Surpassing Gradient Descent Provably: A Cyclic Incremental Method with Linear Convergence Rate

TL;DR

This paper introduces DIAG, a cyclic incremental gradient method that achieves linear convergence and outperforms gradient descent in large-scale convex optimization problems by efficiently approximating the full gradient.

Contribution

The paper proposes a novel cyclic incremental aggregated gradient method (DIAG) with linear convergence, improving upon traditional incremental methods and gradient descent.

Findings

DIAG converges linearly to the optimal solution.

The worst-case performance of DIAG surpasses that of gradient descent.

DIAG reduces computational cost while maintaining fast convergence.

Abstract

Recently, there has been growing interest in developing optimization methods for solving large-scale machine learning problems. Most of these problems boil down to the problem of minimizing an average of a finite set of smooth and strongly convex functions where the number of functions $n$ is large. Gradient descent method (GD) is successful in minimizing convex problems at a fast linear rate; however, it is not applicable to the considered large-scale optimization setting because of the high computational complexity. Incremental methods resolve this drawback of gradient methods by replacing the required gradient for the descent direction with an incremental gradient approximation. They operate by evaluating one gradient per iteration and executing the average of the $n$ available gradients as a gradient approximate. Although, incremental methods reduce the computational cost of GD, their convergence rates do not justify their advantage relative to GD in terms of the total number of gradient evaluations until convergence. In this paper, we introduce a Double Incremental Aggregated Gradient method (DIAG) that computes the gradient of only one function at each iteration, which is chosen based on a cyclic scheme, and uses the aggregated average gradient of all the functions to approximate the full gradient. The iterates of the proposed DIAG method uses averages of both iterates and gradients in oppose to classic incremental methods that utilize gradient averages but do not utilize iterate averages. We prove that not only the proposed DIAG method converges linearly to the optimal solution, but also its linear convergence factor justifies the advantage of incremental methods on GD. In particular, we prove that the worst case performance of DIAG is better than the worst case performance of GD.