Hi, I am learning the Dual Averaging based on the NUTS paper in 2014. Could anyone help to explain why the updates in the equation (6) were defined as attached? What are the exact expressions of the dual or primal problems here? The paper by Nesterov (2009) had very clear expression of the primal or dual problems. The 2014 paper did not mention either. Particularly, why is the **average** of the x_{t+1} is defined as this? why was the update of x_{t+1} defined **not** as the relationship with x_{t} but with the summation of Hi. What is the expression between x_{t+1} and x_{t} then? Why is this update called dual averaging specifically?

Thank you for your help in advance.

Yan