Showing papers by "Michael I. Jordan published in 2019"

Theoretically Principled Trade-off between Robustness and Accuracy

Abstract: 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.

Theoretically Principled Trade-off between Robustness and Accuracy

Abstract: 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.

Transferable Representation Learning with Deep Adaptation Networks

Abstract: Domain adaptation studies learning algorithms that generalize across source domains and target domains that exhibit different distributions Recent studies reveal that deep neural networks can learn transferable features that generalize well to similar novel tasks However, as deep features eventually transition from general to specific along the network, feature transferability drops significantly in higher task-specific layers with increasing domain discrepancy To formally reduce the effects of this discrepancy and enhance feature transferability in task-specific layers, we develop a novel framework for deep adaptation networks that extends deep convolutional neural networks to domain adaptation problems The framework embeds the deep features of all task-specific layers into reproducing kernel Hilbert spaces (RKHSs) and optimally matches different domain distributions The deep features are made more transferable by exploiting low-density separation of target-unlabeled data in very deep architectures, while the domain discrepancy is further reduced via the use of multiple kernel learning that enhances the statistical power of kernel embedding matching The overall framework is cast in a minimax game setting Extensive empirical evidence shows that the proposed networks yield state-of-the-art results on standard visual domain-adaptation benchmarks

Provably Efficient Reinforcement Learning with Linear Function Approximation

Abstract: Modern Reinforcement Learning (RL) is commonly applied to practical problems with an enormous number of states, where function approximation must be deployed to approximate either the value function or the policy. The introduction of function approximation raises a fundamental set of challenges involving computational and statistical efficiency, especially given the need to manage the exploration/exploitation tradeoff. As a result, a core RL question remains open: how can we design provably efficient RL algorithms that incorporate function approximation? This question persists even in a basic setting with linear dynamics and linear rewards, for which only linear function approximation is needed. This paper presents the first provable RL algorithm with both polynomial runtime and polynomial sample complexity in this linear setting, without requiring a "simulator" or additional assumptions. Concretely, we prove that an optimistic modification of Least-Squares Value Iteration (LSVI)---a classical algorithm frequently studied in the linear setting---achieves $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret, where $d$ is the ambient dimension of feature space, $H$ is the length of each episode, and $T$ is the total number of steps. Importantly, such regret is independent of the number of states and actions.

Universal Domain Adaptation

Abstract: Domain adaptation aims to transfer knowledge in the presence of the domain gap. Existing domain adaptation methods rely on rich prior knowledge about the relationship between the label sets of source and target domains, which greatly limits their application in the wild. This paper introduces Universal Domain Adaptation (UDA) that requires no prior knowledge on the label sets. For a given source label set and a target label set, they may contain a common label set and hold a private label set respectively, bringing up an additional category gap. UDA requires a model to either (1) classify the target sample correctly if it is associated with a label in the common label set, or (2) mark it as ``unknown'' otherwise. More importantly, a UDA model should work stably against a wide spectrum of commonness (the proportion of the common label set over the complete label set) so that it can handle real-world problems with unknown target label sets. To solve the universal domain adaptation problem, we propose Universal Adaptation Network (UAN). It quantifies sample-level transferability to discover the common label set and the label sets private to each domain, thereby promoting the adaptation in the automatically discovered common label set and recognizing the ``unknown'' samples successfully. A thorough evaluation shows that UAN outperforms the state of the art closed set, partial and open set domain adaptation methods in the novel UDA setting.

On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems

Abstract: We consider nonconvex-concave minimax problems, $\min_{\mathbf{x}} \max_{\mathbf{y} \in \mathcal{Y}} f(\mathbf{x}, \mathbf{y})$, where $f$ is nonconvex in $\mathbf{x}$ but concave in $\mathbf{y}$ and $\mathcal{Y}$ is a convex and bounded set. One of the most popular algorithms for solving this problem is the celebrated gradient descent ascent (GDA) algorithm, which has been widely used in machine learning, control theory and economics. Despite the extensive convergence results for the convex-concave setting, GDA with equal stepsize can converge to limit cycles or even diverge in a general setting. In this paper, we present the complexity results on two-time-scale GDA for solving nonconvex-concave minimax problems, showing that the algorithm can find a stationary point of the function $\Phi(\cdot) := \max_{\mathbf{y} \in \mathcal{Y}} f(\cdot, \mathbf{y})$ efficiently. To the best our knowledge, this is the first nonasymptotic analysis for two-time-scale GDA in this setting, shedding light on its superior practical performance in training generative adversarial networks (GANs) and other real applications.

Bridging Theory and Algorithm for Domain Adaptation

Abstract: This paper addresses the problem of unsupervised domain adaption from theoretical and algorithmic perspectives. Existing domain adaptation theories naturally imply minimax optimization algorithms, which connect well with the domain adaptation methods based on adversarial learning. However, several disconnections still exist and form the gap between theory and algorithm. We extend previous theories (Mansour et al., 2009c; Ben-David et al., 2010) to multiclass classification in domain adaptation, where classifiers based on the scoring functions and margin loss are standard choices in algorithm design. We introduce Margin Disparity Discrepancy, a novel measurement with rigorous generalization bounds, tailored to the distribution comparison with the asymmetric margin loss, and to the minimax optimization for easier training. Our theory can be seamlessly transformed into an adversarial learning algorithm for domain adaptation, successfully bridging the gap between theory and algorithm. A series of empirical studies show that our algorithm achieves the state of the art accuracies on challenging domain adaptation tasks.

Communication-Efficient Distributed Statistical Inference

Abstract: We present a communication-efficient surrogate likelihood (CSL) framework for solving distributed statistical inference problems. CSL provides a communication-efficient surrogate to the global like...

Transferable Adversarial Training: A General Approach to Adapting Deep Classifiers

Abstract: Domain adaptation enables knowledge transfer from a labeled source domain to an unlabeled target domain. A mainstream approach is adversarial feature adaptation, which learns domain-invariant representations through aligning the feature distributions of both domains. However, a theoretical prerequisite of domain adaptation is the adaptability measured by the expected risk of an ideal joint hypothesis over the source and target domains. In this respect, adversarial feature adaptation may potentially deteriorate the adaptability, since it distorts the original feature distributions when suppressing domain-specific variations. To this end, we propose Transferable Adversarial Training (TAT) to enable the adaptation of deep classifiers. The approach generates transferable examples to fill in the gap between the source and target domains, and adversarially trains the deep classifiers to make consistent predictions over the transferable examples. Without learning domaininvariant representations at the expense of distorting the feature distributions, the adaptability in the theoretical learning bound is algorithmically guaranteed. A series of experiments validate that our approach advances the state of the arts on a variety of domain adaptation tasks in vision and NLP, including object recognition, learning from synthetic to real data, and sentiment classification.

Provably efficient reinforcement learning with linear function approximation

Abstract: Modern Reinforcement Learning (RL) is commonly applied to practical problems with an enormous number of states, where function approximation must be deployed to approximate either the value function or the policy. The introduction of function approximation raises a fundamental set of challenges involving computational and statistical efficiency, especially given the need to manage the exploration/exploitation tradeoff. As a result, a core RL question remains open: how can we design provably efficient RL algorithms that incorporate function approximation? This question persists even in a basic setting with linear dynamics and linear rewards, for which only linear function approximation is needed. This paper presents the first provable RL algorithm with both polynomial runtime and polynomial sample complexity in this linear setting, without requiring a "simulator" or additional assumptions. Concretely, we prove that an optimistic modification of Least-Squares Value Iteration (LSVI)---a classical algorithm frequently studied in the linear setting---achieves $\tilde{\mathcal{O}}(\sqrt{d^3H^3T})$ regret, where $d$ is the ambient dimension of feature space, $H$ is the length of each episode, and $T$ is the total number of steps. Importantly, such regret is independent of the number of states and actions.

Bridging Theory and Algorithm for Domain Adaptation.

Abstract: This paper addresses the problem of unsupervised domain adaption from theoretical and algorithmic perspectives. Existing domain adaptation theories naturally imply minimax optimization algorithms, which connect well with the domain adaptation methods based on adversarial learning. However, several disconnections still exist and form the gap between theory and algorithm. We extend previous theories (Mansour et al., 2009c; Ben-David et al., 2010) to multiclass classification in domain adaptation, where classifiers based on the scoring functions and margin loss are standard choices in algorithm design. We introduce Margin Disparity Discrepancy, a novel measurement with rigorous generalization bounds, tailored to the distribution comparison with the asymmetric margin loss, and to the minimax optimization for easier training. Our theory can be seamlessly transformed into an adversarial learning algorithm for domain adaptation, successfully bridging the gap between theory and algorithm. A series of empirical studies show that our algorithm achieves the state of the art accuracies on challenging domain adaptation tasks.

Artificial Intelligence—The Revolution Hasn’t Happened Yet

Abstract: We praise Jordan for bringing much needed clarity about the current status of Artificial Intelligence (AI)—what it currently is and what it is not—as well as explaining the current challenges lying ahead and outlining what is missing and remains to be done. Jordan makes several claims supported by a list of talking points that we hope will reach a wide audience; ideally, that audience will include academic, university, and governmental leaders, at a time where significant resources are being allocated to AI for research and education.

First-order methods almost always avoid strict saddle points

Abstract: We establish that first-order methods avoid strict saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including (manifold) 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 strict saddle points.

Transferable Normalization: Towards Improving Transferability of Deep Neural Networks

Abstract: Deep neural networks (DNNs) excel at learning representations when trained on large-scale datasets. Pre-trained DNNs also show strong transferability when fine-tuned to other labeled datasets. However, such transferability becomes weak when the target dataset is fully unlabeled as in Unsupervised Domain Adaptation (UDA). We envision that the loss of transferability may stem from the intrinsic limitation of the architecture design of DNNs. In this paper, we delve into the components of DNN architectures and propose Transferable Normalization (TransNorm) in place of existing normalization techniques. TransNorm is an end-to-end trainable layer to make DNNs more transferable across domains. As a general method, TransNorm can be easily applied to various deep neural networks and domain adaption methods, without introducing any extra hyper-parameters or learnable parameters. Empirical results justify that TransNorm not only improves classification accuracies but also accelerates convergence for mainstream DNN-based domain adaptation methods.

On Nonconvex Optimization for Machine Learning: Gradients, Stochasticity, and Saddle Points

Abstract: Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable successes in machine learning have involved nonconvex optimization, and a gap has arisen between theory and practice. Indeed, traditional analyses of GD and SGD show that both algorithms converge to stationary points efficiently. But these analyses do not take into account the possibility of converging to saddle points. More recent theory has shown that GD and SGD can avoid saddle points, but the dependence on dimension in these analyses is polynomial. For modern machine learning, where the dimension can be in the millions, such dependence would be catastrophic. We analyze perturbed versions of GD and SGD and show that they are truly efficient---their dimension dependence is only polylogarithmic. Indeed, these algorithms converge to second-order stationary points in essentially the same time as they take to converge to classical first-order stationary points.

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

Abstract: 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).

What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?

Abstract: Minimax optimization has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs), adversarial training and multi-agent reinforcement learning. As most of these applications involve continuous nonconvex-nonconcave formulations, a very basic question arises---"what is a proper definition of local optima?" Most previous work answers this question using classical notions of equilibria from simultaneous games, where the min-player and the max-player act simultaneously. In contrast, most applications in machine learning, including GANs and adversarial training, correspond to sequential games, where the order of which player acts first is crucial (since minimax is in general not equal to maximin due to the nonconvex-nonconcave nature of the problems). The main contribution of this paper is to propose a proper mathematical definition of local optimality for this sequential setting---local minimax, as well as to present its properties and existence results. Finally, we establish a strong connection to a basic local search algorithm---gradient descent ascent (GDA): under mild conditions, all stable limit points of GDA are exactly local minimax points up to some degenerate points.

A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm

Abstract: In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.

Sampling can be faster than optimization.

Abstract: Optimization algorithms and Monte Carlo sampling algorithms have provided the computational foundations for the rapid growth in applications of statistical machine learning in recent years. There is, however, limited theoretical understanding of the relationships between these 2 kinds of methodology, and limited understanding of relative strengths and weaknesses. Moreover, existing results have been obtained primarily in the setting of convex functions (for optimization) and log-concave functions (for sampling). In this setting, where local properties determine global properties, optimization algorithms are unsurprisingly more efficient computationally than sampling algorithms. We instead examine a class of nonconvex objective functions that arise in mixture modeling and multistable systems. In this nonconvex setting, we find that the computational complexity of sampling algorithms scales linearly with the model dimension while that of optimization algorithms scales exponentially.

Towards Accurate Model Selection in Deep Unsupervised Domain Adaptation.

Abstract: Deep unsupervised domain adaptation (Deep UDA) methods successfully leverage rich labeled data in a source domain to boost the performance on related but unlabeled data in a target domain. However, algorithm comparison is cumbersome in Deep UDA due to the absence of accurate and standardized model selection method, posing an obstacle to further advances in the field. Existing model selection methods for Deep UDA are either highly biased, restricted, unstable, or even controversial (requiring labeled target data). To this end, we propose Deep Embedded Validation (DEV), which embeds adapted feature representation into the validation procedure to obtain unbiased estimation of the target risk with bounded variance. The variance is further reduced by the technique of control variate. The efficacy of the method has been justified both theoretically and empirically.

Minmax Optimization: Stable Limit Points of Gradient Descent Ascent are Locally Optimal.

Abstract: Minmax optimization, especially in its general nonconvex-nonconcave formulation, has found extensive applications in modern machine learning frameworks such as generative adversarial networks (GAN), adversarial training and multi-agent reinforcement learning. Gradient-based algorithms, in particular gradient descent ascent (GDA), are widely used in practice to solve these problems. Despite the practical popularity of GDA, however, its theoretical behavior has been considered highly undesirable. Indeed, apart from possiblity of non-convergence, recent results (Daskalakis and Panageas, 2018; Mazumdar and Ratliff, 2018; Adolphs et al., 2018) show that even when GDA converges, its stable limit points can be points that are not local Nash equilibria, thus not game-theoretically meaningful. In this paper, we initiate a discussion on the proper optimality measures for minmax optimization, and introduce a new notion of local optimality---local minmax---as a more suitable alternative to the notion of local Nash equilibrium. We establish favorable properties of local minmax points, and show, most importantly, that as the ratio of the ascent step size to the descent step size goes to infinity, stable limit points of GDA are exactly local minmax points up to degenerate points, demonstrating that all stable limit points of GDA have a game-theoretic meaning for minmax problems.

How Does Learning Rate Decay Help Modern Neural Networks

Abstract: Learning rate decay (lrDecay) is a \emph{de facto} technique for training modern neural networks. It starts with a large learning rate and then decays it multiple times. It is empirically observed to help both optimization and generalization. Common beliefs in how lrDecay works come from the optimization analysis of (Stochastic) Gradient Descent: 1) an initially large learning rate accelerates training or helps the network escape spurious local minima; 2) decaying the learning rate helps the network converge to a local minimum and avoid oscillation. Despite the popularity of these common beliefs, experiments suggest that they are insufficient in explaining the general effectiveness of lrDecay in training modern neural networks that are deep, wide, and nonconvex. We provide another novel explanation: an initially large learning rate suppresses the network from memorizing noisy data while decaying the learning rate improves the learning of complex patterns. The proposed explanation is validated on a carefully-constructed dataset with tractable pattern complexity. And its implication, that additional patterns learned in later stages of lrDecay are more complex and thus less transferable, is justified in real-world datasets. We believe that this alternative explanation will shed light into the design of better training strategies for modern neural networks.

A Dynamical Systems Perspective on Nesterov Acceleration

Abstract: 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.

Harmonization and Annotation of Single-cell Transcriptomics data with Deep Generative Models

Abstract: As single-cell transcriptomics becomes a mainstream technology, the natural next step is to integrate the accumulating data in order to achieve a common ontology of cell types and states. However, owing to various nuisance factors of variation, it is not straightforward how to compare gene expression levels across data sets and how to automatically assign cell type labels in a new data set based on existing annotations. In this manuscript, we demonstrate that our previously developed method, scVI, provides an effective and fully probabilistic approach for joint representation and analysis of cohorts of single-cell RNA-seq data sets, while accounting for uncertainty caused by biological and measurement noise. We also introduce single-cell ANnotation using Variational Inference (scANVI), a semi-supervised variant of scVI designed to leverage any available cell state annotations — for instance when only one data set in a cohort is annotated, or when only a few cells in a single data set can be labeled using marker genes. We demonstrate that scVI and scANVI compare favorably to the existing methods for data integration and cell state annotation in terms of accuracy, scalability, and adaptability to challenging settings such as a hierarchical structure of cell state labels. We further show that different from existing methods, scVI and scANVI represent the integrated datasets with a single generative model that can be directly used for any probabilistic decision making task, using differential expression as our case study. scVI and scANVI are available as open source software and can be readily used to facilitate cell state annotation and help ensure consistency and reproducibility across studies.

Stochastic Gradient Descent Escapes Saddle Points Efficiently.

Abstract: This paper considers the perturbed stochastic gradient descent algorithm and shows that it finds $\epsilon$-second order stationary points ($\left\| abla f(x)\right\|\leq \epsilon$ and $ abla^2 f(x) \succeq -\sqrt{\epsilon} \mathbf{I}$) in $\tilde{O}(d/\epsilon^4)$ iterations, giving the first result that has linear dependence on dimension for this setting. For the special case, where stochastic gradients are Lipschitz, the dependence on dimension reduces to polylogarithmic. In addition to giving new results, this paper also presents a simplified proof strategy that gives a shorter and more elegant proof of previously known results (Jin et al. 2017) on perturbed gradient descent algorithm.

Acceleration via Symplectic Discretization of High-Resolution Differential Equations

Abstract: We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al. [2018] achieves an accelerated rate for minimizing smooth strongly convex functions. On the other hand, the resulting algorithm either fails to achieve acceleration or is impractical when the scheme is implicit, the ODE is low-resolution, or the scheme is explicit.

Is There an Analog of Nesterov Acceleration for MCMC

Abstract: We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback-Leibler (KL) divergence as the objective functional. We show that an underdamped form of the Langevin algorithm performs accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm.

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

Abstract: 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.

On the Complexity of Approximating Multimarginal Optimal Transport.

Abstract: We study the complexity of approximating the multimarginal optimal transport (MOT) distance, a generalization of the classical optimal transport distance, considered here between $m$ discrete probability distributions supported each on $n$ support points. First, we show that the standard linear programming (LP) representation of the MOT problem is not a minimum-cost flow problem when $m \geq 3$. This negative result implies that some combinatorial algorithms, e.g., network simplex method, are not suitable for approximating the MOT problem, while the worst-case complexity bound for the deterministic interior-point algorithm remains a quantity of $\tilde{O}(n^{3m})$. We then propose two simple and \textit{deterministic} algorithms for approximating the MOT problem. The first algorithm, which we refer to as \textit{multimarginal Sinkhorn} algorithm, is a provably efficient multimarginal generalization of the Sinkhorn algorithm. We show that it achieves a complexity bound of $\tilde{O}(m^3n^m\varepsilon^{-2})$ for a tolerance $\varepsilon \in (0, 1)$. This provides a first \textit{near-linear time} complexity bound guarantee for approximating the MOT problem and matches the best known complexity bound for the Sinkhorn algorithm in the classical OT setting when $m = 2$. The second algorithm, which we refer to as \textit{accelerated multimarginal Sinkhorn} algorithm, achieves the acceleration by incorporating an estimate sequence and the complexity bound is $\tilde{O}(m^3n^{m+1/3}\varepsilon^{-4/3})$. This bound is better than that of the first algorithm in terms of $1/\varepsilon$, and accelerated alternating minimization algorithm~\citep{Tupitsa-2020-Multimarginal} in terms of $n$. Finally, we compare our new algorithms with the commercial LP solver \textsc{Gurobi}. Preliminary results on synthetic data and real images demonstrate the effectiveness and efficiency of our algorithms.

A unified treatment of multiple testing with prior knowledge using the p-filter

Abstract: There is a significant literature on methods for incorporating knowledge into multiple testing procedures so as to improve their power and precision. Some common forms of prior knowledge include (a) beliefs about which hypotheses are null, modeled by nonuniform prior weights; (b) differing importances of hypotheses, modeled by differing penalties for false discoveries; (c) multiple arbitrary partitions of the hypotheses into (possibly overlapping) groups and (d) knowledge of independence, positive or arbitrary dependence between hypotheses or groups, suggesting the use of more aggressive or conservative procedures. We present a unified algorithmic framework called p-filter for global null testing and false discovery rate (FDR) control that allows the scientist to incorporate all four types of prior knowledge (a)–(d) simultaneously, recovering a variety of known algorithms as special cases.