Importance sampling the union of rare events with an application to power systems analysis

We consider importance sampling to estimate the probability $\mu$ of a union of $J$ rare events $H_j$ defined by a random variable $\boldsymbol{x}$. The sampler we study has been used in spatial statistics, genomics and combinatorics going back at least to Karp and Luby (1983). It works by sampling one event at random, then sampling $\boldsymbol{x}$ conditionally on that event happening and it constructs an unbiased estimate of $\mu$ by multiplying an inverse moment of the number of occuring events by the union bound. We prove some variance bounds for this sampler. For a sample size of $n$, it has a variance no larger than $\mu(\bar\mu-\mu)/n$ where $\bar\mu$ is the union bound. It also has a coefficient of variation no larger than $\sqrt{(J+J^{-1}-2)/(4n)}$ regardless of the overlap pattern among the $J$ events. Our motivating problem comes from power system reliability, where the phase differences between connected nodes have a joint Gaussian distribution and the $J$ rare events arise from unacceptably large phase differences. In the grid reliability problems even some events defined by $5772$ constraints in $326$ dimensions, with probability below $10^{-22}$, are estimated with a coefficient of variation of about $0.0024$ with only $n=10{,}000$ sample values

Lectures on Randomized Numerical Linear Algebra

This chapter is based on lectures on Randomized Numerical Linear Algebra from the 2016 Park City Mathematics Institute summer school on The Mathematics of Data.Comment: To appear in the edited volume of lectures from the 2016 PCMI summer schoo

On the Distributed Complexity of Large-Scale Graph Computations

Motivated by the increasing need to understand the distributed algorithmic foundations of large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where $k \geq 2$ machines jointly perform computations on graphs with $n$ nodes (typically, $n \gg k$). The input graph is assumed to be initially randomly partitioned among the $k$ machines, a common implementation in many real-world systems. Communication is point-to-point, and the goal is to minimize the number of communication {\em rounds} of the computation. Our main contribution is the {\em General Lower Bound Theorem}, a theorem that can be used to show non-trivial lower bounds on the round complexity of distributed large-scale data computations. The General Lower Bound Theorem is established via an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic and this theorem can be used in a "cookbook" fashion to show distributed lower bounds in the context of several problems, including non-graph problems. We present two applications by showing (almost) tight lower bounds for the round complexity of two fundamental graph problems, namely {\em PageRank computation} and {\em triangle enumeration}. Our approach, as demonstrated in the case of PageRank, can yield tight lower bounds for problems (including, and especially, under a stochastic partition of the input) where communication complexity techniques are not obvious. Our approach, as demonstrated in the case of triangle enumeration, can yield stronger round lower bounds as well as message-round tradeoffs compared to approaches that use communication complexity techniques