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We present a new local-search algorithm for the $k$-median clustering problem. We show that local optima for this algorithm give a $(2.836+\epsilon)$-approximation; our result improves upon the $(3+\epsilon)$-approximate local-search algorithm of Arya et al. [STOC 01]. Moreover, a computer-aided analysis of a natural extension suggests that this approach may lead to an improvement over the best-known approximation guarantee for the problem. The new ingredient in our algorithm is the use of a potential function based on both the closest and second-closest facilities to each client. Specifically, the potential is the sum over all clients, of the distance of the client to its closest facility, plus (a small constant times) the truncated distance to its second-closest facility. We move from one solution to another only if the latter can be obtained by swapping a constant number of facilities, and has a smaller potential than the former. This refined potential allows us to avoid the bad local optima given by Arya et al. for the local-search algorithm based only on the cost of the solution.

This paper studies a Group Influence with Minimum cost which aims to find a seed set with smallest cost that can influence all target groups, where each user is associated with a cost and a group is influenced if the total score of the influenced users belonging to the group is at least a certain threshold. As the group-influence function is neither submodular nor supermodular, theoretical bounds on the quality of solutions returned by the well-known greedy approach may not be guaranteed. To address this challenge, we propose a bi-criteria polynomial-time approximation algorithm with high certainty. At the heart of the algorithm is a novel group reachable reverse sample concept, which helps speed up the estimation of the group influence function. Finally, extensive experiments conducted on real social networks show that our proposed algorithm outperform the state-of-the-art algorithms in terms of the objective value and the running time.

Linear regression on a set of observations linked by a network has been an essential tool in modeling the relationship between response and covariates with additional network data. Despite its wide range of applications in many areas, such as in social sciences and health-related research, the problem has not been well-studied in statistics so far. Previous methods either lack inference tools or rely on restrictive assumptions on social effects and usually assume that networks are observed without errors, which is unrealistic in many problems. This paper proposes a linear regression model with nonparametric network effects. The model does not assume that the relational data or network structure is exactly observed; thus, the method can be provably robust to a certain network perturbation level. A set of asymptotic inference results is established under a general requirement of the network observational errors, and the robustness of this method is studied in the specific setting when the errors come from random network models. We discover a phase-transition phenomenon of the inference validity concerning the network density when no prior knowledge of the network model is available while also showing a significant improvement achieved by knowing the network model. As a by-product of this analysis, we derive a rate-optimal concentration bound for random subspace projection that may be of independent interest. Extensive simulation studies are conducted to verify these theoretical results and demonstrate the advantage of the proposed method over existing work in terms of accuracy and computational efficiency under different data-generating models. The method is then applied to middle school students' network data to study the effectiveness of educational workshops in reducing school conflicts.

In classical statistics, a statistical experiment consisting of $n$ i.i.d observations from d-dimensional multinomial distributions can be well approximated by a $d-1$ dimensional Gaussian distribution. In a quantum version of the result it has been shown that a collection of $n$ qudits of full rank can be well approximated by a quantum system containing a classical part, which is a $d-1$ dimensional Gaussian distribution, and a quantum part containing an ensemble of $d(d-1)/2$ shifted thermal states. In this paper, we obtain a generalization of this result when the qudits are not of full rank. We show that when the rank of the qudits is $r$, then the limiting experiment consists of an $r-1$ dimensional Gaussian distribution and an ensemble of both shifted pure and shifted thermal states. We also outline a two-stage procedure for the estimation of the low-rank qudit, where we obtain an estimator which is sharp minimax optimal. For the estimation of a linear functional of the quantum state, we construct an estimator, analyze the risk and use quantum LAN to show that our estimator is also optimal in the minimax sense.

A search engine maintains local copies of different web pages to provide quick search results. This local cache is kept up-to-date by a web crawler that frequently visits these different pages to track changes in them. Ideally, the local copy should be updated as soon as a page changes on the web. However, finite bandwidth availability and server restrictions limit how frequently different pages can be crawled. This brings forth the following optimization problem: maximize the freshness of the local cache subject to the crawling frequencies being within prescribed bounds. While tractable algorithms do exist to solve this problem, these either assume the knowledge of exact page change rates or use inefficient methods such as MLE for estimating the same. We address this issue here. We provide three novel schemes for online estimation of page change rates, all of which have extremely low running times per iteration. The first is based on the law of large numbers and the second on stochastic approximation. The third is an extension of the second and includes a heavy-ball momentum term. All these schemes only need partial information about the page change process, i.e., they only need to know if the page has changed or not since the last crawled instance. Our main theoretical results concern asymptotic convergence and convergence rates of these three schemes. In fact, our work is the first to show convergence of the original stochastic heavy-ball method when neither the gradient nor the noise variance is uniformly bounded. We also provide some numerical experiments (based on real and synthetic data) to demonstrate the superiority of our proposed estimators over existing ones such as MLE. We emphasize that our algorithms are also readily applicable to the synchronization of databases and network inventory management.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

Finding approximate Nash equilibria in zero-sum imperfect-information games is challenging when the number of information states is large. Policy Space Response Oracles (PSRO) is a deep reinforcement learning algorithm grounded in game theory that is guaranteed to converge to an approximate Nash equilibrium. However, PSRO requires training a reinforcement learning policy at each iteration, making it too slow for large games. We show through counterexamples and experiments that DCH and Rectified PSRO, two existing approaches to scaling up PSRO, fail to converge even in small games. We introduce Pipeline PSRO (P2SRO), the first scalable general method for finding approximate Nash equilibria in large zero-sum imperfect-information games. P2SRO is able to parallelize PSRO with convergence guarantees by maintaining a hierarchical pipeline of reinforcement learning workers, each training against the policies generated by lower levels in the hierarchy. We show that unlike existing methods, P2SRO converges to an approximate Nash equilibrium, and does so faster as the number of parallel workers increases, across a variety of imperfect information games. We also introduce an open-source environment for Barrage Stratego, a variant of Stratego with an approximate game tree complexity of $10^{50}$. P2SRO is able to achieve state-of-the-art performance on Barrage Stratego and beats all existing bots.

Clustering is one of the most fundamental and wide-spread techniques in exploratory data analysis. Yet, the basic approach to clustering has not really changed: a practitioner hand-picks a task-specific clustering loss to optimize and fit the given data to reveal the underlying cluster structure. Some types of losses---such as k-means, or its non-linear version: kernelized k-means (centroid based), and DBSCAN (density based)---are popular choices due to their good empirical performance on a range of applications. Although every so often the clustering output using these standard losses fails to reveal the underlying structure, and the practitioner has to custom-design their own variation. In this work we take an intrinsically different approach to clustering: rather than fitting a dataset to a specific clustering loss, we train a recurrent model that learns how to cluster. The model uses as training pairs examples of datasets (as input) and its corresponding cluster identities (as output). By providing multiple types of training datasets as inputs, our model has the ability to generalize well on unseen datasets (new clustering tasks). Our experiments reveal that by training on simple synthetically generated datasets or on existing real datasets, we can achieve better clustering performance on unseen real-world datasets when compared with standard benchmark clustering techniques. Our meta clustering model works well even for small datasets where the usual deep learning models tend to perform worse.

In recent years, there has been an exponential growth in the number of complex documents and texts that require a deeper understanding of machine learning methods to be able to accurately classify texts in many applications. Many machine learning approaches have achieved surpassing results in natural language processing. The success of these learning algorithms relies on their capacity to understand complex models and non-linear relationships within data. However, finding suitable structures, architectures, and techniques for text classification is a challenge for researchers. In this paper, a brief overview of text classification algorithms is discussed. This overview covers different text feature extractions, dimensionality reduction methods, existing algorithms and techniques, and evaluations methods. Finally, the limitations of each technique and their application in the real-world problem are discussed.

Recent studies have shown the vulnerability of reinforcement learning (RL) models in noisy settings. The sources of noises differ across scenarios. For instance, in practice, the observed reward channel is often subject to noise (e.g., when observed rewards are collected through sensors), and thus observed rewards may not be credible as a result. Also, in applications such as robotics, a deep reinforcement learning (DRL) algorithm can be manipulated to produce arbitrary errors. In this paper, we consider noisy RL problems where observed rewards by RL agents are generated with a reward confusion matrix. We call such observed rewards as perturbed rewards. We develop an unbiased reward estimator aided robust RL framework that enables RL agents to learn in noisy environments while observing only perturbed rewards. Our framework draws upon approaches for supervised learning with noisy data. The core ideas of our solution include estimating a reward confusion matrix and defining a set of unbiased surrogate rewards. We prove the convergence and sample complexity of our approach. Extensive experiments on different DRL platforms show that policies based on our estimated surrogate reward can achieve higher expected rewards, and converge faster than existing baselines. For instance, the state-of-the-art PPO algorithm is able to obtain 67.5% and 46.7% improvements in average on five Atari games, when the error rates are 10% and 30% respectively.