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Fit a surrogate posterior to a target (unnormalized) log density.
tfp.vi.fit_surrogate_posterior(
target_log_prob_fn,
surrogate_posterior,
optimizer,
num_steps,
convergence_criterion=None,
trace_fn=_trace_loss,
discrepancy_fn=tfp.vi.kl_reverse
,
sample_size=1,
importance_sample_size=1,
trainable_variables=None,
jit_compile=None,
seed=None,
name='fit_surrogate_posterior'
)
Used in the notebooks
The default behavior constructs and minimizes the negative variational evidence lower bound (ELBO), given by
q_samples = surrogate_posterior.sample(num_draws)
elbo_loss = -tf.reduce_mean(
target_log_prob_fn(q_samples) - surrogate_posterior.log_prob(q_samples))
This corresponds to minimizing the 'reverse' Kullback-Liebler divergence
(KL[q||p]
) between the variational distribution and the unnormalized
target_log_prob_fn
, and defines a lower bound on the marginal log
likelihood, log p(x) >= -elbo_loss
. [1]
More generally, this function supports fitting variational distributions that minimize any Csiszar f-divergence.
Args | |
---|---|
target_log_prob_fn
|
Python callable that takes a set of Tensor arguments
and returns a Tensor log-density. Given
q_sample = surrogate_posterior.sample(sample_size) , this
will be called as target_log_prob_fn(*q_sample) if q_sample is a list
or a tuple, target_log_prob_fn(**q_sample) if q_sample is a
dictionary, or target_log_prob_fn(q_sample) if q_sample is a Tensor .
It should support batched evaluation, i.e., should return a result of
shape [sample_size] .
|
surrogate_posterior
|
A tfp.distributions.Distribution
instance defining a variational posterior (could be a
tfd.JointDistribution ). Crucially, the distribution's log_prob and
(if reparameterized) sample methods must directly invoke all ops
that generate gradients to the underlying variables. One way to ensure
this is to use tfp.util.TransformedVariable and/or
tfp.util.DeferredTensor to represent any parameters defined as
transformations of unconstrained variables, so that the transformations
execute at runtime instead of at distribution creation.
|
optimizer
|
Optimizer instance to use. This may be a TF1-style
tf.train.Optimizer , TF2-style tf.optimizers.Optimizer , or any Python
object that implements optimizer.apply_gradients(grads_and_vars) .
|
num_steps
|
Python int number of steps to run the optimizer.
|
convergence_criterion
|
Optional instance of
tfp.optimizer.convergence_criteria.ConvergenceCriterion
representing a criterion for detecting convergence. If None ,
the optimization will run for num_steps steps, otherwise, it will run
for at most num_steps steps, as determined by the provided criterion.
Default value: None .
|
trace_fn
|
Python callable with signature traced_values = trace_fn(
traceable_quantities) , where the argument is an instance of
tfp.math.MinimizeTraceableQuantities and the returned traced_values
may be a Tensor or nested structure of Tensor s. The traced values are
stacked across steps and returned.
The default trace_fn simply returns the loss. In general, trace
functions may also examine the gradients, values of parameters,
the state propagated by the specified convergence_criterion , if any (if
no convergence criterion is specified, this will be None ),
as well as any other quantities captured in the closure of trace_fn ,
for example, statistics of a variational distribution.
Default value: lambda traceable_quantities: traceable_quantities.loss .
|
discrepancy_fn
|
Python callable representing a Csiszar f function in
in log-space. See the docs for tfp.vi.monte_carlo_variational_loss for
examples.
Default value: tfp.vi.kl_reverse .
|
sample_size
|
Python int number of Monte Carlo samples to use
in estimating the variational divergence. Larger values may stabilize
the optimization, but at higher cost per step in time and memory.
Default value: 1 .
|
importance_sample_size
|
Python int number of terms used to define an
importance-weighted divergence. If importance_sample_size > 1 , then the
surrogate_posterior is optimized to function as an importance-sampling
proposal distribution. In this case, posterior expectations should be
approximated by importance sampling, as demonstrated in the example below.
Default value: 1 .
|
trainable_variables
|
Optional list of tf.Variable instances to optimize
with respect to. If None , defaults to the set of all variables accessed
during the computation of the variational bound, i.e., those defining
surrogate_posterior and the model target_log_prob_fn .
Default value: None
|
jit_compile
|
If True, compiles the loss function and gradient update using
XLA. XLA performs compiler optimizations, such as fusion, and attempts to
emit more efficient code. This may drastically improve the performance.
See the docs for tf.function . (In JAX, this will apply jax.jit ).
Default value: None .
|
seed
|
PRNG seed; see tfp.random.sanitize_seed for details.
|
name
|
Python str name prefixed to ops created by this function.
Default value: 'fit_surrogate_posterior'.
|
Examples
Normal-Normal model. We'll first consider a simple model
z ~ N(0, 1)
, x ~ N(z, 1)
, where we suppose we are interested in the
posterior p(z | x=5)
:
import tensorflow_probability as tfp
from tensorflow_probability import distributions as tfd
def log_prob(z, x):
return tfd.Normal(0., 1.).log_prob(z) + tfd.Normal(z, 1.).log_prob(x)
conditioned_log_prob = lambda z: log_prob(z, x=5.)
The posterior is itself normal by conjugacy, and can be computed
analytically (it's N(loc=5/2., scale=1/sqrt(2)
). But suppose we don't want
to bother doing the math: we can use variational inference instead!
q_z = tfp.experimental.util.make_trainable(tfd.Normal, name='q_z')
losses = tfp.vi.fit_surrogate_posterior(
conditioned_log_prob,
surrogate_posterior=q_z,
optimizer=tf.optimizers.Adam(learning_rate=0.1),
num_steps=100)
print(q_z.mean(), q_z.stddev()) # => approximately [2.5, 1/sqrt(2)]
Custom loss function. Suppose we prefer to fit the same model using
the forward KL divergence KL[p||q]
. We can pass a custom discrepancy
function:
losses = tfp.vi.fit_surrogate_posterior(
conditioned_log_prob,
surrogate_posterior=q_z,
optimizer=tf.optimizers.Adam(learning_rate=0.1),
num_steps=100,
discrepancy_fn=tfp.vi.kl_forward)
Note that in practice this may have substantially higher-variance gradients than the reverse KL.
Importance weighting. A surrogate posterior may be corrected by interpreting it as a proposal for an importance sampler. That is, one can use weighted samples from the surrogate to estimate expectations under the true posterior:
zs, q_log_prob = surrogate_posterior.experimental_sample_and_log_prob(
num_samples)
# Naive expectation under the surrogate posterior.
expected_x = tf.reduce_mean(f(zs), axis=0)
# Importance-weighted estimate of the expectation under the true posterior.
self_normalized_log_weights = tf.nn.log_softmax(
target_log_prob_fn(zs) - q_log_prob)
expected_x = tf.reduce_sum(
tf.exp(self_normalized_log_weights) * f(zs),
axis=0)
Any distribution may be used as a proposal, but it is often natural to
consider surrogates that were themselves fit by optimizing an
importance-weighted variational objective [2], which directly optimizes the
surrogate's effectiveness as an proposal distribution. This may be specified
by passing importance_sample_size > 1
. The importance-weighted objective
may favor different characteristics than the original objective.
For example, effective proposals are generally overdispersed, whereas a
surrogate optimizing reverse KL would otherwise tend to be underdispersed.
Although importance sampling is guaranteed to tighten the variational bound, some research has found that this does not necessarily improve the quality of deep generative models, because it also introduces gradient noise that can lead to a weaker training signal [3]. As always, evaluation is important to choose the approach that works best for a particular task.
When using an importance-weighted loss to fit a surrogate, it is also recommended to apply importance sampling when computing expectations under that surrogate.
# Fit `q` with an importance-weighted variational loss.
losses = tfp.vi.fit_surrogate_posterior(
conditioned_log_prob,
surrogate_posterior=q_z,
importance_sample_size=10,
optimizer=tf.optimizers.Adam(learning_rate=0.1),
num_steps=200)
# Estimate posterior statistics with importance sampling.
zs, q_log_prob = q_z.experimental_sample_and_log_prob(1000)
self_normalized_log_weights = tf.nn.log_softmax(
conditioned_log_prob(zs) - q_log_prob)
posterior_mean = tf.reduce_sum(
tf.exp(self_normalized_log_weights) * zs,
axis=0)
posterior_variance = tf.reduce_sum(
tf.exp(self_normalized_log_weights) * (zs - posterior_mean)**2,
axis=0)
Inhomogeneous Poisson Process. For a more interesting example, let's
consider a model with multiple latent variables as well as trainable
parameters in the model itself. Given observed counts y
from spatial
locations X
, consider an inhomogeneous Poisson process model
log_rates = GaussianProcess(index_points=X); y = Poisson(exp(log_rates))
in which the latent (log) rates are spatially correlated following a Gaussian
process. We'll fit a variational model to the latent rates while also
optimizing the GP kernel hyperparameters (largely for illustration; in
practice we might prefer to 'be Bayesian' about these parameters and include
them as latents in our model and variational posterior). First we define
the model, including trainable variables:
# Toy 1D data.
index_points = np.array([-10., -7.2, -4., -0.1, 0.1, 4., 6.2, 9.]).reshape(
[-1, 1]).astype(np.float32)
observed_counts = np.array(
[100, 90, 60, 13, 18, 37, 55, 42]).astype(np.float32)
# Trainable GP hyperparameters.
kernel_log_amplitude = tf.Variable(0., name='kernel_log_amplitude')
kernel_log_lengthscale = tf.Variable(0., name='kernel_log_lengthscale')
observation_noise_log_scale = tf.Variable(
0., name='observation_noise_log_scale')
# Generative model.
Root = tfd.JointDistributionCoroutine.Root
def model_fn():
kernel = tfp.math.psd_kernels.ExponentiatedQuadratic(
amplitude=tf.exp(kernel_log_amplitude),
length_scale=tf.exp(kernel_log_lengthscale))
latent_log_rates = yield Root(tfd.GaussianProcess(
kernel,
index_points=index_points,
observation_noise_variance=tf.exp(observation_noise_log_scale),
name='latent_log_rates'))
y = yield tfd.Independent(tfd.Poisson(log_rate=latent_log_rates, name='y'),
reinterpreted_batch_ndims=1)
model = tfd.JointDistributionCoroutine(model_fn)
Next we define a variational distribution. We incorporate the observations directly into the variational model using the 'trick' of representing them by a deterministic distribution (observe that the true posterior on an observed value is in fact a point mass at the observed value).
logit_locs = tf.Variable(tf.zeros(observed_counts.shape), name='logit_locs')
logit_softplus_scales = tf.Variable(tf.ones(observed_counts.shape) * -4,
name='logit_softplus_scales')
def variational_model_fn():
latent_rates = yield Root(tfd.Independent(
tfd.Normal(loc=logit_locs, scale=tf.nn.softplus(logit_softplus_scales)),
reinterpreted_batch_ndims=1))
y = yield tfd.VectorDeterministic(observed_counts)
q = tfd.JointDistributionCoroutine(variational_model_fn)
Note that here we could apply transforms to variables without using
DeferredTensor
because the JointDistributionCoroutine
argument is a
function, i.e., executed "on demand." (The same is true when
distribution-making functions are supplied to JointDistributionSequential
and JointDistributionNamed
. That is, as long as variables are transformed
within the callable, they will appear on the gradient tape when
q.log_prob()
or q.sample()
are invoked.
Finally, we fit the variational posterior and model variables jointly: by not
explicitly specifying trainable_variables
, the optimization will
automatically include all variables accessed. We'll
use a custom trace_fn
to see how the kernel amplitudes and a set of sampled
latent rates with fixed seed evolve during the course of the optimization:
losses, log_amplitude_path, sample_path = tfp.vi.fit_surrogate_posterior(
target_log_prob_fn=lambda *args: model.log_prob(args),
surrogate_posterior=q,
optimizer=tf.optimizers.Adam(learning_rate=0.1),
sample_size=1,
num_steps=500,
trace_fn=lambda loss, grads, vars: (loss, kernel_log_amplitude,
q.sample(5, seed=42)[0]))
References
[1]: Christopher M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006.
[2] Yuri Burda, Roger Grosse, and Ruslan Salakhutdinov. Importance Weighted Autoencoders. In International Conference on Learning Representations, 2016. https://arxiv.org/abs/1509.00519
[3] Tom Rainforth, Adam R. Kosiorek, Tuan Anh Le, Chris J. Maddison, Maximilian Igl, Frank Wood, and Yee Whye Teh. Tighter Variational Bounds are Not Necessarily Better. In International Conference on Machine Learning (ICML), 2018. https://arxiv.org/abs/1802.04537