Optimizer that implements the Conditional Gradient optimization.

### Used in the notebooks

Used in the tutorials

This optimizer helps handle constraints well.

Currently only supports frobenius norm constraint or nuclear norm constraint. See https://arxiv.org/pdf/1803.06453.pdf

variable -= (1-learning_rate) * (variable + lambda_ * gradient
/ (frobenius_norm(gradient) + epsilon))


Note that lambda_ here refers to the constraint "lambda" in the paper. epsilon is constant with tiny value as compared to the value of frobenius norm of gradient. The purpose of epsilon here is to avoid the case that the value of frobenius norm of gradient is 0.

In this implementation, epsilon defaults to $$10^{-7}$$.

For nucler norm constraint, the formula is as following:

variable -= (1-learning_rate) * (variable
+ lambda_ * top_singular_vector(gradient))


learning_rate A Tensor or a floating point value. or a schedule that is a tf.keras.optimizers.schedules.LearningRateSchedule The learning rate.
lambda_ A Tensor or a floating point value. The constraint.
epsilon A Tensor or a floating point value. A small constant for numerical stability when handling the case of norm of gradient to be zero.
ord Order of the norm. Supported values are 'fro' and 'nuclear'. Default is 'fro', which is frobenius norm.
name Optional name prefix for the operations created when applying gradients. Defaults to 'ConditionalGradient'.
**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.

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.

Check tf.clip_by_global_norm for more details.

iterations Variable. The number of training steps this Optimizer has run.
weights Returns variables of this Optimizer based on the order created.

## Methods

### add_slot

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.

### apply_gradients

Apply gradients to variables.

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)
# Processing aggregated gradients.



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 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
TypeError If grads_and_vars is malformed.
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.

### get_gradients

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
List of gradient tensors.

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.legacy.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)
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 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.
grad_loss (Optional). A Tensor holding the gradient computed for loss.
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.legacy.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)
results = m.fit(data, labels)  # Training.
new_weights = [np.array(10), np.ones([20, 10]), np.zeros([10])]
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" }]