Showing papers by "Michael I. Jordan published in 2021"

Understanding the acceleration phenomenon via high-resolution differential equations

Abstract: 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.

Probabilistic harmonization and annotation of single-cell transcriptomics data with deep generative models.

Abstract: As the number of single-cell transcriptomics datasets grows, the natural next step is to integrate the accumulating data to achieve a common ontology of cell types and states. However, it is not straightforward to compare gene expression levels across datasets and to automatically assign cell type labels in a new dataset 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 scRNA-seq data, 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 existing cell state annotations. We demonstrate that scVI and scANVI compare favorably to state-of-the-art methods for data integration and cell state annotation in terms of accuracy, scalability, and adaptability to challenging settings. In contrast to existing methods, scVI and scANVI integrate multiple datasets with a single generative model that can be directly used for downstream tasks, such as differential expression. Both methods are easily accessible through scvi-tools.

Distribution-Free, Risk-Controlling Prediction Sets

Abstract: While improving prediction accuracy has been the focus of machine learning in recent years, this alone does not suffice for reliable decision-making. Deploying learning systems in consequential settings also requires calibrating and communicating the uncertainty of predictions. To convey instance-wise uncertainty for prediction tasks, we show how to generate set-valued predictions from a black-box predictor that control the expected loss on future test points at a user-specified level. Our approach provides explicit finite-sample guarantees for any dataset by using a holdout set to calibrate the size of the prediction sets. This framework enables simple, distribution-free, rigorous error control for many tasks, and we demonstrate it in five large-scale machine learning problems: (1) classification problems where some mistakes are more costly than others; (2) multi-label classification, where each observation has multiple associated labels; (3) classification problems where the labels have a hierarchical structure; (4) image segmentation, where we wish to predict a set of pixels containing an object of interest; and (5) protein structure prediction. Lastly, we discuss extensions to uncertainty quantification for ranking, metric learning and distributionally robust learning.

scvi-tools: a library for deep probabilistic analysis of single-cell omics data

Abstract: AO_SCPLOWBSTRACTC_SCPLOWProbabilistic models have provided the underpinnings for state-of-the-art performance in many single-cell omics data analysis tasks, including dimensionality reduction, clustering, differential expression, annotation, removal of unwanted variation, and integration across modalities. Many of the models being deployed are amenable to scalable stochastic inference techniques, and accordingly they are able to process single-cell datasets of realistic and growing sizes. However, the community-wide adoption of probabilistic approaches is hindered by a fractured software ecosystem resulting in an array of packages with distinct, and often complex interfaces. To address this issue, we developed scvi-tools (https://scvi-tools.org), a Python package that implements a variety of leading probabilistic methods. These methods, which cover many fundamental analysis tasks, are accessible through a standardized, easy-to-use interface with direct links to Scanpy, Seurat, and Bioconductor workflows. By standardizing the implementations, we were able to develop and reuse novel functionalities across different models, such as support for complex study designs through nonlinear removal of unwanted variation due to multiple covariates and reference-query integration via scArches. The extensible software building blocks that underlie scvi-tools also enable a developer environment in which new probabilistic models for single cell omics can be efficiently developed, benchmarked, and deployed. We demonstrate this through a code-efficient reimplementation of Stereoscope for deconvolution of spatial transcriptomics profiles. By catering to both the end user and developer audiences, we expect scvi-tools to become an essential software dependency and serve to formulate a community standard for probabilistic modeling of single cell omics.

MultiVI: deep generative model for the integration of multi-modal data

Abstract: The ability to jointly profile the transcriptional and chromatin land-scape of single-cells has emerged as a powerful technique to identify cellular populations and shed light on their regulation of gene expression. Current computational methods analyze jointly profiled (paired) or individual data modalities (unpaired), but do not offer a principled method to analyze both paired and unpaired samples jointly. Here we present MultiVI, a probabilistic framework that leverages deep neural networks to jointly analyze scRNA, scATAC and multiomic (scRNA + scATAC) data. MultiVI creates an informative low-dimensional latent space that accurately reflects both chromatin and transcriptional properties of the cells even when one of the modalities is missing. MultiVI accounts for technical effects in both scRNA and scATAC-seq while correcting for batch effects in both data modalities. We use public datasets to demonstrate that MultiVI is stable, easy to use, and outperforms current approaches for the joint analysis of paired and unpaired data. MultiVI is available as an open source package, implemented in the scvi-tools frame-work: https://docs.scvi-tools.org/.

Is there an analog of Nesterov acceleration for gradient-based 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.

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.

High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm

Abstract: We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of $d$-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most $\varepsilon > 0$ in Wasserstein distance from the target distribution in $O\left(\frac{d^{1/4}}{ \varepsilon^{1/2}} \right)$ steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with $\alpha$-th order smoothness, we prove that the mixing time scales as $O \left(\frac{d^{1/4}}{\varepsilon^{1/2}} + \frac{d^{1/2}}{\varepsilon^{1/(\alpha - 1)}} \right)$.

Distribution-Free, Risk-Controlling Prediction Sets

Abstract: While improving prediction accuracy has been the focus of machine learning in recent years, this alone does not suffice for reliable decision-making. Deploying learning systems in consequential settings also requires calibrating and communicating the uncertainty of predictions. To convey instance-wise uncertainty for prediction tasks, we show how to generate set-valued predictions from a black-box predictor that control the expected loss on future test points at a user-specified level. Our approach provides explicit finite-sample guarantees for any dataset by using a holdout set to calibrate the size of the prediction sets. This framework enables simple, distribution-free, rigorous error control for many tasks, and we demonstrate it in five large-scale machine learning problems: (1) classification problems where some mistakes are more costly than others; (2) multi-label classification, where each observation has multiple associated labels; (3) classification problems where the labels have a hierarchical structure; (4) image segmentation, where we wish to predict a set of pixels containing an object of interest; and (5) protein structure prediction. Lastly, we discuss extensions to uncertainty quantification for ranking, metric learning and distributionally robust learning.

Multi-resolution deconvolution of spatial transcriptomics data reveals continuous patterns of inflammation

Abstract: The function of mammalian cells is largely influenced by their tissue microenvironment. Advances in spatial transcriptomics open the way for studying these important determinants of cellular function by enabling a transcriptome-wide evaluation of gene expression in situ. A critical limitation of the current technologies, however, is that their resolution is limited to niches (spots) of sizes well beyond that of a single cell, thus providing measurements for cell aggregates which may mask critical interactions between neighboring cells of different types. While joint analysis with single-cell RNA-sequencing (scRNA-seq) can be leveraged to alleviate this problem, current analyses are limited to a discrete view of cell type proportion inside every spot. This limitation becomes critical in the common case where, even within a cell type, there is a continuum of cell states that cannot be clearly demarcated but reflects important differences in the way cells function and interact with their surroundings. To address this, we developed Deconvolution of Spatial Transcriptomics profiles using Variational Inference (DestVI), a probabilistic method for multi-resolution analysis for spatial transcriptomics that explicitly models continuous variation within cell types. Using simulations, we demonstrate that DestVI is capable of providing higher resolution compared to the existing methods and that it can estimate gene expression by every cell type inside every spot. We then introduce an automated pipeline that uses DestVI for analysis of single tissue slices and comparison between tissues. We apply this pipeline to study the immune crosstalk within lymph nodes to infection and explore the spatial organization of a mouse tumor model. In both cases, we demonstrate that DestVI can provide a high resolution and accurate spatial characterization of the cellular organization of these tissues, and that it is capable of identifying important cell-type-specific changes in gene expression - between different tissue regions or between conditions. DestVI is available as an open-source software package in the scvi-tools codebase (https://scvi-tools.org).

Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence

Abstract: Variational Inference (VI) is a popular alternative to asymptotically exact sampling in Bayesian inference. Its main workhorse is optimization over a reverse Kullback-Leibler divergence (RKL), which typically underestimates the tail of the posterior leading to miscalibration and potential degeneracy. Importance sampling (IS), on the other hand, is often used to fine-tune and de-bias the estimates of approximate Bayesian inference procedures. The quality of IS crucially depends on the choice of the proposal distribution. Ideally, the proposal distribution has heavier tails than the target, which is rarely achievable by minimizing the RKL. We thus propose a novel combination of optimization and sampling techniques for approximate Bayesian inference by constructing an IS proposal distribution through the minimization of a forward KL (FKL) divergence. This approach guarantees asymptotic consistency and a fast convergence towards both the optimal IS estimator and the optimal variational approximation. We empirically demonstrate on real data that our method is competitive with variational boosting and MCMC.

Is Temporal Difference Learning Optimal? An Instance-Dependent Analysis

Abstract: We address the problem of policy evaluation in discounted, tabular Markov decision processes and provide instance-dependent guarantees on the $\ell_\infty$-error under a generative model. We establ...

Generalized Momentum-Based Methods: A Hamiltonian Perspective

Abstract: We take a Hamiltonian-based perspective to generalize Nesterov's accelerated gradient descent and Polyak's heavy ball method to a broad class of momentum methods in the setting of (possibly) constr...

On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification

Abstract: Optimal transport (OT) distances are increasingly used as loss functions for statistical inference, notably in the learning of generative models or supervised learning. Yet, the behavior of minimum Wasserstein estimators is poorly understood, notably in high-dimensional regimes or under model misspecification. In this work we adopt the viewpoint of projection robust (PR) OT, which seeks to maximize the OT cost between two measures by choosing a $k$-dimensional subspace onto which they can be projected. Our first contribution is to establish several fundamental statistical properties of PR Wasserstein distances, complementing and improving previous literature that has been restricted to one-dimensional and well-specified cases. Next, we propose the integral PR Wasserstein (IPRW) distance as an alternative to the PRW distance, by averaging rather than optimizing on subspaces. Our complexity bounds can help explain why both PRW and IPRW distances outperform Wasserstein distances empirically in high-dimensional inference tasks. Finally, we consider parametric inference using the PRW distance. We provide an asymptotic guarantee of two types of minimum PRW estimators and formulate a central limit theorem for max-sliced Wasserstein estimator under model misspecification. To enable our analysis on PRW with projection dimension larger than one, we devise a novel combination of variational analysis and statistical theory.

Representation Matters: Assessing the Importance of Subgroup Allocations in Training Data

Abstract: Collecting more diverse and representative training data is often touted as a remedy for the disparate performance of machine learning predictors across subpopulations. However, a precise framework for understanding how dataset properties like diversity affect learning outcomes is largely lacking. By casting data collection as part of the learning process, we demonstrate that diverse representation in training data is key not only to increasing subgroup performances, but also to achieving population level objectives. Our analysis and experiments describe how dataset compositions influence performance and provide constructive results for using trends in existing data, alongside domain knowledge, to help guide intentional, objective-aware dataset design.

Optimization with Momentum: Dynamical, Control-Theoretic, and Symplectic Perspectives

Abstract: We analyze the convergence rate of various momentum-based optimization algorithms from a dynamical systems point of view. Our analysis exploits fundamental topological properties, such as the continuous dependence of iterates on their initial conditions, to provide a simple characterization of convergence rates. In many cases, closed-form expressions are obtained that relate algorithm parameters to the convergence rate. The analysis encompasses discrete time and continuous time, as well as time-invariant and time-variant formulations, and is not limited to a convex or Euclidean setting. In addition, the article rigorously establishes why symplectic discretization schemes are important for momentum-based optimization algorithms, and provides a characterization of algorithms that exhibit accelerated convergence.

Evaluating Sensitivity to the Stick-Breaking Prior in Bayesian Nonparametrics

Abstract: Bayesian models based on the Dirichlet process and other stick-breaking priors have been proposed as core ingredients for clustering, topic modeling, and other unsupervised learning tasks. Prior specification is, however, relatively difficult for such models, given that their flexibility implies that the consequences of prior choices are often relatively opaque. Moreover, these choices can have a substantial effect on posterior inferences. Thus, considerations of robustness need to go hand in hand with nonparametric modeling. In the current paper, we tackle this challenge by exploiting the fact that variational Bayesian methods, in addition to having computational advantages in fitting complex nonparametric models, also yield sensitivities with respect to parametric and nonparametric aspects of Bayesian models. In particular, we demonstrate how to assess the sensitivity of conclusions to the choice of concentration parameter and stick-breaking distribution for inferences under Dirichlet process mixtures and related mixture models. We provide both theoretical and empirical support for our variational approach to Bayesian sensitivity analysis.

A control-theoretic perspective on optimal high-order optimization

Abstract: We provide a control-theoretic perspective on optimal tensor algorithms for minimizing a convex function in a finite-dimensional Euclidean space. Given a function $$\varPhi : {\mathbb {R}}^d \rightarrow {\mathbb {R}}$$ that is convex and twice continuously differentiable, we study a closed-loop control system that is governed by the operators $$ abla \varPhi $$ and $$ abla ^2 \varPhi $$ together with a feedback control law $$\lambda (\cdot )$$ satisfying the algebraic equation $$(\lambda (t))^p\Vert abla \varPhi (x(t))\Vert ^{p-1} = \theta $$ for some $$\theta \in (0, 1)$$ . Our first contribution is to prove the existence and uniqueness of a local solution to this system via the Banach fixed-point theorem. We present a simple yet nontrivial Lyapunov function that allows us to establish the existence and uniqueness of a global solution under certain regularity conditions and analyze the convergence properties of trajectories. The rate of convergence is $$O(1/t^{(3p+1)/2})$$ in terms of objective function gap and $$O(1/t^{3p})$$ in terms of squared gradient norm. Our second contribution is to provide two algorithmic frameworks obtained from discretization of our continuous-time system, one of which generalizes the large-step A-HPE framework of Monteiro and Svaiter (SIAM J Optim 23(2):1092–1125, 2013) and the other of which leads to a new optimal p-th order tensor algorithm. While our discrete-time analysis can be seen as a simplification and generalization of Monteiro and Svaiter (2013), it is largely motivated by the aforementioned continuous-time analysis, demonstrating the fundamental role that the feedback control plays in optimal acceleration and the clear advantage that the continuous-time perspective brings to algorithmic design. A highlight of our analysis is that we show that all of the p-th order optimal tensor algorithms that we discuss minimize the squared gradient norm at a rate of $$O(k^{-3p})$$ , which complements the recent analysis in Gasnikov et al. (in: COLT, PMLR, pp 1374–1391, 2019), Jiang et al. (in: COLT, PMLR, pp 1799–1801, 2019) and Bubeck et al. (in: COLT, PMLR, pp 492–507, 2019).

Learning from eXtreme Bandit Feedback

Abstract: We study the problem of batch learning from bandit feedback in the setting of extremely large action spaces. Learning from extreme bandit feedback is ubiquitous in recommendation systems, in which billions of decisions are made over sets consisting of millions of choices in a single day, yielding massive observational data. In these large-scale real-world applications, supervised learning frameworks such as eXtreme Multi-label Classification (XMC) are widely used despite the fact that they incur significant biases due to the mismatch between bandit feedback and supervised labels. Such biases can be mitigated by importance sampling techniques, but these techniques suffer from impractical variance when dealing with a large number of actions. In this paper, we introduce a selective importance sampling estimator (sIS) that operates in a significantly more favorable bias-variance regime. The sIS estimator is obtained by performing importance sampling on the conditional expectation of the reward with respect to a small subset of actions for each instance (a form of Rao-Blackwellization). We employ this estimator in a novel algorithmic procedure---named Policy Optimization for eXtreme Models (POXM)---for learning from bandit feedback on XMC tasks. In POXM, the selected actions for the sIS estimator are the top-p actions of the logging policy, where p is adjusted from the data and is significantly smaller than the size of the action space. We use a supervised-to-bandit conversion on three XMC datasets to benchmark our POXM method against three competing methods: BanditNet, a previously applied partial matching pruning strategy, and a supervised learning baseline. Whereas BanditNet sometimes improves marginally over the logging policy, our experiments show that POXM systematically and significantly improves over all baselines.

Uncertainty Sets for Image Classifiers using Conformal Prediction

Abstract: Convolutional image classifiers can achieve high predictive accuracy, but quantifying their uncertainty remains an unresolved challenge, hindering their deployment in consequential settings. Existing uncertainty quantification techniques, such as Platt scaling, attempt to calibrate the network’s probability estimates, but they do not have formal guarantees. We present an algorithm that modifies any classifier to output a predictive set containing the true label with a user-specified probability, such as 90%. The algorithm is simple and fast like Platt scaling, but provides a formal finite-sample coverage guarantee for every model and dataset. Our method modifies an existing conformal prediction algorithm to give more stable predictive sets by regularizing the small scores of unlikely classes after Platt scaling. In experiments on both Imagenet and Imagenet-V2 with ResNet-152 and other classifiers, our scheme outperforms existing approaches, achieving coverage with sets that are often factors of 5 to 10 smaller.

Reconstructing unobserved cellular states from paired single-cell lineage tracing and transcriptomics data

Abstract: AO_SCPLOWBSTRACTC_SCPLOWNovel experimental assays now simultaneously measure lineage relationships and transcriptomic states from single cells, thanks to CRISPR/Cas9-based genome engineering. These multimodal measurements allow researchers not only to build comprehensive phylogenetic models relating all cells but also infer transcriptomic determinants of consequential subclonal behavior. The gene expression data, however, is limited to cells that are currently present ("leaves" of the phylogeny). As a consequence, researchers cannot form hypotheses about unobserved, or "ancestral", states that gave rise to the observed population. To address this, we introduce TreeVAE: a probabilistic framework for estimating ancestral transcriptional states. TreeVAE uses a variational autoencoder (VAE) to model the observed transcriptomic data while accounting for the phylogenetic relationships between cells. Using simulations, we demonstrate that TreeVAE outperforms benchmarks in reconstructing ancestral states on several metrics. TreeVAE also provides a measure of uncertainty, which we demonstrate to correlate well with its prediction accuracy. This estimate therefore potentially provides a data-driven way to estimate how far back in the ancestor chain predictions could be made. Finally, using real data from lung cancer metastasis, we show that accounting for phylogenetic relationship between cells improves goodness of fit. Together, TreeVAE provides a principled framework for reconstructing unobserved cellular states from single cell lineage tracing data.

Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization.

Abstract: The use of min-max optimization in adversarial training of deep neural network classifiers and training of generative adversarial networks has motivated the study of nonconvex-nonconcave optimization objectives, which frequently arise in these applications. Unfortunately, recent results have established that even approximate first-order stationary points of such objectives are intractable, even under smoothness conditions, motivating the study of min-max objectives with additional structure. We introduce a new class of structured nonconvex-nonconcave min-max optimization problems, proposing a generalization of the extragradient algorithm which provably converges to a stationary point. The algorithm applies not only to Euclidean spaces, but also to general $\ell_p$-normed finite-dimensional real vector spaces. We also discuss its stability under stochastic oracles and provide bounds on its sample complexity. Our iteration complexity and sample complexity bounds either match or improve the best known bounds for the same or less general nonconvex-nonconcave settings, such as those that satisfy variational coherence or in which a weak solution to the associated variational inequality problem is assumed to exist.

Asynchronous Online Testing of Multiple Hypotheses

Abstract: We consider the problem of asynchronous online testing, aimed at providing control of the false discovery rate (FDR) during a continual stream of data collection and testing, where each test may be a sequential test that can start and stop at arbitrary times. This setting increasingly characterizes real-world applications in science and industry, where teams of researchers across large organizations may conduct tests of hypotheses in a decentralized manner. The overlap in time and space also tends to induce dependencies among test statistics, a challenge for classical methodology, which either assumes (overly optimistically) independence or (overly pessimistically) arbitrary dependence between test statistics. We present a general framework that addresses both of these issues via a unified computational abstraction that we refer to as "conflict sets." We show how this framework yields algorithms with formal FDR guarantees under a more intermediate, local notion of dependence. We illustrate our algorithms in simulations by comparing to existing algorithms for online FDR control.

The Stereotyping Problem in Collaboratively Filtered Recommender Systems

Abstract: Recommender systems play a crucial role in mediating our access to online information. We show that such algorithms induce a particular kind of stereotyping: if preferences for a set of items are anti-correlated in the general user population, then those items may not be recommended together to a user, regardless of that user’s preferences and rating history. First, we introduce a notion of joint accessibility, which measures the extent to which a set of items can jointly be accessed by users. We then study joint accessibility under the standard factorization-based collaborative filtering framework, and provide theoretical necessary and sufficient conditions when joint accessibility is violated. Moreover, we show that these conditions can easily be violated when the users are represented by a single feature vector. To improve joint accessibility, we further propose an alternative modelling fix, which is designed to capture the diverse multiple interests of each user using a multi-vector representation. We conduct extensive experiments on real and simulated datasets, demonstrating the stereotyping problem with standard single-vector matrix factorization models.

Provable Meta-Learning of Linear Representations

Abstract: Meta-learning, or learning-to-learn, seeks to design algorithms that can utilize previous experience to rapidly learn new skills or adapt to new environments. Representation learning -- a key tool for performing meta-learning -- learns a data representation that can transfer knowledge across multiple tasks, which is essential in regimes where data is scarce. Despite a recent surge of interest in the practice of meta-learning, the theoretical underpinnings of meta-learning algorithms are lacking, especially in the context of learning transferable representations. In this paper, we focus on the problem of multi-task linear regression -- in which multiple linear regression models share a common, low-dimensional linear representation. Here, we provide provably fast, sample-efficient algorithms to address the dual challenges of (1) learning a common set of features from multiple, related tasks, and (2) transferring this knowledge to new, unseen tasks. Both are central to the general problem of meta-learning. Finally, we complement these results by providing information-theoretic lower bounds on the sample complexity of learning these linear features.

A 1.5nJ/cls Unsupervised Online Learning Classifier for Seizure Detection

Abstract: This work presents a 1.5 nJ/classification (nJ/cls) seizure detection classifier which provides unsupervised online updates on an initial offline-trained regression model to achieve >97% average sensitivity and specificity on 27 patient datasets, including three that have >250 hours of continuous recording. The classifier was fabricated in 28nm CMOS and operates at 0.5V supply. Through hardware optimizations and low overall computational complexity and voltage scaling, the online learning classifier achieves 24× better energy per classification and occupies 10x lower area than state-of-the-art.

On the Convergence of Stochastic Extragradient for Bilinear Games with Restarted Iteration Averaging.

Abstract: We study the stochastic bilinear minimax optimization problem, presenting an analysis of the Stochastic ExtraGradient (SEG) method with constant step size, and presenting variations of the method that yield favorable convergence. We first note that the last iterate of the basic SEG method only contracts to a fixed neighborhood of the Nash equilibrium, independent of the step size. This contrasts sharply with the standard setting of minimization where standard stochastic algorithms converge to a neighborhood that vanishes in proportion to the square-root (constant) step size. Under the same setting, however, we prove that when augmented with iteration averaging, SEG provably converges to the Nash equilibrium, and such a rate is provably accelerated by incorporating a scheduled restarting procedure. In the interpolation setting, we achieve an optimal convergence rate up to tight constants. We present numerical experiments that validate our theoretical findings and demonstrate the effectiveness of the SEG method when equipped with iteration averaging and restarting.

Fast Distributionally Robust Learning with Variance Reduced Min-Max Optimization.

Abstract: Distributionally robust supervised learning (DRSL) is emerging as a key paradigm for building reliable machine learning systems for real-world applications -- reflecting the need for classifiers and predictive models that are robust to the distribution shifts that arise from phenomena such as selection bias or nonstationarity. Existing algorithms for solving Wasserstein DRSL -- one of the most popular DRSL frameworks based around robustness to perturbations in the Wasserstein distance -- involve solving complex subproblems or fail to make use of stochastic gradients, limiting their use in large-scale machine learning problems. We revisit Wasserstein DRSL through the lens of min-max optimization and derive scalable and efficiently implementable stochastic extra-gradient algorithms which provably achieve faster convergence rates than existing approaches. We demonstrate their effectiveness on synthetic and real data when compared to existing DRSL approaches. Key to our results is the use of variance reduction and random reshuffling to accelerate stochastic min-max optimization, the analysis of which may be of independent interest.