# Can someone explain how dual averaging helps the No U-Turn Sampler (NUTS) choose step-size adaptively?

I have read both the original NUTS paper and also the dual averaging paper by Nesterov but due to my lack of background knowledge in optimisation, I don't really understand how dual averaging works.

In particular, I don't understand why the update scheme below will cause the step size to converge to a value such that we get the acceptance probability we want.

Any help/intuition will be appreciated. Thank you.

We have an acceptance probability for this, which would be one iff we could exactly run the dynamics. Essentially, this acceptance probability is related to how good the numerical approximation to the Hamiltonian dynamics is. Then, we balance the tradeoff between $$\epsilon$$ and the time it takes to generate any given sample by varying $$\delta$$: either spend more time generating better approximations of the Hamiltonian-dynamics, or alternately spend less time generating crappier ones knowing that we will throw more away.