We quantify the minimax rate for a nonparametric regression model over a convex function class $\mathcal{F}$ with bounded diameter. We obtain a minimax rate of ${\varepsilon^{\ast}}^2\wedge\mathrm{diam}(\mathcal{F})^2$ where \[\varepsilon^{\ast} =\sup\{\varepsilon>0:n\varepsilon^2 \le \log M_{\mathcal{F}}^{\operatorname{loc}}(\varepsilon,c)\},\] where $M_{\mathcal{F}}^{\operatorname{loc}}(\cdot, c)$ is the local metric entropy of $\mathcal{F}$ and our loss function is the squared population $L_2$ distance over our input space $\mathcal{X}$. In contrast to classical works on the topic [cf. Yang and Barron, 1999], our results do not require functions in $\mathcal{F}$ to be uniformly bounded in sup-norm. In addition, we prove that our estimator is adaptive to the true point, and to the best of our knowledge this is the first such estimator in this general setting. This work builds on the Gaussian sequence framework of Neykov [2022] using a similar algorithmic scheme to achieve the minimax rate. Our algorithmic rate also applies with sub-Gaussian noise. We illustrate the utility of this theory with examples including multivariate monotone functions, linear functionals over ellipsoids, and Lipschitz classes.
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We describe a simple deterministic near-linear time approximation scheme for uncapacitated minimum cost flow in undirected graphs with real edge weights, a problem also known as transshipment. Specifically, our algorithm takes as input a (connected) undirected graph $G = (V, E)$, vertex demands $b \in \mathbb{R}^V$ such that $\sum_{v \in V} b(v) = 0$, positive edge costs $c \in \mathbb{R}_{>0}^E$, and a parameter $\varepsilon > 0$. In $O(\varepsilon^{-2} m \log^{O(1)} n)$ time, it returns a flow $f$ such that the net flow out of each vertex is equal to the vertex's demand and the cost of the flow is within a $(1 + \varepsilon)$ factor of optimal. Our algorithm is combinatorial and has no running time dependency on the demands or edge costs. With the exception of a recent result presented at STOC 2022 for polynomially bounded edge weights, all almost- and near-linear time approximation schemes for transshipment relied on randomization to embed the problem instance into low-dimensional space. Our algorithm instead deterministically approximates the cost of routing decisions that would be made if the input were subject to a random tree embedding. To avoid computing the $\Omega(n^2)$ vertex-vertex distances that an approximation of this kind suggests, we also limit the available routing decisions using distances explicitly stored in the well-known Thorup-Zwick distance oracle.
At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered generation that yields an $n^{O(d)}$-delay algorithm listing all minimal transversals of an $n$-vertex hypergraph of degeneracy $d$. Recently at IWOCA 2019, Conte, Kant\'e, Marino, and Uno asked whether this XP-delay algorithm parameterized by $d$ could be made FPT-delay for a weaker notion of degeneracy, or even parameterized by the maximum degree $\Delta$, i.e., whether it can be turned into an algorithm with delay $f(\Delta)\cdot n^{O(1)}$ for some computable function $f$. Moreover, and as a first step toward answering that question, they note that they could not achieve these time bounds even for the particular case of minimal dominating sets enumeration. In this paper, using ordered generation, we show that an FPT-delay algorithm can be devised for minimal transversals enumeration parameterized by the degeneracy and dimension, giving a positive and more general answer to the latter question.
We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish $L^2$ stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The $L^2$ estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.
Although the theoretical properties in the $p_0$ model based on a differentially private bi-degree sequence have been derived, it is still lack of a unified theory for a general class of directed network models with the $p_{0}$ model as a special case. We use the popular Laplace data releasing method to output the bi-degree sequence of directed networks, which satisfies the private standard--differential privacy. The method of moment is used to estimate unknown parameters. We prove that the differentially private estimator is uniformly consistent and asymptotically normal under some conditions. Our results are illustrated by the Probit model. We carry out simulation studies to illustrate theoretical results and provide a real data analysis.
Are asymptotic confidence sequences and anytime $p$-values uniformly valid for a nontrivial class of distributions $\mathcal{P}$? We give a positive answer to this question by deriving distribution-uniform anytime-valid inference procedures. Historically, anytime-valid methods -- including confidence sequences, anytime $p$-values, and sequential hypothesis tests that enable inference at stopping times -- have been justified nonasymptotically. Nevertheless, asymptotic procedures such as those based on the central limit theorem occupy an important part of statistical toolbox due to their simplicity, universality, and weak assumptions. While recent work has derived asymptotic analogues of anytime-valid methods with the aforementioned benefits, these were not shown to be $\mathcal{P}$-uniform, meaning that their asymptotics are not uniformly valid in a class of distributions $\mathcal{P}$. Indeed, the anytime-valid inference literature currently has no central limit theory to draw from that is both uniform in $\mathcal{P}$ and in the sample size $n$. This paper fills that gap by deriving a novel $\mathcal{P}$-uniform strong Gaussian approximation theorem. We apply some of these results to obtain an anytime-valid test of conditional independence without the Model-X assumption, as well as a $\mathcal{P}$-uniform law of the iterated logarithm.
In this work we provide theoretical estimates for the ranks of the power functions $f(k) = k^{-\alpha}$, $\alpha>1$ in the quantized tensor train (QTT) format for $k = 1, 2, 3, \ldots, 2^{d}$. Such functions and their several generalizations (e.~g. $f(k) = k^{-\alpha} \cdot e^{-\lambda k}, \lambda > 0$) play an important role in studies of the asymptotic solutions of the aggregation-fragmentation kinetic equations. In order to support the constructed theory we verify the values of QTT-ranks of these functions in practice with the use of the TTSVD procedure and show an agreement between the numerical and analytical results.
Inferring parameters of a latent variable model can be a daunting task when the conditional distribution of the latent variables given the observed ones is intractable. Variational approaches prove to be computationally efficient but, possibly, lack theoretical guarantees on the estimates, while sampling based solutions are quite the opposite. Starting from already available variational approximations, we define a first Monte Carlo EM algorithm to obtain maximum likelihood estimators, focusing on the Poisson log-normal model which provides a generic framework for the analysis of multivariate count data. We then extend this algorithm to the case of a composite likelihood in order to be able to handle higher dimensional count data.
We consider an observed subcritical Galton Watson process $\{Y_n,\ n\in \mathbb{Z} \}$ with correlated stationary immigration process $\{\epsilon_n,\ n\in \mathbb{Z} \}$. Two situations are presented. The first one is when $\mbox{Cov}(\epsilon_0,\epsilon_k)=0$ for $k$ larger than some $k_0$: a consistent estimator for the reproduction and mean immigration rates is given, and a central limit theorem is proved. The second one is when $\{\epsilon_n,\ n\in \mathbb{Z} \}$ has general correlation structure: under mixing assumptions, we exhibit an estimator for the the logarithm of the reproduction rate and we prove that it converges in quadratic mean with explicit speed. In addition, when the mixing coefficients decrease fast enough, we provide and prove a two terms expansion for the estimator. Numerical illustrations are provided.
We describe a new dependent-rounding algorithmic framework for bipartite graphs. Given a fractional assignment $\vec x$ of values to edges of graph $G = (U \cup V, E)$, the algorithms return an integral solution $\vec X$ such that each right-node $v \in V$ has at most one neighboring edge $f$ with $X_f = 1$, and where the variables $X_e$ also satisfy broad nonpositive-correlation properties. In particular, for any edges $e_1, e_2$ sharing a left-node $u \in U$, the variables $X_{e_1}, X_{e_2}$ have strong negative-correlation properties, i.e. the expectation of $X_{e_1} X_{e_2}$ is significantly below $x_{e_1} x_{e_2}$. This algorithm is based on generating negatively-correlated Exponential random variables and using them in a contention-resolution scheme inspired by an algorithm Im & Shadloo (2020). Our algorithm gives stronger and much more flexible negative correlation properties. Dependent rounding schemes with negative correlation properties have been used for approximation algorithms for job-scheduling on unrelated machines to minimize weighted completion times (Bansal, Srinivasan, & Svensson (2021), Im & Shadloo (2020), Im & Li (2023)). Using our new dependent-rounding algorithm, among other improvements, we obtain a $1.398$-approximation for this problem. This significantly improves over the prior $1.45$-approximation ratio of Im & Li (2023).
We consider the problem of counting 4-cycles ($C_4$) in an undirected graph $G$ of $n$ vertices and $m$ edges (in bipartite graphs, 4-cycles are also often referred to as $\textit{butterflies}$). There have been a number of previous algorithms for this problem based on sorting the graph by degree and using randomized hash tables. These are appealing in theory due to compact storage and fast access on average. But, the performance of hash tables can degrade unpredictably and are also vulnerable to adversarial input. We develop a new simpler algorithm for counting $C_4$ requiring $O(m\bar\delta(G))$ time and $O(n)$ space, where $\bar \delta(G) \leq O(\sqrt{m})$ is the $\textit{average degeneracy}$ parameter introduced by Burkhardt, Faber & Harris (2020). It has several practical improvements over previous algorithms; for example, it is fully deterministic, does not require any sorting of the input graph, and uses only addition and array access in its inner loops. To the best of our knowledge, all previous efficient algorithms for $C_4$ counting have required $\Omega(m)$ space in addition to storing the input graph. Our algorithm is very simple and easily adapted to count 4-cycles incident to each vertex and edge. Empirical tests demonstrate that our array-based approach is $4\times$ -- $7\times$ faster on average compared to popular hash table implementations.
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