arXiv subjects

Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 361 records · Page 20Linked to original sources

arXiv2017

Decoding from Pooled Data: Phase Transitions of Message Passing

We consider the problem of decoding a discrete signal of categorical variables from the observation of several histograms of pooled subsets of it. We present an Approximate Message Passing (AMP) algorithm for recovering the signal in the random dense setting where each observed histogram involves a random subset of entries of size proportional to n. We characterize the performance of the algorithm in the asymptotic regime where the number of observations $m$ tends to infinity proportionally to n, by deriving the corresponding State Evolution (SE) equations and studying their dynamics. We initiate the analysis of the multi-dimensional SE dynamics by proving their convergence to a fixed point, along with some further properties of the iterates. The analysis reveals sharp phase transition phenomena where the behavior of AMP changes from exact recovery to weak correlation with the signal as m/n crosses a threshold. We derive formulae for the threshold in some special cases and show that they accurately match experimental behavior.

cs.IT↗

arXiv2016

Decoding from Pooled Data: Sharp Information-Theoretic Bounds

Consider a population consisting of n individuals, each of whom has one of d types (e.g. their blood type, in which case d=4). We are allowed to query this database by specifying a subset of the population, and in response we observe a noiseless histogram (a d-dimensional vector of counts) of types of the pooled individuals. This measurement model arises in practical situations such as pooling of genetic data and may also be motivated by privacy considerations. We are interested in the number of queries one needs to unambiguously determine the type of each individual. In this paper, we study this information-theoretic question under the random, dense setting where in each query, a random subset of individuals of size proportional to n is chosen. This makes the problem a particular example of a random constraint satisfaction problem (CSP) with a "planted" solution. We establish almost matching upper and lower bounds on the minimum number of queries m such that there is no solution other than the planted one with probability tending to 1 as n tends to infinity. Our proof relies on the computation of the exact "annealed free energy" of this model in the thermodynamic limit, which corresponds to the exponential rate of decay of the expected number of solution to this planted CSP. As a by-product of the analysis, we show an identity of independent interest relating the Gaussian integral over the space of Eulerian flows of a graph to its spanning tree polynomial.

math.PR↗

arXiv2016

Communication-Efficient Distributed Statistical Inference

We present a Communication-efficient Surrogate Likelihood (CSL) framework for solving distributed statistical inference problems. CSL provides a communication-efficient surrogate to the global likelihood that can be used for low-dimensional estimation, high-dimensional regularized estimation and Bayesian inference. For low-dimensional estimation, CSL provably improves upon naive averaging schemes and facilitates the construction of confidence intervals. For high-dimensional regularized estimation, CSL leads to a minimax-optimal estimator with controlled communication cost. For Bayesian inference, CSL can be used to form a communication-efficient quasi-posterior distribution that converges to the true posterior. This quasi-posterior procedure significantly improves the computational efficiency of MCMC algorithms even in a non-distributed setting. We present both theoretical analysis and experiments to explore the properties of the CSL approximation.

stat.ML↗

arXiv2016

Function-Specific Mixing Times and Concentration Away from Equilibrium

Slow mixing is the central hurdle when working with Markov chains, especially those used for Monte Carlo approximations (MCMC). In many applications, it is only of interest to estimate the stationary expectations of a small set of functions, and so the usual definition of mixing based on total variation convergence may be too conservative. Accordingly, we introduce function-specific analogs of mixing times and spectral gaps, and use them to prove Hoeffding-like function-specific concentration inequalities. These results show that it is possible for empirical expectations of functions to concentrate long before the underlying chain has mixed in the classical sense, and we show that the concentration rates we achieve are optimal up to constants. We use our techniques to derive confidence intervals that are sharper than those implied by both classical Markov chain Hoeffding bounds and Berry-Esseen-corrected CLT bounds. For applications that require testing, rather than point estimation, we show similar improvements over recent sequential testing results for MCMC. We conclude by applying our framework to real data examples of MCMC, providing evidence that our theory is both accurate and relevant to practice.

math.ST↗

arXiv2016

Distributed Optimization with Arbitrary Local Solvers

With the growth of data and necessity for distributed optimization methods, solvers that work well on a single machine must be re-designed to leverage distributed computation. Recent work in this area has been limited by focusing heavily on developing highly specific methods for the distributed environment. These special-purpose methods are often unable to fully leverage the competitive performance of their well-tuned and customized single machine counterparts. Further, they are unable to easily integrate improvements that continue to be made to single machine methods. To this end, we present a framework for distributed optimization that both allows the flexibility of arbitrary solvers to be used on each (single) machine locally, and yet maintains competitive performance against other state-of-the-art special-purpose distributed methods. We give strong primal-dual convergence rate guarantees for our framework that hold for arbitrary local solvers. We demonstrate the impact of local solver selection both theoretically and in an extensive experimental comparison. Finally, we provide thorough implementation details for our framework, highlighting areas for practical performance gains.

cs.LG↗

arXiv2016

A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation

We derive a new discrepancy statistic for measuring differences between two probability distributions based on combining Stein's identity with the reproducing kernel Hilbert space theory. We apply our result to test how well a probabilistic model fits a set of observations, and derive a new class of powerful goodness-of-fit tests that are widely applicable for complex and high dimensional distributions, even for those with computationally intractable normalization constants. Both theoretical and empirical properties of our methods are studied thoroughly.

stat.ML↗

arXiv2016

L1-Regularized Distributed Optimization: A Communication-Efficient Primal-Dual Framework

Despite the importance of sparsity in many large-scale applications, there are few methods for distributed optimization of sparsity-inducing objectives. In this paper, we present a communication-efficient framework for L1-regularized optimization in the distributed environment. By viewing classical objectives in a more general primal-dual setting, we develop a new class of methods that can be efficiently distributed and applied to common sparsity-inducing models, such as Lasso, sparse logistic regression, and elastic net-regularized problems. We provide theoretical convergence guarantees for our framework, and demonstrate its efficiency and flexibility with a thorough experimental comparison on Amazon EC2. Our proposed framework yields speedups of up to 50x as compared to current state-of-the-art methods for distributed L1-regularized optimization.

cs.LG↗

arXiv2016

CYCLADES: Conflict-free Asynchronous Machine Learning

We present CYCLADES, a general framework for parallelizing stochastic optimization algorithms in a shared memory setting. CYCLADES is asynchronous during shared model updates, and requires no memory locking mechanisms, similar to HOGWILD!-type algorithms. Unlike HOGWILD!, CYCLADES introduces no conflicts during the parallel execution, and offers a black-box analysis for provable speedups across a large family of algorithms. Due to its inherent conflict-free nature and cache locality, our multi-core implementation of CYCLADES consistently outperforms HOGWILD!-type algorithms on sufficiently sparse datasets, leading to up to 40% speedup gains compared to the HOGWILD! implementation of SGD, and up to 5x gains over asynchronous implementations of variance reduction algorithms.

stat.ML↗

arXiv2016

Posteriors, conjugacy, and exponential families for completely random measures

We demonstrate how to calculate posteriors for general CRM-based priors and likelihoods for Bayesian nonparametric models. We further show how to represent Bayesian nonparametric priors as a sequence of finite draws using a size-biasing approach---and how to represent full Bayesian nonparametric models via finite marginals. Motivated by conjugate priors based on exponential family representations of likelihoods, we introduce a notion of exponential families for CRMs, which we call exponential CRMs. This construction allows us to specify automatic Bayesian nonparametric conjugate priors for exponential CRM likelihoods. We demonstrate that our exponential CRMs allow particularly straightforward recipes for size-biased and marginal representations of Bayesian nonparametric models. Along the way, we prove that the gamma process is a conjugate prior for the Poisson likelihood process and the beta prime process is a conjugate prior for a process we call the odds Bernoulli process. We deliver a size-biased representation of the gamma process and a marginal representation of the gamma process coupled with a Poisson likelihood process.

math.ST↗

arXiv2016

A Linearly-Convergent Stochastic L-BFGS Algorithm

We propose a new stochastic L-BFGS algorithm and prove a linear convergence rate for strongly convex and smooth functions. Our algorithm draws heavily from a recent stochastic variant of L-BFGS proposed in Byrd et al. (2014) as well as a recent approach to variance reduction for stochastic gradient descent from Johnson and Zhang (2013). We demonstrate experimentally that our algorithm performs well on large-scale convex and non-convex optimization problems, exhibiting linear convergence and rapidly solving the optimization problems to high levels of precision. Furthermore, we show that our algorithm performs well for a wide-range of step sizes, often differing by several orders of magnitude.

math.OC↗

arXiv2016

Perturbed Iterate Analysis for Asynchronous Stochastic Optimization

We introduce and analyze stochastic optimization methods where the input to each gradient update is perturbed by bounded noise. We show that this framework forms the basis of a unified approach to analyze asynchronous implementations of stochastic optimization algorithms.In this framework, asynchronous stochastic optimization algorithms can be thought of as serial methods operating on noisy inputs. Using our perturbed iterate framework, we provide new analyses of the Hogwild! algorithm and asynchronous stochastic coordinate descent, that are simpler than earlier analyses, remove many assumptions of previous models, and in some cases yield improved upper bounds on the convergence rates. We proceed to apply our framework to develop and analyze KroMagnon: a novel, parallel, sparse stochastic variance-reduced gradient (SVRG) algorithm. We demonstrate experimentally on a 16-core machine that the sparse and parallel version of SVRG is in some cases more than four orders of magnitude faster than the standard SVRG algorithm.

stat.ML↗

arXiv2016

A Variational Perspective on Accelerated Methods in Optimization

Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. While many generalizations and extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the \emph{Bregman Lagrangian} which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods correspond to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov's technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.

math.OC↗

arXiv2016

Gradient Descent Converges to Minimizers

We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.

stat.ML↗

arXiv2016

Asymptotic behavior of $\ell_p$-based Laplacian regularization in semi-supervised learning

Given a weighted graph with $N$ vertices, consider a real-valued regression problem in a semi-supervised setting, where one observes $n$ labeled vertices, and the task is to label the remaining ones. We present a theoretical study of $\ell_p$-based Laplacian regularization under a $d$-dimensional geometric random graph model. We provide a variational characterization of the performance of this regularized learner as $N$ grows to infinity while $n$ stays constant, the associated optimality conditions lead to a partial differential equation that must be satisfied by the associated function estimate $\hat{f}$. From this formulation we derive several predictions on the limiting behavior the $d$-dimensional function $\hat{f}$, including (a) a phase transition in its smoothness at the threshold $p = d + 1$, and (b) a tradeoff between smoothness and sensitivity to the underlying unlabeled data distribution $P$. Thus, over the range $p \leq d$, the function estimate $\hat{f}$ is degenerate and "spiky," whereas for $p\geq d+1$, the function estimate $\hat{f}$ is smooth. We show that the effect of the underlying density vanishes monotonically with $p$, such that in the limit $p = \infty$, corresponding to the so-called Absolutely Minimal Lipschitz Extension, the estimate $\hat{f}$ is independent of the distribution $P$. Under the assumption of semi-supervised smoothness, ignoring $P$ can lead to poor statistical performance, in particular, we construct a specific example for $d=1$ to demonstrate that $p=2$ has lower risk than $p=\infty$ due to the former penalty adapting to $P$ and the latter ignoring it. We also provide simulations that verify the accuracy of our predictions for finite sample sizes. Together, these properties show that $p = d+1$ is an optimal choice, yielding a function estimate $\hat{f}$ that is both smooth and non-degenerate, while remaining maximally sensitive to $P$.

cs.LG↗

arXiv2016

SparkNet: Training Deep Networks in Spark

Training deep networks is a time-consuming process, with networks for object recognition often requiring multiple days to train. For this reason, leveraging the resources of a cluster to speed up training is an important area of work. However, widely-popular batch-processing computational frameworks like MapReduce and Spark were not designed to support the asynchronous and communication-intensive workloads of existing distributed deep learning systems. We introduce SparkNet, a framework for training deep networks in Spark. Our implementation includes a convenient interface for reading data from Spark RDDs, a Scala interface to the Caffe deep learning framework, and a lightweight multi-dimensional tensor library. Using a simple parallelization scheme for stochastic gradient descent, SparkNet scales well with the cluster size and tolerates very high-latency communication. Furthermore, it is easy to deploy and use with no parameter tuning, and it is compatible with existing Caffe models. We quantify the dependence of the speedup obtained by SparkNet on the number of machines, the communication frequency, and the cluster's communication overhead, and we benchmark our system's performance on the ImageNet dataset.

stat.ML↗

arXiv2015

Optimal prediction for sparse linear models? Lower bounds for coordinate-separable M-estimators

For the problem of high-dimensional sparse linear regression, it is known that an $\ell_0$-based estimator can achieve a $1/n$ "fast" rate on the prediction error without any conditions on the design matrix, whereas in absence of restrictive conditions on the design matrix, popular polynomial-time methods only guarantee the $1/\sqrt{n}$ "slow" rate. In this paper, we show that the slow rate is intrinsic to a broad class of M-estimators. In particular, for estimators based on minimizing a least-squares cost function together with a (possibly non-convex) coordinate-wise separable regularizer, there is always a "bad" local optimum such that the associated prediction error is lower bounded by a constant multiple of $1/\sqrt{n}$. For convex regularizers, this lower bound applies to all global optima. The theory is applicable to many popular estimators, including convex $\ell_1$-based methods as well as M-estimators based on nonconvex regularizers, including the SCAD penalty or the MCP regularizer. In addition, for a broad class of nonconvex regularizers, we show that the bad local optima are very common, in that a broad class of local minimization algorithms with random initialization will typically converge to a bad solution.

math.ST↗

arXiv2015

Learning Halfspaces and Neural Networks with Random Initialization

We study non-convex empirical risk minimization for learning halfspaces and neural networks. For loss functions that are $L$-Lipschitz continuous, we present algorithms to learn halfspaces and multi-layer neural networks that achieve arbitrarily small excess risk $ε>0$. The time complexity is polynomial in the input dimension $d$ and the sample size $n$, but exponential in the quantity $(L/ε^2)\log(L/ε)$. These algorithms run multiple rounds of random initialization followed by arbitrary optimization steps. We further show that if the data is separable by some neural network with constant margin $γ>0$, then there is a polynomial-time algorithm for learning a neural network that separates the training data with margin $Ω(γ)$. As a consequence, the algorithm achieves arbitrary generalization error $ε>0$ with ${\rm poly}(d,1/ε)$ sample and time complexity. We establish the same learnability result when the labels are randomly flipped with probability $η<1/2$.

cs.LG↗

arXiv2015

Asynchronous Complex Analytics in a Distributed Dataflow Architecture

Scalable distributed dataflow systems have recently experienced widespread adoption, with commodity dataflow engines such as Hadoop and Spark, and even commodity SQL engines routinely supporting increasingly sophisticated analytics tasks (e.g., support vector machines, logistic regression, collaborative filtering). However, these systems' synchronous (often Bulk Synchronous Parallel) dataflow execution model is at odds with an increasingly important trend in the machine learning community: the use of asynchrony via shared, mutable state (i.e., data races) in convex programming tasks, which has---in a single-node context---delivered noteworthy empirical performance gains and inspired new research into asynchronous algorithms. In this work, we attempt to bridge this gap by evaluating the use of lightweight, asynchronous state transfer within a commodity dataflow engine. Specifically, we investigate the use of asynchronous sideways information passing (ASIP) that presents single-stage parallel iterators with a Volcano-like intra-operator iterator that can be used for asynchronous information passing. We port two synchronous convex programming algorithms, stochastic gradient descent and the alternating direction method of multipliers (ADMM), to use ASIPs. We evaluate an implementation of ASIPs within on Apache Spark that exhibits considerable speedups as well as a rich set of performance trade-offs in the use of these asynchronous algorithms.

cs.DB↗