npdl.optimizers

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:

Examples

Using SGD to define an update dictionary for a toy example network:

>>> import npdl
>>> from npdl.activations import ReLU
>>> from npdl.activations import Softmax
>>> from npdl.objectives import SCCE
>>> model = npdl.model.Model()
>>> model.add(npdl.layers.Dense(n_out=100, n_in=50, activation=ReLU()))
>>> model.add(npdl.layers.Dense(n_out=200, activation=ReLU()))
>>> model.add(npdl.layers.Dense(n_out=100, activation=ReLU()))
>>> model.add(npdl.layers.Dense(n_out=10, activation=Softmax()))
>>> model.compile(loss=SCCE(), optimizer=npdl.optimizers.SGD(lr=0.005))

Optimizers

SGD Stochastic Gradient Descent (SGD) updates
Momentum Stochastic Gradient Descent (SGD) updates with momentum
NesterovMomentum Stochastic Gradient Descent (SGD) updates with Nesterov momentum
Adagrad Adagrad updates
RMSprop RMSProp updates
Adadelta Adadelta updates
Adam Adam updates
Adamax Adamax updates

Detailed Description

class npdl.optimizers.SGD(lr=0.001, clip=-1)[source]

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

class npdl.optimizers.Momentum(lr=0.01, momentum=0.9)[source]

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

See also

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.

class npdl.optimizers.NesterovMomentum[source]

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

See also

apply_nesterov_momentum
Function applying momentum to 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.

class npdl.optimizers.Adagrad[source]

Adagrad updates

Scale learning rates by dividing with the square root of accumulated squared gradients. See [R33] 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:

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

as such the learning rate is monotonically decreasing.

Epsilon is not included in the typical formula, see [R34].

References

[R33](1, 2) Duchi, J., Hazan, E., & Singer, Y. (2011): Adaptive subgradient methods for online learning and stochastic optimization. JMLR, 12:2121-2159.
[R34](1, 2) Chris Dyer: Notes on AdaGrad. http://www.ark.cs.cmu.edu/cdyer/adagrad.pdf
class npdl.optimizers.RMSprop[source]

RMSProp updates

Scale learning rates by dividing with the moving average of the root mean squared (RMS) gradients. See [R35] 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 \(\eta\) 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

[R35](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)
class npdl.optimizers.Adadelta[source]

Adadelta updates

Scale learning rates by the ratio of accumulated gradients to accumulated updates, see [R36] 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:

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

References

[R36](1, 2) Zeiler, M. D. (2012): ADADELTA: An Adaptive Learning Rate Method. arXiv Preprint arXiv:1212.5701.
class npdl.optimizers.Adam[source]

Adam updates

Adam updates implemented as in [R37].

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 [R37] 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

[R37](1, 2, 3) Kingma, Diederik, and Jimmy Ba (2014): Adam: A Method for Stochastic Optimization. arXiv preprint arXiv:1412.6980.
class npdl.optimizers.Adamax[source]

Adamax updates

Adamax updates implemented as in [R38]. 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

[R38](1, 2) Kingma, Diederik, and Jimmy Ba (2014): Adam: A Method for Stochastic Optimization. arXiv preprint arXiv:1412.6980.