Fastpart Over Parameterized Stochastic Gradient Descent For Sparse Optimisation On Measures

Deep learning has witnessed fast growth and wide application in recent years. One of the essential properties of the modern deep learning model is that it is sufficiently over-parameterized, i.e., it has much more learnable parameters than the number of training examples. The over-parameterization, on one hand, is the core of the superior approximation/representation ability of the neural network function, while, on the other hand, could lead to severe over-fitting issues, according to the conventional wisdom in learning theory. However, this is not consistent with the empirical success in deep learning, where the neural network model, trained by standard optimization algorithms (e.g., stochastic gradient descent, Adam, etc.), can not only perfectly fit the training data (i.e., finding the global solution of the training objective), but also generalizes well on the test data. This dissertation seeks to understand and explain this phenomenon by carefully characterizing the role of optimization algorithms in learning over-parameterized models. We begin the dissertation by studying arguably the simplest model: over-parameterized linear regression problems. In particular, we consider a class of SGD algorithms and prove problem-dependent generalization error bounds accordingly. Based on the established generalization guarantees, we will further characterize the sufficient conditions on the least square problem itself (e.g., conditions on data distribution and ground-truth model parameters) such that the SGD algorithm can generalize. Moreover, motivated by the recent work on the implicit regularization of SGD, we also provide a complete comparison between the SGD solution and the solution of regularized least square (i.e., ridge regression). We demonstrate the benefit of SGD compared to ridge regression for a large class of the least square problems classes, which partially explains the implicit regularization effect of SGD. In the second part, we study the effect of optimization algorithms for learning over-parameterized neural network models. Different from linear models that their optimization guarantees can be easily established, studying the optimization in training deep neural networks is challenging since the training objective is nonconvex or even nonsmooth. Therefore, we first study the optimization in training over-parameterized neural network models and establish global convergence guarantees for both GD and SGD under mild conditions on the data distribution. Based on the optimization analysis, we further establish an algorithm-dependent generalization analysis for SGD/GD. We show that if the data distribution admits certain good separation properties, a deep ReLU network with polylogarithmic width can be successfully trained with a global convergence guarantee and good generalization ability. Finally, we compare the generalization ability of two different optimization algorithms in learning over-parameterized neural networks: GD and Adam, and show that Adam and GD exhibit different algorithmic biases, which consequently leads to different solutions that have different generalization performances. The works covered in this dissertation form exploration in understanding the role of optimization algorithms in learning over-parameterized models, which is an incomplete collection of the recent advances in deep learning theory but shed light on a broader class of future directions in deep learning research.

This volume gathers together the definitive collection of papers describing the 2009 KDD Cup competition, a challenge task designed for the 15th ACM SIGKDD Conference on Knowledge Discovery and Data Mining held in Paris on June 28, 2009. The collection includes a paper summarizing the results of the challenge and contributions from the top ranking entrants. The book is complemented by a web site from which the datasets can be downloaded and at which post-challenge submissions can be made to benchmark new algorithms.

Initial training in pure and applied sciences tends to present problem-solving as the process of elaborating explicit closed-form solutions from basic principles, and then using these solutions in numerical applications. This approach is only applicable to very limited classes of problems that are simple enough for such closed-form solutions to exist. Unfortunately, most real-life problems are too complex to be amenable to this type of treatment. Numerical Methods – a Consumer Guide presents methods for dealing with them. Shifting the paradigm from formal calculus to numerical computation, the text makes it possible for the reader to · discover how to escape the dictatorship of those particular cases that are simple enough to receive a closed-form solution, and thus gain the ability to solve complex, real-life problems; · understand the principles behind recognized algorithms used in state-of-the-art numerical software; · learn the advantages and limitations of these algorithms, to facilitate the choice of which pre-existing bricks to assemble for solving a given problem; and · acquire methods that allow a critical assessment of numerical results. Numerical Methods – a Consumer Guide will be of interest to engineers and researchers who solve problems numerically with computers or supervise people doing so, and to students of both engineering and applied mathematics.

The book includes contributions on the latest model-based methods for the development of personal and commercial vehicle control devices. The main topics treated are: application of simulation and model design to development of driver assistance systems; physical and database model design for engines, motors, powertrain, undercarriage and the whole vehicle; new simulation tools, methods and optimization processes; applications of simulation in function and software development; function and software testing using HiL, MiL and SiL simulation; application of simulation and optimization in application of control devices; automation approaches at all stages of the development process.

This book is a printed edition of the Special Issue "Real-Time Optimization" that was published in Processes

The widespread availability of high-quality, high-frequency data has revolutionised the study of financial markets. By describing not only asset prices, but also market participants' actions and interactions, this wealth of information offers a new window into the inner workings of the financial ecosystem. In this original text, the authors discuss empirical facts of financial markets and introduce a wide range of models, from the micro-scale mechanics of individual order arrivals to the emergent, macro-scale issues of market stability. Throughout this journey, data is king. All discussions are firmly rooted in the empirical behaviour of real stocks, and all models are calibrated and evaluated using recent data from Nasdaq. By confronting theory with empirical facts, this book for practitioners, researchers and advanced students provides a fresh, new, and often surprising perspective on topics as diverse as optimal trading, price impact, the fragile nature of liquidity, and even the reasons why people trade at all.

. . . ????????????????????????????????? ????????????? ????????????,????? ???? ??????????? ???????????????????? ???. THUCYDIDIS HISTORIAE IV:108 C. Hude ed. , Teubner, Lipsiae MCMXIII ???????????,????? ??,? ????????????????? ???????????????????? ?????? ?????? ?????? ??? ????????? ??? ?’ ?????????? ??’ ?????????? ? ??????? ??? ????????????? ???????. ???????????????????:108 ???????????? ?????????????????????? ?. ?????????????. ????????????,????? It being the fashion of men, what they wish to be true to admit even upon an ungrounded hope, and what they wish not, with a magistral kind of arguing to reject. Thucydides (the Peloponnesian War Part I), IV:108 Thomas Hobbes Trans. , Sir W. Molesworth ed. In The English Works of Thomas Hobbes of Malmesbury, Vol. VIII I have been introduced to clock design very early in my professional career when I was tapped right out of school to design and implement the clock generation and distribution of the Alpha 21364 microprocessor. Traditionally, Alpha processors - hibited highly innovative clocking systems, always worthy of ISSCC/JSSC publi- tions and for a while Alpha processors were leading the industry in terms of clock performance. I had huge shoes to ?ll. Obviously, I was overwhelmed, confused and highly con?dent that I would drag the entire project down.

The focus of the paper is the nonparametric estimation of an instrumental regression function f defined by conditional moment restrictions stemming from a structural econometric model: E [Y - f (Z) | W] = 0, and involving endogenous variables Y and Z and instruments W. The function f is the solution of an ill-posed inverse problem and we propose an estimation procedure based on Tikhonov regularization. The paper analyses identification and overidentification of this model and presents asymptotic properties of the estimated nonparametric instrumental regression function.