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
Learning rate
beta1 : float
Exponential decay rate for the first moment estimates.
beta2 : float
Exponential decay rate for the second moment estimates.
epsilon : float
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(0.0))
updates = OrderedDict()
t = t_prev + 1
a_t = learning_rate * T.sqrt(1 - beta2 ** t) / (1 - 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 + (1 - beta1) * g_t
v_t = beta2 * v_prev + (1 - 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 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 = theano.compat.get_value(param, borrow=True)
accu = theano.shared(np.zeros(value.shape, dtype=value.dtype), broadcastable=theano.compat.broadcastable(param))
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()
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 + (1 - rho) * grad ** 2
updates[accu] = accu_new
updates[param] = param - (learning_rate * grad / T.sqrt(accu_new + epsilon))
return updates
def get_output_for(self, input, deterministic=False, **kwargs):
input_mean = input.mean(self.axes)
input_inv_std = T.inv(T.sqrt(input.var(self.axes) + self.epsilon))
# Decide whether to use the stored averages or mini-batch statistics
use_averages = kwargs.get('batch_norm_use_averages',
deterministic)
if use_averages:
mean = self.mean
inv_std = self.inv_std
else:
mean = input_mean
inv_std = input_inv_std
# Decide whether to update the stored averages
update_averages = kwargs.get('batch_norm_update_averages',
not deterministic)
if update_averages:
# Trick: To update the stored statistics, we create memory-aliased
# clones of the stored statistics:
running_mean = theano.clone(self.mean, share_inputs=False)
running_inv_std = theano.clone(self.inv_std, share_inputs=False)
# set a default update for them:
running_mean.default_update = ((1 - self.alpha) * running_mean +
self.alpha * input_mean)
running_inv_std.default_update = ((1 - self.alpha) *
running_inv_std +
self.alpha * input_inv_std)
# and make sure they end up in the graph without participating in
# the computation (this way their default_update will be collected
# and applied, but the computation will be optimized away):
mean += 0 * running_mean
inv_std += 0 * running_inv_std
# prepare dimshuffle pattern inserting broadcastable axes as needed
param_axes = iter(range(input.ndim - len(self.axes)))
pattern = ['x' if input_axis in self.axes
else next(param_axes)
for input_axis in range(input.ndim)]
# apply dimshuffle pattern to all parameters
beta = 0 if self.beta is None else self.beta.dimshuffle(pattern)
gamma = 1 if self.gamma is None else self.gamma.dimshuffle(pattern)
mean = mean.dimshuffle(pattern)
inv_std = inv_std.dimshuffle(pattern)
# normalize
normalized = (input - mean) * (gamma * inv_std) + beta
return normalized
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
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 adadelta(loss_or_grads, params, learning_rate=1.0, rho=0.95, epsilon=1e-6):
""" Adadelta updates
Scale learning rates by a the ratio of accumulated gradients to accumulated
step sizes, 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()
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 + (1 - 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 + (1 - rho) * update ** 2
updates[delta_accu] = delta_accu_new
return updates
