The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

In the last years, Artificial Intelligence (AI) has achieved a notable momentum that may deliver the best of expectations over many application sectors across the field. For this to occur, the entire community stands in front of the barrier of explainability, an inherent problem of AI techniques brought by sub-symbolism (e.g. ensembles or Deep Neural Networks) that were not present in the last hype of AI. Paradigms underlying this problem fall within the so-called eXplainable AI (XAI) field, which is acknowledged as a crucial feature for the practical deployment of AI models. This overview examines the existing literature in the field of XAI, including a prospect toward what is yet to be reached. We summarize previous efforts to define explainability in Machine Learning, establishing a novel definition that covers prior conceptual propositions with a major focus on the audience for which explainability is sought. We then propose and discuss about a taxonomy of recent contributions related to the explainability of different Machine Learning models, including those aimed at Deep Learning methods for which a second taxonomy is built. This literature analysis serves as the background for a series of challenges faced by XAI, such as the crossroads between data fusion and explainability. Our prospects lead toward the concept of Responsible Artificial Intelligence, namely, a methodology for the large-scale implementation of AI methods in real organizations with fairness, model explainability and accountability at its core. Our ultimate goal is to provide newcomers to XAI with a reference material in order to stimulate future research advances, but also to encourage experts and professionals from other disciplines to embrace the benefits of AI in their activity sectors, without any prior bias for its lack of interpretability.

Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI.

Federated learning (FL) is a machine learning setting where many clients (e.g. mobile devices or whole organizations) collaboratively train a model under the orchestration of a central server (e.g. service provider), while keeping the training data decentralized. FL embodies the principles of focused data collection and minimization, and can mitigate many of the systemic privacy risks and costs resulting from traditional, centralized machine learning and data science approaches. Motivated by the explosive growth in FL research, this paper discusses recent advances and presents an extensive collection of open problems and challenges.

Advances and Open Problems in Federated Learning

In recent years, deep neural networks have been successful in both industry and academia, especially for computer vision tasks. The great success of deep learning is mainly due to its scalability to encode large-scale data and to maneuver billions of model parameters. However, it is a challenge to deploy these cumbersome deep models on devices with limited resources, e.g., mobile phones and embedded devices, not only because of the high computational complexity but also the large storage requirements. To this end, a variety of model compression and acceleration techniques have been developed. As a representative type of model compression and acceleration, knowledge distillation effectively learns a small student model from a large teacher model. It has received rapid increasing attention from the community. This paper provides a comprehensive survey of knowledge distillation from the perspectives of knowledge categories, training schemes, teacher-student architecture, distillation algorithms, performance comparison and applications. Furthermore, challenges in knowledge distillation are briefly reviewed and comments on future research are discussed and forwarded.

Knowledge Distillation: A Survey

我的台灣, 看見心靈的故鄉 =2009林磐聳藝術與設計展

Stochastic Gradient Descent (SGD) is a fundamental algorithm in machine learning, representing the optimization backbone for training several classic models, from regression to neural networks. Given the recent practical focus on distributed machine learning, significant work has been dedicated to the convergence properties of this algorithm under the inconsistent and noisy updates arising from execution in a distributed environment. However, surprisingly, the convergence properties of this classic algorithm in the standard shared-memory model are still not well-understood. 
In this work, we address this gap, and provide new convergence bounds for lock-free concurrent stochastic gradient descent, executing in the classic asynchronous shared memory model, against a strong adaptive adversary. Our results give improved upper and lower bounds on the "price of asynchrony" when executing the fundamental SGD algorithm in a concurrent setting. They show that this classic optimization tool can converge faster and with a wider range of parameters than previously known under asynchronous iterations. At the same time, we exhibit a fundamental trade-off between the maximum delay in the system and the rate at which SGD can converge, which governs the set of parameters under which this algorithm can still work efficiently.

The Convergence of Stochastic Gradient Descent in Asynchronous Shared Memory

Second-order information, in the form of Hessian- or Inverse-Hessian-vector products, is a fundamental tool for solving optimization problems. Recently, there has been significant interest in utilizing this information in the context of deep neural networks; however, relatively little is known about the quality of existing approximations in this context. Our work examines this question, identifies issues with existing approaches, and proposes a method called WoodFisher to compute a faithful and efficient estimate of the inverse Hessian. Our main application is to neural network compression, where we build on the classic Optimal Brain Damage/Surgeon framework. We demonstrate that WoodFisher significantly outperforms popular state-of-the-art methods for one-shot pruning. Further, even when iterative, gradual pruning is considered, our method results in a gain in test accuracy over the state-of-the-art approaches, for pruning popular neural networks (like ResNet-50, MobileNetV1) trained on standard image classification datasets such as ImageNet ILSVRC. We examine how our method can be extended to take into account first-order information, as well as illustrate its ability to automatically set layer-wise pruning thresholds and perform compression in the limited-data regime. The code is available at the following link, this https URL.

WoodFisher: Efficient second-order approximations for model compression.

This paper addresses the following question: what is the minimum-sized synchronous window needed to solve consensus in an otherwise asynchronous system? In answer to this question, we present the first optimally-resilient algorithm ASAPthat solves consensus as soon as possiblein an eventually synchronous system, i.e., a system that from some time GSTonwards, delivers messages in a timely fashion. ASAPguarantees that, in an execution with at most ffailures, every process decides no later than round GST+ f+ 2, which is optimal.

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How to Solve Consensus in the Smallest Window of Synchrony

As the size and complexity of models and datasets grow, so does the need for communication-efficient variants of stochastic gradient descent that can be deployed to perform parallel model training. One popular communication-compression method for data-parallel SGD is QSGD (Alistarh et al., 2017), which quantizes and encodes gradients to reduce communication costs. The baseline variant of QSGD provides strong theoretical guarantees, however, for practical purposes, the authors proposed a heuristic variant which we call QSGDinf, which demonstrated impressive empirical gains for distributed training of large neural networks. In this paper, we build on this work to propose a new gradient quantization scheme, and show that it has both stronger theoretical guarantees than QSGD, and matches and exceeds the empirical performance of the QSGDinf heuristic and of other compression methods.

NUQSGD: Provably Communication-efficient Data-parallel SGD via Nonuniform Quantization

Most of today's distributed machine learning systems assume {\em reliable networks}: whenever two machines exchange information (e.g., gradients or models), the network should guarantee the delivery of the message. At the same time, recent work exhibits the impressive tolerance of machine learning algorithms to errors or noise arising from relaxed communication or synchronization. In this paper, we connect these two trends, and consider the following question: {\em Can we design machine learning systems that are tolerant to network unreliability during training?} With this motivation, we focus on a theoretical problem of independent interest---given a standard distributed parameter server architecture, if every communication between the worker and the server has a non-zero probability $p$ of being dropped, does there exist an algorithm that still converges, and at what speed? The technical contribution of this paper is a novel theoretical analysis proving that distributed learning over unreliable network can achieve comparable convergence rate to centralized or distributed learning over reliable networks. Further, we prove that the influence of the packet drop rate diminishes with the growth of the number of \textcolor{black}{parameter servers}. We map this theoretical result onto a real-world scenario, training deep neural networks over an unreliable network layer, and conduct network simulation to validate the system improvement by allowing the networks to be unreliable.

/pdf/distributed-learning-over-unreliable-networks-je0xkjm6g9.pdf

Dan Alistarh

Papers

The Convergence of Stochastic Gradient Descent in Asynchronous Shared Memory

WoodFisher: Efficient second-order approximations for model compression.

How to Solve Consensus in the Smallest Window of Synchrony

NUQSGD: Provably Communication-efficient Data-parallel SGD via Nonuniform Quantization

Distributed Learning over Unreliable Networks