Pareto optimality to this field and directly cast multi-task learning as multi-objective optimization problems, but solutions returned by existing methods are typically finite, sparse, and discrete. We present a novel, efficient method that generates locally continuous Pareto sets and Pareto fronts, which opens up the possibility of continuous analysis of Pareto optimal solutions in machine learning problems. We scale up theoretical results in multi-objective optimization to modern machine learning problems by proposing a sample-based sparse linear system, for which standard Hessian-free solvers in machine learning can be applied. We compare our method to the state-of-the-art algorithms and demonstrate its usage of analyzing local Pareto sets on various multi-task classification and regression problems. The experimental results confirm that our algorithm reveals the primary directions in local Pareto sets for trade-off balancing, finds more solutions with different trade-offs efficiently, and scales well to tasks with millions of parameters.
Recently, the discretization of decision and objective spaces has been discussed in the literature. In some studies, it is shown that the decision space discretization improves the performance of evolutionary multi-objective optimization (EMO) algorithms on continuous multi-objective test problems. In other studies, it is shown that the objective space discretization improves the performance on combinatorial multi-objective problems. However, the effect of the simultaneous discretization of both spaces has not been examined in the literature. In this paper, we examine the effects of the decision space discretization, objective space discretization and simultaneous discretization on the performance of NSGA-II through computational experiments on the DTLZ and WFG problems. Using various settings about the number of decision variables and the number of objectives, our experiments are performed on four types of problems: standard problems, large-scale problems, many-objective problems, and large-scale many-objective problems. We show that the decision space discretization has a positive effect for large-scale problems and the objective space discretization has a positive effect for many-objective problems. We also show the discretization of both spaces is useful for large-scale many-objective problems.
Machine Learning (ML) algorithms are vulnerable to poisoning attacks, where a fraction of the training data is manipulated to deliberately degrade the algorithms' performance. Optimal poisoning attacks, which can be formulated as bilevel optimisation problems, help to assess the robustness of learning algorithms in worst-case scenarios. However, current attacks against algorithms with hyperparameters typically assume that these hyperparameters remain constant ignoring the effect the attack has on them. We show that this approach leads to an overly pessimistic view of the robustness of the algorithms. We propose a novel optimal attack formulation that considers the effect of the attack on the hyperparameters by modelling the attack as a multiobjective bilevel optimisation problem. We apply this novel attack formulation to ML classifiers using $L_2$ regularisation and show that, in contrast to results previously reported, $L_2$ regularisation enhances the stability of the learning algorithms and helps to mitigate the attacks. Our empirical evaluation on different datasets confirms the limitations of previous strategies, evidences the benefits of using $L_2$ regularisation to dampen the effect of poisoning attacks and shows how the regularisation hyperparameter increases with the fraction of poisoning points.
Dominance move (DoM) is a binary quality indicator to compare solution sets in multiobjective optimization. The indicator allows a more natural and intuitive relation when comparing solution sets. It is Pareto compliant and does not demand any parameters or reference sets. In spite of its advantages, the combinatorial calculation nature is a limitation. The original formulation presents an efficient method to calculate it in a biobjective case only. This work presents an assignment formulation to calculate DoM in problems with three objectives or more. Some initial experiments, in the biobjective space, were done to present the model correctness. Next, other experiments, using three dimensions, were also done to show how DoM could be compared with other indicators: inverted generational distance (IGD) and hypervolume (HV). Results show the assignment formulation for DoM is valid for more than three objectives. However, there are some strengths and weaknesses, which are discussed and detailed. Some notes, considerations, and future research paths conclude this work.
With modern requirements, there is an increasing tendency of considering multiple objectives/criteria simultaneously in many Software Engineering (SE) scenarios. Such a multi-objective optimization scenario comes with an important issue -- how to evaluate the outcome of optimization algorithms, which typically is a set of incomparable solutions (i.e., being Pareto non-dominated to each other). This issue can be challenging for the SE community, particularly for practitioners of Search-Based SE (SBSE). On one hand, multi-objective optimization could still be relatively new to SE/SBSE researchers, who may not be able to identify the right evaluation methods for their problems. On the other hand, simply following the evaluation methods for general multi-objective optimization problems may not be appropriate for specific SE problems, especially when the problem nature or decision maker's preferences are explicitly/implicitly available. This has been well echoed in the literature by various inappropriate/inadequate selection and inaccurate/misleading use of evaluation methods. In this paper, we first carry out a systematic and critical review of quality evaluation for multi-objective optimization in SBSE. We survey 717 papers published between 2009 and 2019 from 36 venues in seven repositories, and select 95 prominent studies, through which we identify five important but overlooked issues in the area. We then conduct an in-depth analysis of quality evaluation indicators/methods and general situations in SBSE, which, together with the identified issues, enables us to codify a methodological guidance for selecting and using evaluation methods in different SBSE scenarios.
Recently, a deep reinforcement learning method is proposed to solve multiobjective optimization problem. In this method, the multiobjective optimization problem is decomposed to a number of single-objective optimization subproblems and all the subproblems are optimized in a collaborative manner. Each subproblem is modeled with a pointer network and the model is trained with reinforcement learning. However, when pointer network extracts the features of an instance, it ignores the underlying structure information of the input nodes. Thus, this paper proposes a multiobjective deep reinforcement learning method using decomposition and attention model to solve multiobjective optimization problem. In our method, each subproblem is solved by an attention model, which can exploit the structure features as well as node features of input nodes. The experiment results on multiobjective travelling salesman problem show the proposed algorithm achieves better performance compared with the previous method.
Neural architecture search (NAS) has been very successful at outperforming human-designed convolutional neural networks (CNN) in accuracy, and when hardware information is present, latency as well. However, NAS-designed CNNs typically have a complicated topology, therefore, it may be difficult to design a custom hardware (HW) accelerator for such CNNs. We automate HW-CNN codesign using NAS by including parameters from both the CNN model and the HW accelerator, and we jointly search for the best model-accelerator pair that boosts accuracy and efficiency. We call this Codesign-NAS. In this paper we focus on defining the Codesign-NAS multiobjective optimization problem, demonstrating its effectiveness, and exploring different ways of navigating the codesign search space. For CIFAR-10 image classification, we enumerate close to 4 billion model-accelerator pairs, and find the Pareto frontier within that large search space. This allows us to evaluate three different reinforcement-learning-based search strategies. Finally, compared to ResNet on its most optimal HW accelerator from within our HW design space, we improve on CIFAR-100 classification accuracy by 1.3% while simultaneously increasing performance/area by 41% in just~1000 GPU-hours of running Codesign-NAS.
Dominance move (DoM) is a binary quality indicator that can be used in multiobjective optimization. It can compare solution sets while representing some important features such as convergence, spread, uniformity, and cardinality. DoM has an intuitive concept and considers the minimum move of one set needed to weakly Pareto dominate the other set. Despite the aforementioned properties, DoM is hard to calculate. The original formulation presents an efficient and exact method to calculate it in a biobjective case only. This work presents a new approach to calculate and extend DoM to deal with three or more objectives. The idea is to use a mixed integer programming (MIP) approach to calculate DoM. Some initial experiments, in the biobjective space, were done to verify the model correctness. Furthermore, other experiments, using three, five, and ten objective functions were done to show how the model behaves in higher dimensional cases. Algorithms such as IBEA, MOEAD, NSGAIII, NSGAII, and SPEA2 were used to generate the solution sets, however any other algorithms could be used with DoM indicator. The results have confirmed the effectiveness of the MIP DoM in problems with more than three objective functions. Final notes, considerations, and future research are discussed to exploit some solution sets particularities and improve the model and its use for other situations.
Deep reinforcement learning approaches have shown impressive results in a variety of different domains, however, more complex heterogeneous architectures such as world models require the different neural components to be trained separately instead of end-to-end. While a simple genetic algorithm recently showed end-to-end training is possible, it failed to solve a more complex 3D task. This paper presents a method called Deep Innovation Protection (DIP) that addresses the credit assignment problem in training complex heterogenous neural network models end-to-end for such environments. The main idea behind the approach is to employ multiobjective optimization to temporally reduce the selection pressure on specific components in multi-component network, allowing other components to adapt. We investigate the emergent representations of these evolved networks, which learn to predict properties important for the survival of the agent, without the need for a specific forward-prediction loss.
Multi-task learning is a powerful method for solving multiple correlated tasks simultaneously. However, it is often impossible to find one single solution to optimize all the tasks, since different tasks might conflict with each other. Recently, a novel method is proposed to find one single Pareto optimal solution with good trade-off among different tasks by casting multi-task learning as multiobjective optimization. In this paper, we generalize this idea and propose a novel Pareto multi-task learning algorithm (Pareto MTL) to find a set of well-distributed Pareto solutions which can represent different trade-offs among different tasks. The proposed algorithm first formulates a multi-task learning problem as a multiobjective optimization problem, and then decomposes the multiobjective optimization problem into a set of constrained subproblems with different trade-off preferences. By solving these subproblems in parallel, Pareto MTL can find a set of well-representative Pareto optimal solutions with different trade-off among all tasks. Practitioners can easily select their preferred solution from these Pareto solutions, or use different trade-off solutions for different situations. Experimental results confirm that the proposed algorithm can generate well-representative solutions and outperform some state-of-the-art algorithms on many multi-task learning applications.
Singleton interval type-2 fuzzy logic systems (FLSs)have been widely applied in several real-world applications, whereit was shown that the singleton interval type-2 FLSs outperformtheir singleton type-1 counterparts in applications with high un-certainty levels. However, one of the main criticisms of singletoninterval type-2 FLSs is the fact that they outperform singletontype-1 FLSs solely based on their use of extra degrees of freedom(extra parameters) and that type-1 FLSs with a sufficiently largenumber of parameters may provide the same performance as in-terval type-2 FLSs. In addition, most works on type-2 FLSs onlycompare their results with singleton type-1 FLSs but fail to con-sider nonsingleton type-1 systems. In this paper, we aim to directlyaddress and investigate this criticism. In order to do so, we willperform a comparative study between optimized singleton type-1,nonsingleton type-1, and singleton interval type-2 FLSs under thepresence of noise. We will also present a multiobjective evolution-ary algorithm (MOEA) for the optimization of singleton type-1,nonsingleton type-1, and singleton interval type-2 fuzzy systems forfunction approximation problems. The MOEA will aim to satisfytwo objectives to maximize the accuracy of the FLS and minimizethe number of rules in the FLS, thus improving its interpretability.Furthermore, we will present a methodology to obtain “optimal”consequents for the FLSs. Hence, this paper has two main con-tributions: First, it provides a common methodology to learn thethree types of FLSs (i.e., singleton type-1, nonsingleton type-1, andsingleton interval type-2 FLSs) from data samples. The secondcontribution is the creation of a common framework for the com-parison of type-1 and type-2 FLSs that allows us to address theaforementioned criticism. We provide details of a series of experi-ments and include statistical analysis showing that the type-2 FLSis able to handle higher levels of noise than its nonsingleton andsingleton type-1 counterparts.
The goal of lossy data compression is to reduce the storage cost of a data set $X$ while retaining as much information as possible about something ($Y$) that you care about. For example, what aspects of an image $X$ contain the most information about whether it depicts a cat? Mathematically, this corresponds to finding a mapping $X\to Z\equiv f(X)$ that maximizes the mutual information $I(Z,Y)$ while the entropy $H(Z)$ is kept below some fixed threshold. We present a method for mapping out the Pareto frontier for classification tasks, reflecting the tradeoff between retained entropy and class information. We first show how a random variable $X$ (an image, say) drawn from a class $Y\in\{1,...,n\}$ can be distilled into a vector $W=f(X)\in \mathbb{R}^{n-1}$ losslessly, so that $I(W,Y)=I(X,Y)$; for example, for a binary classification task of cats and dogs, each image $X$ is mapped into a single real number $W$ retaining all information that helps distinguish cats from dogs. For the $n=2$ case of binary classification, we then show how $W$ can be further compressed into a discrete variable $Z=g_\beta(W)\in\{1,...,m_\beta\}$ by binning $W$ into $m_\beta$ bins, in such a way that varying the parameter $\beta$ sweeps out the full Pareto frontier, solving a generalization of the Discrete Information Bottleneck (DIB) problem. We argue that the most interesting points on this frontier are "corners" maximizing $I(Z,Y)$ for a fixed number of bins $m=2,3...$ which can be conveniently be found without multiobjective optimization. We apply this method to the CIFAR-10, MNIST and Fashion-MNIST datasets, illustrating how it can be interpreted as an information-theoretically optimal image clustering algorithm.
The state-of-the-art approaches employ approximate computing to reduce the energy consumption of DNN hardware. Approximate DNNs then require extensive retraining afterwards to recover from the accuracy loss caused by the use of approximate operations. However, retraining of complex DNNs does not scale well. In this paper, we demonstrate that efficient approximations can be introduced into the computational path of DNN accelerators while retraining can completely be avoided. ALWANN provides highly optimized implementations of DNNs for custom low-power accelerators in which the number of computing units is lower than the number of DNN layers. First, a fully trained DNN is converted to operate with 8-bit weights and 8-bit multipliers in convolutional layers. A suitable approximate multiplier is then selected for each computing element from a library of approximate multipliers in such a way that (i) one approximate multiplier serves several layers, and (ii) the overall classification error and energy consumption are minimized. The optimizations including the multiplier selection problem are solved by means of a multiobjective optimization NSGA-II algorithm. In order to completely avoid the computationally expensive retraining of DNN, which is usually employed to improve the classification accuracy, we propose a simple weight updating scheme that compensates the inaccuracy introduced by employing approximate multipliers. The proposed approach is evaluated for two architectures of DNN accelerators with approximate multipliers from the open-source "EvoApprox" library. We report that the proposed approach saves 30% of energy needed for multiplication in convolutional layers of ResNet-50 while the accuracy is degraded by only 0.6%. The proposed technique and approximate layers are available as an open-source extension of TensorFlow at https://github.com/ehw-fit/tf-approximate.
Scalarizing functions have been widely used to convert a multiobjective optimization problem into a single objective optimization problem. However, their use in solving (computationally) expensive multi- and many-objective optimization problems in Bayesian multiobjective optimization is scarce. Scalarizing functions can play a crucial role on the quality and number of evaluations required when doing the optimization. In this article, we study and review 15 different scalarizing functions in the framework of Bayesian multiobjective optimization and build Gaussian process models (as surrogates, metamodels or emulators) on them. We use expected improvement as infill criterion (or acquisition function) to update the models. In particular, we compare different scalarizing functions and analyze their performance on several benchmark problems with different number of objectives to be optimized. The review and experiments on different functions provide useful insights when using and selecting a scalarizing function when using a Bayesian multiobjective optimization method.
Tuning parameters is an important step for the application of metaheuristics to problem classes of interest. In this work we present a tuning framework based on the sequential optimization of perturbed regression models. Besides providing algorithm configurations with good expected performance, the proposed methodology can also provide insights on the relevance of each parameter and their interactions, as well as models of expected algorithm performance for a given problem class, conditional on the parameter values. A test case is presented for the tuning of six parameters of a decomposition-based multiobjective optimization algorithm, in which an instantiation of the proposed framework is compared against the results obtained by the most recent version the Iterated Racing (Irace) procedure. The results suggest that the proposed approach returns solutions that are as good as those of Irace in terms of mean performance, with the advantage of providing more information on the relevance and effect of each parameter on the expected performance of the algorithm.
Most current state-of-the-art text-independent speaker verification systems take probabilistic linear discriminant analysis (PLDA) as their backend classifiers. The parameters of PLDA are often estimated by maximizing the objective function, which focuses on increasing the value of log-likelihood function, but ignoring the distinction between speakers. In order to better distinguish speakers, we propose a multi-objective optimization training for PLDA. Experiment results show that the proposed method has more than 10% relative performance improvement in both EER and MinDCF on the NIST SRE14 i-vector challenge dataset, and about 20% relative performance improvement in EER on the MCE18 dataset.
Given a set of human's decisions that are observed, inverse optimization has been developed and utilized to infer the underlying decision making problem. The majority of existing studies assumes that the decision making problem is with a single objective function, and attributes data divergence to noises, errors or bounded rationality, which, however, could lead to a corrupted inference when decisions are tradeoffs among multiple criteria. In this paper, we take a data-driven approach and design a more sophisticated inverse optimization formulation to explicitly infer parameters of a multiobjective decision making problem from noisy observations. This framework, together with our mathematical analyses and advanced algorithm developments, demonstrates a strong capacity in estimating critical parameters, decoupling "interpretable" components from noises or errors, deriving the denoised \emph{optimal} decisions, and ensuring statistical significance. In particular, for the whole decision maker population, if suitable conditions hold, we will be able to understand the overall diversity and the distribution of their preferences over multiple criteria, which is important when a precise inference on every single decision maker is practically unnecessary or infeasible. Numerical results on a large number of experiments are reported to confirm the effectiveness of our unique inverse optimization model and the computational efficacy of the developed algorithms.
The processes occurring in climatic change evolution and their variations play a major role in environmental engineering. Different techniques are used to model the relationship between temperatures, dew point and relative humidity. Gene expression programming is capable of modelling complex realities with great accuracy, allowing, at the same time, the extraction of knowledge from the evolved models compared to other learning algorithms. This research aims to use Gene Expression Programming for modelling of dew point. Generally, accuracy of the model is the only objective used by selection mechanism of GEP. This will evolve large size models with low training error. To avoid this situation, use of multiple objectives, like accuracy and size of the model are preferred by Genetic Programming practitioners. Multi-objective problem finds a set of solutions satisfying the objectives given by decision maker. Multiobjective based GEP will be used to evolve simple models. Various algorithms widely used for multi objective optimization like NSGA II and SPEA 2 are tested for different test cases. The results obtained thereafter gives idea that SPEA 2 is better algorithm compared to NSGA II based on the features like execution time, number of solutions obtained and convergence rate. Thus compared to models obtained by GEP, multi-objective algorithms fetch better solutions considering the dual objectives of fitness and size of the equation. These simple models can be used to predict dew point.
The present survey provides the state-of-the-art of research, copiously devoted to Evolutionary Approach (EAs) for clustering exemplified with a diversity of evolutionary computations. The Survey provides a nomenclature that highlights some aspects that are very important in the context of evolutionary data clustering. The paper missions the clustering trade-offs branched out with wide-ranging Multi Objective Evolutionary Approaches (MOEAs) methods. Finally, this study addresses the potential challenges of MOEA design and data clustering, along with conclusions and recommendations for novice and researchers by positioning most promising paths of future research. MOEAs have substantial success across a variety of MOP applications, from pedagogical multifunction optimization to real-world engineering design. The survey paper noticeably organizes the developments witnessed in the past three decades for EAs based metaheuristics to solve multiobjective optimization problems (MOP) and to derive significant progression in ruling high quality elucidations in a single run. Data clustering is an exigent task, whose intricacy is caused by a lack of unique and precise definition of a cluster. The discrete optimization problem uses the cluster space to derive a solution for Multiobjective data clustering. Discovery of a majority or all of the clusters (of illogical shapes) present in the data is a long-standing goal of unsupervised predictive learning problems or exploratory pattern analysis.