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We consider Bayesian best arm identification in the multi-armed bandit problem. Assuming certain continuity conditions of the prior, we characterize the rate of the Bayesian simple regret. Differing from Bayesian regret minimization (Lai, 1987), the leading factor in Bayesian simple regret derives from the region where the gap between optimal and sub-optimal arms is smaller than $\sqrt{\frac{\log T}{T}}$. We propose a simple and easy-to-compute algorithm with its leading factor matches with the lower bound up to a constant factor; simulation results support our theoretical findings.

We study the hierarchy of communities in real-world networks under a generic stochastic block model, in which the connection probabilities are structured in a binary tree. Under such model, a standard recursive bi-partitioning algorithm is dividing the network into two communities based on the Fiedler vector of the unnormalized graph Laplacian and repeating the split until a stopping rule indicates no further community structures. We prove the strong consistency of this method under a wide range of model parameters, which include sparse networks with node degrees as small as $O(\log n)$. In addition, unlike most of existing work, our theory covers multiscale networks where the connection probabilities may differ by orders of magnitude, which comprise an important class of models that are practically relevant but technically challenging to deal with. Finally we demonstrate the performance of our algorithm on synthetic data and real-world examples.

We propose the first near-optimal quantum algorithm for estimating in Euclidean norm the mean of a vector-valued random variable with finite mean and covariance. Our result aims at extending the theory of multivariate sub-Gaussian estimators to the quantum setting. Unlike classically, where any univariate estimator can be turned into a multivariate estimator with at most a logarithmic overhead in the dimension, no similar result can be proved in the quantum setting. Indeed, Heinrich ruled out the existence of a quantum advantage for the mean estimation problem when the sample complexity is smaller than the dimension. Our main result is to show that, outside this low-precision regime, there is a quantum estimator that outperforms any classical estimator. Our approach is substantially more involved than in the univariate setting, where most quantum estimators rely only on phase estimation. We exploit a variety of additional algorithmic techniques such as amplitude amplification, the Bernstein-Vazirani algorithm, and quantum singular value transformation. Our analysis also uses concentration inequalities for multivariate truncated statistics. We develop our quantum estimators in two different input models that showed up in the literature before. The first one provides coherent access to the binary representation of the random variable and it encompasses the classical setting. In the second model, the random variable is directly encoded into the phases of quantum registers. This model arises naturally in many quantum algorithms but it is often incomparable to having classical samples. We adapt our techniques to these two settings and we show that the second model is strictly weaker for solving the mean estimation problem. Finally, we describe several applications of our algorithms, notably in measuring the expectation values of commuting observables and in the field of machine learning.

In directional statistics, the von Mises distribution is a key element in the analysis of circular data. While there is a general agreement regarding the estimation of its location parameter $\mu$, several methods have been proposed to estimate the concentration parameter $\kappa$. We here provide a thorough evaluation of the behavior of 12 such estimators for datasets of size $N$ ranging from 2 to 8\,192 generated with a $\kappa$ ranging from 0 to 100. We provide detailed results as well as a global analysis of the results, showing that (1) for a given $\kappa$, most estimators have behaviors that are very similar for large datasets ($N \geq 16$) and more variable for small datasets, and (2) for a given estimator, results are very similar if we consider the mean absolute error for $\kappa \leq 1$ and the mean relative absolute error for $\kappa \geq 1$.

We study fast rates of convergence in the setting of nonparametric online regression, namely where regret is defined with respect to an arbitrary function class which has bounded complexity. Our contributions are two-fold: - In the realizable setting of nonparametric online regression with the absolute loss, we propose a randomized proper learning algorithm which gets a near-optimal mistake bound in terms of the sequential fat-shattering dimension of the hypothesis class. In the setting of online classification with a class of Littlestone dimension $d$, our bound reduces to $d \cdot {\rm poly} \log T$. This result answers a question as to whether proper learners could achieve near-optimal mistake bounds; previously, even for online classification, the best known mistake bound was $\tilde O( \sqrt{dT})$. Further, for the real-valued (regression) setting, the optimal mistake bound was not even known for improper learners, prior to this work. - Using the above result, we exhibit an independent learning algorithm for general-sum binary games of Littlestone dimension $d$, for which each player achieves regret $\tilde O(d^{3/4} \cdot T^{1/4})$. This result generalizes analogous results of Syrgkanis et al. (2015) who showed that in finite games the optimal regret can be accelerated from $O(\sqrt{T})$ in the adversarial setting to $O(T^{1/4})$ in the game setting. To establish the above results, we introduce several new techniques, including: a hierarchical aggregation rule to achieve the optimal mistake bound for real-valued classes, a multi-scale extension of the proper online realizable learner of Hanneke et al. (2021), an approach to show that the output of such nonparametric learning algorithms is stable, and a proof that the minimax theorem holds in all online learnable games.

We propose a diffusion approximation method to the continuous-state Markov Decision Processes (MDPs) that can be utilized to address autonomous navigation and control in unstructured off-road environments. In contrast to most decision-theoretic planning frameworks that assume fully known state transition models, we design a method that eliminates such a strong assumption that is often extremely difficult to engineer in reality. We first take the second-order Taylor expansion of the value function. The Bellman optimality equation is then approximated by a partial differential equation, which only relies on the first and second moments of the transition model. By combining the kernel representation of the value function, we then design an efficient policy iteration algorithm whose policy evaluation step can be represented as a linear system of equations characterized by a finite set of supporting states. We first validate the proposed method through extensive simulations in $2D$ obstacle avoidance and $2.5D$ terrain navigation problems. The results show that the proposed approach leads to a much superior performance over several baselines. We then develop a system that integrates our decision-making framework with onboard perception and conduct real-world experiments in both cluttered indoor and unstructured outdoor environments. The results from the physical systems further demonstrate the applicability of our method in challenging real-world environments.

Optimal control problems can be solved by first applying the Pontryagin maximum principle, followed by computing a solution of the corresponding unconstrained Hamiltonian dynamical system. In this paper, and to achieve a balance between robustness and efficiency, we learn a reduced Hamiltonian of the unconstrained Hamiltonian. This reduced Hamiltonian is learned by going backward in time and by minimizing the loss function resulting from application of the Pontryagin maximum principle conditions. The robustness of our learning process is then further improved by progressively learning a posterior distribution of reduced Hamiltonians. This leads to a more efficient sampling of the generalized coordinates (position, velocity) of our phase space. Our solution framework applies to not only optimal control problems with finite-dimensional phase (state) spaces but also the infinite dimensional case.

The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series $(\y_n)_{n \in \mathbb{Z}}$ with independent components is studied under the asymptotic regime where the sample size $N$ converges towards $+\infty$ while the dimension $M$ of $\y$ and the smoothing span of the estimator grow to infinity at the same rate in such a way that $\frac{M}{N} \rightarrow 0$. It is established that, at each frequency, the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically $\mathcal{N}_{\mathbb{C}}(0,\I_M)$ distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitly. Using concentration inequalities, it is shown that the order of magnitude of the supremum over the frequencies of the deviation of each LSS from its deterministic approximation is of the order of $\frac{1}{M} + \frac{\sqrt{M}}{N}+ (\frac{M}{N})^{3}$ where $N$ is the sample size. Numerical simulations supports our results.

In the present paper, progression of non-alcoholic fatty liver disease (NAFLD) process is modeled by Continuous time Markov chains (CTMC) with 4 states .The transition intensities among the states are estimated using maximum likelihood estimation (MLE) method. The transition probabilities are also calculated. The mean sojourn time and its variance are estimated as well as the state probability distribution and its asymptotic covariance matrix. The life expectancy of the patient, one of the important statistical indices, is also obtained. The paper illustrates the new approach of using MLE to compensate for missing values in the follow up periods of patients in the longitudinal studies. This new approach also yields that the estimated rates among states are approximately equals to the observed rates.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.