We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich (2019) shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters satisfy the summability condition $\sum_{j=1}^{\infty}\gamma_j^{1/\alpha}<\infty$. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces. In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition $\sum_{j=1}^{\infty}\gamma_j^{1/\alpha}<\infty$ holds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively.
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We analyze the impact of transient and Byzantine faults on the construction of a maximal independent set in a general network. We adapt the self-stabilizing algorithm presented by Turau \cite{turau2007linear} for computing such a vertex set. Our algorithm is self-stabilizing and also works under the more difficult context of arbitrary Byzantine faults. Byzantine nodes can prevent nodes close to them from taking part in the independent set for an arbitrarily long time. We give boundaries to their impact by focusing on the set of all nodes excluding nodes at distance 1 or less of Byzantine nodes, and excluding some of the nodes at distance 2. As far as we know, we present the first algorithm tolerating both transient and Byzantine faults under the fair distributed daemon. We prove that this algorithm converges in $ \mathcal O(\Delta n)$ rounds w.h.p., where $n$ and $\Delta$ are the size and the maximum degree of the network, resp. Additionally, we present a modified version of this algorithm for anonymous systems under the adversarial distributed daemon that converges in $ \mathcal O(n^{2})$ expected number of steps.
We study a class of weakly identifiable location-scale mixture models for which the maximum likelihood estimates based on $n$ i.i.d. samples are known to have lower accuracy than the classical $n^{- \frac{1}{2}}$ error. We investigate whether the Expectation-Maximization (EM) algorithm also converges slowly for these models. We provide a rigorous characterization of EM for fitting a weakly identifiable Gaussian mixture in a univariate setting where we prove that the EM algorithm converges in order $n^{\frac{3}{4}}$ steps and returns estimates that are at a Euclidean distance of order ${ n^{- \frac{1}{8}}}$ and ${ n^{-\frac{1} {4}}}$ from the true location and scale parameter respectively. Establishing the slow rates in the univariate setting requires a novel localization argument with two stages, with each stage involving an epoch-based argument applied to a different surrogate EM operator at the population level. We demonstrate several multivariate ($d \geq 2$) examples that exhibit the same slow rates as the univariate case. We also prove slow statistical rates in higher dimensions in a special case, when the fitted covariance is constrained to be a multiple of the identity.
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.
This paper presents some new results on maximum likelihood of incomplete data. Finite sample properties of conditional observed information matrices are established. In particular, they possess the same Loewner partial ordering properties as the expected information matrices do. In its new form, the observed Fisher information (OFI) simplifies conditional expectation of outer product of the complete-data score function appearing in the Louis (1982) general matrix formula. It verifies positive definiteness and consistency to the expected Fisher information as the sample size increases. Furthermore, it shows a resulting information loss presented in the incomplete data. For this reason, the OFI may not be the right (consistent and efficient) estimator to derive the standard error (SE) of maximum likelihood estimates (MLE) for incomplete data. A sandwich estimator of covariance matrix is developed to provide consistent and efficient estimates of SE. The proposed sandwich estimator coincides with the Huber sandwich estimator for model misspecification under complete data (Huber, 1967; Freedman, 2006; Little and Rubin, 2020). However, in contrast to the latter, the new estimator does not involve OFI which notably gives an appealing feature for application. Recursive algorithms for the MLE, the observed information and the sandwich estimator are presented. Application to parameter estimation of a regime switching conditional Markov jump process is considered to verify the results. The recursive equations for the inverse OFI generalizes the algorithm of Hero and Fessler (1994). The simulation study confirms that the MLEs are accurate and consistent having asymptotic normality. The sandwich estimator produces standard error of the MLE close to their analytic values compared to those overestimated by the OFI.
A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies $d_S(U,W):=2k-2\dim(U\cap W)\ge d$ for all pairs of different codewords $U$, $W$. Constant dimension codes have applications in e.g.\ random linear network coding, cryptography, and distributed storage. Bounds for $A_q(n,d;k)$ are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show up the potential for further improvements. As examples we give improved constructions for the cases $A_q(10,4;5)$, $A_q(11,4;4)$, $A_q(12,6;6)$, and $A_q(15,4;4)$. We also derive general upper bounds for subcodes arising in those constructions.
We give a new algorithm for the estimation of the cross-covariance matrix $\mathbb{E} XY'$ of two large dimensional signals $X\in\mathbb{R}^n$, $Y\in \mathbb{R}^p$ in the context where the number $T$ of observations of the pair $(X,Y)$ is itself large, but with $T/n$ and $T/p$ not supposed to be small. In the asymptotic regime where $n,p,T$ are large, with high probability, this algorithm is optimal for the Frobenius norm among rotationally invariant estimators, i.e. estimators derived from the empirical estimator by cleaning the singular values, while letting singular vectors unchanged.
Randomization has been shown to be beneficial to iterative methods for solving large-scale systems of linear equations, notably the Kaczmarz algorithm. We analyze the convergence of a broad class of pursuit algorithms that at each step pick $n$ members at random, from a system of linear equations, and update the iterate using the orthogonal projection to the intersection of the hyperplanes they represent. We identify, in this context, a specific degree-$n$ polynomial that non-linearly transforms the singular values of the system. This transformation to singular values and the corresponding condition number then characterizes the convergence rate, in expectation, of the pursuit. As a consequence, our results specify the convergence rate of the stochastic gradient descent algorithm, in terms of the mini-batch size $n$, when used for solving systems of linear equations.
In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand's projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand's theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand's theorem using the theory of computing.
In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.
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.
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