A minibatch stochastic Quasi-Newton method adapted for nonconvex deep learning problems

In this study, we develop a limited memory nonconvex Quasi-Newton (QN) method, tailored to deep learning (DL) applications. Since the stochastic nature of (sampled) function information in minibatch processing can affect the performance of QN methods, three strategies are utilized to overcome this issue. These involve a novel progressive trust-region radius update (suitable for stochastic … Read more

Quasi-Newton Methods for Deep Learning: Forget the Past, Just Sample

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

An Accelerated Communication-Efficient Primal-Dual Optimization Framework for Structured Machine Learning

Distributed optimization algorithms are essential for training machine learning models on very large-scale datasets. However, they often suffer from communication bottlenecks. Confronting this issue, a communication-efficient primal-dual coordinate ascent framework (CoCoA) and its improved variant CoCoA+ have been proposed, achieving a convergence rate of $\mathcal{O}(1/t)$ for solving empirical risk minimization problems with Lipschitz continuous losses. … Read more

Underestimate Sequences via Quadratic Averaging

In this work we introduce the concept of an Underestimate Sequence (UES), which is a natural extension of Nesterov’s estimate sequence. Our definition of a UES utilizes three sequences, one of which is a lower bound (or under-estimator) of the objective function. The question of how to construct an appropriate sequence of lower bounds is … Read more

A Robust Multi-Batch L-BFGS Method for Machine Learning

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

Parallel Coordinate Descent Methods for Big Data Optimization

In this work we show that randomized (block) coordinate descent methods can be accelerated by parallelization when applied to the problem of minimizing the sum of a partially separable smooth convex function and a simple separable convex function. The theoretical speedup, as compared to the serial method, and referring to the number of iterations needed … Read more

Efficient Serial and Parallel Coordinate Descent Methods for Huge-Scale Truss Topology Design

In this work we propose solving huge-scale instances of the truss topology design problem with coordinate descent methods. We develop four efficient codes: serial and parallel implementations of randomized and greedy rules for the selection of the variable (potential bar) to be updated in the next iteration. Both serial methods enjoy an O(n/k) iteration complexity … Read more

Iteration Complexity of Randomized Block-Coordinate Descent Methods for Minimizing a Composite Function

In this paper we develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an $\epsilon$-accurate solution with probability at least $1-\rho$ in at most $O(\tfrac{n}{\epsilon} \log \tfrac{1}{\rho})$ iterations, where $n$ is the number of blocks. For strongly convex functions … Read more