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Adapts the inner kernel's step_size
based on log_accept_prob
.
Inherits From: TransitionKernel
tfp.mcmc.DualAveragingStepSizeAdaptation(
inner_kernel, num_adaptation_steps, target_accept_prob=0.75,
exploration_shrinkage=0.05, shrinkage_target=None, step_count_smoothing=10,
decay_rate=0.75, step_size_setter_fn=hmc_like_step_size_setter_fn,
step_size_getter_fn=hmc_like_step_size_getter_fn,
log_accept_prob_getter_fn=hmc_like_log_accept_prob_getter_fn,
reduce_fn=tfp.math.reduce_logmeanexp, validate_args=False, name=None
)
Used in the notebooks
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 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 itsC0
andC1
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 itsC0
dimension. This means that you will learn a shared step size based on the mean acceptance probability across chains that share the coordinate across theC1
dimension. This can be useful when theC1
dimension indexes different distributions, whileC0
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 itsC1
dimension. This means that you will learn a shared step size based on the mean acceptance probability across chains that share the coordinate across theC0
dimension. This can be useful when theC0
dimension indexes different distributions, whileC1
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
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
[4]: Hoffman, M., Radul, A., & Sountsov, P. An Adaptive MCMC Scheme for Setting Trajectory Lengths in Hamiltonian Monte Carlo, 2020. In preparation.
Args | |
---|---|
inner_kernel
|
TransitionKernel -like object.
|
num_adaptation_steps
|
Scalar int 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][1]).
|
exploration_shrinkage
|
Floating point scalar Tensor . How strongly the
exploration rate is biased towards the shrinkage target.
|
shrinkage_target
|
Tensor or list of tensors. Value the exploration
step size(s) is/are biased towards.
As num_adaptation_steps --> infinity , this bias goes to zero.
Defaults to 10 times the initial step size.
|
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. A value of 0 gives weight only to the most recent iteration.
|
step_size_setter_fn
|
A callable 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 Tensor s 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 Tensor s.
|
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.
|
reduce_fn
|
A callable with signature (input_tensor, axis, keepdims) ->
tensor that returns a log-reduction of log_accept_prob , typically
some sort of mean. By default, this performs an arithmetic mean.
|
validate_args
|
Python bool . When True kernel parameters are checked
for validity. When False invalid inputs may silently render incorrect
outputs.
|
name
|
Python str name prefixed to Ops created by this function.
Default value: None (i.e., 'dual_averaging_step_size_adaptation').
|
Attributes | |
---|---|
inner_kernel
|
|
is_calibrated
|
Returns True if Markov chain converges to specified distribution.
|
name
|
|
num_adaptation_steps
|
|
parameters
|
Return dict of __init__ arguments and their values.
|
Methods
bootstrap_results
bootstrap_results(
init_state
)
Returns an object with the same type as returned by one_step(...)[1]
.
Args | |
---|---|
init_state
|
Tensor or Python list of Tensor s representing the
initial state(s) of the Markov chain(s).
|
Returns | |
---|---|
kernel_results
|
A (possibly nested) tuple , namedtuple or list of
Tensor s representing internal calculations made within this function.
|
copy
copy(
**override_parameter_kwargs
)
Non-destructively creates a deep copy of the kernel.
Args | |
---|---|
**override_parameter_kwargs
|
Python String/value dictionary of
initialization arguments to override with new values.
|
Returns | |
---|---|
new_kernel
|
TransitionKernel object of same type as self ,
initialized with the union of self.parameters and
override_parameter_kwargs, with any shared keys overridden by the
value of override_parameter_kwargs, i.e.,
dict(self.parameters, **override_parameters_kwargs) .
|
log_accept_prob_getter_fn
log_accept_prob_getter_fn(
kernel_results
)
one_step
one_step(
current_state, previous_kernel_results, seed=None
)
Takes one step of the TransitionKernel.
Must be overridden by subclasses.
Args | |
---|---|
current_state
|
Tensor or Python list of Tensor s representing the
current state(s) of the Markov chain(s).
|
previous_kernel_results
|
A (possibly nested) tuple , namedtuple or
list of Tensor s representing internal calculations made within the
previous call to this function (or as returned by bootstrap_results ).
|
seed
|
Optional, a seed for reproducible sampling. |
Returns | |
---|---|
next_state
|
Tensor or Python list of Tensor s representing the
next state(s) of the Markov chain(s).
|
kernel_results
|
A (possibly nested) tuple , namedtuple or list of
Tensor s representing internal calculations made within this function.
|
reduce_fn
reduce_fn(
input_tensor, axis, keepdims
)
step_size_getter_fn
step_size_getter_fn(
kernel_results
)
step_size_setter_fn
step_size_setter_fn(
kernel_results, new_step_size
)