Randomized Sketches of Convex Programs with Sharp Guarantees

29 Apr 2014  ·  , ·

Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by the solution of a lower-dimensional problem... Such dimensionality reduction is essential in computation-limited settings, since the complexity of general convex programming can be quite high (e.g., cubic for quadratic programs, and substantially higher for semidefinite programs). In addition to computational savings, random projection is also useful for reducing memory usage, and has useful properties for privacy-sensitive optimization. We prove that the approximation ratio of this procedure can be bounded in terms of the geometry of constraint set. For a broad class of random projections, including those based on various sub-Gaussian distributions as well as randomized Hadamard and Fourier transforms, the data matrix defining the cost function can be projected down to the statistical dimension of the tangent cone of the constraints at the original solution, which is often substantially smaller than the original dimension. We illustrate consequences of our theory for various cases, including unconstrained and $\ell_1$-constrained least squares, support vector machines, low-rank matrix estimation, and discuss implications on privacy-sensitive optimization and some connections with de-noising and compressed sensing. read more

PDF Abstract

Code Add Remove Mark official

No code implementations yet. Submit your code now

Datasets

Add Datasets introduced or used in this paper

Results from the Paper Edit

Submit results from this paper to get state-of-the-art GitHub badges and help the community compare results to other papers.