Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-mzfmx Total loading time: 0.338 Render date: 2022-08-14T07:43:42.031Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

4 - Gradient Methods Using Momentum

Published online by Cambridge University Press:  31 March 2022

Stephen J. Wright
Affiliation:
University of Wisconsin, Madison
Benjamin Recht
Affiliation:
University of California, Berkeley
Get access

Summary

This chapter describes methods based on gradient information that achieve faster rates than basic algorithms such as those described in Chapter 3. These accelerated gradient methods, most notably the heavy-ball method and Nesterov’s optimal method, use the concept of momentum which means that each step combines information from recent gradient values but also earlier steps. These methods are described and analyzed using an analysis based on Lyapunov functions. The cases of convex and strongly convex functions are analyzed separately. We motivate these methods using continuous-time limits, which link gradient methods to dynamical systems described by differential equations. We mention also the conjugate gradient method, which was developed separately from the other method but which also makes use of momentum. Finally, we discuss the concept of lower bounds on algorithmic complexity, introducing a function on which no method based on gradients can attain convergence faster than a certain given rate.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×