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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 String. The name to use for momentum accumulator weights created by the optimizer.
gradient_aggregator The function to use to aggregate gradients across devices (when using tf.distribute.Strategy). If None, defaults to summing the gradients across devices. The function should accept and return a list of (gradient, variable) tuples.
gradient_transformers Optional. List of functions to use to transform gradients before applying updates to Variables. The functions are applied after gradient_aggregator. The functions should accept and return a list of (gradient, variable) tuples.
**kwargs keyword arguments. Allowed arguments are clipvalue, clipnorm, global_clipnorm. If clipvalue (float) is set, the gradient of each weight is clipped to be no higher than this value. If clipnorm (float) is set, the gradient of each weight is individually clipped so that its norm is no higher than this value. If global_clipnorm (float) is set the gradient of all weights is clipped so that their global norm is no higher than this value.

clipnorm float or None. If set, clips gradients to a maximum norm.
clipvalue float or None. If set, clips gradients to a maximum value.
global_clipnorm float or None. If set, clips gradients to a maximum norm.
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 a new slot variable for var.

A slot variable is an additional variable associated with var to train. It is allocated and managed by optimizers, e.g. Adam.

Args
var a Variable object.
slot_name name of the slot variable.
initializer initializer of the slot variable
shape (Optional) shape of the slot variable. If not set, it will default to the shape of var.

Returns
A slot variable.

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:

Args
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 presence 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
ValueError If none of the variables have gradients.
RuntimeError If called in a cross-replica context.

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.

Args
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.

Returns gradients of loss with respect to params.

Should be used only in legacy v1 graph mode.

Args
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.

Args
loss Tensor or callable. If a callable, loss should take no arguments and return the value to minimize. If a Tensor, the tape argument must be passed.
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.
name (Optional) str. Name for the returned operation.
tape (Optional) tf.GradientTape. If loss is provided as a Tensor, the tape that computed the loss must be provided.

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>

Args
weights weight values as a list of numpy arrays.

variables

Returns variables of this Optimizer based on the order created.

[{ "type": "thumb-down", "id": "missingTheInformationINeed", "label":"Missing the information I need" },{ "type": "thumb-down", "id": "tooComplicatedTooManySteps", "label":"Too complicated / too many steps" },{ "type": "thumb-down", "id": "outOfDate", "label":"Out of date" },{ "type": "thumb-down", "id": "samplesCodeIssue", "label":"Samples / code issue" },{ "type": "thumb-down", "id": "otherDown", "label":"Other" }]
[{ "type": "thumb-up", "id": "easyToUnderstand", "label":"Easy to understand" },{ "type": "thumb-up", "id": "solvedMyProblem", "label":"Solved my problem" },{ "type": "thumb-up", "id": "otherUp", "label":"Other" }]