9 research outputs found
Smooth Primal-Dual Coordinate Descent Algorithms for Nonsmooth Convex Optimization
We propose a new randomized coordinate descent method for a convex
optimization template with broad applications. Our analysis relies on a novel
combination of four ideas applied to the primal-dual gap function: smoothing,
acceleration, homotopy, and coordinate descent with non-uniform sampling. As a
result, our method features the first convergence rate guarantees among the
coordinate descent methods, that are the best-known under a variety of common
structure assumptions on the template. We provide numerical evidence to support
the theoretical results with a comparison to state-of-the-art algorithms.Comment: NIPS 201
A generic coordinate descent solver for nonsmooth convex optimization
International audienceWe present a generic coordinate descent solver for the minimization of a nonsmooth convex objective with structure. The method can deal in particular with problems with linear constraints. The implementation makes use of efficient residual updates and automatically determines which dual variables should be duplicated. A list of basic functional atoms is pre-compiled for efficiency and a modelling language in Python allows the user to combine them at run time. So, the algorithm can be used to solve a large variety of problems including Lasso, sparse multinomial logistic regression, linear and quadratic programs
A Conditional Gradient Framework for Composite Convex Minimization with Applications to Semidefinite Programming
We propose a conditional gradient framework for a composite convex minimization template with broad applications. Our approach combines the notions of smoothing and homotopy under the CGM framework, and provably achieves the optimal O(1/sqrt(k)) convergence rate. We demonstrate that the same rate holds if the linear subproblems are solved approximately with additive or multiplicative error. Specific applications of the framework include the non-smooth minimization semidefinite programming, minimization with linear inclusion constraints over a compact domain. We provide numerical evidence to demonstrate the benefits of the new framework
On the convergence of stochastic primal-dual hybrid gradient
In this paper, we analyze the recently proposed stochastic primal-dual hybrid
gradient (SPDHG) algorithm and provide new theoretical results. In particular,
we prove almost sure convergence of the iterates to a solution and linear
convergence with standard step sizes, independent of strong convexity
constants. Our assumption for linear convergence is metric subregularity, which
is satisfied for smooth and strongly convex problems in addition to many
nonsmooth and/or nonstrongly convex problems, such as linear programs, Lasso,
and support vector machines. In the general convex case, we prove optimal
sublinear rates for the ergodic sequence, without bounded domain assumptions.
We also provide numerical evidence showing that SPDHG with standard step sizes
shows favorable and robust practical performance against its specialized
strongly convex variant SPDHG- and other state-of-the-art algorithms
including variance reduction methods and stochastic dual coordinate ascent