Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

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Convex Analysisの二,三の進展について

On robust estimation of the location parameter

This monograph is about a class of optimization algorithms called proximal algorithms. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Here, we discuss the many different interpretations of proximal operators and algorithms, describe their connections to many other topics in optimization and applied mathematics, survey some popular algorithms, and provide a large number of examples of proximal operators that commonly arise in practice.

Proximal Algorithms

Neural net classifiers trained on data with annotated class labels can also capture apparent visual similarity among categories without being directed to do so. We study whether this observation can be extended beyond the conventional domain of supervised learning: Can we learn a good feature representation that captures apparent similarity among instances, instead of classes, by merely asking the feature to be discriminative of individual instances? We formulate this intuition as a non-parametric classification problem at the instance-level, and use noise-contrastive estimation to tackle the computational challenges imposed by the large number of instance classes. Our experimental results demonstrate that, under unsupervised learning settings, our method surpasses the state-of-the-art on ImageNet classification by a large margin. Our method is also remarkable for consistently improving test performance with more training data and better network architectures. By fine-tuning the learned feature, we further obtain competitive results for semi-supervised learning and object detection tasks. Our non-parametric model is highly compact: With 128 features per image, our method requires only 600MB storage for a million images, enabling fast nearest neighbour retrieval at the run time.

https://openaccess.thecvf.com/content_cvpr_2018/papers/Wu_Unsupervised_Feature_Learning_CVPR_2018_paper.pdf

Unsupervised Feature Learning via Non-parametric Instance Discrimination

Many problems of recent interest in statistics and machine learning can be posed in the framework of convex optimization. Due to the explosion in size and complexity of modern datasets, it is increasingly important to be able to solve problems with a very large number of features or training examples. As a result, both the decentralized collection or storage of these datasets as well as accompanying distributed solution methods are either necessary or at least highly desirable. In this review, we argue that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas. The method was developed in the 1970s, with roots in the 1950s, and is equivalent or closely related to many other algorithms, such as dual decomposition, the method of multipliers, Douglas–Rachford splitting, Spingarn's method of partial inverses, Dykstra's alternating projections, Bregman iterative algorithms for l1 problems, proximal methods, and others. After briefly surveying the theory and history of the algorithm, we discuss applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. We also discuss general distributed optimization, extensions to the nonconvex setting, and efficient implementation, including some details on distributed MPI and Hadoop MapReduce implementations.

/pdf/distributed-optimization-and-statistical-learning-via-the-8s9gtddw0d.pdf

Distributed Optimization and Statistical Learning Via the Alternating Direction Method of Multipliers

We introduce a first-order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone. This approach has several favorable properties. Compared to interior-point methods, first-order methods scale to very large problems, at the cost of requiring more time to reach very high accuracy. Compared to other first-order methods for cone programs, our approach finds both primal and dual solutions when available or a certificate of infeasibility or unboundedness otherwise, is parameter free, and the per-iteration cost of the method is the same as applying a splitting method to the primal or dual alone. We discuss efficient implementation of the method in detail, including direct and indirect methods for computing projection onto the subspace, scaling the original problem data, and stopping criteria. We describe an open-source implementation, which handles the usual (symmetric) nonnegative, second-order, and semidefinite cones as well as the (non-self-dual) exponential and power cones and their duals. We report numerical results that show speedups over interior-point cone solvers for large problems, and scaling to very large general cone programs.

/pdf/conic-optimization-via-operator-splitting-and-homogeneous-1fxr7pcmul.pdf

Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding

This paper describes a general purpose method for solving convex optimization problems in a distributed computing environment. In particular, if the problem data includes a large linear operator or matrix $$A$$
 , the method allows for handling each sub-block of $$A$$
 on a separate machine. The approach works as follows. First, we define a canonical problem form called graph form, in which we have two sets of variables related by a linear operator $$A$$
 , such that the objective function is separable across these two sets of variables. Many types of problems are easily expressed in graph form, including cone programs and a wide variety of regularized loss minimization problems from statistics, like logistic regression, the support vector machine, and the lasso. Next, we describe graph projection splitting, a form of Douglas–Rachford splitting or the alternating direction method of multipliers, to solve graph form problems serially. Finally, we derive a distributed block splitting algorithm based on graph projection splitting. In a statistical or machine learning context, this allows for training models exactly with a huge number of both training examples and features, such that each processor handles only a subset of both. To the best of our knowledge, this is the only general purpose method with this property. We present several numerical experiments in both the serial and distributed settings.

/pdf/block-splitting-for-distributed-optimization-2s5hd9mo6d.pdf

Block splitting for distributed optimization

This paper describes a framework for generating easily verifiable code to solve convex optimization problems in embedded applications by transforming them into equivalent second-order cone programs. In embedded applications, it is critical to be able to verify code correctness, but it is also desirable to be able to rapidly prototype and deploy high-performance solvers for different problems. To balance these two requirements, we propose a code generation system that takes high-level descriptions of convex optimization problems and generates code that maps the parameters in the original problem to data in an equivalent second-order cone program, which is then solved by a single, external solver that can be verified once and for all. A novel aspect is that we restrict the parameters in the original problem to only appear in affine functions, which lets us map the parameters to problem data without performing any floating point operations. As a result, the generated code is lightweight, fast, and trivial to verify. The approach thus marries the benefits of high-level parser/solvers with custom, high-performance, high-reliability solvers for embedded applications.

/pdf/code-generation-for-embedded-second-order-cone-programming-z7r7zgxge8.pdf

Neal Parikh

Papers

Distributed Optimization and Statistical Learning Via the Alternating Direction Method of Multipliers

Proximal Algorithms

Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding

Block splitting for distributed optimization

Code generation for embedded second-order cone programming