arXiv subjects

Christopher De Sa

Publications and source records attributed to Christopher De Sa.

85 records · Page 5Linked to original sources

arXiv2018

Gaussian Quadrature for Kernel Features

Kernel methods have recently attracted resurgent interest, showing performance competitive with deep neural networks in tasks such as speech recognition. The random Fourier features map is a technique commonly used to scale up kernel machines, but employing the randomized feature map means that $O(ε^{-2})$ samples are required to achieve an approximation error of at most $ε$. We investigate some alternative schemes for constructing feature maps that are deterministic, rather than random, by approximating the kernel in the frequency domain using Gaussian quadrature. We show that deterministic feature maps can be constructed, for any $γ> 0$, to achieve error $ε$ with $O(e^{e^γ} + ε^{-1/γ})$ samples as $ε$ goes to 0. Our method works particularly well with sparse ANOVA kernels, which are inspired by the convolutional layer of CNNs. We validate our methods on datasets in different domains, such as MNIST and TIMIT, showing that deterministic features are faster to generate and achieve accuracy comparable to the state-of-the-art kernel methods based on random Fourier features.

cs.LG↗

arXiv2017

A Two Pronged Progress in Structured Dense Matrix Multiplication

Matrix-vector multiplication is one of the most fundamental computing primitives. Given a matrix $A\in\mathbb{F}^{N\times N}$ and a vector $b$, it is known that in the worst case $Θ(N^2)$ operations over $\mathbb{F}$ are needed to compute $Ab$. A broad question is to identify classes of structured dense matrices that can be represented with $O(N)$ parameters, and for which matrix-vector multiplication can be performed sub-quadratically. One such class of structured matrices is the orthogonal polynomial transforms, whose rows correspond to a family of orthogonal polynomials. Other well known classes include the Toeplitz, Hankel, Vandermonde, Cauchy matrices and their extensions that are all special cases of a ldisplacement rank property. In this paper, we make progress on two fronts: 1. We introduce the notion of recurrence width of matrices. For matrices with constant recurrence width, we design algorithms to compute $Ab$ and $A^Tb$ with a near-linear number of operations. This notion of width is finer than all the above classes of structured matrices and thus we can compute multiplication for all of them using the same core algorithm. 2. We additionally adapt this algorithm to an algorithm for a much more general class of matrices with displacement structure: those with low displacement rank with respect to quasiseparable matrices. This class includes Toeplitz-plus-Hankel-like matrices, Discrete Cosine/Sine Transforms, and more, and captures all previously known matrices with displacement structure that we are aware of under a unified parametrization and algorithm. Our work unifies, generalizes, and simplifies existing state-of-the-art results in structured matrix-vector multiplication. Finally, we show how applications in areas such as multipoint evaluations of multivariate polynomials can be reduced to problems involving low recurrence width matrices.

cs.DS↗

arXiv2017

Socratic Learning: Augmenting Generative Models to Incorporate Latent Subsets in Training Data

A challenge in training discriminative models like neural networks is obtaining enough labeled training data. Recent approaches use generative models to combine weak supervision sources, like user-defined heuristics or knowledge bases, to label training data. Prior work has explored learning accuracies for these sources even without ground truth labels, but they assume that a single accuracy parameter is sufficient to model the behavior of these sources over the entire training set. In particular, they fail to model latent subsets in the training data in which the supervision sources perform differently than on average. We present Socratic learning, a paradigm that uses feedback from a corresponding discriminative model to automatically identify these subsets and augments the structure of the generative model accordingly. Experimentally, we show that without any ground truth labels, the augmented generative model reduces error by up to 56.06% for a relation extraction task compared to a state-of-the-art weak supervision technique that utilizes generative models.

cs.LG↗

arXiv2017

Accelerated Stochastic Power Iteration

Principal component analysis (PCA) is one of the most powerful tools in machine learning. The simplest method for PCA, the power iteration, requires $\mathcal O(1/Δ)$ full-data passes to recover the principal component of a matrix with eigen-gap $Δ$. Lanczos, a significantly more complex method, achieves an accelerated rate of $\mathcal O(1/\sqrtΔ)$ passes. Modern applications, however, motivate methods that only ingest a subset of available data, known as the stochastic setting. In the online stochastic setting, simple algorithms like Oja's iteration achieve the optimal sample complexity $\mathcal O(σ^2/Δ^2)$. Unfortunately, they are fully sequential, and also require $\mathcal O(σ^2/Δ^2)$ iterations, far from the $\mathcal O(1/\sqrtΔ)$ rate of Lanczos. We propose a simple variant of the power iteration with an added momentum term, that achieves both the optimal sample and iteration complexity. In the full-pass setting, standard analysis shows that momentum achieves the accelerated rate, $\mathcal O(1/\sqrtΔ)$. We demonstrate empirically that naively applying momentum to a stochastic method, does not result in acceleration. We perform a novel, tight variance analysis that reveals the "breaking-point variance" beyond which this acceleration does not occur. By combining this insight with modern variance reduction techniques, we construct stochastic PCA algorithms, for the online and offline setting, that achieve an accelerated iteration complexity $\mathcal O(1/\sqrtΔ)$. Due to the embarassingly parallel nature of our methods, this acceleration translates directly to wall-clock time if deployed in a parallel environment. Our approach is very general, and applies to many non-convex optimization problems that can now be accelerated using the same technique.

math.OC↗

arXiv2017

Data Programming: Creating Large Training Sets, Quickly

Large labeled training sets are the critical building blocks of supervised learning methods and are key enablers of deep learning techniques. For some applications, creating labeled training sets is the most time-consuming and expensive part of applying machine learning. We therefore propose a paradigm for the programmatic creation of training sets called data programming in which users express weak supervision strategies or domain heuristics as labeling functions, which are programs that label subsets of the data, but that are noisy and may conflict. We show that by explicitly representing this training set labeling process as a generative model, we can "denoise" the generated training set, and establish theoretically that we can recover the parameters of these generative models in a handful of settings. We then show how to modify a discriminative loss function to make it noise-aware, and demonstrate our method over a range of discriminative models including logistic regression and LSTMs. Experimentally, on the 2014 TAC-KBP Slot Filling challenge, we show that data programming would have led to a new winning score, and also show that applying data programming to an LSTM model leads to a TAC-KBP score almost 6 F1 points over a state-of-the-art LSTM baseline (and into second place in the competition). Additionally, in initial user studies we observed that data programming may be an easier way for non-experts to create machine learning models when training data is limited or unavailable.

stat.ML↗

arXiv2016

Parallel SGD: When does averaging help?

Consider a number of workers running SGD independently on the same pool of data and averaging the models every once in a while -- a common but not well understood practice. We study model averaging as a variance-reducing mechanism and describe two ways in which the frequency of averaging affects convergence. For convex objectives, we show the benefit of frequent averaging depends on the gradient variance envelope. For non-convex objectives, we illustrate that this benefit depends on the presence of multiple globally optimal points. We complement our findings with multicore experiments on both synthetic and real data.

stat.ML↗

arXiv2016

Ensuring Rapid Mixing and Low Bias for Asynchronous Gibbs Sampling

Gibbs sampling is a Markov chain Monte Carlo technique commonly used for estimating marginal distributions. To speed up Gibbs sampling, there has recently been interest in parallelizing it by executing asynchronously. While empirical results suggest that many models can be efficiently sampled asynchronously, traditional Markov chain analysis does not apply to the asynchronous case, and thus asynchronous Gibbs sampling is poorly understood. In this paper, we derive a better understanding of the two main challenges of asynchronous Gibbs: bias and mixing time. We show experimentally that our theoretical results match practical outcomes.

cs.LG↗

arXiv2016

Scan Order in Gibbs Sampling: Models in Which it Matters and Bounds on How Much

Gibbs sampling is a Markov Chain Monte Carlo sampling technique that iteratively samples variables from their conditional distributions. There are two common scan orders for the variables: random scan and systematic scan. Due to the benefits of locality in hardware, systematic scan is commonly used, even though most statistical guarantees are only for random scan. While it has been conjectured that the mixing times of random scan and systematic scan do not differ by more than a logarithmic factor, we show by counterexample that this is not the case, and we prove that that the mixing times do not differ by more than a polynomial factor under mild conditions. To prove these relative bounds, we introduce a method of augmenting the state space to study systematic scan using conductance.

cs.LG↗

arXiv2015

Generating Configurable Hardware from Parallel Patterns

In recent years the computing landscape has seen an in- creasing shift towards specialized accelerators. Field pro- grammable gate arrays (FPGAs) are particularly promising as they offer significant performance and energy improvements compared to CPUs for a wide class of applications and are far more flexible than fixed-function ASICs. However, FPGAs are difficult to program. Traditional programming models for reconfigurable logic use low-level hardware description languages like Verilog and VHDL, which have none of the pro- ductivity features of modern software development languages but produce very efficient designs, and low-level software lan- guages like C and OpenCL coupled with high-level synthesis (HLS) tools that typically produce designs that are far less efficient. Functional languages with parallel patterns are a better fit for hardware generation because they both provide high-level abstractions to programmers with little experience in hard- ware design and avoid many of the problems faced when gen- erating hardware from imperative languages. In this paper, we identify two optimizations that are important when using par- allel patterns to generate hardware: tiling and metapipelining. We present a general representation of tiled parallel patterns, and provide rules for automatically tiling patterns and gen- erating metapipelines. We demonstrate experimentally that these optimizations result in speedups up to 40x on a set of benchmarks from the data analytics domain.

cs.DC↗

arXiv2015

Taming the Wild: A Unified Analysis of Hogwild!-Style Algorithms

Stochastic gradient descent (SGD) is a ubiquitous algorithm for a variety of machine learning problems. Researchers and industry have developed several techniques to optimize SGD's runtime performance, including asynchronous execution and reduced precision. Our main result is a martingale-based analysis that enables us to capture the rich noise models that may arise from such techniques. Specifically, we use our new analysis in three ways: (1) we derive convergence rates for the convex case (Hogwild!) with relaxed assumptions on the sparsity of the problem; (2) we analyze asynchronous SGD algorithms for non-convex matrix problems including matrix completion; and (3) we design and analyze an asynchronous SGD algorithm, called Buckwild!, that uses lower-precision arithmetic. We show experimentally that our algorithms run efficiently for a variety of problems on modern hardware.

cs.LG↗

arXiv2015

Rapidly Mixing Gibbs Sampling for a Class of Factor Graphs Using Hierarchy Width

Gibbs sampling on factor graphs is a widely used inference technique, which often produces good empirical results. Theoretical guarantees for its performance are weak: even for tree structured graphs, the mixing time of Gibbs may be exponential in the number of variables. To help understand the behavior of Gibbs sampling, we introduce a new (hyper)graph property, called hierarchy width. We show that under suitable conditions on the weights, bounded hierarchy width ensures polynomial mixing time. Our study of hierarchy width is in part motivated by a class of factor graph templates, hierarchical templates, which have bounded hierarchy width---regardless of the data used to instantiate them. We demonstrate a rich application from natural language processing in which Gibbs sampling provably mixes rapidly and achieves accuracy that exceeds human volunteers.

cs.LG↗

arXiv2015

Incremental Knowledge Base Construction Using DeepDive

Populating a database with unstructured information is a long-standing problem in industry and research that encompasses problems of extraction, cleaning, and integration. Recent names used for this problem include dealing with dark data and knowledge base construction (KBC). In this work, we describe DeepDive, a system that combines database and machine learning ideas to help develop KBC systems, and we present techniques to make the KBC process more efficient. We observe that the KBC process is iterative, and we develop techniques to incrementally produce inference results for KBC systems. We propose two methods for incremental inference, based respectively on sampling and variational techniques. We also study the tradeoff space of these methods and develop a simple rule-based optimizer. DeepDive includes all of these contributions, and we evaluate DeepDive on five KBC systems, showing that it can speed up KBC inference tasks by up to two orders of magnitude with negligible impact on quality.

cs.DB↗

arXiv2015

Global Convergence of Stochastic Gradient Descent for Some Non-convex Matrix Problems

Stochastic gradient descent (SGD) on a low-rank factorization is commonly employed to speed up matrix problems including matrix completion, subspace tracking, and SDP relaxation. In this paper, we exhibit a step size scheme for SGD on a low-rank least-squares problem, and we prove that, under broad sampling conditions, our method converges globally from a random starting point within $O(ε^{-1} n \log n)$ steps with constant probability for constant-rank problems. Our modification of SGD relates it to stochastic power iteration. We also show experiments to illustrate the runtime and convergence of the algorithm.

cs.LG↗