# Copyright 2018 The TensorFlow Probability Authors.
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"""Utilities for fitting variational distributions."""

from __future__ import absolute_import
from __future__ import division
# [internal] enable type annotations
from __future__ import print_function

import functools

from tensorflow_probability.python import math as tfp_math
from tensorflow_probability.python.vi import csiszar_divergence

_trace_loss = lambda traceable_quantities: traceable_quantities.loss

# Silent fallback to score-function gradients leads to difficult-to-debug
# failures, so we force reparameterization gradients by default.
_reparameterized_elbo = functools.partial(
    csiszar_divergence.monte_carlo_variational_loss,
    discrepancy_fn=csiszar_divergence.kl_reverse,
    use_reparameterization=True)


def fit_surrogate_posterior(target_log_prob_fn,
                            surrogate_posterior,
                            optimizer,
                            num_steps,
                            convergence_criterion=None,
                            trace_fn=_trace_loss,
                            variational_loss_fn=_reparameterized_elbo,
                            sample_size=1,
                            trainable_variables=None,
                            jit_compile=None,
                            seed=None,
                            name='fit_surrogate_posterior'):
  """Fit a surrogate posterior to a target (unnormalized) log density.

  The default behavior constructs and minimizes the negative variational
  evidence lower bound (ELBO), given by

  ```python
  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](https://en.wikipedia.org/wiki/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`.
    variational_loss_fn: Python `callable` with signature
      `loss = variational_loss_fn(target_log_prob_fn, surrogate_posterior,
       sample_size, seed)` defining a variational loss function. The default is
       a Monte Carlo approximation to the standard evidence lower bound (ELBO),
       equivalent to minimizing the 'reverse' `KL[q||p]` divergence between the
       surrogate `q` and true posterior `p`. [1]
       Default value: `functools.partial(
         tfp.vi.monte_carlo_variational_loss,
         discrepancy_fn=tfp.vi.kl_reverse,
         use_reparameterization=True)`.
    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`.
    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'.

  Returns:
    results: `Tensor` or nested structure of `Tensor`s, according to the return
      type of `trace_fn`. Each `Tensor` has an added leading dimension of size
      `num_steps`, packing the trajectory of the result over the course of the
      optimization.

  #### 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)`:

  ```python
  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](
  https://en.wikipedia.org/wiki/Conjugate_prior), 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!

  ```python
  q_z = tfd.Normal(loc=tf.Variable(0., name='q_z_loc'),
                   scale=tfp.util.TransformedVariable(1., tfb.Softplus(),
                                                      name='q_z_scale'),
                   name='q_z')
  losses = tfp.vi.fit_surrogate_posterior(
      conditioned_log_prob,
      surrogate_posterior=q,
      optimizer=tf.optimizers.Adam(learning_rate=0.1),
      num_steps=100)
  print(q_z.mean(), q_z.stddev())  # => approximately [2.5, 1/sqrt(2)]
  ```

  Note that we ensure positive scale by using a softplus transformation of
  the underlying variable, invoked via `TransformedVariable`. Deferring the
  transformation causes it to be applied upon evaluation of the distribution's
  methods, creating a gradient to the underlying variable. If we
  had simply specified `scale=tf.nn.softplus(scale_var)` directly,
  without the `TransformedVariable`, fitting would fail because calls to
  `q.log_prob` and `q.sample` would never access the underlying variable. In
  general, transformations of trainable parameters must be deferred to runtime,
  using either `TransformedVariable` or `DeferredTensor` or by the callable
  mechanisms available in joint distribution classes (demonstrated below).

  **Custom loss function**. Suppose we prefer to fit the same model using
    the forward KL divergence `KL[p||q]`. We can pass a custom loss function:

  ```python
    import functools
    forward_kl_loss = functools.partial(
      tfp.vi.monte_carlo_variational_loss, discrepancy_fn=tfp.vi.kl_forward)
    losses = tfp.vi.fit_surrogate_posterior(
        conditioned_log_prob,
        surrogate_posterior=q,
        optimizer=tf.optimizers.Adam(learning_rate=0.1),
        num_steps=100,
        variational_loss_fn=forward_kl_loss)
  ```

  Note that in practice this may have substantially higher-variance gradients
  than the reverse KL.

  **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:

  ```python
  # 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:

  ```python
  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]: Bishop, Christopher M. Pattern Recognition and Machine Learning.
       Springer, 2006.
  """

  def complete_variational_loss_fn(seed=None):
    return variational_loss_fn(
        target_log_prob_fn,
        surrogate_posterior,
        sample_size=sample_size,
        seed=seed)

  return tfp_math.minimize(complete_variational_loss_fn,
                           num_steps=num_steps,
                           optimizer=optimizer,
                           convergence_criterion=convergence_criterion,
                           trace_fn=trace_fn,
                           trainable_variables=trainable_variables,
                           jit_compile=jit_compile,
                           seed=seed,
                           name=name)
