[Minor] Enable gradient for base inputs

[frankaging]

Description

Allow gradient on the base input. Minor change on initial temp setting for intervention.

Testing Done

TBD

Checklist:

• [x] My PR title strictly follows the format: [Your Priority] Your Title
• [x] I have attached the testing log above
• [x] I provide enough comments to my code
• [x] I have changed documentations
• [x] I have added tests for my changes