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Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 307 records · Page 17Linked to original sources
arXiv2019

Quantitative Weak Convergence for Discrete Stochastic Processes

In this paper, we quantitative convergence in $W_2$ for a family of Langevin-like stochastic processes that includes stochastic gradient descent and related gradient-based algorithms. Under certain regularity assumptions, we show that the iterates of these stochastic processes converge to an invariant distribution at a rate of $\tilde{O}\lrp{1/\sqrt{k}}$ where $k$ is the number of steps; this rate is provably tight up to log factors. Our result reduces to a quantitative form of the classical Central Limit Theorem in the special case when the potential is quadratic.

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arXiv2019

Theoretically Principled Trade-off between Robustness and Accuracy

We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. In this work, we decompose the prediction error for adversarial examples (robust error) as the sum of the natural (classification) error and boundary error, and provide a differentiable upper bound using the theory of classification-calibrated loss, which is shown to be the tightest possible upper bound uniform over all probability distributions and measurable predictors. Inspired by our theoretical analysis, we also design a new defense method, TRADES, to trade adversarial robustness off against accuracy. Our proposed algorithm performs well experimentally in real-world datasets. The methodology is the foundation of our entry to the NeurIPS 2018 Adversarial Vision Challenge in which we won the 1st place out of ~2,000 submissions, surpassing the runner-up approach by $11.41\%$ in terms of mean $\ell_2$ perturbation distance.

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arXiv2019

ML-LOO: Detecting Adversarial Examples with Feature Attribution

Deep neural networks obtain state-of-the-art performance on a series of tasks. However, they are easily fooled by adding a small adversarial perturbation to input. The perturbation is often human imperceptible on image data. We observe a significant difference in feature attributions of adversarially crafted examples from those of original ones. Based on this observation, we introduce a new framework to detect adversarial examples through thresholding a scale estimate of feature attribution scores. Furthermore, we extend our method to include multi-layer feature attributions in order to tackle the attacks with mixed confidence levels. Through vast experiments, our method achieves superior performances in distinguishing adversarial examples from popular attack methods on a variety of real data sets among state-of-the-art detection methods. In particular, our method is able to detect adversarial examples of mixed confidence levels, and transfer between different attacking methods.

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arXiv2019

A Dynamical Systems Perspective on Nesterov Acceleration

We present a dynamical system framework for understanding Nesterov's accelerated gradient method. In contrast to earlier work, our derivation does not rely on a vanishing step size argument. We show that Nesterov acceleration arises from discretizing an ordinary differential equation with a semi-implicit Euler integration scheme. We analyze both the underlying differential equation as well as the discretization to obtain insights into the phenomenon of acceleration. The analysis suggests that a curvature-dependent damping term lies at the heart of the phenomenon. We further establish connections between the discretized and the continuous-time dynamics.

math.OC↗
arXiv2019

Less than a Single Pass: Stochastically Controlled Stochastic Gradient Method

We develop and analyze a procedure for gradient-based optimization that we refer to as stochastically controlled stochastic gradient (SCSG). As a member of the SVRG family of algorithms, SCSG makes use of gradient estimates at two scales, with the number of updates at the faster scale being governed by a geometric random variable. Unlike most existing algorithms in this family, both the computation cost and the communication cost of SCSG do not necessarily scale linearly with the sample size $n$; indeed, these costs are independent of $n$ when the target accuracy is low. An experimental evaluation on real datasets confirms the effectiveness of SCSG.

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arXiv2019

Non-convex Finite-Sum Optimization Via SCSG Methods

We develop a class of algorithms, as variants of the stochastically controlled stochastic gradient (SCSG) methods (Lei and Jordan, 2016), for the smooth non-convex finite-sum optimization problem. Assuming the smoothness of each component, the complexity of SCSG to reach a stationary point with $\mathbb{E} \|\nabla f(x)\|^{2}\le ε$ is $O\left (\min\{ε^{-5/3}, ε^{-1}n^{2/3}\}\right)$, which strictly outperforms the stochastic gradient descent. Moreover, SCSG is never worse than the state-of-the-art methods based on variance reduction and it significantly outperforms them when the target accuracy is low. A similar acceleration is also achieved when the functions satisfy the Polyak-Lojasiewicz condition. Empirical experiments demonstrate that SCSG outperforms stochastic gradient methods on training multi-layers neural networks in terms of both training and validation loss.

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arXiv2019

Rao-Blackwellized Stochastic Gradients for Discrete Distributions

We wish to compute the gradient of an expectation over a finite or countably infinite sample space having $K \leq \infty$ categories. When $K$ is indeed infinite, or finite but very large, the relevant summation is intractable. Accordingly, various stochastic gradient estimators have been proposed. In this paper, we describe a technique that can be applied to reduce the variance of any such estimator, without changing its bias---in particular, unbiasedness is retained. We show that our technique is an instance of Rao-Blackwellization, and we demonstrate the improvement it yields on a semi-supervised classification problem and a pixel attention task.

stat.ML↗
arXiv2019

A joint model of unpaired data from scRNA-seq and spatial transcriptomics for imputing missing gene expression measurements

Spatial studies of transcriptome provide biologists with gene expression maps of heterogeneous and complex tissues. However, most experimental protocols for spatial transcriptomics suffer from the need to select beforehand a small fraction of genes to be quantified over the entire transcriptome. Standard single-cell RNA sequencing (scRNA-seq) is more prevalent, easier to implement and can in principle capture any gene but cannot recover the spatial location of the cells. In this manuscript, we focus on the problem of imputation of missing genes in spatial transcriptomic data based on (unpaired) standard scRNA-seq data from the same biological tissue. Building upon domain adaptation work, we propose gimVI, a deep generative model for the integration of spatial transcriptomic data and scRNA-seq data that can be used to impute missing genes. After describing our generative model and an inference procedure for it, we compare gimVI to alternative methods from computational biology or domain adaptation on real datasets and outperform Seurat Anchors, Liger and CORAL to impute held-out genes.

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arXiv2019

LS-Tree: Model Interpretation When the Data Are Linguistic

We study the problem of interpreting trained classification models in the setting of linguistic data sets. Leveraging a parse tree, we propose to assign least-squares based importance scores to each word of an instance by exploiting syntactic constituency structure. We establish an axiomatic characterization of these importance scores by relating them to the Banzhaf value in coalitional game theory. Based on these importance scores, we develop a principled method for detecting and quantifying interactions between words in a sentence. We demonstrate that the proposed method can aid in interpretability and diagnostics for several widely-used language models.

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arXiv2019

On Finding Local Nash Equilibria (and Only Local Nash Equilibria) in Zero-Sum Games

We propose local symplectic surgery, a two-timescale procedure for finding local Nash equilibria in two-player zero-sum games. We first show that previous gradient-based algorithms cannot guarantee convergence to local Nash equilibria due to the existence of non-Nash stationary points. By taking advantage of the differential structure of the game, we construct an algorithm for which the local Nash equilibria are the only attracting fixed points. We also show that the algorithm exhibits no oscillatory behaviors in neighborhoods of equilibria and show that it has the same per-iteration complexity as other recently proposed algorithms. We conclude by validating the algorithm on two numerical examples: a toy example with multiple Nash equilibria and a non-Nash equilibrium, and the training of a small generative adversarial network (GAN).

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arXiv2018

Conditional Adversarial Domain Adaptation

Adversarial learning has been embedded into deep networks to learn disentangled and transferable representations for domain adaptation. Existing adversarial domain adaptation methods may not effectively align different domains of multimodal distributions native in classification problems. In this paper, we present conditional adversarial domain adaptation, a principled framework that conditions the adversarial adaptation models on discriminative information conveyed in the classifier predictions. Conditional domain adversarial networks (CDANs) are designed with two novel conditioning strategies: multilinear conditioning that captures the cross-covariance between feature representations and classifier predictions to improve the discriminability, and entropy conditioning that controls the uncertainty of classifier predictions to guarantee the transferability. With theoretical guarantees and a few lines of codes, the approach has exceeded state-of-the-art results on five datasets.

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arXiv2018

DAGGER: A sequential algorithm for FDR control on DAGs

We propose a linear-time, single-pass, top-down algorithm for multiple testing on directed acyclic graphs (DAGs), where nodes represent hypotheses and edges specify a partial ordering in which hypotheses must be tested. The procedure is guaranteed to reject a sub-DAG with bounded false discovery rate (FDR) while satisfying the logical constraint that a rejected node's parents must also be rejected. It is designed for sequential testing settings, when the DAG structure is known a priori, but the $p$-values are obtained selectively (such as in a sequence of experiments), but the algorithm is also applicable in non-sequential settings when all $p$-values can be calculated in advance (such as variable/model selection). Our DAGGER algorithm, shorthand for Greedily Evolving Rejections on DAGs, provably controls the false discovery rate under independence, positive dependence or arbitrary dependence of the $p$-values. The DAGGER procedure specializes to known algorithms in the special cases of trees and line graphs, and simplifies to the classical Benjamini-Hochberg procedure when the DAG has no edges. We explore the empirical performance of DAGGER using simulations, as well as a real dataset corresponding to a gene ontology, showing favorable performance in terms of time and power.

stat.ME↗
arXiv2018

Information Constraints on Auto-Encoding Variational Bayes

Parameterizing the approximate posterior of a generative model with neural networks has become a common theme in recent machine learning research. While providing appealing flexibility, this approach makes it difficult to impose or assess structural constraints such as conditional independence. We propose a framework for learning representations that relies on Auto-Encoding Variational Bayes and whose search space is constrained via kernel-based measures of independence. In particular, our method employs the $d$-variable Hilbert-Schmidt Independence Criterion (dHSIC) to enforce independence between the latent representations and arbitrary nuisance factors. We show how to apply this method to a range of problems, including the problems of learning invariant representations and the learning of interpretable representations. We also present a full-fledged application to single-cell RNA sequencing (scRNA-seq). In this setting the biological signal is mixed in complex ways with sequencing errors and sampling effects. We show that our method out-performs the state-of-the-art in this domain.

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arXiv2018

Understanding the Acceleration Phenomenon via High-Resolution Differential Equations

Gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms---Nesterov's accelerated gradient method for strongly convex functions (NAG-SC) and Polyak's heavy-ball method---we study an alternative limiting process that yields high-resolution ODEs. We show that these ODEs permit a general Lyapunov function framework for the analysis of convergence in both continuous and discrete time. We also show that these ODEs are more accurate surrogates for the underlying algorithms; in particular, they not only distinguish between NAG-SC and Polyak's heavy-ball method, but they allow the identification of a term that we refer to as "gradient correction" that is present in NAG-SC but not in the heavy-ball method and is responsible for the qualitative difference in convergence of the two methods. We also use the high-resolution ODE framework to study Nesterov's accelerated gradient method for (non-strongly) convex functions, uncovering a hitherto unknown result---that NAG-C minimizes the squared gradient norm at an inverse cubic rate. Finally, by modifying the high-resolution ODE of NAG-C, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates of NAG-C for smooth convex functions.

math.OC↗
arXiv2018

Probabilistic Multilevel Clustering via Composite Transportation Distance

We propose a novel probabilistic approach to multilevel clustering problems based on composite transportation distance, which is a variant of transportation distance where the underlying metric is Kullback-Leibler divergence. Our method involves solving a joint optimization problem over spaces of probability measures to simultaneously discover grouping structures within groups and among groups. By exploiting the connection of our method to the problem of finding composite transportation barycenters, we develop fast and efficient optimization algorithms even for potentially large-scale multilevel datasets. Finally, we present experimental results with both synthetic and real data to demonstrate the efficiency and scalability of the proposed approach.

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arXiv2018

Kernel Feature Selection via Conditional Covariance Minimization

We propose a method for feature selection that employs kernel-based measures of independence to find a subset of covariates that is maximally predictive of the response. Building on past work in kernel dimension reduction, we show how to perform feature selection via a constrained optimization problem involving the trace of the conditional covariance operator. We prove various consistency results for this procedure, and also demonstrate that our method compares favorably with other state-of-the-art algorithms on a variety of synthetic and real data sets.

stat.ML↗
arXiv2018

Covariances, Robustness, and Variational Bayes

Mean-field Variational Bayes (MFVB) is an approximate Bayesian posterior inference technique that is increasingly popular due to its fast runtimes on large-scale datasets. However, even when MFVB provides accurate posterior means for certain parameters, it often mis-estimates variances and covariances. Furthermore, prior robustness measures have remained undeveloped for MFVB. By deriving a simple formula for the effect of infinitesimal model perturbations on MFVB posterior means, we provide both improved covariance estimates and local robustness measures for MFVB, thus greatly expanding the practical usefulness of MFVB posterior approximations. The estimates for MFVB posterior covariances rely on a result from the classical Bayesian robustness literature relating derivatives of posterior expectations to posterior covariances and include the Laplace approximation as a special case. Our key condition is that the MFVB approximation provides good estimates of a select subset of posterior means---an assumption that has been shown to hold in many practical settings. In our experiments, we demonstrate that our methods are simple, general, and fast, providing accurate posterior uncertainty estimates and robustness measures with runtimes that can be an order of magnitude faster than MCMC.

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