arXiv subjects

Christopher Re

Publications and source records attributed to Christopher Re.

27 records · Page 2Linked to original sources

arXiv2019

Sparse Recovery for Orthogonal Polynomial Transforms

In this paper we consider the following sparse recovery problem. We have query access to a vector $\vx \in \R^N$ such that $\vhx = \vF \vx$ is $k$-sparse (or nearly $k$-sparse) for some orthogonal transform $\vF$. The goal is to output an approximation (in an $\ell_2$ sense) to $\vhx$ in sublinear time. This problem has been well-studied in the special case that $\vF$ is the Discrete Fourier Transform (DFT), and a long line of work has resulted in sparse Fast Fourier Transforms that run in time $O(k \cdot \mathrm{polylog} N)$. However, for transforms $\vF$ other than the DFT (or closely related transforms like the Discrete Cosine Transform), the question is much less settled. In this paper we give sublinear-time algorithms---running in time $\poly(k \log(N))$---for solving the sparse recovery problem for orthogonal transforms $\vF$ that arise from orthogonal polynomials. More precisely, our algorithm works for any $\vF$ that is an orthogonal polynomial transform derived from Jacobi polynomials. The Jacobi polynomials are a large class of classical orthogonal polynomials (and include Chebyshev and Legendre polynomials as special cases), and show up extensively in applications like numerical analysis and signal processing. One caveat of our work is that we require an assumption on the sparsity structure of the sparse vector, although we note that vectors with random support have this property with high probability. Our approach is to give a very general reduction from the $k$-sparse sparse recovery problem to the $1$-sparse sparse recovery problem that holds for any flat orthogonal polynomial transform; then we solve this one-sparse recovery problem for transforms derived from Jacobi polynomials.

cs.DS↗

arXiv2019

A Formal Framework For Probabilistic Unclean Databases

Most theoretical frameworks that focus on data errors and inconsistencies follow logic-based reasoning. Yet, practical data cleaning tools need to incorporate statistical reasoning to be effective in real-world data cleaning tasks. Motivated by these empirical successes, we propose a formal framework for unclean databases, where two types of statistical knowledge are incorporated: The first represents a belief of how intended (clean) data is generated, and the second represents a belief of how noise is introduced in the actual observed database instance. To capture this noisy channel model, we introduce the concept of a Probabilistic Unclean Database (PUD), a triple that consists of a probabilistic database that we call the intention, a probabilistic data transformator that we call the realization and captures how noise is introduced, and a dirty observed database instance that we call the observation. We define three computational problems in the PUD framework: cleaning (infer the most probable clean instance given a PUD), probabilistic query answering (compute the probability of an answer tuple over the unclean observed instance), and learning (estimate the most likely intention and realization models of a PUD given a collection of training data). We illustrate the PUD framework on concrete representations of the intention and realization, show that they generalize traditional concepts of repairs such as cardinality and value repairs, draw connection to consistent query answering, and prove tractability results. We further show that parameters can be learned in practical instantiations, and in fact, prove that under certain conditions we can learn a PUD directly from a single dirty database instance without any need for clean examples.

cs.DB↗

arXiv2017

Infrastructure for Usable Machine Learning: The Stanford DAWN Project

Despite incredible recent advances in machine learning, building machine learning applications remains prohibitively time-consuming and expensive for all but the best-trained, best-funded engineering organizations. This expense comes not from a need for new and improved statistical models but instead from a lack of systems and tools for supporting end-to-end machine learning application development, from data preparation and labeling to productionization and monitoring. In this document, we outline opportunities for infrastructure supporting usable, end-to-end machine learning applications in the context of the nascent DAWN (Data Analytics for What's Next) project at Stanford.

cs.LG↗

arXiv2015

It's all a matter of degree: Using degree information to optimize multiway joins

We optimize multiway equijoins on relational tables using degree information. We give a new bound that uses degree information to more tightly bound the maximum output size of a query. On real data, our bound on the number of triangles in a social network can be up to $95$ times tighter than existing worst case bounds. We show that using only a constant amount of degree information, we are able to obtain join algorithms with a running time that has a smaller exponent than existing algorithms--{\em for any database instance}. We also show that this degree information can be obtained in nearly linear time, which yields asymptotically faster algorithms in the serial setting and lower communication algorithms in the MapReduce setting. In the serial setting, the data complexity of join processing can be expressed as a function $O(\IN^x + \OUT)$ in terms of input size $\IN$ and output size $\OUT$ in which $x$ depends on the query. An upper bound for $x$ is given by fractional hypertreewidth. We are interested in situations in which we can get algorithms for which $x$ is strictly smaller than the fractional hypertreewidth. We say that a join can be processed in subquadratic time if $x < 2$. Building on the AYZ algorithm for processing cycle joins in quadratic time, for a restricted class of joins which we call $1$-series-parallel graphs, we obtain a complete decision procedure for identifying subquadratic solvability (subject to the $3$-SUM problem requiring quadratic time). Our $3$-SUM based quadratic lower bound is tight, making it the only known tight bound for joins that does not require any assumption about the matrix multiplication exponent $ω$. We also give a MapReduce algorithm that meets our improved communication bound and handles essentially optimal parallelism.

cs.DB↗

arXiv2014

Exploiting Correlations for Expensive Predicate Evaluation

User Defined Function(UDFs) are used increasingly to augment query languages with extra, application dependent functionality. Selection queries involving UDF predicates tend to be expensive, either in terms of monetary cost or latency. In this paper, we study ways to efficiently evaluate selection queries with UDF predicates. We provide a family of techniques for processing queries at low cost while satisfying user-specified precision and recall constraints. Our techniques are applicable to a variety of scenarios including when selection probabilities of tuples are available beforehand, when this information is available but noisy, or when no such prior information is available. We also generalize our techniques to more complex queries. Finally, we test our techniques on real datasets, and show that they achieve significant savings in cost of up to $80\%$, while incurring only a small reduction in accuracy.

cs.DB↗

arXiv2013

Skew Strikes Back: New Developments in the Theory of Join Algorithms

Evaluating the relational join is one of the central algorithmic and most well-studied problems in database systems. A staggering number of variants have been considered including Block-Nested loop join, Hash-Join, Grace, Sort-merge for discussions of more modern issues). Commercial database engines use finely tuned join heuristics that take into account a wide variety of factors including the selectivity of various predicates, memory, IO, etc. In spite of this study of join queries, the textbook description of join processing is suboptimal. This survey describes recent results on join algorithms that have provable worst-case optimality runtime guarantees. We survey recent work and provide a simpler and unified description of these algorithms that we hope is useful for theory-minded readers, algorithm designers, and systems implementors.

cs.DB↗

arXiv2013

Factoring nonnegative matrices with linear programs

This paper describes a new approach, based on linear programming, for computing nonnegative matrix factorizations (NMFs). The key idea is a data-driven model for the factorization where the most salient features in the data are used to express the remaining features. More precisely, given a data matrix X, the algorithm identifies a matrix C such that X approximately equals CX and some linear constraints. The constraints are chosen to ensure that the matrix C selects features; these features can then be used to find a low-rank NMF of X. A theoretical analysis demonstrates that this approach has guarantees similar to those of the recent NMF algorithm of Arora et al. (2012). In contrast with this earlier work, the proposed method extends to more general noise models and leads to efficient, scalable algorithms. Experiments with synthetic and real datasets provide evidence that the new approach is also superior in practice. An optimized C++ implementation can factor a multigigabyte matrix in a matter of minutes.

math.OC↗

arXiv2012

Beneath the valley of the noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences

Randomized algorithms that base iteration-level decisions on samples from some pool are ubiquitous in machine learning and optimization. Examples include stochastic gradient descent and randomized coordinate descent. This paper makes progress at theoretically evaluating the difference in performance between sampling with- and without-replacement in such algorithms. Focusing on least means squares optimization, we formulate a noncommutative arithmetic-geometric mean inequality that would prove that the expected convergence rate of without-replacement sampling is faster than that of with-replacement sampling. We demonstrate that this inequality holds for many classes of random matrices and for some pathological examples as well. We provide a deterministic worst-case bound on the gap between the discrepancy between the two sampling models, and explore some of the impediments to proving this inequality in full generality. We detail the consequences of this inequality for stochastic gradient descent and the randomized Kaczmarz algorithm for solving linear systems.

math.OC↗

arXiv2011

HOGWILD!: A Lock-Free Approach to Parallelizing Stochastic Gradient Descent

Stochastic Gradient Descent (SGD) is a popular algorithm that can achieve state-of-the-art performance on a variety of machine learning tasks. Several researchers have recently proposed schemes to parallelize SGD, but all require performance-destroying memory locking and synchronization. This work aims to show using novel theoretical analysis, algorithms, and implementation that SGD can be implemented without any locking. We present an update scheme called HOGWILD! which allows processors access to shared memory with the possibility of overwriting each other's work. We show that when the associated optimization problem is sparse, meaning most gradient updates only modify small parts of the decision variable, then HOGWILD! achieves a nearly optimal rate of convergence. We demonstrate experimentally that HOGWILD! outperforms alternative schemes that use locking by an order of magnitude.

math.OC↗