[WIP] add SGHMC, SGLD trajectories

[sivapvarma]

Just placeholders for now, still getting a feeling for how things are organized in AHMC, at the same time wanted to get started.

Goal is to fix Issue #60 .

Comments are welcome.

[xukai92]

We probably only need to change this line to implmenet SGHMC.

[xukai92]

We probably only need to change this line to implmenet SGLD.

[xukai92]

I left some comments on how to proceed.

[xukai92]

The commit history is messed up. Please rebase and force-push.

[sivapvarma]

@xukai92 I have added the updates for SGHMC.

I am still stuck about how we compute the stochastic gradients. My main source of confusion is that there are many AD frameworks supported by Turing ( ForwardDiff, ReverseDiff, Zygote, Tracker). I know Zygote is the way forward. But I keep getting lost in how all of them are handled by Turing. Any pointers to AD documentation for Turing would help.

Furthermore, we need to compute gradients on minibatches, so it is still more details of how the interface should be designed.

[xukai92]

I am still stuck about how we compute the stochastic gradients. My main source of confusion is that there are many AD frameworks supported by Turing ( ForwardDiff, ReverseDiff, Zygote, Tracker). I know Zygote is the way forward. But I keep getting lost in how all of them are handled by Turing. Any pointers to AD documentation for Turing would help.

I think the design here is Turing/AD-agonistic. All we need to assume is that Hamiltonian.∂ℓπ∂θ returns a tuple of value and gradient.

Furthermore, we need to compute gradients on minibatches, so it is still more details of how the interface should be designed.

I guess there are two options:

1. Assume the gradient is scaled correctly by users when implementing ∂ℓπ∂θ or
2. Include the batch size (M) and whole dataset size (N) in SGLD/SGHMC and we scale it inside AHMC