An optimizer module for stochastic gradient Langevin dynamics.

### Used in the notebooks

This implements the preconditioned Stochastic Gradient Langevin Dynamics optimizer [(Li et al., 2016)]. The optimization variable is regarded as a sample from the posterior under Stochastic Gradient Langevin Dynamics with noise rescaled in each dimension according to RMSProp.

#### Examples

##### Optimizing energy of a 3D-Gaussian distribution

This example demonstrates that for a fixed step size SGLD works as an approximate version of MALA (tfp.mcmc.MetropolisAdjustedLangevinAlgorithm).

``````import tensorflow as tf
import tensorflow_probability as tfp
import numpy as np

tfd = tfp.distributions
dtype = np.float32

with tf.Session(graph=tf.Graph()) as sess:
# Set up random seed for the optimizer
tf.random.set_seed(42)
true_mean = dtype([0, 0, 0])
true_cov = dtype([[1, 0.25, 0.25], [0.25, 1, 0.25], [0.25, 0.25, 1]])
# Loss is defined through the Cholesky decomposition
chol = tf.linalg.cholesky(true_cov)

var_1 = tf.Variable(name='var_1', initial_value=[1., 1.])
var_2 = tf.Variable(name='var_2', initial_value=[1.])

def loss_fn():
var = tf.concat([var_1, var_2], axis=-1)
loss_part = tf.linalg.cholesky_solve(chol, var[..., tf.newaxis])
return tf.linalg.matvec(loss_part, var, transpose_a=True)

# Set up the learning rate with a polynomial decay
step = tf.Variable(0, dtype=tf.int64)
starter_learning_rate = .3
end_learning_rate = 1e-4
decay_steps = 1e4
learning_rate = tf.compat.v1.train.polynomial_decay(
starter_learning_rate,
step,
decay_steps,
end_learning_rate,
power=1.)

# Set up the optimizer
learning_rate=learning_rate, preconditioner_decay_rate=0.99)
optimizer_kernel.iterations = step
optimizer = optimizer_kernel.minimize(loss_fn, var_list=[var_1, var_2])

# Number of training steps
training_steps = 5000
# Record the steps as and treat them as samples
samples = [np.zeros([training_steps, 2]), np.zeros([training_steps, 1])]
sess.run(tf.compat.v1.global_variables_initializer())
for step in range(training_steps):
sess.run(optimizer)
sample = [sess.run(var_1), sess.run(var_2)]
samples[step, :] = sample
samples[step, :] = sample

samples_ = np.concatenate(samples, axis=-1)
sample_mean = np.mean(samples_, 0)
print('sample mean', sample_mean)
``````

Args: learning_rate: Scalar `float`-like `Tensor`. The base learning rate for the optimizer. Must be tuned to the specific function being minimized. preconditioner_decay_rate: Scalar `float`-like `Tensor`. The exponential decay rate of the rescaling of the preconditioner (RMSprop). (This is "alpha" in Li et al. (2016)). Should be smaller than but nearly `1` to approximate sampling from the posterior. (Default: `0.95`) data_size: Scalar `int`-like `Tensor`. The effective number of points in the data set. Assumes that the loss is taken as the mean over a minibatch. Otherwise if the sum was taken, divide this number by the batch size. If a prior is included in the loss function, it should be normalized by `data_size`. Default value: `1`. burnin: Scalar `int`-like `Tensor`. The number of iterations to collect gradient statistics to update the preconditioner before starting to draw noisy samples. (Default: `25`) diagonal_bias: Scalar `float`-like `Tensor`. Term added to the diagonal of the preconditioner to prevent the preconditioner from degenerating. (Default: `1e-8`) name: Python `str` describing ops managed by this function. (Default: `"StochasticGradientLangevinDynamics"`) parallel_iterations: the number of coordinates for which the gradients of the preconditioning matrix can be computed in parallel. Must be a positive integer.

`InvalidArgumentError` If preconditioner_decay_rate is a `Tensor` not in `(0,1]`.
`NotImplementedError` If eager execution is enabled.

#### References

: Chunyuan Li, Changyou Chen, David Carlson, and Lawrence Carin. Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks. In Association for the Advancement of Artificial Intelligence, 2016. https://arxiv.org/abs/1512.07666

`name` A non-empty string. The name to use for accumulators created for the optimizer.
`**kwargs` keyword arguments. Allowed to be {`clipnorm`, `clipvalue`, `lr`, `decay`}. `clipnorm` is clip gradients by norm; `clipvalue` is clip gradients by value, `decay` is included for backward compatibility to allow time inverse decay of learning rate. `lr` is included for backward compatibility, recommended to use `learning_rate` instead.

`ValueError` If name is malformed.

`iterations` Variable. The number of training steps this Optimizer has run.
`variable_scope` Variable scope of all calls to `tf.get_variable`.
`weights` Returns variables of this Optimizer based on the order created.

## Methods

### `add_slot`

Add a new slot variable for `var`.

### `apply_gradients`

This is the second part of `minimize()`. It returns an `Operation` that applies gradients.

The method sums gradients from all replicas in the presence of `tf.distribute.Strategy` by default. You can aggregate gradients yourself by passing `experimental_aggregate_gradients=False`.

#### Example:

``````grads = tape.gradient(loss, vars)

``````

Args
`grads_and_vars` List of (gradient, variable) pairs.
`name` Optional name for the returned operation. Default to the name passed to the `Optimizer` constructor.
`experimental_aggregate_gradients` Whether to sum gradients from different replicas in the presense of `tf.distribute.Strategy`. If False, it's user responsibility to aggregate the gradients. Default to True.

Returns
An `Operation` that applies the specified gradients. The `iterations` will be automatically increased by 1.

Raises
`TypeError` If `grads_and_vars` is malformed.
`ValueError` If none of the variables have gradients.

### `from_config`

Creates an optimizer from its config.

This method is the reverse of `get_config`, capable of instantiating the same optimizer from the config dictionary.

Arguments
`config` A Python dictionary, typically the output of get_config.
`custom_objects` A Python dictionary mapping names to additional Python objects used to create this optimizer, such as a function used for a hyperparameter.

Returns
An optimizer instance.

### `get_config`

View source

Returns the config of the optimizer.

An optimizer config is a Python dictionary (serializable) containing the configuration of an optimizer. The same optimizer can be reinstantiated later (without any saved state) from this configuration.

Returns
Python dictionary.

### `get_gradients`

Returns gradients of `loss` with respect to `params`.

Arguments
`loss` Loss tensor.
`params` List of variables.

Returns

Raises
`ValueError` In case any gradient cannot be computed (e.g. if gradient function not implemented).

### `get_slot_names`

A list of names for this optimizer's slots.

### `get_weights`

Returns the current weights of the optimizer.

The weights of an optimizer are its state (ie, variables). This function returns the weight values associated with this optimizer as a list of Numpy arrays. The first value is always the iterations count of the optimizer, followed by the optimizer's state variables in the order they were created. The returned list can in turn be used to load state into similarly parameterized optimizers.

For example, the RMSprop optimizer for this simple model returns a list of three values-- the iteration count, followed by the root-mean-square value of the kernel and bias of the single Dense layer:

````opt = tf.keras.optimizers.RMSprop()`
`m = tf.keras.models.Sequential([tf.keras.layers.Dense(10)])`
`m.compile(opt, loss='mse')`
`data = np.arange(100).reshape(5, 20)`
`labels = np.zeros(5)`
`print('Training'); results = m.fit(data, labels)`
`Training ...`
`len(opt.get_weights())`
`3`
```

Returns
Weights values as a list of numpy arrays.

### `minimize`

Minimize `loss` by updating `var_list`.

This method simply computes gradient using `tf.GradientTape` and calls `apply_gradients()`. If you want to process the gradient before applying then call `tf.GradientTape` and `apply_gradients()` explicitly instead of using this function.

Args
`loss` A callable taking no arguments which returns the value to minimize.
`var_list` list or tuple of `Variable` objects to update to minimize `loss`, or a callable returning the list or tuple of `Variable` objects. Use callable when the variable list would otherwise be incomplete before `minimize` since the variables are created at the first time `loss` is called.
`grad_loss` Optional. A `Tensor` holding the gradient computed for `loss`.
`name` Optional name for the returned operation.

Returns
An `Operation` that updates the variables in `var_list`. The `iterations` will be automatically increased by 1.

Raises
`ValueError` If some of the variables are not `Variable` objects.

### `set_weights`

Set the weights of the optimizer.

The weights of an optimizer are its state (ie, variables). This function takes the weight values associated with this optimizer as a list of Numpy arrays. The first value is always the iterations count of the optimizer, followed by the optimizer's state variables in the order they are created. The passed values are used to set the new state of the optimizer.

For example, the RMSprop optimizer for this simple model takes a list of three values-- the iteration count, followed by the root-mean-square value of the kernel and bias of the single Dense layer:

````opt = tf.keras.optimizers.RMSprop()`
`m = tf.keras.models.Sequential([tf.keras.layers.Dense(10)])`
`m.compile(opt, loss='mse')`
`data = np.arange(100).reshape(5, 20)`
`labels = np.zeros(5)`
`print('Training'); results = m.fit(data, labels)`
`Training ...`
`new_weights = [np.array(10), np.ones([20, 10]), np.zeros()]`
`opt.set_weights(new_weights)`
`opt.iterations`
`<tf.Variable 'RMSprop/iter:0' shape=() dtype=int64, numpy=10>`
```

Arguments
`weights` weight values as a list of numpy arrays.

### `variables`

Returns variables of this Optimizer based on the order created.