Primal-Dual Rates and Certificates

We propose an algorithm-independent framework to equip existing optimization methods with primal-dual certificates. Such certificates and corresponding rate of convergence guarantees are important for practitioners to diagnose progress, in particular in machine learning applications. We obtain new primal-dual convergence rates, e.g., for the Lasso as well as many L1, Elastic Net, group Lasso and TV-regularized problems. The theory applies to any norm-regularized generalized linear model. Our approach provides efficiently computable duality gaps which are globally defined, without modifying the original problems in the region of interest.Comment: appearing at ICML 2016 - Proceedings of the 33rd International Conference on Machine Learning, New York, NY, USA, 2016. JMLR: W&CP volume 4

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators on Domains Given by Linear Minimization Oracles

The majority of First Order methods for large-scale convex-concave saddle point problems and variational inequalities with monotone operators are proximal algorithms which at every iteration need to minimize over problem's domain X the sum of a linear form and a strongly convex function. To make such an algorithm practical, X should be proximal-friendly -- admit a strongly convex function with easy to minimize linear perturbations. As a byproduct, X admits a computationally cheap Linear Minimization Oracle (LMO) capable to minimize over X linear forms. There are, however, important situations where a cheap LMO indeed is available, but X is not proximal-friendly, which motivates search for algorithms based solely on LMO's. For smooth convex minimization, there exists a classical LMO-based algorithm -- Conditional Gradient. In contrast, known to us LMO-based techniques for other problems with convex structure (nonsmooth convex minimization, convex-concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper is to develop an alternative (and seemingly much simpler) LMO-based decomposition techniques for bilinear saddle point problems and for variational inequalities with affine monotone operators

L1-Regularized Distributed Optimization: A Communication-Efficient Primal-Dual Framework

Despite the importance of sparsity in many large-scale applications, there are few methods for distributed optimization of sparsity-inducing objectives. In this paper, we present a communication-efficient framework for L1-regularized optimization in the distributed environment. By viewing classical objectives in a more general primal-dual setting, we develop a new class of methods that can be efficiently distributed and applied to common sparsity-inducing models, such as Lasso, sparse logistic regression, and elastic net-regularized problems. We provide theoretical convergence guarantees for our framework, and demonstrate its efficiency and flexibility with a thorough experimental comparison on Amazon EC2. Our proposed framework yields speedups of up to 50x as compared to current state-of-the-art methods for distributed L1-regularized optimization

CoCoA: A General Framework for Communication-Efficient Distributed Optimization

The scale of modern datasets necessitates the development of efficient distributed optimization methods for machine learning. We present a general-purpose framework for distributed computing environments, CoCoA, that has an efficient communication scheme and is applicable to a wide variety of problems in machine learning and signal processing. We extend the framework to cover general non-strongly-convex regularizers, including L1-regularized problems like lasso, sparse logistic regression, and elastic net regularization, and show how earlier work can be derived as a special case. We provide convergence guarantees for the class of convex regularized loss minimization objectives, leveraging a novel approach in handling non-strongly-convex regularizers and non-smooth loss functions. The resulting framework has markedly improved performance over state-of-the-art methods, as we illustrate with an extensive set of experiments on real distributed datasets

OSQP: An Operator Splitting Solver for Quadratic Programs

We present a general-purpose solver for convex quadratic programs based on the alternating direction method of multipliers, employing a novel operator splitting technique that requires the solution of a quasi-definite linear system with the same coefficient matrix at almost every iteration. Our algorithm is very robust, placing no requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. It can be configured to be division-free once an initial matrix factorization is carried out, making it suitable for real-time applications in embedded systems. In addition, our technique is the first operator splitting method for quadratic programs able to reliably detect primal and dual infeasible problems from the algorithm iterates. The method also supports factorization caching and warm starting, making it particularly efficient when solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint, is library-free, and has been extensively tested on many problem instances from a wide variety of application areas. It is typically ten times faster than competing interior-point methods, and sometimes much more when factorization caching or warm start is used. OSQP has already shown a large impact with tens of thousands of users both in academia and in large corporations

Faster Coordinate Descent via Adaptive Importance Sampling

Coordinate descent methods employ random partial updates of decision variables in order to solve huge-scale convex optimization problems. In this work, we introduce new adaptive rules for the random selection of their updates. By adaptive, we mean that our selection rules are based on the dual residual or the primal-dual gap estimates and can change at each iteration. We theoretically characterize the performance of our selection rules and demonstrate improvements over the state-of-the-art, and extend our theory and algorithms to general convex objectives. Numerical evidence with hinge-loss support vector machines and Lasso confirm that the practice follows the theory.Comment: appearing at AISTATS 201

Comparing Experiments to the Fault-Tolerance Threshold

Achieving error rates that meet or exceed the fault-tolerance threshold is a central goal for quantum computing experiments, and measuring these error rates using randomized benchmarking is now routine. However, direct comparison between measured error rates and thresholds is complicated by the fact that benchmarking estimates average error rates while thresholds reflect worst-case behavior when a gate is used as part of a large computation. These two measures of error can differ by orders of magnitude in the regime of interest. Here we facilitate comparison between the experimentally accessible average error rates and the worst-case quantities that arise in current threshold theorems by deriving relations between the two for a variety of physical noise sources. Our results indicate that it is coherent errors that lead to an enormous mismatch between average and worst case, and we quantify how well these errors must be controlled to ensure fair comparison between average error probabilities and fault-tolerance thresholds.Comment: 5 pages, 2 figures, 13 page appendi