This paper managed to induce probability theory (sigma system) and possibility theory (max system) respectively from randomness and fuzziness, through which the premature theory of possibility is expected to be well founded. Such an objective is achieved by addressing three open key issues: a) the lack of clear mathematical definitions of randomness and fuzziness; b) the lack of intuitive mathematical definition of possibility; c) the lack of abstraction procedure of the axiomatic definitions of probability/possibility from their intuitive definitions. Especially, the last issue involves the question why the key axiom of "maxitivity" is adopted for possibility measure. By taking advantage of properties of the well-defined randomness and fuzziness, we derived the important conclusion that "max" is the only but un-strict disjunctive operator that is applicable across the fuzzy event space, and is an exact operator for fuzzy feature extraction that assures the max inference is an exact mechanism. It is fair to claim that the long-standing problem of lack of consensus to the foundation of possibility theory is well resolved, which would facilitate wider adoption of possibility theory in practice and promote cross prosperity of the two uncertainty theories of probability and possibility. Randomness and fuzziness are well recognized as two kinds of fundamental uncertainties of this world. It remains as an open topic on how to correctly comprehend these uncertainties and effectively handle them in practice. For modeling of random uncertainty, probability theory and the derivative subjects of statistics and stochastic process are no doubt the classic tool set. Probability theory, which satisfies the key axiom of "additivity" [18,23], has grown up to be mature, upon which nearly the whole building of information sciences is based and applications of which could be found over a great diversity of communities [22, 29,41,42,52,53].

We study the model-based reward-free reinforcement learning with linear function approximation for episodic Markov decision processes (MDPs). In this setting, the agent works in two phases. In the exploration phase, the agent interacts with the environment and collects samples without the reward. In the planning phase, the agent is given a specific reward function and uses samples collected from the exploration phase to learn a good policy. We propose a new provably efficient algorithm, called UCRL-RFE under the Linear Mixture MDP assumption, where the transition probability kernel of the MDP can be parameterized by a linear function over certain feature mappings defined on the triplet of state, action, and next state. We show that to obtain an $\epsilon$-optimal policy for arbitrary reward function, UCRL-RFE needs to sample at most $\tilde O(H^5d^2\epsilon^{-2})$ episodes during the exploration phase. Here, $H$ is the length of the episode, $d$ is the dimension of the feature mapping. We also propose a variant of UCRL-RFE using Bernstein-type bonus and show that it needs to sample at most $\tilde O(H^4d(H + d)\epsilon^{-2})$ to achieve an $\epsilon$-optimal policy. By constructing a special class of linear Mixture MDPs, we also prove that for any reward-free algorithm, it needs to sample at least $\tilde \Omega(H^2d\epsilon^{-2})$ episodes to obtain an $\epsilon$-optimal policy. Our upper bound matches the lower bound in terms of the dependence on $\epsilon$ and the dependence on $d$ if $H \ge d$.

Data collected by multiple methods or from multiple sources is called multi-view data. To make full use of the multi-view data, multi-view learning plays an increasingly important role. Traditional multi-view learning methods rely on a large number of labeled and completed multi-view data. However, it is expensive and time-consuming to obtain a large number of labeled multi-view data in real-world applications. Moreover, multi-view data is often incomplete because of data collection failures, self-deficiency, or other reasons. Therefore, we may have to face the problem of fewer labeled and incomplete multi-view data in real application scenarios. In this paper, a transductive semi-supervised incomplete multi-view TSK fuzzy system modeling method (SSIMV_TSK) is proposed to address these challenges. First, in order to alleviate the dependency on labeled data and keep the model interpretable, the proposed method integrates missing view imputation, pseudo label learning of unlabeled data, and fuzzy system modeling into a single process to yield a model with interpretable fuzzy rules. Then, two new mechanisms, i.e. the bidirectional structural preservation of instance and label, as well as the adaptive multiple alignment collaborative learning, are proposed to improve the robustness of the model. The proposed method has the following distinctive characteristics: 1) it can deal with the incomplete and few labeled multi-view data simultaneously; 2) it integrates the missing view imputation and model learning as a single process, which is more efficient than the traditional two-step strategy; 3) attributed to the interpretable fuzzy inference rules, this method is more interpretable. Experimental results on real datasets show that the proposed method significantly outperforms the state-of-the-art methods.

Recently, deep reinforcement learning (RL) has achieved remarkable empirical success by integrating deep neural networks into RL frameworks. However, these algorithms often require a large number of training samples and admit little theoretical understanding. To mitigate these issues, we propose a theoretically principled nearest neighbor (NN) function approximator that can improve the value networks in deep RL methods. Inspired by human similarity judgments, the NN approximator estimates the action values using rollouts on past observations and can provably obtain a small regret bound that depends only on the intrinsic complexity of the environment. We present (1) Nearest Neighbor Actor-Critic (NNAC), an online policy gradient algorithm that demonstrates the practicality of combining function approximation with deep RL, and (2) a plug-and-play NN update module that aids the training of existing deep RL methods. Experiments on classical control and MuJoCo locomotion tasks show that the NN-accelerated agents achieve higher sample efficiency and stability than the baseline agents. Based on its theoretical benefits, we believe that the NN approximator can be further applied to other complex domains to speed-up learning.

Rule-based models, e.g., decision trees, are widely used in scenarios demanding high model interpretability for their transparent inner structures and good model expressivity. However, rule-based models are hard to optimize, especially on large data sets, due to their discrete parameters and structures. Ensemble methods and fuzzy/soft rules are commonly used to improve performance, but they sacrifice the model interpretability. To obtain both good scalability and interpretability, we propose a new classifier, named Rule-based Representation Learner (RRL), that automatically learns interpretable non-fuzzy rules for data representation and classification. To train the non-differentiable RRL effectively, we project it to a continuous space and propose a novel training method, called Gradient Grafting, that can directly optimize the discrete model using gradient descent. An improved design of logical activation functions is also devised to increase the scalability of RRL and enable it to discretize the continuous features end-to-end. Exhaustive experiments on nine small and four large data sets show that RRL outperforms the competitive interpretable approaches and can be easily adjusted to obtain a trade-off between classification accuracy and model complexity for different scenarios. Our code is available at: https://github.com/12wang3/rrl.

When the sizes of the state and action spaces are large, solving MDPs can be computationally prohibitive even if the probability transition matrix is known. So in practice, a number of techniques are used to approximately solve the dynamic programming problem, including lookahead, approximate policy evaluation using an m-step return, and function approximation. In a recent paper, (Efroni et al. 2019) studied the impact of lookahead on the convergence rate of approximate dynamic programming. In this paper, we show that these convergence results change dramatically when function approximation is used in conjunction with lookout and approximate policy evaluation using an m-step return. Specifically, we show that when linear function approximation is used to represent the value function, a certain minimum amount of lookahead and multi-step return is needed for the algorithm to even converge. And when this condition is met, we characterize the finite-time performance of policies obtained using such approximate policy iteration. Our results are presented for two different procedures to compute the function approximation: linear least-squares regression and gradient descent.

Natural language inference (NLI) aims to determine the logical relationship between two sentences among the target labels Entailment, Contradiction, and Neutral. In recent years, deep learning models have become a prevailing approach to NLI, but they lack interpretability and explainability. In this work, we address the explainability for NLI by weakly supervised logical reasoning, and propose an Explainable Phrasal Reasoning (EPR) approach. Our model first detects phrases as the semantic unit and aligns corresponding phrases. Then, the model predicts the NLI label for the aligned phrases, and induces the sentence label by fuzzy logic formulas. Our EPR is almost everywhere differentiable and thus the system can be trained end-to-end in a weakly supervised manner. We annotated a corpus and developed a set of metrics to evaluate phrasal reasoning. Results show that our EPR yields much more meaningful explanations in terms of F scores than previous studies. To the best of our knowledge, we are the first to develop a weakly supervised phrasal reasoning model for the NLI task.

A number of numeric measures like rough inclusion functions (RIFs) are used in general rough sets and soft computing. But these are often intrusive by definition, and amount to making unjustified assumptions about the data. The contamination problem is also about recognizing the domains of discourses involved in this, specifying errors and reducing data intrusion relative to them. In this research, weak quasi rough inclusion functions (wqRIFs) are generalized to general granular operator spaces with scope for limiting contamination. New algebraic operations are defined over collections of such functions, and are studied by the present author. It is shown by her that the algebras formed by the generalized wqRIFs are ordered hemirings with additional operators. By contrast, the generalized rough inclusion functions lack similar structure. This potentially contributes to improving the selection (possibly automatic) of such functions, training methods, and reducing contamination (and data intrusion) in applications. The underlying framework and associated concepts are explained in some detail, as they are relatively new.

In this paper, an interpretable classifier using an interval type-2 fuzzy neural network for detecting patients suffering from Parkinson's Disease (PD) based on analyzing the gait cycle is presented. The proposed method utilizes clinical features extracted from the vertical Ground Reaction Force (vGRF), measured by 16 wearable sensors placed in the soles of subjects' shoes and learns interpretable fuzzy rules. Therefore, experts can verify the decision made by the proposed method based on investigating the firing strength of interpretable fuzzy rules. Moreover, experts can utilize the extracted fuzzy rules for patient diagnosing or adjust them based on their knowledge. To improve the robustness of the proposed method against uncertainty and noisy sensor measurements, Interval Type-2 Fuzzy Logic is applied. To learn fuzzy rules, two paradigms are proposed: 1- A batch learning approach based on clustering available samples is applied to extract initial fuzzy rules, 2- A complementary online learning is proposed to improve the rule base encountering new labeled samples. The performance of the method is evaluated for classifying patients and healthy subjects in different conditions including the presence of noise or observing new instances. Moreover, the performance of the model is compared to some previous supervised and unsupervised machine learning approaches. The final Accuracy, Precision, Recall, and F1 Score of the proposed method are 88.74%, 89.41%, 95.10%, and 92.16%. Finally, the extracted fuzzy sets for each feature are reported.

A novel procedure to perform fuzzy clustering of multivariate time series generated from different dependence models is proposed. Different amounts of dissimilarity between the generating models or changes on the dynamic behaviours over time are some arguments justifying a fuzzy approach, where each series is associated to all the clusters with specific membership levels. Our procedure considers quantile-based cross-spectral features and consists of three stages: (i) each element is characterized by a vector of proper estimates of the quantile cross-spectral densities, (ii) principal component analysis is carried out to capture the main differences reducing the effects of the noise, and (iii) the squared Euclidean distance between the first retained principal components is used to perform clustering through the standard fuzzy C-means and fuzzy C-medoids algorithms. The performance of the proposed approach is evaluated in a broad simulation study where several types of generating processes are considered, including linear, nonlinear and dynamic conditional correlation models. Assessment is done in two different ways: by directly measuring the quality of the resulting fuzzy partition and by taking into account the ability of the technique to determine the overlapping nature of series located equidistant from well-defined clusters. The procedure is compared with the few alternatives suggested in the literature, substantially outperforming all of them whatever the underlying process and the evaluation scheme. Two specific applications involving air quality and financial databases illustrate the usefulness of our approach.