arXiv subjects

Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 181 records · Page 10Linked to original sources

arXiv2022

Continuous-time Analysis for Variational Inequalities: An Overview and Desiderata

Algorithms that solve zero-sum games, multi-objective agent objectives, or, more generally, variational inequality (VI) problems are notoriously unstable on general problems. Owing to the increasing need for solving such problems in machine learning, this instability has been highlighted in recent years as a significant research challenge. In this paper, we provide an overview of recent progress in the use of continuous-time perspectives in the analysis and design of methods targeting the broad VI problem class. Our presentation draws parallels between single-objective problems and multi-objective problems, highlighting the challenges of the latter. We also formulate various desiderata for algorithms that apply to general VIs and we argue that achieving these desiderata may profit from an understanding of the associated continuous-time dynamics.

stat.ML↗

arXiv2022

NumS: Scalable Array Programming for the Cloud

Scientists increasingly rely on Python tools to perform scalable distributed memory array operations using rich, NumPy-like expressions. However, many of these tools rely on dynamic schedulers optimized for abstract task graphs, which often encounter memory and network bandwidth-related bottlenecks due to sub-optimal data and operator placement decisions. Tools built on the message passing interface (MPI), such as ScaLAPACK and SLATE, have better scaling properties, but these solutions require specialized knowledge to use. In this work, we present NumS, an array programming library which optimizes NumPy-like expressions on task-based distributed systems. This is achieved through a novel scheduler called Load Simulated Hierarchical Scheduling (LSHS). LSHS is a local search method which optimizes operator placement by minimizing maximum memory and network load on any given node within a distributed system. Coupled with a heuristic for load balanced data layouts, our approach is capable of attaining communication lower bounds on some common numerical operations, and our empirical study shows that LSHS enhances performance on Ray by decreasing network load by a factor of 2x, requiring 4x less memory, and reducing execution time by 10x on the logistic regression problem. On terabyte-scale data, NumS achieves competitive performance to SLATE on DGEMM, up to 20x speedup over Dask on a key operation for tensor factorization, and a 2x speedup on logistic regression compared to Dask ML and Spark's MLlib.

cs.DC↗

arXiv2022

Mechanisms that Incentivize Data Sharing in Federated Learning

Federated learning is typically considered a beneficial technology which allows multiple agents to collaborate with each other, improve the accuracy of their models, and solve problems which are otherwise too data-intensive / expensive to be solved individually. However, under the expectation that other agents will share their data, rational agents may be tempted to engage in detrimental behavior such as free-riding where they contribute no data but still enjoy an improved model. In this work, we propose a framework to analyze the behavior of such rational data generators. We first show how a naive scheme leads to catastrophic levels of free-riding where the benefits of data sharing are completely eroded. Then, using ideas from contract theory, we introduce accuracy shaping based mechanisms to maximize the amount of data generated by each agent. These provably prevent free-riding without needing any payment mechanism.

cs.GT↗

arXiv2022

No-Regret Learning in Partially-Informed Auctions

Auctions with partially-revealed information about items are broadly employed in real-world applications, but the underlying mechanisms have limited theoretical support. In this work, we study a machine learning formulation of these types of mechanisms, presenting algorithms that are no-regret from the buyer's perspective. Specifically, a buyer who wishes to maximize his utility interacts repeatedly with a platform over a series of $T$ rounds. In each round, a new item is drawn from an unknown distribution and the platform publishes a price together with incomplete, "masked" information about the item. The buyer then decides whether to purchase the item. We formalize this problem as an online learning task where the goal is to have low regret with respect to a myopic oracle that has perfect knowledge of the distribution over items and the seller's masking function. When the distribution over items is known to the buyer and the mask is a SimHash function mapping $\mathbb{R}^d$ to $\{0,1\}^{\ell}$, our algorithm has regret $\tilde O((Td\ell)^{1/2})$. In a fully agnostic setting when the mask is an arbitrary function mapping to a set of size $n$ and the prices are stochastic, our algorithm has regret $\tilde O((Tn)^{1/2})$.

cs.LG↗

arXiv2022

Recommendation Systems with Distribution-Free Reliability Guarantees

When building recommendation systems, we seek to output a helpful set of items to the user. Under the hood, a ranking model predicts which of two candidate items is better, and we must distill these pairwise comparisons into the user-facing output. However, a learned ranking model is never perfect, so taking its predictions at face value gives no guarantee that the user-facing output is reliable. Building from a pre-trained ranking model, we show how to return a set of items that is rigorously guaranteed to contain mostly good items. Our procedure endows any ranking model with rigorous finite-sample control of the false discovery rate (FDR), regardless of the (unknown) data distribution. Moreover, our calibration algorithm enables the easy and principled integration of multiple objectives in recommender systems. As an example, we show how to optimize for recommendation diversity subject to a user-specified level of FDR control, circumventing the need to specify ad hoc weights of a diversity loss against an accuracy loss. Throughout, we focus on the problem of learning to rank a set of possible recommendations, evaluating our methods on the Yahoo! Learning to Rank and MSMarco datasets.

cs.IR↗

arXiv2022

Robust Calibration with Multi-domain Temperature Scaling

Uncertainty quantification is essential for the reliable deployment of machine learning models to high-stakes application domains. Uncertainty quantification is all the more challenging when training distribution and test distribution are different, even the distribution shifts are mild. Despite the ubiquity of distribution shifts in real-world applications, existing uncertainty quantification approaches mainly study the in-distribution setting where the train and test distributions are the same. In this paper, we develop a systematic calibration model to handle distribution shifts by leveraging data from multiple domains. Our proposed method -- multi-domain temperature scaling -- uses the heterogeneity in the domains to improve calibration robustness under distribution shift. Through experiments on three benchmark data sets, we find our proposed method outperforms existing methods as measured on both in-distribution and out-of-distribution test sets.

cs.LG↗

arXiv2022

Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm

We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of $m$ discrete probability measures supported on a finite metric space of size $n$. We show first that the constraint matrix arising from the standard linear programming (LP) representation of the FS-WBP is \textit{not totally unimodular} when $m \geq 3$ and $n \geq 3$. This result resolves an open question pertaining to the relationship between the FS-WBP and the minimum-cost flow (MCF) problem since it proves that the FS-WBP in the standard LP form is not an MCF problem when $m \geq 3$ and $n \geq 3$. We also develop a provably fast \textit{deterministic} variant of the celebrated iterative Bregman projection (IBP) algorithm, named \textsc{FastIBP}, with a complexity bound of $\tilde{O}(mn^{7/3}\varepsilon^{-4/3})$, where $\varepsilon \in (0, 1)$ is the desired tolerance. This complexity bound is better than the best known complexity bound of $\tilde{O}(mn^2\varepsilon^{-2})$ for the IBP algorithm in terms of $\varepsilon$, and that of $\tilde{O}(mn^{5/2}\varepsilon^{-1})$ from accelerated alternating minimization algorithm or accelerated primal-dual adaptive gradient algorithm in terms of $n$. Finally, we conduct extensive experiments with both synthetic data and real images and demonstrate the favorable performance of the \textsc{FastIBP} algorithm in practice.

cs.CC↗

arXiv2022

Online Nonsubmodular Minimization with Delayed Costs: From Full Information to Bandit Feedback

Motivated by applications to online learning in sparse estimation and Bayesian optimization, we consider the problem of online unconstrained nonsubmodular minimization with delayed costs in both full information and bandit feedback settings. In contrast to previous works on online unconstrained submodular minimization, we focus on a class of nonsubmodular functions with special structure, and prove regret guarantees for several variants of the online and approximate online bandit gradient descent algorithms in static and delayed scenarios. We derive bounds for the agent's regret in the full information and bandit feedback setting, even if the delay between choosing a decision and receiving the incurred cost is unbounded. Key to our approach is the notion of $(α, β)$-regret and the extension of the generic convex relaxation model from~\citet{El-2020-Optimal}, the analysis of which is of independent interest. We conduct and showcase several simulation studies to demonstrate the efficacy of our algorithms.

cs.LG↗

arXiv2022

Taming Nonconvexity in Kernel Feature Selection -- Favorable Properties of the Laplace Kernel

Kernel-based feature selection is an important tool in nonparametric statistics. Despite many practical applications of kernel-based feature selection, there is little statistical theory available to support the method. A core challenge is the objective function of the optimization problems used to define kernel-based feature selection are nonconvex. The literature has only studied the statistical properties of the \emph{global optima}, which is a mismatch, given that the gradient-based algorithms available for nonconvex optimization are only able to guarantee convergence to local minima. Studying the full landscape associated with kernel-based methods, we show that feature selection objectives using the Laplace kernel (and other $\ell_1$ kernels) come with statistical guarantees that other kernels, including the ubiquitous Gaussian kernel (or other $\ell_2$ kernels) do not possess. Based on a sharp characterization of the gradient of the objective function, we show that $\ell_1$ kernels eliminate unfavorable stationary points that appear when using an $\ell_2$ kernel. Armed with this insight, we establish statistical guarantees for $\ell_1$ kernel-based feature selection which do not require reaching the global minima. In particular, we establish model-selection consistency of $\ell_1$-kernel-based feature selection in recovering main effects and hierarchical interactions in the nonparametric setting with $n \sim \log p$ samples.

math.ST↗

arXiv2022

On the Efficiency of Entropic Regularized Algorithms for Optimal Transport

We present several new complexity results for the entropic regularized algorithms that approximately solve the optimal transport (OT) problem between two discrete probability measures with at most $n$ atoms. First, we improve the complexity bound of a greedy variant of Sinkhorn, known as \textit{Greenkhorn}, from $\widetilde{O}(n^2\varepsilon^{-3})$ to $\widetilde{O}(n^2\varepsilon^{-2})$. Notably, our result can match the best known complexity bound of Sinkhorn and help clarify why Greenkhorn significantly outperforms Sinkhorn in practice in terms of row/column updates as observed by~\citet{Altschuler-2017-Near}. Second, we propose a new algorithm, which we refer to as \textit{APDAMD} and which generalizes an adaptive primal-dual accelerated gradient descent (APDAGD) algorithm~\citep{Dvurechensky-2018-Computational} with a prespecified mirror mapping $ϕ$. We prove that APDAMD achieves the complexity bound of $\widetilde{O}(n^2\sqrtδ\varepsilon^{-1})$ in which $δ>0$ stands for the regularity of $ϕ$. In addition, we show by a counterexample that the complexity bound of $\widetilde{O}(\min\{n^{9/4}\varepsilon^{-1}, n^2\varepsilon^{-2}\})$ proved for APDAGD before is invalid and give a refined complexity bound of $\widetilde{O}(n^{5/2}\varepsilon^{-1})$. Further, we develop a \textit{deterministic} accelerated variant of Sinkhorn via appeal to estimated sequence and prove the complexity bound of $\widetilde{O}(n^{7/3}\varepsilon^{-4/3})$. As such, we see that accelerated variant of Sinkhorn outperforms Sinkhorn and Greenkhorn in terms of $1/\varepsilon$ and APDAGD and accelerated alternating minimization (AAM)~\citep{Guminov-2021-Combination} in terms of $n$. Finally, we conduct the experiments on synthetic and real data and the numerical results show the efficiency of Greenkhorn, APDAMD and accelerated Sinkhorn in practice.

cs.DS↗

arXiv2022

SOUL: An Energy-Efficient Unsupervised Online Learning Seizure Detection Classifier

Implantable devices that record neural activity and detect seizures have been adopted to issue warnings or trigger neurostimulation to suppress epileptic seizures. Typical seizure detection systems rely on high-accuracy offline-trained machine learning classifiers that require manual retraining when seizure patterns change over long periods of time. For an implantable seizure detection system, a low power, at-the-edge, online learning algorithm can be employed to dynamically adapt to the neural signal drifts, thereby maintaining high accuracy without external intervention. This work proposes SOUL: Stochastic-gradient-descent-based Online Unsupervised Logistic regression classifier. After an initial offline training phase, continuous online unsupervised classifier updates are applied in situ, which improves sensitivity in patients with drifting seizure features. SOUL was tested on two human electroencephalography (EEG) datasets: the CHB-MIT scalp EEG dataset, and a long (>100 hours) NeuroVista intracranial EEG dataset. It was able to achieve an average sensitivity of 97.5% and 97.9% for the two datasets respectively, at >95% specificity. Sensitivity improved by at most 8.2% on long-term data when compared to a typical seizure detection classifier. SOUL was fabricated in TSMC's 28 nm process occupying 0.1 mm2 and achieves 1.5 nJ/classification energy efficiency, which is at least 24x more efficient than state-of-the-art.

eess.SP↗

arXiv2022

The Sky Above The Clouds

Technology ecosystems often undergo significant transformations as they mature. For example, telephony, the Internet, and PCs all started with a single provider, but in the United States each is now served by a competitive market that uses comprehensive and universal technology standards to provide compatibility. This white paper presents our view on how the cloud ecosystem, barely over fifteen years old, could evolve as it matures.

cs.DC↗

arXiv2022

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

We study the stochastic bilinear minimax optimization problem, presenting an analysis of the same-sample Stochastic ExtraGradient (SEG) method with constant step size, and presenting variations of the method that yield favorable convergence. In sharp contrasts with the basic SEG method whose last iterate only contracts to a fixed neighborhood of the Nash equilibrium, SEG augmented with iteration averaging provably converges to the Nash equilibrium under the same standard settings, and such a rate is further improved by incorporating a scheduled restarting procedure. In the interpolation setting where noise vanishes at the Nash equilibrium, 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.

math.OC↗

arXiv2022

Tactical Optimism and Pessimism for Deep Reinforcement Learning

In recent years, deep off-policy actor-critic algorithms have become a dominant approach to reinforcement learning for continuous control. One of the primary drivers of this improved performance is the use of pessimistic value updates to address function approximation errors, which previously led to disappointing performance. However, a direct consequence of pessimism is reduced exploration, running counter to theoretical support for the efficacy of optimism in the face of uncertainty. So which approach is best? In this work, we show that the most effective degree of optimism can vary both across tasks and over the course of learning. Inspired by this insight, we introduce a novel deep actor-critic framework, Tactical Optimistic and Pessimistic (TOP) estimation, which switches between optimistic and pessimistic value learning online. This is achieved by formulating the selection as a multi-arm bandit problem. We show in a series of continuous control tasks that TOP outperforms existing methods which rely on a fixed degree of optimism, setting a new state of the art in challenging pixel-based environments. Since our changes are simple to implement, we believe these insights can easily be incorporated into a multitude of off-policy algorithms.

cs.LG↗

arXiv2022

ElegantRL-Podracer: Scalable and Elastic Library for Cloud-Native Deep Reinforcement Learning

Deep reinforcement learning (DRL) has revolutionized learning and actuation in applications such as game playing and robotic control. The cost of data collection, i.e., generating transitions from agent-environment interactions, remains a major challenge for wider DRL adoption in complex real-world problems. Following a cloud-native paradigm to train DRL agents on a GPU cloud platform is a promising solution. In this paper, we present a scalable and elastic library ElegantRL-podracer for cloud-native deep reinforcement learning, which efficiently supports millions of GPU cores to carry out massively parallel training at multiple levels. At a high-level, ElegantRL-podracer employs a tournament-based ensemble scheme to orchestrate the training process on hundreds or even thousands of GPUs, scheduling the interactions between a leaderboard and a training pool with hundreds of pods. At a low-level, each pod simulates agent-environment interactions in parallel by fully utilizing nearly 7,000 GPU CUDA cores in a single GPU. Our ElegantRL-podracer library features high scalability, elasticity and accessibility by following the development principles of containerization, microservices and MLOps. Using an NVIDIA DGX SuperPOD cloud, we conduct extensive experiments on various tasks in locomotion and stock trading and show that ElegantRL-podracer substantially outperforms RLlib. Our codes are available on GitHub.

cs.LG↗

arXiv2022

Instability, Computational Efficiency and Statistical Accuracy

Many statistical estimators are defined as the fixed point of a data-dependent operator, with estimators based on minimizing a cost function being an important special case. The limiting performance of such estimators depends on the properties of the population-level operator in the idealized limit of infinitely many samples. We develop a general framework that yields bounds on statistical accuracy based on the interplay between the deterministic convergence rate of the algorithm at the population level, and its degree of (in)stability when applied to an empirical object based on $n$ samples. Using this framework, we analyze both stable forms of gradient descent and some higher-order and unstable algorithms, including Newton's method and its cubic-regularized variant, as well as the EM algorithm. We provide applications of our general results to several concrete classes of models, including Gaussian mixture estimation, non-linear regression models, and informative non-response models. We exhibit cases in which an unstable algorithm can achieve the same statistical accuracy as a stable algorithm in exponentially fewer steps -- namely, with the number of iterations being reduced from polynomial to logarithmic in sample size $n$.

cs.LG↗

arXiv2022

Post-Selection Inference via Algorithmic Stability

When the target of statistical inference is chosen in a data-driven manner, the guarantees provided by classical theories vanish. We propose a solution to the problem of inference after selection by building on the framework of algorithmic stability, in particular its branch with origins in the field of differential privacy. Stability is achieved via randomization of selection and it serves as a quantitative measure that is sufficient to obtain non-trivial post-selection corrections for classical confidence intervals. Importantly, the underpinnings of algorithmic stability translate directly into computational efficiency -- our method computes simple corrections for selective inference without recourse to Markov chain Monte Carlo sampling.

math.ST↗

arXiv2022

Convergence Rates for Gaussian Mixtures of Experts

We provide a theoretical treatment of over-specified Gaussian mixtures of experts with covariate-free gating networks. We establish the convergence rates of the maximum likelihood estimation (MLE) for these models. Our proof technique is based on a novel notion of \emph{algebraic independence} of the expert functions. Drawing on optimal transport theory, we establish a connection between the algebraic independence and a certain class of partial differential equations (PDEs). Exploiting this connection allows us to derive convergence rates and minimax lower bounds for parameter estimation.

math.ST↗