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Register a new userRapid Approximate Aggregation with Distribution-Sensitive Interval Guarantees
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Added by
Stephen Macke
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Aggregating data is fundamental to data analytics, data exploration, and OLAP. Approximate query processing (AQP) techniques are often used to accelerate computation of aggregates using samples, for which confidence intervals (CIs) are widely used to quantify the associated error. CIs used in practice fall into two categories: techniques that are tight but not correct, i.e., they yield tight intervals but only offer asymptotic guarantees, making them unreliable, or techniques that are correct but not tight, i.e., they offer rigorous guarantees, but are overly conservative, leading to confidence intervals that are too loose to be useful. In this paper, we develop a CI technique that is both correct and tighter than traditional approaches. Starting from conservative CIs, we identify two issues they often face: pessimistic mass allocation (PMA) and phantom outlier sensitivity (PHOS). By developing a novel range-trimming technique for eliminating PHOS and pairing it with known CI techniques without PMA, we develop a technique for computing CIs with strong guarantees that requires fewer samples for the same width. We implement our techniques underneath a sampling-optimized in-memory column store and show how to accelerate queries involving aggregates on a real dataset with speedups of up to 124x over traditional AQP-with-guarantees and more than 1000x over exact methods.
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Researchers and industry analysts are increasingly interested in computing aggregation queries over large, unstructured datasets with selective predicates that are computed using expensive deep neural networks (DNNs). As these DNNs are expensive and because many applications can tolerate approximate answers, analysts are interested in accelerating these queries via approximations. Unfortunately, standard approximate query processing techniques to accelerate such queries are not applicable because they assume the result of the predicates are available ahead of time. Furthermore, recent work using cheap approximations (i.e., proxies) do not support aggregation queries with predicates. To accelerate aggregation queries with expensive predicates, we develop and analyze a query processing algorithm that leverages proxies (ABae). ABae must account for the key challenge that it may sample records that do not satisfy the predicate. To address this challenge, we first use the proxy to group records into strata so that records satisfying the predicate are ideally grouped into few strata. Given these strata, ABae uses pilot sampling and plugin estimates to sample according to the optimal allocation. We show that ABae converges at an optimal rate in a novel analysis of stratified sampling with draws that may not satisfy the predicate. We further show that ABae outperforms on baselines on six real-world datasets, reducing labeling costs by up to 2.3x.
Sample-based approximate query processing (AQP) suffers from many pitfalls such as the inability to answer very selective queries and unreliable confidence intervals when sample sizes are small. Recent research presented an intriguing solution of combining materialized, pre-computed aggregates with sampling for accurate and more reliable AQP. We explore this solution in detail in this work and propose an AQP physical design called PASS, or Precomputation-Assisted Stratified Sampling. PASS builds a tree of partial aggregates that cover different partitions of the dataset. The leaf nodes of this tree form the strata for stratified samples. Aggregate queries whose predicates align with the partitions (or unions of partitions) are exactly answered with a depth-first search, and any partial overlaps are approximated with the stratified samples. We propose an algorithm for optimally partitioning the data into such a data structure with various practical approximation techniques.
Parallel aggregation is a ubiquitous operation in data analytics that is expressed as GROUP BY in SQL, reduce in Hadoop, or segment in TensorFlow. Parallel aggregation starts with an optional local pre-aggregation step and then repartitions the intermediate result across the network. While local pre-aggregation works well for low-cardinality aggregations, the network communication cost remains significant for high-cardinality aggregations even after local pre-aggregation. The problem is that the repartition-based algorithm for high-cardinality aggregation does not fully utilize the network. In this work, we first formulate a mathematical model that captures the performance of parallel aggregation. We prove that finding optimal aggregation plans from a known data distribution is NP-hard, assuming the Small Set Expansion conjecture. We propose GRASP, a GReedy Aggregation Scheduling Protocol that decomposes parallel aggregation into phases. GRASP is distribution-aware as it aggregates the most similar partitions in each phase to reduce the transmitted data size in subsequent phases. In addition, GRASP takes the available network bandwidth into account when scheduling aggregations in each phase to maximize network utilization. The experimental evaluation on real data shows that GRASP outperforms repartition-based aggregation by 3.5x and LOOM by 2.0x.
Recently, invariant risk minimization (IRM) (Arjovsky et al.) was proposed as a promising solution to address out-of-distribution (OOD) generalization. In Ahuja et al., it was shown that solving for the Nash equilibria of a new class of ensemble-games is equivalent to solving IRM. In this work, we extend the framework in Ahuja et al. for linear regressions by projecting the ensemble-game on an $ell_{infty}$ ball. We show that such projections help achieve non-trivial OOD guarantees despite not achieving perfect invariance. For linear models with confounders, we prove that Nash equilibria of these games are closer to the ideal OOD solutions than the standard empirical risk minimization (ERM) and we also provide learning algorithms that provably converge to these Nash Equilibria. Empirical comparisons of the proposed approach with the state-of-the-art show consistent gains in achieving OOD solutions in several settings involving anti-causal variables and confounders.
In the dictionary learning (or sparse coding) problem, we are given a collection of signals (vectors in $mathbb{R}^d$), and the goal is to find a basis in which the signals have a sparse (approximate) representation. The problem has received a lot of attention in signal processing, learning, and theoretical computer science. The problem is formalized as factorizing a matrix $X (d times n)$ (whose columns are the signals) as $X = AY$, where $A$ has a prescribed number $m$ of columns (typically $m ll n$), and $Y$ has columns that are $k$-sparse (typically $k ll d$). Most of the known theoretical results involve assuming that the columns of the unknown $A$ have certain incoherence properties, and that the coefficient matrix $Y$ has random (or partly random) structure. The goal of our work is to understand what can be said in the absence of such assumptions. Can we still find $A$ and $Y$ such that $X approx AY$? We show that this is possible, if we allow violating the bounds on $m$ and $k$ by appropriate factors that depend on $k$ and the desired approximation. Our results rely on an algorithm for what we call the threshold correlation problem, which turns out to be related to hypercontractive norms of matrices. We also show that our algorithmic ideas apply to a setting in which some of the columns of $X$ are outliers, thus giving similar guarantees even in this challenging setting.
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