arXiv subjects

Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 343 records · Page 19Linked to original sources

arXiv2018

Underdamped Langevin MCMC: A non-asymptotic analysis

We study the underdamped Langevin diffusion when the log of the target distribution is smooth and strongly concave. We present a MCMC algorithm based on its discretization and show that it achieves $\varepsilon$ error (in 2-Wasserstein distance) in $\mathcal{O}(\sqrt{d}/\varepsilon)$ steps. This is a significant improvement over the best known rate for overdamped Langevin MCMC, which is $\mathcal{O}(d/\varepsilon^2)$ steps under the same smoothness/concavity assumptions. The underdamped Langevin MCMC scheme can be viewed as a version of Hamiltonian Monte Carlo (HMC) which has been observed to outperform overdamped Langevin MCMC methods in a number of application areas. We provide quantitative rates that support this empirical wisdom.

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arXiv2017

Stochastic Cubic Regularization for Fast Nonconvex Optimization

This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(ε^{-3.5})$ stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the $\mathcal{\tilde{O}}(ε^{-4})$ rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima without requiring any delicate acceleration or variance-reduction techniques.

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arXiv2017

Measuring Cluster Stability for Bayesian Nonparametrics Using the Linear Bootstrap

Clustering procedures typically estimate which data points are clustered together, a quantity of primary importance in many analyses. Often used as a preliminary step for dimensionality reduction or to facilitate interpretation, finding robust and stable clusters is often crucial for appropriate for downstream analysis. In the present work, we consider Bayesian nonparametric (BNP) models, a particularly popular set of Bayesian models for clustering due to their flexibility. Because of its complexity, the Bayesian posterior often cannot be computed exactly, and approximations must be employed. Mean-field variational Bayes forms a posterior approximation by solving an optimization problem and is widely used due to its speed. An exact BNP posterior might vary dramatically when presented with different data. As such, stability and robustness of the clustering should be assessed. A popular mean to assess stability is to apply the bootstrap by resampling the data, and rerun the clustering for each simulated data set. The time cost is thus often very expensive, especially for the sort of exploratory analysis where clustering is typically used. We propose to use a fast and automatic approximation to the full bootstrap called the "linear bootstrap", which can be seen by local data perturbation. In this work, we demonstrate how to apply this idea to a data analysis pipeline, consisting of an MFVB approximation to a BNP clustering posterior of time course gene expression data. We show that using auto-differentiation tools, the necessary calculations can be done automatically, and that the linear bootstrap is a fast but approximate alternative to the bootstrap.

stat.ME↗

arXiv2017

Accelerated Gradient Descent Escapes Saddle Points Faster than Gradient Descent

Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in $\tilde{O}(1/ε^{7/4})$ iterations, faster than the $\tilde{O}(1/ε^{2})$ iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization.

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arXiv2017

Gradient Descent Can Take Exponential Time to Escape Saddle Points

Although gradient descent (GD) almost always escapes saddle points asymptotically [Lee et al., 2016], this paper shows that even with fairly natural random initialization schemes and non-pathological functions, GD can be significantly slowed down by saddle points, taking exponential time to escape. On the other hand, gradient descent with perturbations [Ge et al., 2015, Jin et al., 2017] is not slowed down by saddle points - it can find an approximate local minimizer in polynomial time. This result implies that GD is inherently slower than perturbed GD, and justifies the importance of adding perturbations for efficient non-convex optimization. While our focus is theoretical, we also present experiments that illustrate our theoretical findings.

math.OC↗

arXiv2017

Fast Black-box Variational Inference through Stochastic Trust-Region Optimization

We introduce TrustVI, a fast second-order algorithm for black-box variational inference based on trust-region optimization and the reparameterization trick. At each iteration, TrustVI proposes and assesses a step based on minibatches of draws from the variational distribution. The algorithm provably converges to a stationary point. We implemented TrustVI in the Stan framework and compared it to two alternatives: Automatic Differentiation Variational Inference (ADVI) and Hessian-free Stochastic Gradient Variational Inference (HFSGVI). The former is based on stochastic first-order optimization. The latter uses second-order information, but lacks convergence guarantees. TrustVI typically converged at least one order of magnitude faster than ADVI, demonstrating the value of stochastic second-order information. TrustVI often found substantially better variational distributions than HFSGVI, demonstrating that our convergence theory can matter in practice.

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arXiv2017

First-order Methods Almost Always Avoid Saddle Points

We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical systems perspective in which appropriate instantiations of the Stable Manifold Theorem allow for a global stability analysis. Thus, neither access to second-order derivative information nor randomness beyond initialization is necessary to provably avoid saddle points.

stat.ML↗

arXiv2017

Finite Size Corrections and Likelihood Ratio Fluctuations in the Spiked Wigner Model

In this paper we study principal components analysis in the regime of high dimensionality and high noise. Our model of the problem is a rank-one deformation of a Wigner matrix where the signal-to-noise ratio (SNR) is of constant order, and we are interested in the fundamental limits of detection of the spike. Our main goal is to gain a fine understanding of the asymptotics for the log-likelihood ratio process, also known as the free energy, as a function of the SNR. Our main results are twofold. We first prove that the free energy has a finite-size correction to its limit---the replica-symmetric formula---which we explicitly compute. This provides a formula for the Kullback-Leibler divergence between the planted and null models. Second, we prove that below the reconstruction threshold, where it becomes impossible to reconstruct the spike, the log-likelihood ratio has fluctuations of constant order and converges in distribution to a Gaussian under both the planted and (under restrictions) the null model. As a consequence, we provide a general proof of contiguity between these two distributions that holds up to the reconstruction threshold, and is valid for an arbitrary separable prior on the spike. Formulae for the total variation distance, and the Type-I and Type-II errors of the optimal test are also given. Our proofs are based on Gaussian interpolation methods and a rigorous incarnation of the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington--Kirkpatrick spin-glass model.

math.ST↗

arXiv2017

Online control of the false discovery rate with decaying memory

In the online multiple testing problem, p-values corresponding to different null hypotheses are observed one by one, and the decision of whether or not to reject the current hypothesis must be made immediately, after which the next p-value is observed. Alpha-investing algorithms to control the false discovery rate (FDR), formulated by Foster and Stine, have been generalized and applied to many settings, including quality-preserving databases in science and multiple A/B or multi-armed bandit tests for internet commerce. This paper improves the class of generalized alpha-investing algorithms (GAI) in four ways: (a) we show how to uniformly improve the power of the entire class of monotone GAI procedures by awarding more alpha-wealth for each rejection, giving a win-win resolution to a recent dilemma raised by Javanmard and Montanari, (b) we demonstrate how to incorporate prior weights to indicate domain knowledge of which hypotheses are likely to be non-null, (c) we allow for differing penalties for false discoveries to indicate that some hypotheses may be more important than others, (d) we define a new quantity called the decaying memory false discovery rate (mem-FDR) that may be more meaningful for truly temporal applications, and which alleviates problems that we describe and refer to as "piggybacking" and "alpha-death". Our GAI++ algorithms incorporate all four generalizations simultaneously, and reduce to more powerful variants of earlier algorithms when the weights and decay are all set to unity. Finally, we also describe a simple method to derive new online FDR rules based on an estimated false discovery proportion.

stat.ME↗

arXiv2017

Breaking Locality Accelerates Block Gauss-Seidel

Recent work by Nesterov and Stich showed that momentum can be used to accelerate the rate of convergence for block Gauss-Seidel in the setting where a fixed partitioning of the coordinates is chosen ahead of time. We show that this setting is too restrictive, constructing instances where breaking locality by running non-accelerated Gauss-Seidel with randomly sampled coordinates substantially outperforms accelerated Gauss-Seidel with any fixed partitioning. Motivated by this finding, we analyze the accelerated block Gauss-Seidel algorithm in the random coordinate sampling setting. Our analysis captures the benefit of acceleration with a new data-dependent parameter which is well behaved when the matrix sub-blocks are well-conditioned. Empirically, we show that accelerated Gauss-Seidel with random coordinate sampling provides speedups for large scale machine learning tasks when compared to non-accelerated Gauss-Seidel and the classical conjugate-gradient algorithm.

math.OC↗

arXiv2017

Deep Transfer Learning with Joint Adaptation Networks

Deep networks have been successfully applied to learn transferable features for adapting models from a source domain to a different target domain. In this paper, we present joint adaptation networks (JAN), which learn a transfer network by aligning the joint distributions of multiple domain-specific layers across domains based on a joint maximum mean discrepancy (JMMD) criterion. Adversarial training strategy is adopted to maximize JMMD such that the distributions of the source and target domains are made more distinguishable. Learning can be performed by stochastic gradient descent with the gradients computed by back-propagation in linear-time. Experiments testify that our model yields state of the art results on standard datasets.

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arXiv2017

Partial Transfer Learning with Selective Adversarial Networks

Adversarial learning has been successfully embedded into deep networks to learn transferable features, which reduce distribution discrepancy between the source and target domains. Existing domain adversarial networks assume fully shared label space across domains. In the presence of big data, there is strong motivation of transferring both classification and representation models from existing big domains to unknown small domains. This paper introduces partial transfer learning, which relaxes the shared label space assumption to that the target label space is only a subspace of the source label space. Previous methods typically match the whole source domain to the target domain, which are prone to negative transfer for the partial transfer problem. We present Selective Adversarial Network (SAN), which simultaneously circumvents negative transfer by selecting out the outlier source classes and promotes positive transfer by maximally matching the data distributions in the shared label space. Experiments demonstrate that our models exceed state-of-the-art results for partial transfer learning tasks on several benchmark datasets.

cs.LG↗

arXiv2017

On kernel methods for covariates that are rankings

Permutation-valued features arise in a variety of applications, either in a direct way when preferences are elicited over a collection of items, or an indirect way in which numerical ratings are converted to a ranking. To date, there has been relatively limited study of regression, classification, and testing problems based on permutation-valued features, as opposed to permutation-valued responses. This paper studies the use of reproducing kernel Hilbert space methods for learning from permutation-valued features. These methods embed the rankings into an implicitly defined function space, and allow for efficient estimation of regression and test functions in this richer space. Our first contribution is to characterize both the feature spaces and spectral properties associated with two kernels for rankings, the Kendall and Mallows kernels. Using tools from representation theory, we explain the limited expressive power of the Kendall kernel by characterizing its degenerate spectrum, and in sharp contrast, we prove that Mallows' kernel is universal and characteristic. We also introduce families of polynomial kernels that interpolate between the Kendall (degree one) and Mallows' (infinite degree) kernels. We show the practical effectiveness of our methods via applications to Eurobarometer survey data as well as a Movielens ratings dataset.

stat.ML↗

arXiv2017

Real-Time Machine Learning: The Missing Pieces

Machine learning applications are increasingly deployed not only to serve predictions using static models, but also as tightly-integrated components of feedback loops involving dynamic, real-time decision making. These applications pose a new set of requirements, none of which are difficult to achieve in isolation, but the combination of which creates a challenge for existing distributed execution frameworks: computation with millisecond latency at high throughput, adaptive construction of arbitrary task graphs, and execution of heterogeneous kernels over diverse sets of resources. We assert that a new distributed execution framework is needed for such ML applications and propose a candidate approach with a proof-of-concept architecture that achieves a 63x performance improvement over a state-of-the-art execution framework for a representative application.

cs.DC↗

arXiv2017

Optimal Rates and Tradeoffs in Multiple Testing

Multiple hypothesis testing is a central topic in statistics, but despite abundant work on the false discovery rate (FDR) and the corresponding Type-II error concept known as the false non-discovery rate (FNR), a fine-grained understanding of the fundamental limits of multiple testing has not been developed. Our main contribution is to derive a precise non-asymptotic tradeoff between FNR and FDR for a variant of the generalized Gaussian sequence model. Our analysis is flexible enough to permit analyses of settings where the problem parameters vary with the number of hypotheses $n$, including various sparse and dense regimes (with $o(n)$ and $\mathcal{O}(n)$ signals). Moreover, we prove that the Benjamini-Hochberg algorithm as well as the Barber-Candès algorithm are both rate-optimal up to constants across these regimes.

math.ST↗

arXiv2017

Trust Region Policy Optimization

We describe an iterative procedure for optimizing policies, with guaranteed monotonic improvement. By making several approximations to the theoretically-justified procedure, we develop a practical algorithm, called Trust Region Policy Optimization (TRPO). This algorithm is similar to natural policy gradient methods and is effective for optimizing large nonlinear policies such as neural networks. Our experiments demonstrate its robust performance on a wide variety of tasks: learning simulated robotic swimming, hopping, and walking gaits; and playing Atari games using images of the screen as input. Despite its approximations that deviate from the theory, TRPO tends to give monotonic improvement, with little tuning of hyperparameters.

cs.LG↗

arXiv2017

How to Escape Saddle Points Efficiently

This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost "dimension-free"). The convergence rate of this procedure matches the well-known convergence rate of gradient descent to first-order stationary points, up to log factors. When all saddle points are non-degenerate, all second-order stationary points are local minima, and our result thus shows that perturbed gradient descent can escape saddle points almost for free. Our results can be directly applied to many machine learning applications, including deep learning. As a particular concrete example of such an application, we show that our results can be used directly to establish sharp global convergence rates for matrix factorization. Our results rely on a novel characterization of the geometry around saddle points, which may be of independent interest to the non-convex optimization community.

cs.LG↗

arXiv2017

Unsupervised Domain Adaptation with Residual Transfer Networks

The recent success of deep neural networks relies on massive amounts of labeled data. For a target task where labeled data is unavailable, domain adaptation can transfer a learner from a different source domain. In this paper, we propose a new approach to domain adaptation in deep networks that can jointly learn adaptive classifiers and transferable features from labeled data in the source domain and unlabeled data in the target domain. We relax a shared-classifier assumption made by previous methods and assume that the source classifier and target classifier differ by a residual function. We enable classifier adaptation by plugging several layers into deep network to explicitly learn the residual function with reference to the target classifier. We fuse features of multiple layers with tensor product and embed them into reproducing kernel Hilbert spaces to match distributions for feature adaptation. The adaptation can be achieved in most feed-forward models by extending them with new residual layers and loss functions, which can be trained efficiently via back-propagation. Empirical evidence shows that the new approach outperforms state of the art methods on standard domain adaptation benchmarks.

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