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Stochastic gradient descent and momentum optimizer.

Inherits From: Optimizer

The update rule for $\theta$ with gradient $g$ when momentum is 0.0:

$$\theta_t = \theta_{t-1} - \mathrm{learning\_rate} * g_t$$

The update rule when momentum is larger than 0.0:

$$v_t = \mathrm{momentum} * v_{t-1} - \mathrm{learning\_rate} * g_t$$
$$\theta_t = \theta_{t-1} + v_t$$

if nesterov is False, gradient is evaluated at $\theta_t$. if nesterov is True, gradient is evaluated at $\theta_t + momentum * v_t$, and the variables always store $\theta + m v$ instead of $theta$


opt = tf.keras.optimizers.SGD(learning_rate=0.1)
var = tf.Variable(1.0)
loss = lambda: (var ** 2)/2.0         # d(loss)/d(var1) = var1
step_count = opt.minimize(loss, [var]).numpy()
# Step is `-learning_rate*grad`
opt = tf.keras.optimizers.SGD(learning_rate=0.1, momentum=0.9)
var = tf.Variable(1.0)
val0 = var.value()
loss = lambda: (var ** 2)/2.0         # d(loss)/d(var1) = var1
# First step is `-learning_rate*grad`
step_count = opt.minimize(loss, [var]).numpy()
val1 = var.value()
(val0 - val1).numpy()
# On later steps, step-size increases because of momentum
step_count = opt.minimize(loss, [var]).numpy()
val2 = var.value()
(val1 - val2).numpy()

Some of the args below are hyperparameters, where a hyperparameter is defined as a scalar Tensor, a regular Python value, or a callable (which will be evaluated when apply_gradients is called) returning a scalar Tensor or a Python value.


nesterov = True, See [Sutskever et al., 2013](

learning_rate A Tensor, floating point value, or a schedule that is a tf.keras.optimizers.schedules.LearningRateSchedule, or a callable that takes no arguments and returns the actual value to use. The learning rate. Defaults to 0.01.
momentum float hyperparameter >= 0 that accelerates SGD in the relevant direction and dampens oscillations. Defaults to 0.0, i.e., SGD.
nesterov boolean. Whether to apply Nesterov momentum. Defaults to False.
name Optional name prefix for the operations created when applying gradients. Defaults to 'SGD'.
**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.

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



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Add a new slot variable for var.


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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.


grads = tape.gradient(loss, vars)
grads = tf.distribute.get_replica_context().all_reduce('sum', grads)
# Processing aggregated gradients.
optimizer.apply_gradients(zip(grads, vars),

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.

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

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


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Creates an optimizer from its config.

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

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.

An optimizer instance.


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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.

Python dictionary.


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Returns gradients of loss with respect to params.

loss Loss tensor.
params List of variables.

List of gradient tensors.

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


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A list of names for this optimizer's slots.


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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 =, labels)
Training ...

Weights values as a list of numpy arrays.


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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.

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.

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

ValueError If some of the variables are not Variable objects.


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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 mode