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Register a new userResonator Networks outperform optimization methods at solving high-dimensional vector factorization
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Added by
Spencer Kent
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We develop theoretical foundations of Resonator Networks, a new type of recurrent neural network introduced in Frady et al. (2020) to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dimensional vectors, a Resonator Network can efficiently decompose the composite into these factors. We compare the performance of Resonator Networks against optimization-based methods, including Alternating Least Squares and several gradient-based algorithms, showing that Resonator Networks are superior in several important ways. This advantage is achieved by leveraging a combination of nonlinear dynamics and searching in superposition, by which estimates of the correct solution are formed from a weighted superposition of all possible solutions. While the alternative methods also search in superposition, the dynamics of Resonator Networks allow them to strike a more effective balance between exploring the solution space and exploiting local information to drive the network toward probable solutions. Resonator Networks are not guaranteed to converge, but within a particular regime they almost always do. In exchange for relaxing this guarantee of global convergence, Resonator Networks are dramatically more effective at finding factorizations than all alternative approaches considered.
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For a certain scaling of the initialization of stochastic gradient descent (SGD), wide neural networks (NN) have been shown to be well approximated by reproducing kernel Hilbert space (RKHS) methods. Recent empirical work showed that, for some classification tasks, RKHS methods can replace NNs without a large loss in performance. On the other hand, two-layers NNs are known to encode richer smoothness classes than RKHS and we know of special examples for which SGD-trained NN provably outperform RKHS. This is true even in the wide network limit, for a different scaling of the initialization. How can we reconcile the above claims? For which tasks do NNs outperform RKHS? If feature vectors are nearly isotropic, RKHS methods suffer from the curse of dimensionality, while NNs can overcome it by learning the best low-dimensional representation. Here we show that this curse of dimensionality becomes milder if the feature vectors display the same low-dimensional structure as the target function, and we precisely characterize this tradeoff. Building on these results, we present a model that can capture in a unified framework both behaviors observed in earlier work. We hypothesize that such a latent low-dimensional structure is present in image classification. We test numerically this hypothesis by showing that specific perturbations of the training distribution degrade the performances of RKHS methods much more significantly than NNs.
The main feature of the Dynamic Multi-objective Optimization Problems (DMOPs) is that optimization objective functions will change with times or environments. One of the promising approaches for solving the DMOPs is reusing the obtained Pareto optimal set (POS) to train prediction models via machine learning approaches. In this paper, we train an Incremental Support Vector Machine (ISVM) classifier with the past POS, and then the solutions of the DMOP we want to solve at the next moment are filtered through the trained ISVM classifier. A high-quality initial population will be generated by the ISVM classifier, and a variety of different types of population-based dynamic multi-objective optimization algorithms can benefit from the population. To verify this idea, we incorporate the proposed approach into three evolutionary algorithms, the multi-objective particle swarm optimization(MOPSO), Nondominated Sorting Genetic Algorithm II (NSGA-II), and the Regularity Model-based multi-objective estimation of distribution algorithm(RE-MEDA). We employ experiments to test these algorithms, and experimental results show the effectiveness.
Echo State Networks (ESNs) are recurrent neural networks that only train their output layer, thereby precluding the need to backpropagate gradients through time, which leads to significant computational gains. Nevertheless, a common issue in ESNs is determining its hyperparameters, which are crucial in instantiating a well performing reservoir, but are often set manually or using heuristics. In this work we optimize the ESN hyperparameters using Bayesian optimization which, given a limited budget of function evaluations, outperforms a grid search strategy. In the context of large volumes of time series data, such as light curves in the field of astronomy, we can further reduce the optimization cost of ESNs. In particular, we wish to avoid tuning hyperparameters per individual time series as this is costly; instead, we want to find ESNs with hyperparameters that perform well not just on individual time series but rather on groups of similar time series without sacrificing predictive performance significantly. This naturally leads to a notion of clusters, where each cluster is represented by an ESN tuned to model a group of time series of similar temporal behavior. We demonstrate this approach both on synthetic datasets and real world light curves from the MACHO survey. We show that our approach results in a significant reduction in the number of ESN models required to model a whole dataset, while retaining predictive performance for the series in each cluster.
Dynamic Multi-objective Optimization Problems (DMOPs) refer to optimization problems that objective functions will change with time. Solving DMOPs implies that the Pareto Optimal Set (POS) at different moments can be accurately found, and this is a very difficult job due to the dynamics of the optimization problems. The POS that have been obtained in the past can help us to find the POS of the next time more quickly and accurately. Therefore, in this paper we present a Support Vector Machine (SVM) based Dynamic Multi-Objective Evolutionary optimization Algorithm, called SVM-DMOEA. The algorithm uses the POS that has been obtained to train a SVM and then take the trained SVM to classify the solutions of the dynamic optimization problem at the next moment, and thus it is able to generate an initial population which consists of different individuals recognized by the trained SVM. The initial populuation can be fed into any population based optimization algorithm, e.g., the Nondominated Sorting Genetic Algorithm II (NSGA-II), to get the POS at that moment. The experimental results show the validity of our proposed approach.
Recently, increasing works have proposed to drive evolutionary algorithms using machine learning models. Usually, the performance of such model based evolutionary algorithms is highly dependent on the training qualities of the adopted models. Since it usually requires a certain amount of data (i.e. the candidate solutions generated by the algorithms) for model training, the performance deteriorates rapidly with the increase of the problem scales, due to the curse of dimensionality. To address this issue, we propose a multi-objective evolutionary algorithm driven by the generative adversarial networks (GANs). At each generation of the proposed algorithm, the parent solutions are first classified into real and fake samples to train the GANs; then the offspring solutions are sampled by the trained GANs. Thanks to the powerful generative ability of the GANs, our proposed algorithm is capable of generating promising offspring solutions in high-dimensional decision space with limited training data. The proposed algorithm is tested on 10 benchmark problems with up to 200 decision variables. Experimental results on these test problems demonstrate the effectiveness of the proposed algorithm.
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