Nuit Blanche: Sub-Sampled Newton Methods

Monday, January 25, 2016

Sub-Sampled Newton Methods

Newton'smethod aided by RandNLA:

Sub-Sampled Newton Methods I: Globally Convergent Algorithms by Farbod Roosta-Khorasani, Michael W. Mahoney

Large scale optimization problems are ubiquitous in machine learning and data analysis and there is a plethora of algorithms for solving such problems. Many of these algorithms employ sub-sampling, as a way to either speed up the computations and/or to implicitly implement a form of statistical regularization. In this paper, we consider second-order iterative optimization algorithms and we provide bounds on the convergence of the variants of Newton's method that incorporate uniform sub-sampling as a means to estimate the gradient and/or Hessian. Our bounds are non-asymptotic and quantitative. Our algorithms are global and are guaranteed to converge from any initial iterate.
Using random matrix concentration inequalities, one can sub-sample the Hessian to preserve the curvature information. Our first algorithm incorporates Hessian sub-sampling while using the full gradient. We also give additional convergence results for when the sub-sampled Hessian is regularized by modifying its spectrum or ridge-type regularization. Next, in addition to Hessian sub-sampling, we also consider sub-sampling the gradient as a way to further reduce the computational complexity per iteration. We use approximate matrix multiplication results from randomized numerical linear algebra to obtain the proper sampling strategy. In all these algorithms, computing the update boils down to solving a large scale linear system, which can be computationally expensive. As a remedy, for all of our algorithms, we also give global convergence results for the case of inexact updates where such linear system is solved only approximately.
This paper has a more advanced companion paper, [42], in which we demonstrate that, by doing a finer-grained analysis, we can get problem-independent bounds for local convergence of these algorithms and explore trade-offs to improve upon the basic results of the present paper.

Sub-Sampled Newton Methods II: Local Convergence Rates by Farbod Roosta-Khorasani, Michael W. Mahoney

Many data-fitting applications require the solution of an optimization problem involving a sum of large number of functions of high dimensional parameter. Here, we consider the problem of minimizing a sum of n functions over a convex constraint set X⊆Rp where both n and p are large. In such problems, sub-sampling as a way to reduce n can offer great amount of computational efficiency, while maintaining their original convergence properties.
Within the context of second order methods, we first give quantitative local convergence results for variants of Newton's method where the Hessian is uniformly sub-sampled. Using random matrix concentration inequalities, one can sub-sample in a way that the curvature information is preserved. Using such sub-sampling strategy, we establish locally Q-linear and Q-superlinear convergence rates. We also give additional convergence results for when the sub-sampled Hessian is regularized by modifying its spectrum or Levenberg-type regularization.
Finally, in addition to Hessian sub-sampling, we consider sub-sampling the gradient as way to further reduce the computational complexity per iteration. We use approximate matrix multiplication results from randomized numerical linear algebra (RandNLA) to obtain the proper sampling strategy and we establish locally R-linear convergence rates. In such a setting, we also show that a very aggressive sample size increase results in a R-superlinearly convergent algorithm.
While the sample size depends on the condition number of the problem, our convergence rates are problem-independent, i.e., they do not depend on the quantities related to the problem. Hence, our analysis here can be used to complement the results of our basic framework from the companion paper, [38], by exploring algorithmic trade-offs that are important in practice.