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"""
Functions to generate Theano update dictionaries for training.
The update functions implement different methods to control the learning
rate for use with stochastic gradient descent.
Update functions take a loss expression or a list of gradient expressions and
a list of parameters as input and return an ordered dictionary of updates:
.. autosummary::
:nosignatures:
sgd
momentum
nesterov_momentum
adagrad
rmsprop
adadelta
adam
adamax
amsgrad
Two functions can be used to further modify the updates to include momentum:
.. autosummary::
:nosignatures:
apply_momentum
apply_nesterov_momentum
Finally, we provide two helper functions to constrain the norm of tensors:
.. autosummary::
:nosignatures:
norm_constraint
total_norm_constraint
:func:`norm_constraint()` can be used to constrain the norm of parameters
(as an alternative to weight decay), or for a form of gradient clipping.
:func:`total_norm_constraint()` constrain the total norm of a list of tensors.
This is often used when training recurrent neural networks.
Examples
--------
Using :func:`nesterov_momentum` to define an update dictionary for a toy
example network:
>>> import lasagne
>>> import theano.tensor as T
>>> import theano
>>> from lasagne.nonlinearities import softmax
>>> from lasagne.layers import InputLayer, DenseLayer, get_output
>>> from lasagne.updates import nesterov_momentum
>>> l_in = InputLayer((100, 20))
>>> l1 = DenseLayer(l_in, num_units=3, nonlinearity=softmax)
>>> x = T.matrix('x') # shp: num_batch x num_features
>>> y = T.ivector('y') # shp: num_batch
>>> l_out = get_output(l1, x)
>>> params = lasagne.layers.get_all_params(l1)
>>> loss = T.mean(T.nnet.categorical_crossentropy(l_out, y))
>>> updates = nesterov_momentum(loss, params, learning_rate=1e-4, momentum=.9)
>>> train_fn = theano.function([x, y], updates=updates)
With :func:`apply_momentum` and :func:`apply_nesterov_momentum`, we can add
momentum to optimization schemes that do not usually support this:
>>> updates = lasagne.updates.rmsprop(loss, params, learning_rate=0.0001)
>>> updates = lasagne.updates.apply_momentum(updates, params, momentum=0.9)
All optimization schemes support symbolic variables for their hyperparameters,
such as shared variables. This allows to vary hyperparameters during training
without recompiling the training function. Note that the dtypes must match the
dtypes of the network parameters, which follow Theano's ``floatX`` setting.
In the following example, we use :func:`lasagne.utils.floatX` to ensure this:
>>> eta = theano.shared(lasagne.utils.floatX(0.001))
>>> updates = lasagne.updates.adam(loss, params, learning_rate=eta)
>>> train_fn = theano.function([x, y], updates=updates)
>>> # we can now modify the learning rate at any time during training:
>>> eta.set_value(lasagne.utils.floatX(eta.get_value() * 0.1))
"""
from collections import OrderedDict
import numpy as np
import theano
import theano.tensor as T
from . import utils
__all__ = [
"sgd",
"apply_momentum",
"momentum",
"apply_nesterov_momentum",
"nesterov_momentum",
"adagrad",
"rmsprop",
"adadelta",
"adam",
"adamax",
"amsgrad",
"norm_constraint",
"total_norm_constraint"
]
def get_or_compute_grads(loss_or_grads, params):
"""Helper function returning a list of gradients
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to return the gradients for
Returns
-------
list of expressions
If `loss_or_grads` is a list, it is assumed to be a list of
gradients and returned as is, unless it does not match the length
of `params`, in which case a `ValueError` is raised.
Otherwise, `loss_or_grads` is assumed to be a cost expression and
the function returns `theano.grad(loss_or_grads, params)`.
Raises
------
ValueError
If `loss_or_grads` is a list of a different length than `params`, or if
any element of `params` is not a shared variable (while we could still
compute its gradient, we can never update it and want to fail early).
"""
if any(not isinstance(p, theano.compile.SharedVariable) for p in params):
raise ValueError("params must contain shared variables only. If it "
"contains arbitrary parameter expressions, then "
"lasagne.utils.collect_shared_vars() may help you.")
if isinstance(loss_or_grads, list):
if not len(loss_or_grads) == len(params):
raise ValueError("Got %d gradient expressions for %d parameters" %
(len(loss_or_grads), len(params)))
return loss_or_grads
else:
return theano.grad(loss_or_grads, params)
def sgd(loss_or_grads, params, learning_rate):
"""Stochastic Gradient Descent (SGD) updates
Generates update expressions of the form:
* ``param := param - learning_rate * gradient``
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to generate update expressions for
learning_rate : float or symbolic scalar
The learning rate controlling the size of update steps
Returns
-------
OrderedDict
A dictionary mapping each parameter to its update expression
"""
grads = get_or_compute_grads(loss_or_grads, params)
updates = OrderedDict()
for param, grad in zip(params, grads):
updates[param] = param - learning_rate * grad
return updates
def apply_momentum(updates, params=None, momentum=0.9):
"""Returns a modified update dictionary including momentum
Generates update expressions of the form:
* ``velocity := momentum * velocity + updates[param] - param``
* ``param := param + velocity``
Parameters
----------
updates : OrderedDict
A dictionary mapping parameters to update expressions
params : iterable of shared variables, optional
The variables to apply momentum to. If omitted, will apply
momentum to all `updates.keys()`.
momentum : float or symbolic scalar, optional
The amount of momentum to apply. Higher momentum results in
smoothing over more update steps. Defaults to 0.9.
Returns
-------
OrderedDict
A copy of `updates` with momentum updates for all `params`.
Notes
-----
Higher momentum also results in larger update steps. To counter that,
you can optionally scale your learning rate by `1 - momentum`.
See Also
--------
momentum : Shortcut applying momentum to SGD updates
"""
if params is None:
params = updates.keys()
updates = OrderedDict(updates)
for param in params:
value = param.get_value(borrow=True)
velocity = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
x = momentum * velocity + updates[param]
updates[velocity] = x - param
updates[param] = x
return updates
def momentum(loss_or_grads, params, learning_rate, momentum=0.9):
"""Stochastic Gradient Descent (SGD) updates with momentum
Generates update expressions of the form:
* ``velocity := momentum * velocity - learning_rate * gradient``
* ``param := param + velocity``
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to generate update expressions for
learning_rate : float or symbolic scalar
The learning rate controlling the size of update steps
momentum : float or symbolic scalar, optional
The amount of momentum to apply. Higher momentum results in
smoothing over more update steps. Defaults to 0.9.
Returns
-------
OrderedDict
A dictionary mapping each parameter to its update expression
Notes
-----
Higher momentum also results in larger update steps. To counter that,
you can optionally scale your learning rate by `1 - momentum`.
See Also
--------
apply_momentum : Generic function applying momentum to updates
nesterov_momentum : Nesterov's variant of SGD with momentum
"""
updates = sgd(loss_or_grads, params, learning_rate)
return apply_momentum(updates, momentum=momentum)
def apply_nesterov_momentum(updates, params=None, momentum=0.9):
"""Returns a modified update dictionary including Nesterov momentum
Generates update expressions of the form:
* ``velocity := momentum * velocity + updates[param] - param``
* ``param := param + momentum * velocity + updates[param] - param``
Parameters
----------
updates : OrderedDict
A dictionary mapping parameters to update expressions
params : iterable of shared variables, optional
The variables to apply momentum to. If omitted, will apply
momentum to all `updates.keys()`.
momentum : float or symbolic scalar, optional
The amount of momentum to apply. Higher momentum results in
smoothing over more update steps. Defaults to 0.9.
Returns
-------
OrderedDict
A copy of `updates` with momentum updates for all `params`.
Notes
-----
Higher momentum also results in larger update steps. To counter that,
you can optionally scale your learning rate by `1 - momentum`.
The classic formulation of Nesterov momentum (or Nesterov accelerated
gradient) requires the gradient to be evaluated at the predicted next
position in parameter space. Here, we use the formulation described at
https://github.com/lisa-lab/pylearn2/pull/136#issuecomment-10381617,
which allows the gradient to be evaluated at the current parameters.
See Also
--------
nesterov_momentum : Shortcut applying Nesterov momentum to SGD updates
"""
if params is None:
params = updates.keys()
updates = OrderedDict(updates)
for param in params:
value = param.get_value(borrow=True)
velocity = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
x = momentum * velocity + updates[param] - param
updates[velocity] = x
updates[param] = momentum * x + updates[param]
return updates
def nesterov_momentum(loss_or_grads, params, learning_rate, momentum=0.9):
"""Stochastic Gradient Descent (SGD) updates with Nesterov momentum
Generates update expressions of the form:
* ``velocity := momentum * velocity - learning_rate * gradient``
* ``param := param + momentum * velocity - learning_rate * gradient``
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to generate update expressions for
learning_rate : float or symbolic scalar
The learning rate controlling the size of update steps
momentum : float or symbolic scalar, optional
The amount of momentum to apply. Higher momentum results in
smoothing over more update steps. Defaults to 0.9.
Returns
-------
OrderedDict
A dictionary mapping each parameter to its update expression
Notes
-----
Higher momentum also results in larger update steps. To counter that,
you can optionally scale your learning rate by `1 - momentum`.
The classic formulation of Nesterov momentum (or Nesterov accelerated
gradient) requires the gradient to be evaluated at the predicted next
position in parameter space. Here, we use the formulation described at
https://github.com/lisa-lab/pylearn2/pull/136#issuecomment-10381617,
which allows the gradient to be evaluated at the current parameters.
See Also
--------
apply_nesterov_momentum : Function applying momentum to updates
"""
updates = sgd(loss_or_grads, params, learning_rate)
return apply_nesterov_momentum(updates, momentum=momentum)
def adagrad(loss_or_grads, params, learning_rate=1.0, epsilon=1e-6):
"""Adagrad updates
Scale learning rates by dividing with the square root of accumulated
squared gradients. See [1]_ for further description.
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to generate update expressions for
learning_rate : float or symbolic scalar
The learning rate controlling the size of update steps
epsilon : float or symbolic scalar
Small value added for numerical stability
Returns
-------
OrderedDict
A dictionary mapping each parameter to its update expression
Notes
-----
Using step size eta Adagrad calculates the learning rate for feature i at
time step t as:
.. math:: \\eta_{t,i} = \\frac{\\eta}
{\\sqrt{\\sum^t_{t^\\prime} g^2_{t^\\prime,i}+\\epsilon}} g_{t,i}
as such the learning rate is monotonically decreasing.
Epsilon is not included in the typical formula, see [2]_.
References
----------
.. [1] Duchi, J., Hazan, E., & Singer, Y. (2011):
Adaptive subgradient methods for online learning and stochastic
optimization. JMLR, 12:2121-2159.
.. [2] Chris Dyer:
Notes on AdaGrad. http://www.ark.cs.cmu.edu/cdyer/adagrad.pdf
"""
grads = get_or_compute_grads(loss_or_grads, params)
updates = OrderedDict()
for param, grad in zip(params, grads):
value = param.get_value(borrow=True)
accu = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
accu_new = accu + grad ** 2
updates[accu] = accu_new
updates[param] = param - (learning_rate * grad /
T.sqrt(accu_new + epsilon))
return updates
def rmsprop(loss_or_grads, params, learning_rate=1.0, rho=0.9, epsilon=1e-6):
"""RMSProp updates
Scale learning rates by dividing with the moving average of the root mean
squared (RMS) gradients. See [1]_ for further description.
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to generate update expressions for
learning_rate : float or symbolic scalar
The learning rate controlling the size of update steps
rho : float or symbolic scalar
Gradient moving average decay factor
epsilon : float or symbolic scalar
Small value added for numerical stability
Returns
-------
OrderedDict
A dictionary mapping each parameter to its update expression
Notes
-----
`rho` should be between 0 and 1. A value of `rho` close to 1 will decay the
moving average slowly and a value close to 0 will decay the moving average
fast.
Using the step size :math:`\\eta` and a decay factor :math:`\\rho` the
learning rate :math:`\\eta_t` is calculated as:
.. math::
r_t &= \\rho r_{t-1} + (1-\\rho)*g^2\\\\
\\eta_t &= \\frac{\\eta}{\\sqrt{r_t + \\epsilon}}
References
----------
.. [1] Tieleman, T. and Hinton, G. (2012):
Neural Networks for Machine Learning, Lecture 6.5 - rmsprop.
Coursera. http://www.youtube.com/watch?v=O3sxAc4hxZU (formula @5:20)
"""
grads = get_or_compute_grads(loss_or_grads, params)
updates = OrderedDict()
# Using theano constant to prevent upcasting of float32
one = T.constant(1)
for param, grad in zip(params, grads):
value = param.get_value(borrow=True)
accu = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
accu_new = rho * accu + (one - rho) * grad ** 2
updates[accu] = accu_new
updates[param] = param - (learning_rate * grad /
T.sqrt(accu_new + epsilon))
return updates
def adadelta(loss_or_grads, params, learning_rate=1.0, rho=0.95, epsilon=1e-6):
""" Adadelta updates
Scale learning rates by the ratio of accumulated gradients to accumulated
updates, see [1]_ and notes for further description.
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to generate update expressions for
learning_rate : float or symbolic scalar
The learning rate controlling the size of update steps
rho : float or symbolic scalar
Squared gradient moving average decay factor
epsilon : float or symbolic scalar
Small value added for numerical stability
Returns
-------
OrderedDict
A dictionary mapping each parameter to its update expression
Notes
-----
rho should be between 0 and 1. A value of rho close to 1 will decay the
moving average slowly and a value close to 0 will decay the moving average
fast.
rho = 0.95 and epsilon=1e-6 are suggested in the paper and reported to
work for multiple datasets (MNIST, speech).
In the paper, no learning rate is considered (so learning_rate=1.0).
Probably best to keep it at this value.
epsilon is important for the very first update (so the numerator does
not become 0).
Using the step size eta and a decay factor rho the learning rate is
calculated as:
.. math::
r_t &= \\rho r_{t-1} + (1-\\rho)*g^2\\\\
\\eta_t &= \\eta \\frac{\\sqrt{s_{t-1} + \\epsilon}}
{\sqrt{r_t + \epsilon}}\\\\
s_t &= \\rho s_{t-1} + (1-\\rho)*(\\eta_t*g)^2
References
----------
.. [1] Zeiler, M. D. (2012):
ADADELTA: An Adaptive Learning Rate Method.
arXiv Preprint arXiv:1212.5701.
"""
grads = get_or_compute_grads(loss_or_grads, params)
updates = OrderedDict()
# Using theano constant to prevent upcasting of float32
one = T.constant(1)
for param, grad in zip(params, grads):
value = param.get_value(borrow=True)
# accu: accumulate gradient magnitudes
accu = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
# delta_accu: accumulate update magnitudes (recursively!)
delta_accu = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
# update accu (as in rmsprop)
accu_new = rho * accu + (one - rho) * grad ** 2
updates[accu] = accu_new
# compute parameter update, using the 'old' delta_accu
update = (grad * T.sqrt(delta_accu + epsilon) /
T.sqrt(accu_new + epsilon))
updates[param] = param - learning_rate * update
# update delta_accu (as accu, but accumulating updates)
delta_accu_new = rho * delta_accu + (one - rho) * update ** 2
updates[delta_accu] = delta_accu_new
return updates
def adam(loss_or_grads, params, learning_rate=0.001, beta1=0.9,
beta2=0.999, epsilon=1e-8):
"""Adam updates
Adam updates implemented as in [1]_.
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to generate update expressions for
learning_rate : float or symbolic scalar
Learning rate
beta1 : float or symbolic scalar
Exponential decay rate for the first moment estimates.
beta2 : float or symbolic scalar
Exponential decay rate for the second moment estimates.
epsilon : float or symbolic scalar
Constant for numerical stability.
Returns
-------
OrderedDict
A dictionary mapping each parameter to its update expression
Notes
-----
The paper [1]_ includes an additional hyperparameter lambda. This is only
needed to prove convergence of the algorithm and has no practical use
(personal communication with the authors), it is therefore omitted here.
References
----------
.. [1] Kingma, Diederik, and Jimmy Ba (2014):
Adam: A Method for Stochastic Optimization.
arXiv preprint arXiv:1412.6980.
"""
all_grads = get_or_compute_grads(loss_or_grads, params)
t_prev = theano.shared(utils.floatX(0.))
updates = OrderedDict()
# Using theano constant to prevent upcasting of float32
one = T.constant(1)
t = t_prev + 1
a_t = learning_rate*T.sqrt(one-beta2**t)/(one-beta1**t)
for param, g_t in zip(params, all_grads):
value = param.get_value(borrow=True)
m_prev = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
v_prev = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
m_t = beta1*m_prev + (one-beta1)*g_t
v_t = beta2*v_prev + (one-beta2)*g_t**2
step = a_t*m_t/(T.sqrt(v_t) + epsilon)
updates[m_prev] = m_t
updates[v_prev] = v_t
updates[param] = param - step
updates[t_prev] = t
return updates
def adamax(loss_or_grads, params, learning_rate=0.002, beta1=0.9,
beta2=0.999, epsilon=1e-8):
"""Adamax updates
Adamax updates implemented as in [1]_. This is a variant of of the Adam
algorithm based on the infinity norm.
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to generate update expressions for
learning_rate : float or symbolic scalar
Learning rate
beta1 : float or symbolic scalar
Exponential decay rate for the first moment estimates.
beta2 : float or symbolic scalar
Exponential decay rate for the weighted infinity norm estimates.
epsilon : float or symbolic scalar
Constant for numerical stability.
Returns
-------
OrderedDict
A dictionary mapping each parameter to its update expression
References
----------
.. [1] Kingma, Diederik, and Jimmy Ba (2014):
Adam: A Method for Stochastic Optimization.
arXiv preprint arXiv:1412.6980.
"""
all_grads = get_or_compute_grads(loss_or_grads, params)
t_prev = theano.shared(utils.floatX(0.))
updates = OrderedDict()
# Using theano constant to prevent upcasting of float32
one = T.constant(1)
t = t_prev + 1
a_t = learning_rate/(one-beta1**t)
for param, g_t in zip(params, all_grads):
value = param.get_value(borrow=True)
m_prev = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
u_prev = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
m_t = beta1*m_prev + (one-beta1)*g_t
u_t = T.maximum(beta2*u_prev, abs(g_t))
step = a_t*m_t/(u_t + epsilon)
updates[m_prev] = m_t
updates[u_prev] = u_t
updates[param] = param - step
updates[t_prev] = t
return updates
def amsgrad(loss_or_grads, params, learning_rate=0.001, beta1=0.9,
beta2=0.999, epsilon=1e-8):
"""AMSGrad updates
AMSGrad updates implemented as in [1]_.
Parameters
----------
loss_or_grads : symbolic expression or list of expressions
A scalar loss expression, or a list of gradient expressions
params : list of shared variables
The variables to generate update expressions for
learning_rate : float or symbolic scalar
Learning rate
beta1 : float or symbolic scalar
Exponential decay rate for the first moment estimates.
beta2 : float or symbolic scalar
Exponential decay rate for the second moment estimates.
epsilon : float or symbolic scalar
Constant for numerical stability.
Returns
-------
OrderedDict
A dictionary mapping each parameter to its update expression
References
----------
.. [1] https://openreview.net/forum?id=ryQu7f-RZ
"""
all_grads = get_or_compute_grads(loss_or_grads, params)
t_prev = theano.shared(utils.floatX(0.))
updates = OrderedDict()
# Using theano constant to prevent upcasting of float32
one = T.constant(1)
t = t_prev + 1
a_t = learning_rate*T.sqrt(one-beta2**t)/(one-beta1**t)
for param, g_t in zip(params, all_grads):
value = param.get_value(borrow=True)
m_prev = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
v_prev = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
v_hat_prev = theano.shared(np.zeros(value.shape, dtype=value.dtype),
broadcastable=param.broadcastable)
m_t = beta1*m_prev + (one-beta1)*g_t
v_t = beta2*v_prev + (one-beta2)*g_t**2
v_hat_t = T.maximum(v_hat_prev, v_t)
step = a_t*m_t/(T.sqrt(v_hat_t) + epsilon)
updates[m_prev] = m_t
updates[v_prev] = v_t
updates[v_hat_prev] = v_hat_t
updates[param] = param - step
updates[t_prev] = t
return updates
def norm_constraint(tensor_var, max_norm, norm_axes=None, epsilon=1e-7):
"""Max weight norm constraints and gradient clipping
This takes a TensorVariable and rescales it so that incoming weight
norms are below a specified constraint value. Vectors violating the
constraint are rescaled so that they are within the allowed range.
Parameters
----------
tensor_var : TensorVariable
Theano expression for update, gradient, or other quantity.
max_norm : scalar
This value sets the maximum allowed value of any norm in
`tensor_var`.
norm_axes : sequence (list or tuple)
The axes over which to compute the norm. This overrides the
default norm axes defined for the number of dimensions
in `tensor_var`. When this is not specified and `tensor_var` is a
matrix (2D), this is set to `(0,)`. If `tensor_var` is a 3D, 4D or
5D tensor, it is set to a tuple listing all axes but axis 0. The
former default is useful for working with dense layers, the latter
is useful for 1D, 2D and 3D convolutional layers.
(Optional)
epsilon : scalar, optional
Value used to prevent numerical instability when dividing by
very small or zero norms.
Returns
-------
TensorVariable
Input `tensor_var` with rescaling applied to weight vectors
that violate the specified constraints.
Examples
--------
>>> param = theano.shared(
... np.random.randn(100, 200).astype(theano.config.floatX))
>>> update = param + 100
>>> update = norm_constraint(update, 10)
>>> func = theano.function([], [], updates=[(param, update)])
>>> # Apply constrained update
>>> _ = func()
>>> from lasagne.utils import compute_norms
>>> norms = compute_norms(param.get_value())
>>> np.isclose(np.max(norms), 10)
True
Notes
-----
When `norm_axes` is not specified, the axes over which the norm is
computed depend on the dimensionality of the input variable. If it is
2D, it is assumed to come from a dense layer, and the norm is computed
over axis 0. If it is 3D, 4D or 5D, it is assumed to come from a
convolutional layer and the norm is computed over all trailing axes
beyond axis 0. For other uses, you should explicitly specify the axes
over which to compute the norm using `norm_axes`.
"""
ndim = tensor_var.ndim
if norm_axes is not None:
sum_over = tuple(norm_axes)
elif ndim == 2: # DenseLayer
sum_over = (0,)
elif ndim in [3, 4, 5]: # Conv{1,2,3}DLayer
sum_over = tuple(range(1, ndim))
else:
raise ValueError(
"Unsupported tensor dimensionality {}."
"Must specify `norm_axes`".format(ndim)
)
dtype = np.dtype(theano.config.floatX).type
norms = T.sqrt(T.sum(T.sqr(tensor_var), axis=sum_over, keepdims=True))
target_norms = T.clip(norms, 0, dtype(max_norm))
constrained_output = \
(tensor_var * (target_norms / (dtype(epsilon) + norms)))
return constrained_output
def total_norm_constraint(tensor_vars, max_norm, epsilon=1e-7,
return_norm=False):
"""Rescales a list of tensors based on their combined norm
If the combined norm of the input tensors exceeds the threshold then all
tensors are rescaled such that the combined norm is equal to the threshold.
Scaling the norms of the gradients is often used when training recurrent
neural networks [1]_.
Parameters
----------
tensor_vars : List of TensorVariables.
Tensors to be rescaled.
max_norm : float
Threshold value for total norm.
epsilon : scalar, optional
Value used to prevent numerical instability when dividing by
very small or zero norms.
return_norm : bool
If true the total norm is also returned.
Returns
-------
tensor_vars_scaled : list of TensorVariables
The scaled tensor variables.
norm : Theano scalar
The combined norms of the input variables prior to rescaling,
only returned if ``return_norms=True``.
Examples
--------
>>> from lasagne.layers import InputLayer, DenseLayer
>>> import lasagne
>>> from lasagne.updates import sgd, total_norm_constraint
>>> x = T.matrix()
>>> y = T.ivector()
>>> l_in = InputLayer((5, 10))
>>> l1 = DenseLayer(l_in, num_units=7, nonlinearity=T.nnet.softmax)
>>> output = lasagne.layers.get_output(l1, x)
>>> cost = T.mean(T.nnet.categorical_crossentropy(output, y))
>>> all_params = lasagne.layers.get_all_params(l1)
>>> all_grads = T.grad(cost, all_params)
>>> scaled_grads = total_norm_constraint(all_grads, 5)
>>> updates = sgd(scaled_grads, all_params, learning_rate=0.1)
Notes
-----
The total norm can be used to monitor training.
References
----------
.. [1] Sutskever, I., Vinyals, O., & Le, Q. V. (2014): Sequence to sequence
learning with neural networks. In Advances in Neural Information
Processing Systems (pp. 3104-3112).
"""
norm = T.sqrt(sum(T.sum(tensor**2) for tensor in tensor_vars))
dtype = np.dtype(theano.config.floatX).type
target_norm = T.clip(norm, 0, dtype(max_norm))
multiplier = target_norm / (dtype(epsilon) + norm)
tensor_vars_scaled = [step*multiplier for step in tensor_vars]
if return_norm:
return tensor_vars_scaled, norm
else:
return tensor_vars_scaled