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Variational methods are extremely popular in the analysis of network data. Statistical guarantees obtained for these methods typically provide asymptotic normality for the problem of estimation of global model parameters under the stochastic block model. In the present work, we consider the case of networks with missing links that is important in application and show that the variational approximation to the maximum likelihood estimator converges at the minimax rate. This provides the first minimax optimal and tractable estimator for the problem of parameter estimation for the stochastic block model with missing links. We complement our results with numerical studies of simulated and real networks, which confirm the advantages of this estimator over current methods.

This paper considers the Fourier transform over the slice of the Boolean hypercube. We prove a relationship between the Fourier coefficients of a function over the slice, and the Fourier coefficients of its restrictions. As an application, we prove a Goldreich-Levin theorem for functions on the slice based on the Kushilevitz-Mansour algorithm for the Boolean hypercube.

We examine one-hidden-layer neural networks with random weights. It is well-known that in the limit of infinitely many neurons they simplify to Gaussian processes. For networks with a polynomial activation, we demonstrate that the rate of this convergence in 2-Wasserstein metric is $O(n^{-\frac{1}{2}})$, where $n$ is the number of hidden neurons. We suspect this rate is asymptotically sharp. We improve the known convergence rate for other activations, to power-law in $n$ for ReLU and inverse-square-root up to logarithmic factors for erf. We explore the interplay between spherical harmonics, Stein kernels and optimal transport in the non-isotropic setting.

In this study, we propose a clustering-based approach on time-series data to capture COVID-19 spread patterns in the early period of the pandemic. We analyze the spread dynamics based on the early and post stages of COVID-19 for different countries based on different geographical locations. Furthermore, we investigate the confinement policies and the effect they made on the spread. We found that implementations of the same confinement policies exhibit different results in different countries. Specifically, lockdowns become less effective in densely populated regions, because of the reluctance to comply with social distancing measures. Lack of testing, contact tracing, and social awareness in some countries forestall people from self-isolation and maintaining social distance. Large labor camps with unhealthy living conditions also aid in high community transmissions in countries depending on foreign labor. Distrust in government policies and fake news instigate the spread in both developed and under-developed countries. Large social gatherings play a vital role in causing rapid outbreaks almost everywhere. While some countries were able to contain the spread by implementing strict and widely adopted confinement policies, some others contained the spread with the help of social distancing measures and rigorous testing capacity. An early and rapid response at the beginning of the pandemic is necessary to contain the spread, yet it is not always sufficient.

Time series forecasting is essential for decision making in many domains. In this work, we address the challenge of predicting prices evolution among multiple potentially interacting financial assets. A solution to this problem has obvious importance for governments, banks, and investors. Statistical methods such as Auto Regressive Integrated Moving Average (ARIMA) are widely applied to these problems. In this paper, we propose to approach economic time series forecasting of multiple financial assets in a novel way via video prediction. Given past prices of multiple potentially interacting financial assets, we aim to predict the prices evolution in the future. Instead of treating the snapshot of prices at each time point as a vector, we spatially layout these prices in 2D as an image, such that we can harness the power of CNNs in learning a latent representation for these financial assets. Thus, the history of these prices becomes a sequence of images, and our goal becomes predicting future images. We build on a state-of-the-art video prediction method for forecasting future images. Our experiments involve the prediction task of the price evolution of nine financial assets traded in U.S. stock markets. The proposed method outperforms baselines including ARIMA, Prophet, and variations of the proposed method, demonstrating the benefits of harnessing the power of CNNs in the problem of economic time series forecasting.

Spatio-temporal forecasting has numerous applications in analyzing wireless, traffic, and financial networks. Many classical statistical models often fall short in handling the complexity and high non-linearity present in time-series data. Recent advances in deep learning allow for better modelling of spatial and temporal dependencies. While most of these models focus on obtaining accurate point forecasts, they do not characterize the prediction uncertainty. In this work, we consider the time-series data as a random realization from a nonlinear state-space model and target Bayesian inference of the hidden states for probabilistic forecasting. We use particle flow as the tool for approximating the posterior distribution of the states, as it is shown to be highly effective in complex, high-dimensional settings. Thorough experimentation on several real world time-series datasets demonstrates that our approach provides better characterization of uncertainty while maintaining comparable accuracy to the state-of-the art point forecasting methods.

In this monograph, I introduce the basic concepts of Online Learning through a modern view of Online Convex Optimization. Here, online learning refers to the framework of regret minimization under worst-case assumptions. I present first-order and second-order algorithms for online learning with convex losses, in Euclidean and non-Euclidean settings. All the algorithms are clearly presented as instantiation of Online Mirror Descent or Follow-The-Regularized-Leader and their variants. Particular attention is given to the issue of tuning the parameters of the algorithms and learning in unbounded domains, through adaptive and parameter-free online learning algorithms. Non-convex losses are dealt through convex surrogate losses and through randomization. The bandit setting is also briefly discussed, touching on the problem of adversarial and stochastic multi-armed bandits. These notes do not require prior knowledge of convex analysis and all the required mathematical tools are rigorously explained. Moreover, all the proofs have been carefully chosen to be as simple and as short as possible.

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.

In this paper, we develop the continuous time dynamic topic model (cDTM). The cDTM is a dynamic topic model that uses Brownian motion to model the latent topics through a sequential collection of documents, where a "topic" is a pattern of word use that we expect to evolve over the course of the collection. We derive an efficient variational approximate inference algorithm that takes advantage of the sparsity of observations in text, a property that lets us easily handle many time points. In contrast to the cDTM, the original discrete-time dynamic topic model (dDTM) requires that time be discretized. Moreover, the complexity of variational inference for the dDTM grows quickly as time granularity increases, a drawback which limits fine-grained discretization. We demonstrate the cDTM on two news corpora, reporting both predictive perplexity and the novel task of time stamp prediction.