Adapts the inner kernel's step_size based on log_accept_prob.

The dual averaging policy uses a noisy step size for exploration, while averaging over tuning steps to provide a smoothed estimate of an optimal value. It is based on section 3.2 of Hoffman and Gelman (2013), which modifies the [stochastic convex optimization scheme of Nesterov (2009). The modified algorithm applies extra weight to recent iterations while keeping the convergence guarantees of Robbins-Monro, and takes care not to make the step size too small too quickly when maintaining a constant trajectory length, to avoid expensive early iterations. A good target acceptance probability depends on the inner kernel. If this kernel is HamiltonianMonteCarlo, then 0.6-0.9 is a good range to aim for. For RandomWalkMetropolis this should be closer to 0.25. See the individual kernels' docstrings for guidance.

mcmc_dual_averaging_step_size_adaptation(
  inner_kernel,
  num_adaptation_steps,
  target_accept_prob = 0.75,
  exploration_shrinkage = 0.05,
  step_count_smoothing = 10,
  decay_rate = 0.75,
  step_size_setter_fn = NULL,
  step_size_getter_fn = NULL,
  log_accept_prob_getter_fn = NULL,
  validate_args = FALSE,
  name = NULL
)

Arguments

inner_kernel	TransitionKernel-like object.
num_adaptation_steps	Scalar integer Tensor number of initial steps to during which to adjust the step size. This may be greater, less than, or equal to the number of burnin steps.
target_accept_prob	A floating point Tensor representing desired acceptance probability. Must be a positive number less than 1. This can either be a scalar, or have shape [num_chains]. Default value: 0.75 (the center of asymptotically optimal rate for HMC).
exploration_shrinkage	Floating point scalar Tensor. How strongly the exploration rate is biased towards the shrinkage target.
step_count_smoothing	Int32 scalar Tensor. Number of "pseudo-steps" added to the number of steps taken to prevents noisy exploration during the early samples.
decay_rate	Floating point scalar Tensor. How much to favor recent iterations over earlier ones. A value of 1 gives equal weight to all history.
step_size_setter_fn	A function with the signature (kernel_results, new_step_size) -> new_kernel_results where kernel_results are the results of the inner_kernel, new_step_size is a Tensor or a nested collection of Tensors with the same structure as returned by the step_size_getter_fn, and new_kernel_results are a copy of kernel_results with the step size(s) set.
step_size_getter_fn	A callable with the signature (kernel_results) -> step_size where kernel_results are the results of the inner_kernel, and step_size is a floating point Tensor or a nested collection of such Tensors.
log_accept_prob_getter_fn	A callable with the signature (kernel_results) -> log_accept_prob where kernel_results are the results of the inner_kernel, and log_accept_prob is a floating point Tensor. log_accept_prob can either be a scalar, or have shape [num_chains]. If it's the latter, step_size should also have the same leading dimension.
validate_args	logical. When TRUE kernel parameters are checked for validity. When FALSE invalid inputs may silently render incorrect outputs.
name	name prefixed to Ops created by this function. Default value: NULL (i.e., 'dual_averaging_step_size_adaptation').

Value

a Monte Carlo sampling kernel

Details

In general, adaptation prevents the chain from reaching a stationary distribution, so obtaining consistent samples requires num_adaptation_steps be set to a value somewhat smaller than the number of burnin steps. However, it may sometimes be helpful to set num_adaptation_steps to a larger value during development in order to inspect the behavior of the chain during adaptation. The step size is assumed to broadcast with the chain state, potentially having leading dimensions corresponding to multiple chains. When there are fewer of those leading dimensions than there are chain dimensions, the corresponding dimensions in the log_accept_prob are averaged (in the direct space, rather than the log space) before being used to adjust the step size. This means that this kernel can do both cross-chain adaptation, or per-chain step size adaptation, depending on the shape of the step size. For example, if your problem has a state with shape [S], your chain state has shape [C0, C1, S] (meaning that there are C0 * C1 total chains) and log_accept_prob has shape [C0, C1] (one acceptance probability per chain), then depending on the shape of the step size, the following will happen:

• Step size has shape [], [S] or [1], the log_accept_prob will be averaged across its C0 and C1 dimensions. This means that you will learn a shared step size based on the mean acceptance probability across all chains. This can be useful if you don't have a lot of steps to adapt and want to average away the noise.

• Step size has shape [C1, 1] or [C1, S], the log_accept_prob will be averaged across its C0 dimension. This means that you will learn a shared step size based on the mean acceptance probability across chains that share the coordinate across the C1 dimension. This can be useful when the C1 dimension indexes different distributions, while C0 indexes replicas of a single distribution, all sampled in parallel.

• Step size has shape [C0, C1, 1] or [C0, C1, S], then no averaging will happen. This means that each chain will learn its own step size. This can be useful when all chains are sampling from different distributions. Even when all chains are for the same distribution, this can help during the initial warmup period.

• Step size has shape [C0, 1, 1] or [C0, 1, S], the log_accept_prob will be averaged across its C1 dimension. This means that you will learn a shared step size based on the mean acceptance probability across chains that share the coordinate across the C0 dimension. This can be useful when the C0 dimension indexes different distributions, while C1 indexes replicas of a single distribution, all sampled in parallel.

References

• Matthew D. Hoffman, Andrew Gelman. The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. In Journal of Machine Learning Research, 15(1):1593-1623, 2014.

• Yurii Nesterov. Primal-dual subgradient methods for convex problems. Mathematical programming 120.1 (2009): 221-259

• https://statmodeling.stat.columbia.edu/2017/12/15/burn-vs-warm-iterative-simulation-algorithms/

See also

For an example how to use see mcmc_no_u_turn_sampler().

Other mcmc_kernels: mcmc_hamiltonian_monte_carlo(), mcmc_metropolis_adjusted_langevin_algorithm(), mcmc_metropolis_hastings(), mcmc_no_u_turn_sampler(), mcmc_random_walk_metropolis(), mcmc_replica_exchange_mc(), mcmc_simple_step_size_adaptation(), mcmc_slice_sampler(), mcmc_transformed_transition_kernel(), mcmc_uncalibrated_hamiltonian_monte_carlo(), mcmc_uncalibrated_langevin(), mcmc_uncalibrated_random_walk()