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:

Two functions can be used to further modify the updates to include momentum:

 apply_momentum Returns a modified update dictionary including momentum apply_nesterov_momentum Returns a modified update dictionary including Nesterov momentum

Finally, we provide two helper functions to constrain the norm of tensors:

 norm_constraint Max weight norm constraints and gradient clipping total_norm_constraint Rescales a list of tensors based on their combined norm

norm_constraint() can be used to constrain the norm of parameters (as an alternative to weight decay), or for a form of gradient clipping. total_norm_constraint() constrain the total norm of a list of tensors. This is often used when training recurrent neural networks.

## Examples¶

Using 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
>>> 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)


With apply_momentum() and 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)


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 lasagne.utils.floatX() to ensure this:

>>> eta = theano.shared(lasagne.utils.floatX(0.001))
>>> # we can now modify the learning rate at any time during training:
>>> eta.set_value(lasagne.utils.floatX(eta.get_value() * 0.1))


## Update functions¶

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 OrderedDict A dictionary mapping each parameter to its update expression

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. OrderedDict A dictionary mapping each parameter to its update expression

apply_momentum
Generic function applying momentum to updates
nesterov_momentum
Nesterov’s variant of SGD with momentum

Notes

Higher momentum also results in larger update steps. To counter that, you can optionally scale your learning rate by 1 - 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. OrderedDict A dictionary mapping each parameter to its update expression

apply_nesterov_momentum

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.

Scale learning rates by dividing with the square root of accumulated squared gradients. See [R167168] 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 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:

$\eta_{t,i} = \frac{\eta} {\sqrt{\sum^t_{t^\prime} g^2_{t^\prime,i}+\epsilon}} g_{t,i}$

as such the learning rate is monotonically decreasing.

Epsilon is not included in the typical formula, see [R168168].

References

 [R167168] (1, 2) Duchi, J., Hazan, E., & Singer, Y. (2011): Adaptive subgradient methods for online learning and stochastic optimization. JMLR, 12:2121-2159.

Scale learning rates by dividing with the moving average of the root mean squared (RMS) gradients. See [R171171] 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 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 $$\eta$$ and a decay factor $$\rho$$ the learning rate $$\eta_t$$ is calculated as:

$\begin{split}r_t &= \rho r_{t-1} + (1-\rho)*g^2\\ \eta_t &= \frac{\eta}{\sqrt{r_t + \epsilon}}\end{split}$

References

 [R171171] (1, 2) 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)

Scale learning rates by the ratio of accumulated gradients to accumulated updates, see [R173173] 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 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:

$\begin{split}r_t &= \rho r_{t-1} + (1-\rho)*g^2\\ \eta_t &= \eta \frac{\sqrt{s_{t-1} + \epsilon}} {\sqrt{r_t + \epsilon}}\\ s_t &= \rho s_{t-1} + (1-\rho)*(\eta_t*g)^2\end{split}$

References

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. OrderedDict A dictionary mapping each parameter to its update expression

Notes

The paper [R175175] 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

 [R175175] (1, 2, 3) Kingma, Diederik, and Jimmy Ba (2014): Adam: A Method for Stochastic Optimization. arXiv preprint arXiv:1412.6980.

Adamax updates implemented as in [R177177]. 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. OrderedDict A dictionary mapping each parameter to its update expression

References

 [R177177] (1, 2) Kingma, Diederik, and Jimmy Ba (2014): Adam: A Method for Stochastic Optimization. arXiv preprint arXiv:1412.6980.

## Update modification functions¶

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. OrderedDict A copy of updates with momentum updates for all params.

momentum
Shortcut applying momentum to SGD updates

Notes

Higher momentum also results in larger update steps. To counter that, you can optionally scale your learning rate by 1 - momentum.

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. OrderedDict A copy of updates with momentum updates for all params.

nesterov_momentum
Shortcut applying Nesterov momentum to SGD updates

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.

## Helper functions¶

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. TensorVariable Input tensor_var with rescaling applied to weight vectors that violate the specified constraints.

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.

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


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 [R179179].

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. 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.

Notes

The total norm can be used to monitor training.

References

 [R179179] (1, 2) 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).

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)