We consider the problem of correctly identifying the mode of a discrete distribution $\mathcal{P}$ with sufficiently high probability by observing a sequence of i.i.d. samples drawn according to $\mathcal{P}$. This problem reduces to the estimation of a single parameter when $\mathcal{P}$ has a support set of size $K = 2$. Noting the efficiency of prior-posterior-ratio (PPR) martingale confidence sequences for handling this special case, we propose a generalisation to mode estimation, in which $\mathcal{P}$ may take $K \geq 2$ values. We observe that the "one-versus-one" principle yields a more efficient generalisation than the "one-versus-rest" alternative. Our resulting stopping rule, denoted PPR-ME, is optimal in its sample complexity up to a logarithmic factor. Moreover, PPR-ME empirically outperforms several other competing approaches for mode estimation. We demonstrate the gains offered by PPR-ME in two practical applications: (1) sample-based forecasting of the winner in indirect election systems, and (2) efficient verification of smart contracts in permissionless blockchains.
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The Gomory-Hu tree or cut tree (Gomory and Hu, 1961) is a classic data structure for reporting $(s,t)$ mincuts (and by duality, the values of $(s,t)$ maxflows) for all pairs of vertices $s$ and $t$ in an undirected graph. Gomory and Hu showed that it can be computed using $n-1$ exact maxflow computations. Surprisingly, this remains the best algorithm for Gomory-Hu trees more than 50 years later, *even for approximate mincuts*. In this paper, we break this longstanding barrier and give an algorithm for computing a $(1+\epsilon)$-approximate Gomory-Hu tree using $\text{polylog}(n)$ maxflow computations. Specifically, we obtain the runtime bounds we describe below. We obtain a randomized (Monte Carlo) algorithm for undirected, weighted graphs that runs in $\tilde O(m + n^{3/2})$ time and returns a $(1+\epsilon)$-approximate Gomory-Hu tree algorithm w.h.p. Previously, the best running time known was $\tilde O(n^{5/2})$, which is obtained by running Gomory and Hu's original algorithm on a cut sparsifier of the graph. Next, we obtain a randomized (Monte Carlo) algorithm for undirected, unweighted graphs that runs in $m^{4/3+o(1)}$ time and returns a $(1+\epsilon)$-approximate Gomory-Hu tree algorithm w.h.p. This improves on our first result for sparse graphs, namely $m = o(n^{9/8})$. Previously, the best running time known for unweighted graphs was $\tilde O(mn)$ for an exact Gomory-Hu tree (Bhalgat et al., STOC 2007); no better result was known if approximations are allowed. As a consequence of our Gomory-Hu tree algorithms, we also solve the $(1+\epsilon)$-approximate all pairs mincut and single source mincut problems in the same time bounds. (These problems are simpler in that the goal is to only return the $(s,t)$ mincut values, and not the mincuts.) This improves on the recent algorithm for these problems in $\tilde O(n^2)$ time due to Abboud et al. (FOCS 2020).
In principal modern detectors, the task of object localization is implemented by the box subnet which concentrates on bounding box regression. The box subnet customarily predicts the position of the object by regressing box center position and scaling factors. Although this approach is frequently adopted, we observe that the result of localization remains defective, which makes the performance of the detector unsatisfactory. In this paper, we prove the flaws in the previous method through theoretical analysis and experimental verification and propose a novel solution to detect objects precisely. Rather than plainly focusing on center and size, our approach refines the edges of the bounding box on previous localization results by estimating the distribution at the boundary of the object. Experimental results have shown the potentiality and generalization of our proposed method.
Simultaneous inference for high-dimensional non-Gaussian time series is always considered to be a challenging problem. Such tasks require not only robust estimation of the coefficients in the random process, but also deriving limiting distribution for a sum of dependent variables. In this paper, we propose a multiplier bootstrap procedure to conduct simultaneous inference for the transition coefficients in high-dimensional non-Gaussian vector autoregressive (VAR) models. This bootstrap-assisted procedure allows the dimension of the time series to grow exponentially fast in the number of observations. As a test statistic, a de-biased estimator is constructed for simultaneous inference. Unlike the traditional de-biased/de-sparsifying Lasso estimator, robust convex loss function and normalizing weight function are exploited to avoid any unfavorable behavior at the tail of the distribution. We develop Gaussian approximation theory for VAR model to derive the asymptotic distribution of the de-biased estimator and propose a multiplier bootstrap-assisted procedure to obtain critical values under very mild moment conditions on the innovations. As an important tool in the convergence analysis of various estimators, we establish a Bernstein-type probabilistic concentration inequality for bounded VAR models. Numerical experiments verify the validity and efficiency of the proposed method.
We overcome two major bottlenecks in the study of low rank approximation by assuming the low rank factors themselves are sparse. Specifically, (1) for low rank approximation with spectral norm error, we show how to improve the best known $\mathsf{nnz}(\mathbf A) k / \sqrt{\varepsilon}$ running time to $\mathsf{nnz}(\mathbf A)/\sqrt{\varepsilon}$ running time plus low order terms depending on the sparsity of the low rank factors, and (2) for streaming algorithms for Frobenius norm error, we show how to bypass the known $\Omega(nk/\varepsilon)$ memory lower bound and obtain an $s k (\log n)/ \mathrm{poly}(\varepsilon)$ memory bound, where $s$ is the number of non-zeros of each low rank factor. Although this algorithm is inefficient, as it must be under standard complexity theoretic assumptions, we also present polynomial time algorithms using $\mathrm{poly}(s,k,\log n,\varepsilon^{-1})$ memory that output rank $k$ approximations supported on a $O(sk/\varepsilon)\times O(sk/\varepsilon)$ submatrix. Both the prior $\mathsf{nnz}(\mathbf A) k / \sqrt{\varepsilon}$ running time and the $nk/\varepsilon$ memory for these problems were long-standing barriers; our results give a natural way of overcoming them assuming sparsity of the low rank factors.
In the era of open data, Poisson and other count regression models are increasingly important. Still, conventional Poisson regression has remaining issues in terms of identifiability and computational efficiency. Especially, due to an identification problem, Poisson regression can be unstable for small samples with many zeros. Provided this, we develop a closed-form inference for an over-dispersed Poisson regression including Poisson additive mixed models. The approach is derived via mode-based log-Gaussian approximation. The resulting method is fast, practical, and free from the identification problem. Monte Carlo experiments demonstrate that the estimation error of the proposed method is a considerably smaller estimation error than the closed-form alternatives and as small as the usual Poisson regressions. For counts with many zeros, our approximation has better estimation accuracy than conventional Poisson regression. We obtained similar results in the case of Poisson additive mixed modeling considering spatial or group effects. The developed method was applied for analyzing COVID-19 data in Japan. This result suggests that influences of pedestrian density, age, and other factors on the number of cases change over periods.
We are interested in the intersection of approximation algorithms and complexity theory, in particular focusing on the complexity class APX. Informally, APX $\subseteq$ NPO is the complexity class comprising optimization problems where the ratio $\frac{OPT(I)}{ALG(I)} \leq c$ for all instances I. We will do a deep dive into studying APX as a complexity class, in particular, investigating how researchers have defined PTAS and L reductions, as well as the notion of APX-completeness, thereby clarifying where APX lies on the polynomial hierarchy. We will discuss the relationship of this class with FPTAS, PTAS, APX, log-APX and poly-APX). We will sketch the proof that Max 3-SAT is APX-hard, and compare this complexity class in relation to $BPP$, $ZPP$ to elucidate whether randomization is powerful enough to achieve certain approximation guarantees and introduce techniques that complement the design of approximation algorithms such as through \textit{primal-dual} analysis, \textit{local search} and \textit{semi-definite programming}. Through the PCP theorem, we will explore the fundamental relationship between hardness of approximation and randomness, and will recast the way we look at the complexity class NP. We will finish by looking at the \textit{"real world"} applications of this material in Economics. Finally, we will touch upon recent breakthroughs in the Metric Travelling Salesman and asymmetric travelling salesman problem, as well original directions for future research, such as quantifying the amount of additional compute power that access to an APX oracle provides, elucidating fundamental combinatorial properties of log-APX problems and unique ways to attack the problem of whether the minimum set-cover problem is self-improvable.
We consider the problem of estimating a $d$-dimensional discrete distribution from its samples observed under a $b$-bit communication constraint. In contrast to most previous results that largely focus on the global minimax error, we study the local behavior of the estimation error and provide \emph{pointwise} bounds that depend on the target distribution $p$. In particular, we show that the $\ell_2$ error decays with $O\left(\frac{\lVert p\rVert_{1/2}}{n2^b}\vee \frac{1}{n}\right)$ (In this paper, we use $a\vee b$ and $a \wedge b$ to denote $\max(a, b)$ and $\min(a,b)$ respectively.) when $n$ is sufficiently large, hence it is governed by the \emph{half-norm} of $p$ instead of the ambient dimension $d$. For the achievability result, we propose a two-round sequentially interactive estimation scheme that achieves this error rate uniformly over all $p$. Our scheme is based on a novel local refinement idea, where we first use a standard global minimax scheme to localize $p$ and then use the remaining samples to locally refine our estimate. We also develop a new local minimax lower bound with (almost) matching $\ell_2$ error, showing that any interactive scheme must admit a $\Omega\left( \frac{\lVert p \rVert_{{(1+\delta)}/{2}}}{n2^b}\right)$ $\ell_2$ error for any $\delta > 0$. The lower bound is derived by first finding the best parametric sub-model containing $p$, and then upper bounding the quantized Fisher information under this model. Our upper and lower bounds together indicate that the $\mathcal{H}_{1/2}(p) = \log(\lVert p \rVert_{{1}/{2}})$ bits of communication is both sufficient and necessary to achieve the optimal (centralized) performance, where $\mathcal{H}_{{1}/{2}}(p)$ is the R\'enyi entropy of order $2$. Therefore, under the $\ell_2$ loss, the correct measure of the local communication complexity at $p$ is its R\'enyi entropy.
In real-world decision making tasks, it is critical for data-driven reinforcement learning methods to be both stable and sample efficient. On-policy methods typically generate reliable policy improvement throughout training, while off-policy methods make more efficient use of data through sample reuse. In this work, we combine the theoretically supported stability benefits of on-policy algorithms with the sample efficiency of off-policy algorithms. We develop policy improvement guarantees that are suitable for the off-policy setting, and connect these bounds to the clipping mechanism used in Proximal Policy Optimization. This motivates an off-policy version of the popular algorithm that we call Generalized Proximal Policy Optimization with Sample Reuse. We demonstrate both theoretically and empirically that our algorithm delivers improved performance by effectively balancing the competing goals of stability and sample efficiency.
Given access to a single long trajectory generated by an unknown irreducible Markov chain $M$, we simulate an $\alpha$-lazy version of $M$ which is ergodic. This enables us to generalize recent results on estimation and identity testing that were stated for ergodic Markov chains in a way that allows fully empirical inference. In particular, our approach shows that the pseudo spectral gap introduced by Paulin [2015] and defined for ergodic Markov chains may be given a meaning already in the case of irreducible but possibly periodic Markov chains.
With the rapid development of new anti-cancer agents which are cytostatic, new endpoints are needed to better measure treatment efficacy in phase II trials. For this purpose, Von Hoff (1998) proposed the growth modulation index (GMI), i.e. the ratio between times to progression or progression-free survival times in two successive treatment lines. An essential task in studies using GMI as an endpoint is to estimate the distribution of GMI. Traditional methods for survival data have been used for estimating the GMI distribution because censoring is common for GMI data. However, we point out that the independent censoring assumption required by traditional survival methods is always violated for GMI, which may lead to severely biased results. In this paper, we construct nonparametric estimators for the distribution of GMI, accounting for the dependent censoring of GMI. We prove that the proposed estimators are consistent and converge weakly to zero-mean Gaussian processes upon proper normalization. Extensive simulation studies show that our estimators perform well in practical situations and outperform traditional methods. A phase II clinical trial using GMI as the primary endpoint is provided for illustration.
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