"Fuzzy Logic is basically a multivalued logic that allows intermediate values to be defined between conventional evaluations like yes/no, true/false, black/white, etc. Notions like rather warm or pretty cold can be formulated mathematically and processed by computers."
– Peter Bauer, Stephan Nouak, and Roman Winkler. A Brief Course in Fuzzy Logic and Fuzzy Control. Available from ESRU [Energy Systems Research Unit], Department of Mechanical Engineering, University of Strathclyde. 1996.
Supervised learning in machine learning can be described in terms of function approximation. Given a dataset comprised of inputs and outputs, we assume that there is an unknown underlying function that is consistent in mapping inputs to outputs in the target domain and resulted in the dataset. We then use supervised learning algorithms to approximate this function. Neural networks are an example of a supervised machine learning algorithm that is perhaps best understood in the context of function approximation. This can be demonstrated with examples of neural networks approximating simple one-dimensional functions that aid in developing the intuition for what is being learned by the model.
Graph neural networks (GNNs) have achieved lots of success on graph-structured data. In light of this, there has been increasing interest in studying their representation power. One line of work focuses on the universal approximation of permutation-invariant functions by certain classes of GNNs, and another demonstrates the limitation of GNNs via graph isomorphism tests. Our work connects these two perspectives and proves their equivalence. We further develop a framework of the representation power of GNNs with the language of sigma-algebra, which incorporates both viewpoints.
We developed a Nonlinear Level-set Learning (NLL) method for dimensionality reduction in high-dimensional function approximation with small data. This work is motivated by a variety of design tasks in real-world engineering applications, where practitioners would replace their computationally intensive physical models (e.g., high-resolution fluid simulators) with fast-to-evaluate predictive machine learning models, so as to accelerate the engineering design processes. There are two major challenges in constructing such predictive models: (a) high-dimensional inputs (e.g., many independent design parameters) and (b) small training data, generated by running extremely time-consuming simulations. Thus, reducing the input dimension is critical to alleviate the over-fitting issue caused by data insufficiency. Existing methods, including sliced inverse regression and active subspace approaches, reduce the input dimension by learning a linear coordinate transformation; our main contribution is to extend the transformation approach to a nonlinear regime.
SARSA is an on-policy algorithm to learn a Markov decision process policy in reinforcement learning. We investigate the SARSA algorithm with linear function approximation under the non-i.i.d.\ setting, where a single sample trajectory is available. With a Lipschitz continuous policy improvement operator that is smooth enough, SARSA has been shown to converge asymptotically. However, its non-asymptotic analysis is challenging and remains unsolved due to the non-i.i.d. In this paper, we develop a novel technique to explicitly characterize the stochastic bias of a type of stochastic approximation procedures with time-varying Markov transition kernels.
Q-learning with function approximation is one of the most popular methods in reinforcement learning. Though the idea of using function approximation was proposed at least 60 years ago, even in the simplest setup, i.e, approximating Q-functions with linear functions, it is still an open problem how to design a provably efficient algorithm that learns a near-optimal policy. The key challenges are how to efficiently explore the state space and how to decide when to stop exploring in conjunction with the function approximation scheme. The current paper presents a provably efficient algorithm for Q-learning with linear function approximation. Under certain regularity assumptions, our algorithm, Difference Maximization Q-learning, combined with linear function approximation, returns a near-optimal policy using polynomial number of trajectories.
Policy evaluation with smooth and nonlinear function approximation has shown great potential for reinforcement learning. Compared to linear function approxi- mation, it allows for using a richer class of approximation functions such as the neural networks. Traditional algorithms are based on two timescales stochastic approximation whose convergence rate is often slow. This paper focuses on an offline setting where a trajectory of $m$ state-action pairs are observed. We formulate the policy evaluation problem as a non-convex primal-dual, finite-sum optimization problem, whose primal sub-problem is non-convex and dual sub-problem is strongly concave.
To effectively train Takagi-Sugeno-Kang (TSK) fuzzy systems for regression problems, a Mini-Batch Gradient Descent with Regularization, DropRule, and AdaBound (MBGD-RDA) algorithm was recently proposed. It has demonstrated superior performances; however, there are also some limitations, e.g., it does not allow the user to specify the number of rules directly, and only Gaussian MFs can be used. This paper proposes two variants of MBGD-RDA to remedy these limitations, and show that they outperform the original MBGD-RDA and the classical ANFIS algorithms with the same number of rules. Furthermore, we also propose a rule pruning algorithm for TSK fuzzy systems, which can reduce the number of rules without significantly sacrificing the regression performance. Experiments showed that the rules obtained from pruning are generally better than training them from scratch directly, especially when Gaussian MFs are used.
Fuzzy c-means based clustering algorithms are frequently used for Takagi-Sugeno-Kang (TSK) fuzzy classifier antecedent parameter estimation. One rule is initialized from each cluster. However, most of these clustering algorithms are unsupervised, which waste valuable label information in the training data. This paper proposes a supervised enhanced soft subspace clustering (SESSC) algorithm, which considers simultaneously the within-cluster compactness, between-cluster separation, and label information in clustering. It can effectively deal with high-dimensional data, be used as a classifier alone, or be integrated into a TSK fuzzy classifier to further improve its performance. Experiments on nine UCI datasets from various application domains demonstrated that SESSC based initialization outperformed other clustering approaches, especially when the number of rules is small.
Approaches based on computing with words find good applicability in decision making systems. Predominantly finding their basis in type-1 fuzzy sets, computing with words approaches employ type-1 fuzzy sets as semantics of the linguistic terms. However, type-2 fuzzy sets have been proven to be scientifically more appropriate to represent linguistic information in practical systems. They take into account both the intra-uncertainty as well as the inter-uncertainty in cases where the linguistic information comes from a group of experts. Hence in this paper, we propose to introduce linguistic terms whose semantics are denoted by interval type-2 fuzzy sets within the hesitant fuzzy linguistic term set framework, resulting in type-2 fuzzy sets based hesitant fuzzy linguistic term sets. We also introduce a novel method of computing type-2 fuzzy envelopes out of multiple interval type-2 fuzzy sets with trapezoidal membership functions. Furthermore, the proposed framework with interval type-2 fuzzy sets is applied on a supplier performance evaluation scenario. Since humans are predominantly involved in the entire process of supply chain, their feedback is crucial while deciding many factors. Towards the end of the paper, we compare our presented model with various existing models and demonstrate the advantages of the former.
Novel applications of artificial intelligence for tuning the parameters of industrial machines for optimal performance are emerging at a fast pace. Tuning the combine harvesters and improving the machine performance can dramatically minimize the wastes during harvesting, and it is also beneficial to machine maintenance. Literature includes several soft computing, machine learning and optimization methods that had been used to model the function of harvesters of various crops. Due to the complexity of the problem, machine learning methods had been recently proposed to predict the optimal performance with promising results. In this paper, through proposing a novel hybrid machine learning model based on artificial neural networks integrated with particle swarm optimization (ANN-PSO), the performance analysis of a common combine harvester is presented. The hybridization of machine learning methods with soft computing techniques has recently shown promising results to improve the performance of the combine harvesters. This research aims at improving the results further by providing more stable models with higher accuracy.