Michael I. Jordan

Bio: Michael I. Jordan is an academic researcher from University of California, Berkeley. The author has contributed to research in topics: Computer science & Inference. The author has an hindex of 176, co-authored 1016 publications receiving 216204 citations. Previous affiliations of Michael I. Jordan include Stanford University & Princeton University.

SAFFRON: an adaptive algorithm for online control of the false discovery rate

Abstract: In the online false discovery rate (FDR) problem, one observes a possibly infinite sequence of $p$-values $P_1,P_2,\dots$, each testing a different null hypothesis, and an algorithm must pick a sequence of rejection thresholds $\alpha_1,\alpha_2,\dots$ in an online fashion, effectively rejecting the $k$-th null hypothesis whenever $P_k \leq \alpha_k$. Importantly, $\alpha_k$ must be a function of the past, and cannot depend on $P_k$ or any of the later unseen $p$-values, and must be chosen to guarantee that for any time $t$, the FDR up to time $t$ is less than some pre-determined quantity $\alpha \in (0,1)$. In this work, we present a powerful new framework for online FDR control that we refer to as SAFFRON. Like older alpha-investing (AI) algorithms, SAFFRON starts off with an error budget, called alpha-wealth, that it intelligently allocates to different tests over time, earning back some wealth on making a new discovery. However, unlike older methods, SAFFRON's threshold sequence is based on a novel estimate of the alpha fraction that it allocates to true null hypotheses. In the offline setting, algorithms that employ an estimate of the proportion of true nulls are called adaptive methods, and SAFFRON can be seen as an online analogue of the famous offline Storey-BH adaptive procedure. Just as Storey-BH is typically more powerful than the Benjamini-Hochberg (BH) procedure under independence, we demonstrate that SAFFRON is also more powerful than its non-adaptive counterparts, such as LORD and other generalized alpha-investing algorithms. Further, a monotone version of the original AI algorithm is recovered as a special case of SAFFRON, that is often more stable and powerful than the original. Lastly, the derivation of SAFFRON provides a novel template for deriving new online FDR rules.

On Learning Rates and Schrödinger Operators

Abstract: The learning rate is perhaps the single most important parameter in the training of neural networks and, more broadly, in stochastic (nonconvex) optimization. Accordingly, there are numerous effective, but poorly understood, techniques for tuning the learning rate, including learning rate decay, which starts with a large initial learning rate that is gradually decreased. In this paper, we present a general theoretical analysis of the effect of the learning rate in stochastic gradient descent (SGD). Our analysis is based on the use of a learning-rate-dependent stochastic differential equation (lr-dependent SDE) that serves as a surrogate for SGD. For a broad class of objective functions, we establish a linear rate of convergence for this continuous-time formulation of SGD, highlighting the fundamental importance of the learning rate in SGD, and contrasting to gradient descent and stochastic gradient Langevin dynamics. Moreover, we obtain an explicit expression for the optimal linear rate by analyzing the spectrum of the Witten-Laplacian, a special case of the Schrodinger operator associated with the lr-dependent SDE. Strikingly, this expression clearly reveals the dependence of the linear convergence rate on the learning rate -- the linear rate decreases rapidly to zero as the learning rate tends to zero for a broad class of nonconvex functions, whereas it stays constant for strongly convex functions. Based on this sharp distinction between nonconvex and convex problems, we provide a mathematical interpretation of the benefits of using learning rate decay for nonconvex optimization.

Dynamical, symplectic and stochastic perspectives on gradient-based optimization

Abstract: Our topic is the relationship between dynamical systems and optimization. This is a venerable, vast area in mathematics, counting among its many historical threads the study of gradient flow and the variational perspective on mechanics. We aim to build some new connections in this general area, studying aspects of gradient-based optimization from a continuous-time, variational point of view. We go beyond classical gradient flow to focus on second-order dynamics, aiming to show the relevance of such dynamics to optimization algorithms that not only converge, but converge quickly. Although our focus is theoretical, it is important to motivate the work by considering the applied context from which it has emerged. Modern statistical data analysis often involves very large data sets and very large parameter spaces, so that computational efficiency is of paramount importance in practical applications. In such settings, the notion of efficiency is more stringent than that of classical computational complexity theory, where the distinction between polynomial complexity and exponential complexity has been a useful focus. In large-scale data analysis, algorithms need to be not merely polynomial, but linear, or nearly linear, in relevant problem parameters. Optimization theory has provided both practical and theoretical support for this endeavor. It has supplied computationally-efficient algorithms, as well as analysis tools that allow rates of convergence to be determined as explicit functions of problem parameters. The dictum of efficiency has led to a focus on algorithms that are based principally on gradients of objective functions, or on estimates of gradients, given that Hessians incur quadratic or cubic complexity in the dimension of the configuration space (Bottou, 2010; Nesterov, 2012). More broadly, the blending of inferential and computational ideas is one of the major intellectual trends of the current century—one currently referred to by terms such as “data science” and “machine learning.” It is a trend that inspires the search for new mathematical concepts that allow computational and inferential desiderata to be studied jointly. For example, one would like to impose runtime budgets on data-analysis algorithms as a function of statistical

Task Decomposition Through Competition in a Modular Connectionist

Abstract: A NOVEL MODULAR CONNECTIONIST ARCHITECTURE IS PRESENTED IN WHICH THE NETWORKS COMPOSING THE ARCHITECTURE COMPETE TO LEARN THE TRAINING PATTERNS. AN OUTCOME OF THE COMPETITION IS THAT DIFFERENT NETWORKS LEARN DIFFERENT TRAINING PATTERNS AND, THUS, LEARN TO COMPUTE DIFFERENT FUNCTIONS. THE ARCHITECTURE PERFORMS TASK DECOMPOSITION IN THE SENSE THAT IT LEARNS TO PARTITION A TASK INTO TWO OR MORE FUNCTIONALLY INDEPENDENT TASKS AND ALLO- CATES DISTINCT NETWORKS TO LEARN EACH TASK. IN ADDITION, THE ARCHITECTURE TENDS TO ALLOCATE TO EACH TASK THE NETWORK WHOSE TOPOLOGY IS MORE APPROPRI- ATE TO THAT TASK. THE ARCHITECTURE''S PERFORMANCE ON "WHAT" AND "WHERE" VISION TASKS IS PRESENTED AND COMPARED WITH THE PERFORMANCE OF TWO MULTI- LAYER NETWORKS. FINALLY, IT IS NOTED THAT FUNCTION DECOMPOSITION IS AN UNDERCONSTRAINED PROBLEM AND, THUS, DIFFERENT MODULAR ARCHITECTURES MAY DECOMPOSE A FUNCTION IN DIFFERENT WAYS. WE ARGUE THAT A DESIRABLE DECOM- POSITION CAN BE ACHIEVED IF THE ARCHITECTURE IS SUITABLY RESTRICTED IN THE TYPES OF FUNCTIONS THAT IT CAN COMPUTE. APPROPRIATE RESTRICTIONS CAN BE FOUND THROUGH THE APPLICATION OF DOMAIN KNOWLEDGE. A STRENGTH OF THE MOD- ULAR ARCHITECTURE IS THAT ITS STRUCTURE IS WELL-SUITED FOR INCORPORATING DOMAIN KNOWLEDGE.

Optimistic Concurrency Control for Distributed Unsupervised Learning

Abstract: Research on distributed machine learning algorithms has focused primarily on one of two extremes - algorithms that obey strict concurrency constraints or algorithms that obey few or no such constraints. We consider an intermediate alternative in which algorithms optimistically assume that conflicts are unlikely and if conflicts do arise a conflict-resolution protocol is invoked. We view this "optimistic concurrency control" paradigm as particularly appropriate for large-scale machine learning algorithms, particularly in the unsupervised setting. We demonstrate our approach in three problem areas: clustering, feature learning and online facility location. We evaluate our methods via large-scale experiments in a cluster computing environment.

Going deeper with convolutions

Abstract: We propose a deep convolutional neural network architecture codenamed Inception that achieves the new state of the art for classification and detection in the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC14). The main hallmark of this architecture is the improved utilization of the computing resources inside the network. By a carefully crafted design, we increased the depth and width of the network while keeping the computational budget constant. To optimize quality, the architectural decisions were based on the Hebbian principle and the intuition of multi-scale processing. One particular incarnation used in our submission for ILSVRC14 is called GoogLeNet, a 22 layers deep network, the quality of which is assessed in the context of classification and detection.

Deep Learning

Abstract: Deep learning is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts. Because the computer gathers knowledge from experience, there is no need for a human computer operator to formally specify all the knowledge that the computer needs. The hierarchy of concepts allows the computer to learn complicated concepts by building them out of simpler ones; a graph of these hierarchies would be many layers deep. This book introduces a broad range of topics in deep learning. The text offers mathematical and conceptual background, covering relevant concepts in linear algebra, probability theory and information theory, numerical computation, and machine learning. It describes deep learning techniques used by practitioners in industry, including deep feedforward networks, regularization, optimization algorithms, convolutional networks, sequence modeling, and practical methodology; and it surveys such applications as natural language processing, speech recognition, computer vision, online recommendation systems, bioinformatics, and videogames. Finally, the book offers research perspectives, covering such theoretical topics as linear factor models, autoencoders, representation learning, structured probabilistic models, Monte Carlo methods, the partition function, approximate inference, and deep generative models. Deep Learning can be used by undergraduate or graduate students planning careers in either industry or research, and by software engineers who want to begin using deep learning in their products or platforms. A website offers supplementary material for both readers and instructors.

Reinforcement Learning: An Introduction

Abstract: Reinforcement learning, one of the most active research areas in artificial intelligence, is a computational approach to learning whereby an agent tries to maximize the total amount of reward it receives when interacting with a complex, uncertain environment. In Reinforcement Learning, Richard Sutton and Andrew Barto provide a clear and simple account of the key ideas and algorithms of reinforcement learning. Their discussion ranges from the history of the field's intellectual foundations to the most recent developments and applications. The only necessary mathematical background is familiarity with elementary concepts of probability. The book is divided into three parts. Part I defines the reinforcement learning problem in terms of Markov decision processes. Part II provides basic solution methods: dynamic programming, Monte Carlo methods, and temporal-difference learning. Part III presents a unified view of the solution methods and incorporates artificial neural networks, eligibility traces, and planning; the two final chapters present case studies and consider the future of reinforcement learning.

Latent dirichlet allocation

Abstract: We describe latent Dirichlet allocation (LDA), a generative probabilistic model for collections of discrete data such as text corpora. LDA is a three-level hierarchical Bayesian model, in which each item of a collection is modeled as a finite mixture over an underlying set of topics. Each topic is, in turn, modeled as an infinite mixture over an underlying set of topic probabilities. In the context of text modeling, the topic probabilities provide an explicit representation of a document. We present efficient approximate inference techniques based on variational methods and an EM algorithm for empirical Bayes parameter estimation. We report results in document modeling, text classification, and collaborative filtering, comparing to a mixture of unigrams model and the probabilistic LSI model.

Latent Dirichlet Allocation

Abstract: We propose a generative model for text and other collections of discrete data that generalizes or improves on several previous models including naive Bayes/unigram, mixture of unigrams [6], and Hof-mann's aspect model, also known as probabilistic latent semantic indexing (pLSI) [3]. In the context of text modeling, our model posits that each document is generated as a mixture of topics, where the continuous-valued mixture proportions are distributed as a latent Dirichlet random variable. Inference and learning are carried out efficiently via variational algorithms. We present empirical results on applications of this model to problems in text modeling, collaborative filtering, and text classification.