Weighted Reservoir Sampling from Distributed Streams

We consider message-efficient continuous random sampling from a distributed stream, where the probability of inclusion of an item in the sample is proportional to a weight associated with the item. The unweighted version, where all weights are equal, is well studied, and admits tight upper and lower bounds on message complexity. For weighted sampling with replacement, there is a simple reduction to unweighted sampling with replacement. However, in many applications the stream has only a few heavy items which may dominate a random sample when chosen with replacement. Weighted sampling \textit{without replacement} (weighted SWOR) eludes this issue, since such heavy items can be sampled at most once. In this work, we present the first message-optimal algorithm for weighted SWOR from a distributed stream. Our algorithm also has optimal space and time complexity. As an application of our algorithm for weighted SWOR, we derive the first distributed streaming algorithms for tracking \textit{heavy hitters with residual error}. Here the goal is to identify stream items that contribute significantly to the residual stream, once the heaviest items are removed. Residual heavy hitters generalize the notion of $\ell_1$ heavy hitters and are important in streams that have a skewed distribution of weights. In addition to the upper bound, we also provide a lower bound on the message complexity that is nearly tight up to a $\log(1/\epsilon)$ factor. Finally, we use our weighted sampling algorithm to improve the message complexity of distributed $L_1$ tracking, also known as count tracking, which is a widely studied problem in distributed streaming. We also derive a tight message lower bound, which closes the message complexity of this fundamental problem.Comment: To appear in PODS 201

Efficient Modeling of Random Sampling-Based LRU Cache

The Miss Ratio Curve (MRC) is an important metric and effective tool for caching system performance prediction and optimization. Since the Least Recently Used (LRU) replacement policy is the de facto policy for many existing caching systems, most previous studies on efficient MRC construction are predominantly focused on the LRU replacement policy. Recently, the random sampling-based replacement mechanism, as opposed to replacement relying on the rigid LRU data structure, gains more popularity due to its lightweight and flexibility. To approximate LRU, at replacement times, the system randomly selects K objects and replaces the least recently used object among the sample. Redis implements this approximated LRU policy. We observe that there can exist a significant miss ratio gap between exact LRU and random sampling-based LRU under different sampling size K; therefore existing LRU MRC construction techniques cannot be directly applied to random sampling based LRU cache without loss of accuracy. In this thesis, we present a new probabilistic stack algorithm named KRR which can be used to accurately model random sampling based-LRU cache with arbitrary sampling size K. We propose two efficient stack update algorithms which reduce the expected running time of KRR from O(NM) to O(Nlog^2M) and O(NlogM), respectively, where N is the workload length and M is the number of distinct objects. Our implementation generates accurate miss ratio curves for both fixed and variable block size cache. Furthermore, we adopt spatial sampling which further reduces the running time of KRR by several orders of magnitude, and thus enables practical, low overhead online application of KRR

Parallel Weighted Random Sampling

Data structures for efficient sampling from a set of weighted items are an important building block of many applications. However, few parallel solutions are known. We close many of these gaps both for shared-memory and distributed-memory machines. We give efficient, fast, and practicable algorithms for sampling single items, k items with/without replacement, permutations, subsets, and reservoirs. We also give improved sequential algorithms for alias table construction and for sampling with replacement. Experiments on shared-memory parallel machines with up to 158 threads show near linear speedups both for construction and queries

Subsampling MCMC - An introduction for the survey statistician

The rapid development of computing power and efficient Markov Chain Monte Carlo (MCMC) simulation algorithms have revolutionized Bayesian statistics, making it a highly practical inference method in applied work. However, MCMC algorithms tend to be computationally demanding, and are particularly slow for large datasets. Data subsampling has recently been suggested as a way to make MCMC methods scalable on massively large data, utilizing efficient sampling schemes and estimators from the survey sampling literature. These developments tend to be unknown by many survey statisticians who traditionally work with non-Bayesian methods, and rarely use MCMC. Our article explains the idea of data subsampling in MCMC by reviewing one strand of work, Subsampling MCMC, a so called pseudo-marginal MCMC approach to speeding up MCMC through data subsampling. The review is written for a survey statistician without previous knowledge of MCMC methods since our aim is to motivate survey sampling experts to contribute to the growing Subsampling MCMC literature.Comment: Accepted for publication in Sankhya A. Previous uploaded version contained a bug in generating the figures and reference