Convergence rates of sub-sampled Newton methods

TL;DR

This paper introduces a new randomized sub-sampled Newton method with low-rank approximation that achieves fast convergence rates and reduced computational costs for large-scale convex optimization problems.

Contribution

The paper proposes a novel sub-sampled Newton algorithm combining low-rank approximation, with proven quadratic and linear convergence phases, improving efficiency over existing methods.

Findings

Achieves convergence rates comparable to classical Newton's method.

Demonstrates robustness to initial conditions and step size.

Shows improved performance on real datasets.

Abstract

We consider the problem of minimizing a sum of $n$ functions over a convex parameter set $\mathcal{C} \subset \mathbb{R}^p$ where $n\gg p\gg 1$. In this regime, algorithms which utilize sub-sampling techniques are known to be effective. In this paper, we use sub-sampling techniques together with low-rank approximation to design a new randomized batch algorithm which possesses comparable convergence rate to Newton's method, yet has much smaller per-iteration cost. The proposed algorithm is robust in terms of starting point and step size, and enjoys a composite convergence rate, namely, quadratic convergence at start and linear convergence when the iterate is close to the minimizer. We develop its theoretical analysis which also allows us to select near-optimal algorithm parameters. Our theoretical results can be used to obtain convergence rates of previously proposed sub-sampling based algorithms as well. We demonstrate how our results apply to well-known machine learning problems. Lastly, we evaluate the performance of our algorithm on several datasets under various scenarios.