Adaptive Newton Sketch: Linear-time Optimization with Quadratic Convergence and Effective Hessian Dimensionality

by   Jonathan Lacotte, et al.

We propose a randomized algorithm with quadratic convergence rate for convex optimization problems with a self-concordant, composite, strongly convex objective function. Our method is based on performing an approximate Newton step using a random projection of the Hessian. Our first contribution is to show that, at each iteration, the embedding dimension (or sketch size) can be as small as the effective dimension of the Hessian matrix. Leveraging this novel fundamental result, we design an algorithm with a sketch size proportional to the effective dimension and which exhibits a quadratic rate of convergence. This result dramatically improves on the classical linear-quadratic convergence rates of state-of-the-art sub-sampled Newton methods. However, in most practical cases, the effective dimension is not known beforehand, and this raises the question of how to pick a sketch size as small as the effective dimension while preserving a quadratic convergence rate. Our second and main contribution is thus to propose an adaptive sketch size algorithm with quadratic convergence rate and which does not require prior knowledge or estimation of the effective dimension: at each iteration, it starts with a small sketch size, and increases it until quadratic progress is achieved. Importantly, we show that the embedding dimension remains proportional to the effective dimension throughout the entire path and that our method achieves state-of-the-art computational complexity for solving convex optimization programs with a strongly convex component.



There are no comments yet.


page 1

page 2

page 3

page 4


Fast Convex Quadratic Optimization Solvers with Adaptive Sketching-based Preconditioners

We consider least-squares problems with quadratic regularization and pro...

Newton Sketch: A Linear-time Optimization Algorithm with Linear-Quadratic Convergence

We propose a randomized second-order method for optimization known as th...

On the Robustness of CountSketch to Adaptive Inputs

CountSketch is a popular dimensionality reduction technique that maps ve...

Curvature-aided Incremental Aggregated Gradient Method

We propose a new algorithm for finite sum optimization which we call the...

Convergence Analysis of the Randomized Newton Method with Determinantal Sampling

We analyze the convergence rate of the Randomized Newton Method (RNM) in...

Effective Dimension Adaptive Sketching Methods for Faster Regularized Least-Squares Optimization

We propose a new randomized algorithm for solving L2-regularized least-s...

Faster Least Squares Optimization

We investigate randomized methods for solving overdetermined linear leas...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.