Optimization formulations and methods are proving to be vital in designing algorithms to extract essential knowledge from huge volumes of data. Machine learning, however, is not simply a consumer of optimization technology but a rapidly evolving field that is itself generating new optimization ideas. This book captures the state of the art of the interaction between optimization and machine learning in a way that is accessible to researchers in both fields.
Pdf Online Optimization Of Large Scale Systems
Optimization approaches have enjoyed prominence in machine learning because of their wide applicability and attractive theoretical properties. The increasing complexity, size, and variety of today's machine learning models call for the reassessment of existing assumptions. This book starts the process of reassessment. It describes the resurgence in novel contexts of established frameworks such as first-order methods, stochastic approximations, convex relaxations, interior-point methods, and proximal methods.
It also devotes attention to newer themes such as regularized optimization, robust optimization, gradient and subgradient methods, splitting techniques, and second-order methods. Many of these techniques draw inspiration from other fields, including operations research, theoretical computer science, and subfields of optimization.
- Real-time optimization of large scale systems.
- Optimization of Large‐Scale Systems.
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The book will enrich the ongoing cross-fertilization between the machine learning community and these other fields, and within the broader optimization community. Schmidt, M.
Abstract We consider projected Newton-type methods for solving large-scale optimization problems arising in machine learning and related fields. We first introduce an algorithmic framework for projected Newton-type methods by reviewing a canonical projected quasi- Newton method. This method, while conceptually pleasing, has a high computation cost per iteration. Thus, we discuss two variants that are more scalable, namely, two-metric projection and inexact projection methods. Finally, we show how to apply the Newton-type framework to handle non-smooth objectives.
Examples are provided throughout the chapter to illustrate machine learning applications of our framework. Abstract In this paper, we analyze the convergence of two general classes of optimization algorithms for regularized kernel methods with convex loss function and quadratic norm regularization. The first methodology is a new class of algorithms based on fixed-point iterations that are well-suited for a parallel implementation and can be used with any convex loss function. The second methodology is based on coordinate descent, and generalizes some techniques previously proposed for linear support vector machines.
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It exploits the structure of additively separable loss functions to compute solutions of line searches in closed form. The two methodologies are both very easy to implement. In this paper, we also show how to remove non-differentiability of the objective functional by exactly reformulating a convex regularization problem as an unconstrained differentiable stabilization problem.
Dinuzzo, F. Abstract We analyze a family of probability distributions that are characterized by an embedded combinatorial structure. This family includes models having arbitrary treewidth and arbitrary sized factors. Abstract Most results for online decision problems with structured concepts, such as trees or cuts, assume linear costs.
In many settings, however, nonlinear costs are more realistic. Owing to their non-separability, these lead to much harder optimization problems. Going beyond linearity, we address online approximation algorithms for structured concepts that allow the cost to be submodular, i. In particular, we show regret bounds for three Hannan-consistent strategies that capture different settings.
Our results also tighten a regret bound for unconstrained online submodular minimization. Online submodular minimization for combinatorial structures In pages: , Editors: Getoor, L. Abstract We propose a new family of non-submodular global energy functions that still use submodularity internally to couple edges in a graph cut. We demonstrate the advantages of edge coupling in a natural setting, namely image segmentation.
Abstract Numerous scientific applications across a variety of fields depend on box-constrained convex optimization. Box-constrained problems therefore continue to attract research interest. We address box-constrained strictly convex problems by deriving two new quasi-Newton algorithms.
Our algorithms are positioned between the projected-gradient [J. Rosen, J. SIAM, 8 , pp. Control Optim. We also prove their convergence under a simple Armijo step-size rule. For both NNLS and NNKL our algorithms perform competitively as compared to well-established methods on medium-sized problems; for larger problems our approach frequently outperforms the competition.
Stochastic Optimization of Large-Scale Complex Systems
Abstract We present a new algorithm for minimizing a convex loss-function subject to regularization. Our approach is based on the trust-region framework with nonsmooth objectives, which allows us to build on known results to provide convergence analysis. We avoid the computational overheads associated with the conventional Hessian approximation used by trust-region methods by instead using a simple separable quadratic approximation. This approximation also enables use of proximity operators for tackling nonsmooth regularizers.
We illustrate the versatility of our resulting algorithm by specializing it to three mixed-norm regression problems: group lasso , group logistic regression , and multi-task lasso . We experiment with both synthetic and real-world large-scale data—our method is seen to be competitive, robust, and scalable. Abstract We develop an algorithm for efficient range search when the notion of dissimilarity is given by a Bregman divergence.
The range search task is to return all points in a potentially large database that are within some specified distance of a query. It arises in many learning algorithms such as locally-weighted regression, kernel density estimation, neighborhood graph-based algorithms, and in tasks like outlier detection and information retrieval. In metric spaces, efficient range search-like algorithms based on spatial data structures have been deployed on a variety of statistical tasks. Here we describe an algorithm for range search for an arbitrary Bregman divergence.
This broad class of dissimilarity measures includes the relative entropy, Mahalanobis distance, Itakura-Saito divergence, and a variety of matrix divergences.
Metric methods cannot be directly applied since Bregman divergences do not in general satisfy the triangle inequality. We derive geometric properties of Bregman divergences that yield an efficient algorithm for range search based on a recently proposed space decomposition for Bregman divergences. Schuurmans, J. Lafferty, C. Williams, A. Kulis, B. Abstract Many important machine learning problems are modeled and solved via semidefinite programs; examples include metric learning, nonlinear embedding, and certain clustering problems.
Often, off-the-shelf software is invoked for the associated optimization, which can be inappropriate due to excessive computational and storage requirements. In this paper, we introduce the use of convex perturbations for solving semidefinite programs SDPs , and for a specific perturbation we derive an algorithm that has several advantages over existing techniques: a it is simple, requiring only a few lines of Matlab, b it is a first-order method, and thereby scalable, and c it can easily exploit the structure of a given SDP e.
A pleasant byproduct of our method is a fast, kernelized version of the large-margin nearest neighbor metric learning algorithm.
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We demonstrate that our algorithm is effective in finding fast approximations to large-scale SDPs arising in some machine learning applications. Our goal is to understand the principles of Perception, Action and Learning in autonomous systems that successfully interact with complex environments and to use this understanding to design future systems.
Toggle navigation. It is highly parallelizable, and can therefore easily speed up problem solving on standard digital computer through parallel computation. As current large-scale computational systems can be used as is, there is no need to install new equipment, making it easy to scale up at a low cost. For example, by using field-programmable gate arrays FPGAs , a good solution to an optimization problem with 2, fully connected variables approximately 2 million connections can be obtained in just 0.
This is approximately 10 times faster than the laser-based quantum computer recognized as the world's fastest can solve the same problem. In addition, using a cluster of eight GPUs, Toshiba obtained a good solution for a large-scale problem involving , fully connected variables about 5 billion connections in only a few seconds.
These results open up new ways of solving large-scale combinatorial optimization problems in many different areas of application. The Simulated Bifurcation Algorithm harnesses bifurcation phenomena, adiabatic processes, and ergodic processes in classical mechanics to rapidly find highly accurate solutions. Toshiba derived the principle from a theory of a quantum computer proposed by the company itself. This discovery in classical mechanics inspired by quantum mechanics is an academically interesting, highly novel result that suggests the existence of unknown mathematical theorems.
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