Albert S. Berahas In this paper, we analyze several methods for approximating gradients of noisy functions using only function values. These methods include finite differences, linear interpolation, Gaussian smoothing and smoothing on a unit sphere. The methods differ in the number of functions sampled, the choice of the sample points, and the way in which the gradient approximations … Read more
A displacement aggregation strategy is proposed for the curvature pairs stored in a limited-memory BFGS (a.k.a. L-BFGS) method such that the resulting (inverse) Hessian approximations are equal to those that would be derived from a full-memory BFGS method. This means that, if a sufficiently large number of pairs are stored, then an optimization algorithm employing … Read more
We present two sampled quasi-Newton methods: sampled LBFGS and sampled LSR1. Contrary to the classical variants of these methods that sequentially build (inverse) Hessian approximations as the optimization progresses, our proposed methods sample points randomly around the current iterate to produce these approximations. As a result, the approximations constructed make use of more reliable (recent … Read more
This paper presents a finite difference quasi-Newton method for the minimization of noisy functions. The method takes advantage of the scalability and power of BFGS updating, and employs an adaptive procedure for choosing the differencing interval h based on the noise estimation techniques of Hamming (2012) and Moré and Wild (2011). This noise estimation procedure … Read more
Methods for distributed optimization have received significant attention in recent years owing to their wide applicability in various domains including machine learning, robotics and sensor networks. A distributed optimization method typically consists of two key components: communication and computation. More specifically, at every iteration (or every several iterations) of a distributed algorithm, each node in … Read more
This paper describes an implementation of the L-BFGS method designed to deal with two adversarial situations. The first occurs in distributed computing environments where some of the computational nodes devoted to the evaluation of the function and gradient are unable to return results on time. A similar challenge occurs in a multi-batch approach in which … Read more
Sketching, a dimensionality reduction technique, has received much attention in the statistics community. In this paper, we study sketching in the context of Newton’s method for solving finite-sum optimization problems in which the number of variables and data points are both large. We study two forms of sketching that perform dimensionality reduction in data space: … Read more