On Learning Continuous Pairwise Markov Random Fields

We consider learning a sparse pairwise Markov Random Field (MRF) with continuous-valued variables from i.i.d samples. We adapt the algorithm of Vuffray et al. (2019) to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise MRFs with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of the optimization criterion employed in Vuffray et al. (2019) can be interpreted as local maximum likelihood estimation (MLE). As part of our analysis, we introduce a robust variation of sparse linear regression a` la Lasso, which may be of interest in its own right.

Model Agnostic Time Series Analysis via Matrix Estimation

We propose an algorithm to impute and forecast a time series by transforming the observed time series into a matrix, utilizing matrix estimation to recover missing values and de-noise observed entries, and performing linear regression to make predictions. At the core of our analysis is a representation result, which states that for a large model class, the transformed time series matrix is (approximately) low-rank. In effect, this generalizes the widely used Singular Spectrum Analysis (SSA) in time series literature, and allows us to establish a rigorous link between time series analysis and matrix estimation. The key to establishing this link is constructing a Page matrix with non-overlapping entries rather than a Hankel matrix as is commonly done in the literature (e.g., SSA). This particular matrix structure allows us to provide finite sample analysis for imputation and prediction, and prove the asymptotic consistency of our method. Another salient feature of our algorithm is that it is model agnostic with respect to both the underlying time dynamics and the noise distribution in the observations. The noise agnostic property of our approach allows us to recover the latent states when only given access to noisy and partial observations a la a Hidden Markov Model; e.g., recovering the time-varying parameter of a Poisson process without knowing that the underlying process is Poisson. Furthermore, since our forecasting algorithm requires regression with noisy features, our approach suggests a matrix estimation based method - coupled with a novel, non-standard matrix estimation error metric - to solve the error-in-variable regression problem, which could be of interest in its own right. Through synthetic and real-world datasets, we demonstrate that our algorithm outperforms standard software packages (including R libraries) in the presence of missing data as well as high levels of noise.

Robust Synthetic Control

We present a robust generalization of the synthetic control method for comparative case studies. Like the classical method, we present an algorithm to estimate the unobservable counterfactual of a treatment unit. A distinguishing feature of our algorithm is that of de-noising the data matrix via singular value thresholding, which renders our approach robust in multiple facets: it automatically identifies a good subset of donors, overcomes the challenges of missing data, and continues to work well in settings where covariate information may not be provided. To begin, we establish the condition under which the fundamental assumption in synthetic control-like approaches holds, i.e. when the linear relationship between the treatment unit and the donor pool prevails in both the pre- and post-intervention periods. We provide the first finite sample analysis for a broader class of models, the Latent Variable Model, in contrast to Factor Models previously considered in the literature. Further, we show that our de-noising procedure accurately imputes missing entries, producing a consistent estimator of the underlying signal matrix provided $p = Omega( T^{-1 + zeta})$ for some $zeta > 0$; here, $p$ is the fraction of observed data and $T$ is the time interval of interest. Under the same setting, we prove that the mean-squared-error (MSE) in our prediction estimation scales as $O(sigma^2/p + 1/sqrt{T})$, where $sigma^2$ is the noise variance. Using a data aggregation method, we show that the MSE can be made as small as $O(T^{-1/2+gamma})$ for any $gamma in (0, 1/2)$, leading to a consistent estimator. We also introduce a Bayesian framework to quantify the model uncertainty through posterior probabilities. Our experiments, using both real-world and synthetic datasets, demonstrate that our robust generalization yields an improvement over the classical synthetic control method.

On the Flow-level Dynamics of a Packet-switched Network

The packet is the fundamental unit of transportation in modern communication networks such as the Internet. Physical layer scheduling decisions are made at the level of packets, and packet-level models with exogenous arrival processes have long been employed to study network performance, as well as design scheduling policies that more efficiently utilize network resources. On the other hand, a user of the network is more concerned with end-to-end bandwidth, which is allocated through congestion control policies such as TCP. Utility-based flow-level models have played an important role in understanding congestion control protocols. In summary, these two classes of models have provided separate insights for flow-level and packet-level dynamics of a network.