class_docstr: 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)][1], which
  modifies the [stochastic convex optimization scheme of Nesterov (2009)][2].
  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.
  
  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][3] 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.
  
  By default, the averaging function used above is the arithmetic mean, which is
  not robust to stuck chains (e.g. average of one chain with `p_accept = 0` and
  three chains with `p_accept = 1` will result in an average `p_accept = 0.75`,
  which will cause this kernel keep the step size roughly the same rather than
  reducing it to unstick the stuck chain). A more robust choice would be to set
  `reduce_fn` argument to `tfp.math.reduce_log_harmonic_mean_exp` [4]. Note,
  however, that the harmonic mean of a set of numbers is usually smaller than
  the arithmetic mean, so its use will typically produce smaller than optimal
  step sizes even for well behaved target distributions.
  
  #### Examples
  
  ```python
  import tensorflow as tf
  import tensorflow_probability as tfp
  tfd = tfp.distributions
  
  target_log_prob_fn = tfd.Normal(loc=0., scale=1.).log_prob
  num_burnin_steps = 500
  num_results = 500
  num_chains = 64
  step_size = 0.1
  # Or, if you want per-chain step size:
  # step_size = tf.fill([num_chains], step_size)
  
  kernel = tfp.mcmc.HamiltonianMonteCarlo(
      target_log_prob_fn=target_log_prob_fn,
      num_leapfrog_steps=2,
      step_size=step_size)
  kernel = tfp.mcmc.DualAveragingStepSizeAdaptation(
      inner_kernel=kernel, num_adaptation_steps=int(num_burnin_steps * 0.8))
  
  # The chain will be stepped for num_results + num_burnin_steps, adapting for
  # the first num_adaptation_steps.
  samples, [step_size, log_accept_ratio] = tfp.mcmc.sample_chain(
      num_results=num_results,
      num_burnin_steps=num_burnin_steps,
      current_state=tf.zeros(num_chains),
      kernel=kernel,
      trace_fn=lambda _, pkr: [pkr.inner_results.accepted_results.step_size,
                               pkr.inner_results.log_accept_ratio])
  
  # ~0.75
  p_accept = tf.math.exp(tfp.math.reduce_logmeanexp(tf.minimum(
      log_accept_ratio, 0.)))
  ```
  #### References
  
  [1]: 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.
      http://jmlr.org/papers/volume15/hoffman14a/hoffman14a.pdf
  
  [2] Yurii Nesterov. Primal-dual subgradient methods for convex problems.
      Mathematical programming 120.1 (2009): 221-259
      https://link.springer.com/article/10.1007/s10107-007-0149-x
  
  [3]:
  http://andrewgelman.com/2017/12/15/burn-vs-warm-iterative-simulation-algorithms/#comment-627745
  
  [4]: Hoffman, M., Radul, A., & Sountsov, P. An Adaptive MCMC Scheme
       for Setting Trajectory Lengths in Hamiltonian Monte Carlo, 2020. In
       preparation.
docstr: Compute new step sizes for each step size part.
  
  If step size part has smaller rank than the corresponding state part, then
  the difference is averaged away in the log accept prob.
  
  Example:
  
    state_part has shape      [2, 3, 4, 5]
    step_size_part has shape     [1, 4, 1]
    log_accept_prob has shape [2, 3, 4]
  
  Since step size has 1 rank fewer than the state, we reduce away the leading
  dimension of `log_accept_prob` to get a Tensor with shape [3, 4]. Next,
  since `log_accept_prob` must broadcast into step_size_part on the left, we
  reduce the dimensions where their shapes differ, to get a Tensor with shape
  [1, 4], which now is compatible with the leading dimensions of
  step_size_part.
  
  There is a subtlety here in that `step_size_parts` might be a length-1 list,
  which means that we'll be "structure-broadcasting" it for all the state
  parts (see logic in, e.g., hmc.py). In this case we must assume that that
  the lone step size provided broadcasts with the event dims of each state
  part. This means that either step size has no dimensions corresponding to
  chain dimensions, or all states are of the same shape. For the former, we
  want to reduce over all chain dimensions. For the later, we want to use
  the same logic as in the non-structure-broadcasted case.
  
  It turns out we can compute the reduction dimensions for both cases
  uniformly by taking the rank of any state part. This obviously works in
  the second case (where all state ranks are the same). In the first case,
  all state parts have the rank L + D_i + B, where L is the rank of
  log_accept_prob, D_i is the non-shared dimensions amongst all states, and
  B are the shared dimensions of all the states, which are equal to the step
  size. When we subtract B, we will always get a number >= L, which means
  we'll get the full reduction we want.
  
  Args:
    step_size: Previous step's step_size.
    state: Previous step's state value.
    error_sum: Previous step's error accumulator.
    log_averaging_step: Previous step's log_averaging_step.
    log_shrinkage_target: Floating point scalar `Tensor`. Logarithm of value
      the exploration step size is biased towards.
    log_accept_prob_rank: Rank of log_accept_prob.
    log_accept_prob: Floating point scalar `Tensor`. Target accept
      probability.
    target_accept_prob: A floating point `Tensor` representing desired
      acceptance probability. Must be a positive number less than 1.
    previous_kernel_results: Results struct from previous step.
  
  Returns:
    new_step_size: Updated `step_size`.
    new_log_averaging_step: Updated `log_averaging_step`.
    new_error_sum: Updated `error_sum`.
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code: def _one_step_part(
      self,
      step_size,
      state,
      error_sum,
      log_averaging_step,
      log_shrinkage_target,
      log_accept_prob_rank=None,
      log_accept_prob=None,
      target_accept_prob=None,
      previous_kernel_results=None,
  ):
      """Compute new step sizes for each step size part.
  
      If step size part has smaller rank than the corresponding state part, then
      the difference is averaged away in the log accept prob.
  
      Example:
  
        state_part has shape      [2, 3, 4, 5]
        step_size_part has shape     [1, 4, 1]
        log_accept_prob has shape [2, 3, 4]
  
      Since step size has 1 rank fewer than the state, we reduce away the leading
      dimension of `log_accept_prob` to get a Tensor with shape [3, 4]. Next,
      since `log_accept_prob` must broadcast into step_size_part on the left, we
      reduce the dimensions where their shapes differ, to get a Tensor with shape
      [1, 4], which now is compatible with the leading dimensions of
      step_size_part.
  
      There is a subtlety here in that `step_size_parts` might be a length-1 list,
      which means that we'll be "structure-broadcasting" it for all the state
      parts (see logic in, e.g., hmc.py). In this case we must assume that that
      the lone step size provided broadcasts with the event dims of each state
      part. This means that either step size has no dimensions corresponding to
      chain dimensions, or all states are of the same shape. For the former, we
      want to reduce over all chain dimensions. For the later, we want to use
      the same logic as in the non-structure-broadcasted case.
  
      It turns out we can compute the reduction dimensions for both cases
      uniformly by taking the rank of any state part. This obviously works in
      the second case (where all state ranks are the same). In the first case,
      all state parts have the rank L + D_i + B, where L is the rank of
      log_accept_prob, D_i is the non-shared dimensions amongst all states, and
      B are the shared dimensions of all the states, which are equal to the step
      size. When we subtract B, we will always get a number >= L, which means
      we'll get the full reduction we want.
  
      Args:
        step_size: Previous step's step_size.
        state: Previous step's state value.
        error_sum: Previous step's error accumulator.
        log_averaging_step: Previous step's log_averaging_step.
        log_shrinkage_target: Floating point scalar `Tensor`. Logarithm of value
          the exploration step size is biased towards.
        log_accept_prob_rank: Rank of log_accept_prob.
        log_accept_prob: Floating point scalar `Tensor`. Target accept
          probability.
        target_accept_prob: A floating point `Tensor` representing desired
          acceptance probability. Must be a positive number less than 1.
        previous_kernel_results: Results struct from previous step.
  
      Returns:
        new_step_size: Updated `step_size`.
        new_log_averaging_step: Updated `log_averaging_step`.
        new_error_sum: Updated `error_sum`.
      """
      num_reduce_dims = ps.minimum(
          log_accept_prob_rank, (ps.rank(state) - ps.rank(step_size))
      )
      reduced_log_accept_prob = self.reduce_fn(
          log_accept_prob,
          axis=ps.range(num_reduce_dims),
          keepdims=False,
          experimental_named_axis=self.experimental_reduce_chain_axis_names,
      )
  
      # reduced_log_accept_prob must broadcast into step_size on the
      # left, so we do an additional reduction over dimensions where their
      # shapes differ.
      reduce_indices = get_differing_dims(
          reduced_log_accept_prob, step_size
      )
      reduced_log_accept_prob = self.reduce_fn(
          reduced_log_accept_prob,
          axis=reduce_indices,
          keepdims=True,
          experimental_named_axis=self.experimental_reduce_chain_axis_names,
      )
      new_error_sum = (
          error_sum
          + target_accept_prob
          - tf.math.exp(reduced_log_accept_prob)
      )
      num_ones_to_pad = ps.maximum(
          ps.rank(log_shrinkage_target) - ps.rank(new_error_sum), 0
      )
      new_error_sum_extend = tf.reshape(
          new_error_sum,
          shape=ps.pad(
              ps.shape(new_error_sum),
              paddings=[[0, num_ones_to_pad]],
              constant_values=1,
          ),
      )
  
      step_count_smoothing = (
          previous_kernel_results.step_count_smoothing
      )
      step = (
          tf.cast(
              previous_kernel_results.step, step_count_smoothing.dtype
          )
          + 1.0
      )
      soft_t = step_count_smoothing + step
  
      new_log_step = log_shrinkage_target - (
          (
              tf.cast(new_error_sum_extend, step.dtype)
              * tf.math.sqrt(step)
          )
          / (soft_t * previous_kernel_results.exploration_shrinkage)
      )
  
      eta = step ** (-previous_kernel_results.decay_rate)
      new_log_averaging_step = (
          eta * new_log_step + (1.0 - eta) * log_averaging_step
      )
  
      # - If still adapting, return an exploring step size,
      # - If just finished, return the averaging step size
      # - Otherwise, do not update
      num_adaptation_steps = (
          previous_kernel_results.num_adaptation_steps
      )
      step = previous_kernel_results.step + 1
      new_step_size = tf.where(
          step < num_adaptation_steps,
          tf.math.exp(new_log_step),
          tf.where(
              step > num_adaptation_steps,
              step_size,
              tf.math.exp(new_log_averaging_step),
          ),
      )
      new_log_averaging_step = tf.where(
          step > num_adaptation_steps,
          log_averaging_step,
          new_log_averaging_step,
      )
      return new_step_size, new_log_averaging_step, new_error_sum
  
imports: import collections
  import functools
  import numpy
  import tensorflow.compat.v2
  from tensorflow_probability.python.internal import assert_util
  from tensorflow_probability.python.internal import dtype_util
  from tensorflow_probability.python.internal import prefer_static
  from tensorflow_probability.python.math.generic import reduce_logmeanexp
  from tensorflow_probability.python.mcmc import kernel
  from tensorflow_probability.python.mcmc.internal import util
  from tensorflow_probability.python.mcmc.simple_step_size_adaptation import get_differing_dims
  from tensorflow_probability.python.mcmc.simple_step_size_adaptation import hmc_like_log_accept_prob_getter_fn
  from tensorflow_probability.python.mcmc.simple_step_size_adaptation import hmc_like_step_size_getter_fn
  from tensorflow_probability.python.mcmc.simple_step_size_adaptation import hmc_like_step_size_setter_fn
label_desc: Use image node_id 10 for calling the DualAveragingStepSizeAdaptation obj's underlying member method code with example usage: obj._one_step_part(step_size, state, error_sum, log_averaging_step, log_shrinkage_target, log_accept_prob_rank, log_accept_prob, target_accept_prob, previous_kernel_results) and returns: new_step_size,  new_log_averaging_step,  new_error_sum