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The goals of this paper are twofold: (1) to present a new method that is able to find linear laws governing the time evolution of Markov chains and (2) to apply this method for anomaly detection in Bitcoin prices. To accomplish these goals, first, the linear laws of Markov chains are derived by using the time embedding of their (categorical) autocorrelation function. Then, a binary series is generated from the first difference of Bitcoin exchange rate (against the United States Dollar). Finally, the minimum number of parameters describing the linear laws of this series is identified through stepped time windows. Based on the results, linear laws typically became more complex (containing an additional third parameter that indicates hidden Markov property) in two periods: before the crash of cryptocurrency markets inducted by the COVID-19 pandemic (12 March 2020), and before the record-breaking surge in the price of Bitcoin (Q4 2020 - Q1 2021). In addition, the locally high values of this third parameter are often related to short-term price peaks, which suggests price manipulation.

We develop a new efficient methodology for Bayesian global sensitivity analysis for large-scale multivariate data. The focus is on computationally demanding models with correlated variables. A multivariate Gaussian process is used as a surrogate model to replace the expensive computer model. To improve the computational efficiency and performance of the model, compactly supported correlation functions are used. The goal is to generate sparse matrices, which give crucial advantages when dealing with large datasets, where we use cross-validation to determine the optimal degree of sparsity. This method was combined with a robust adaptive Metropolis algorithm coupled with a parallel implementation to speed up the convergence to the target distribution. The method was applied to a multivariate dataset from the IMPRESSIONS Integrated Assessment Platform (IAP2), an extension of the CLIMSAVE IAP, which has been widely applied in climate change impact, adaptation and vulnerability assessments. Our empirical results on synthetic and IAP2 data show that the proposed methods are efficient and accurate for global sensitivity analysis of complex models.

The generalization capacity of various machine learning models exhibits different phenomena in the under- and over-parameterized regimes. In this paper, we focus on regression models such as feature regression and kernel regression and analyze a generalized weighted least-squares optimization method for computational learning and inversion with noisy data. The highlight of the proposed framework is that we allow weighting in both the parameter space and the data space. The weighting scheme encodes both a priori knowledge on the object to be learned and a strategy to weight the contribution of different data points in the loss function. Here, we characterize the impact of the weighting scheme on the generalization error of the learning method, where we derive explicit generalization errors for the random Fourier feature model in both the under- and over-parameterized regimes. For more general feature maps, error bounds are provided based on the singular values of the feature matrix. We demonstrate that appropriate weighting from prior knowledge can improve the generalization capability of the learned model.

Running machine learning algorithms on large and rapidly growing volumes of data is often computationally expensive, one common trick to reduce the size of a data set, and thus reduce the computational cost of machine learning algorithms, is \emph{probability sampling}. It creates a sampled data set by including each data point from the original data set with a known probability. Although the benefit of running machine learning algorithms on the reduced data set is obvious, one major concern is that the performance of the solution obtained from samples might be much worse than that of the optimal solution when using the full data set. In this paper, we examine the performance loss caused by probability sampling in the context of adaptive submodular maximization. We consider a simple probability sampling method which selects each data point with probability at least $r\in[0,1]$. If we set $r=1$, our problem reduces to finding a solution based on the original full data set. We define sampling gap as the largest ratio between the optimal solution obtained from the full data set and the optimal solution obtained from the samples, over independence systems. Our main contribution is to show that if the sampling probability of each data point is at least $r$ and the utility function is policywise submodular, then the sampling gap is both upper bounded and lower bounded by $1/r$. We show that the property of policywise submodular can be found in a wide range of real-world applications, including pool-based active learning and adaptive viral marketing.

The purpose of this paper is to introduce a notion of causality in Markov decision processes based on the probability-raising principle and to analyze its algorithmic properties. The latter includes algorithms for checking cause-effect relationships and the existence of probability-raising causes for given effect scenarios. Inspired by concepts of statistical analysis, we study quality measures (recall, coverage ratio and f-score) for causes and develop algorithms for their computation. Finally, the computational complexity for finding optimal causes with respect to these measures is analyzed.

We study the Bahadur efficiency of several weighted L2--type goodness--of--fit tests based on the empirical characteristic function. The methods considered are for normality and exponentiality testing, and for testing goodness--of--fit to the logistic distribution. Our results are helpful in deciding which specific test a potential practitioner should apply. For the celebrated BHEP and energy tests for normality we obtain novel efficiency results, with some of them in the multivariate case, while in the case of the logistic distribution this is the first time that efficiencies are computed for any composite goodness--of--fit test.

Deep reinforcement learning (DRL) has attracted much attention as an approach to solve sequential decision making problems without mathematical models of systems or environments. In general, a constraint may be imposed on the decision making. In this study, we consider the optimal decision making problems with constraints to complete temporal high-level tasks in the continuous state-action domain. We describe the constraints using signal temporal logic (STL), which is useful for time sensitive control tasks since it can specify continuous signals within a bounded time interval. To deal with the STL constraints, we introduce an extended constrained Markov decision process (CMDP), which is called a $\tau$-CMDP. We formulate the STL constrained optimal decision making problem as the $\tau$-CMDP and propose a two-phase constrained DRL algorithm using the Lagrangian relaxation method. Through simulations, we also demonstrate the learning performance of the proposed algorithm.

In this paper we study the convergence of generative adversarial networks (GANs) from the perspective of the informativeness of the gradient of the optimal discriminative function. We show that GANs without restriction on the discriminative function space commonly suffer from the problem that the gradient produced by the discriminator is uninformative to guide the generator. By contrast, Wasserstein GAN (WGAN), where the discriminative function is restricted to $1$-Lipschitz, does not suffer from such a gradient uninformativeness problem. We further show in the paper that the model with a compact dual form of Wasserstein distance, where the Lipschitz condition is relaxed, also suffers from this issue. This implies the importance of Lipschitz condition and motivates us to study the general formulation of GANs with Lipschitz constraint, which leads to a new family of GANs that we call Lipschitz GANs (LGANs). We show that LGANs guarantee the existence and uniqueness of the optimal discriminative function as well as the existence of a unique Nash equilibrium. We prove that LGANs are generally capable of eliminating the gradient uninformativeness problem. According to our empirical analysis, LGANs are more stable and generate consistently higher quality samples compared with WGAN.

Deep reinforcement learning has recently shown many impressive successes. However, one major obstacle towards applying such methods to real-world problems is their lack of data-efficiency. To this end, we propose the Bottleneck Simulator: a model-based reinforcement learning method which combines a learned, factorized transition model of the environment with rollout simulations to learn an effective policy from few examples. The learned transition model employs an abstract, discrete (bottleneck) state, which increases sample efficiency by reducing the number of model parameters and by exploiting structural properties of the environment. We provide a mathematical analysis of the Bottleneck Simulator in terms of fixed points of the learned policy, which reveals how performance is affected by four distinct sources of error: an error related to the abstract space structure, an error related to the transition model estimation variance, an error related to the transition model estimation bias, and an error related to the transition model class bias. Finally, we evaluate the Bottleneck Simulator on two natural language processing tasks: a text adventure game and a real-world, complex dialogue response selection task. On both tasks, the Bottleneck Simulator yields excellent performance beating competing approaches.

Quantum machine learning is expected to be one of the first potential general-purpose applications of near-term quantum devices. A major recent breakthrough in classical machine learning is the notion of generative adversarial training, where the gradients of a discriminator model are used to train a separate generative model. In this work and a companion paper, we extend adversarial training to the quantum domain and show how to construct generative adversarial networks using quantum circuits. Furthermore, we also show how to compute gradients -- a key element in generative adversarial network training -- using another quantum circuit. We give an example of a simple practical circuit ansatz to parametrize quantum machine learning models and perform a simple numerical experiment to demonstrate that quantum generative adversarial networks can be trained successfully.