How to set step-size in Hamiltonian Monte Carlo?

I'm using Hamiltonian Monte Carlo (HMC) implementation in Edward, a probabilistic programming library built on top of TensorFlow. One of the hyper-parameters of HMC is the step size:

inference = ed.HMC({w: qw, b: qb}, data={X: X_train, y: y_train})
inference.initialize(step_size=0.5/float(N))


I know there's an adaptive version of the algorithm: the No-U-Turn Sampler (NUTS). However, I'm wondering how to initialize the step-size in the case of HMC?

There's a section Radford Neal's Handbook on HMC that discusses how to set the discretisation length $\epsilon$ and the number of leapfrog steps $L$ appropriately: http://www.mcmchandbook.net/HandbookChapter5.pdf.

Here I can summarise the key points:

The overall distance moved is $\epsilon L$ so both have to be considered.

Setting $\epsilon$:

• The reason proposals are rejected in HMC is purely due to discretisation error (otherwise the dynamics perfectly preserve probability density/energy).

• If $\epsilon$ is too large, then there will be large discretisation error and low acceptance, if $\epsilon$ is too small then more expensive leapfrog steps will be required to move large distances.

• Ideally we want the largest possible value of $\epsilon$ that gives reasonable acceptance probability. Unfortunately this may vary for different values of the target variable.

• A simple heuristic to set this may be to do a preliminary run with fixed $L$, gradually increasing $\epsilon$ until the acceptance probability is at an appropriate level.

Setting $L$:

Neal says:

"Setting the trajectory length by trial and error therefore seems necessary. For a problem thought to be fairly difficult, a trajectory with L = 100 might be a suitable starting point. If preliminary runs (with a suitable ε; see above) show that HMC reaches a nearly independent point after only one iteration, a smaller value of L might be tried next. (Unless these “preliminary” runs are actually sufficient, in which case there is of course no need to do more runs.) If instead there is high autocorrelation in the run with L = 100, runs with L = 1000 might be tried next"

It may also be advisable to randomly sample $\epsilon$ and $L$ form suitable ranges to avoid the possibility of having paths that are close to periodic as this would slow mixing.

• there's an interesting paper that combines a simple neural network with HMC to solve some of the heuristics associated with the algorithm: arxiv.org/abs/1711.09268 Apr 5, 2018 at 17:47
• That's true but my experience with it has been that the training time of the sampler is itself often a bottle neck and trying to get the right hyper-paramters/temperature annealing schedule for the learned sampler is tricky. I still think NUTS which is implemented in STAN maybe the right way to go if you dont want to tune these. Aug 14, 2018 at 8:50
• @R.Habib; Neal said that stepwise $\epsilon$ should be scaled as $d^{1/4}$ where $d$ is dimension but he seems didn't mention anything on how to scale trajectory length Sep 9, 2019 at 3:54
• Thank you so much @R.Habib!
– amc
Mar 21, 2021 at 20:00
• @ElleryL the trajectory length can be tuned with NUTS. arxiv.org/abs/1111.4246
– amc
Mar 21, 2021 at 20:07