"""
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 :class:`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))
"""
import copy
import numpy as np
from .initializations import _zero
class Optimizer(object):
"""Abstract optimizer base class.
Note: this is the parent class of all optimizers, not an actual optimizer
that can be used for training models.
Parameters
----------
clip : float
If smaller than 0, do not apply parameter clip.
lr : float
The learning rate controlling the size of update steps
decay : float
Decay parameter for the moving average. Must lie in [0, 1) where
lower numbers means a shorter “memory”.
lr_min : float
When adapting step rates, do not move below this value. Default is 0.
lr_max : float
When adapting step rates, do not move above this value. Default is inf.
"""
def __init__(self, lr=0.001, clip=-1, decay=0., lr_min=0., lr_max=np.inf):
self.lr = lr
self.clip = clip
self.decay = decay
self.lr_min = lr_min
self.lr_max = lr_max
self.iterations = 0
def update(self, params, grads):
"""Update parameters.
Parameters
----------
params : list
A list of parameters in model.
grads : list
A list of gradients in model.
"""
self.iterations += 1
self.lr *= (1. / 1 + self.decay * self.iterations)
self.lr = np.clip(self.lr, self.lr_min, self.lr_max)
def __str__(self):
return self.__class__.__name__
[docs]class SGD(Optimizer):
"""Stochastic Gradient Descent (SGD) updates
Generates update expressions of the form:
* ``param := param - learning_rate * gradient``
"""
def __init__(self, *args, **kwargs):
super(SGD, self).__init__(*args, **kwargs)
[docs] def update(self, params, grads):
for p, g in zip(params, grads):
p -= self.lr * npdl_clip(g, self.clip)
super(SGD, self).update(params, grads)
[docs]class Momentum(Optimizer):
"""Stochastic Gradient Descent (SGD) updates with momentum
Generates update expressions of the form:
* ``velocity := momentum * velocity - learning_rate * gradient``
* ``param := param + velocity``
Parameters
----------
momentum : float
The amount of momentum to apply. Higher momentum results in
smoothing over more update steps. Defaults to 0.9.
Notes
-----
Higher momentum also results in larger update steps. To counter that,
you can optionally scale your learning rate by `1 - momentum`.
"""
def __init__(self, momentum=0.9, *args, **kwargs):
super(Momentum, self).__init__(*args, **kwargs)
self.momentum = momentum
self.velocity = None
[docs] def update(self, params, grads):
# init the velocities
if self.velocity is None:
self.velocity = [_zero(p.shape) for p in params]
# update the parameters
for i, (v, p, g) in enumerate(zip(self.velocity, params, grads)):
v = self.momentum * v - self.lr * g
p += v
self.velocity[i] = v
super(Momentum, self).update(params, grads)
[docs]class NesterovMomentum(Optimizer):
"""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
----------
momentum : float
The amount of momentum to apply. Higher momentum results in
smoothing over more update steps. Defaults to 0.9.
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.
"""
def __init__(self, momentum=0.9, *args, **kwargs):
super(NesterovMomentum, self).__init__(*args, **kwargs)
self.momentum = momentum
self.velocity = None
[docs] def update(self, params, grads):
# init the velocities
if self.velocity is None:
self.velocity = [_zero(p.shape) for p in params]
# update the parameters
for i, (v, p, g) in enumerate(zip(self.velocity, params, grads)):
v = self.momentum * v - self.lr * g
p += (self.momentum * v - self.lr * g)
self.velocity[i] = v
super(NesterovMomentum, self).update(params, grads)
[docs]class Adagrad(Optimizer):
"""Adagrad updates
Scale learning rates by dividing with the square root of accumulated
squared gradients. See [1]_ for further description.
Parameters
----------
epsilon : float
Small value added for numerical stability.
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
"""
def __init__(self, epsilon=1e-6, *args, **kwargs):
super(Adagrad, self).__init__(*args, **kwargs)
self.epsilon = epsilon
self.cache = None
[docs] def update(self, params, grads):
# init cache
if self.cache is None:
self.cache = [_zero(g.shape) for g in grads]
# update parameters
for i, (c, p, g) in enumerate(zip(self.cache, params, grads)):
c += np.power(g, 2)
p -= self.lr * g / (np.sqrt(c) + self.epsilon)
self.cache[i] = c
super(Adagrad, self).update(params, grads)
[docs]class RMSprop(Optimizer):
"""RMSProp updates
Scale learning rates by dividing with the moving average of the root mean
squared (RMS) gradients. See [1]_ for further description.
Parameters
----------
rho : float
Gradient moving average decay factor.
epsilon : float
Small value added for numerical stability.
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)
"""
def __init__(self, rho=0.9, epsilon=1e-6, *args, **kwargs):
super(RMSprop, self).__init__(*args, **kwargs)
self.rho = rho
self.epsilon = epsilon
self.cache = None
self.iterations = 0
[docs] def update(self, params, grads):
# init cache
if self.cache is None:
self.cache = [_zero(p.shape) for p in params]
# update parameters
for i, (c, p, g) in enumerate(zip(self.cache, params, grads)):
c = self.rho * c + (1 - self.rho) * np.power(g, 2)
p -= (self.lr * g / np.sqrt(c + self.epsilon))
self.cache[i] = c
[docs]class Adadelta(Optimizer):
""" Adadelta updates
Scale learning rates by the ratio of accumulated gradients to accumulated
updates, see [1]_ and notes for further description.
Parameters
----------
rho : float
Gradient moving average decay factor.
epsilon : float
Small value added for numerical stability.
decay : float
Decay parameter for the moving average.
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.
"""
def __init__(self, rho=0.9, epsilon=1e-6, *args, **kwargs):
super(Adadelta, self).__init__(*args, **kwargs)
self.rho = rho
self.epsilon = epsilon
self.cache = None
self.delta = None
[docs] def update(self, params, grads):
# init cache and delta
if self.cache is None:
self.cache = [_zero(p.shape) for p in params]
if self.delta is None:
self.delta = [_zero(p.shape) for p in params]
# update parameters
for i, (c, d, p, g) in enumerate(zip(self.cache, self.delta, params, grads)):
c = self.rho * c + (1 - self.rho) * np.power(g, 2)
update = g * np.sqrt(d + self.epsilon) / np.sqrt(c + self.epsilon)
p -= self.lr * update
d = self.rho * d + (1 - self.rho) * np.power(update, 2)
self.cache[i] = c
self.delta[i] = d
[docs]class Adam(Optimizer):
"""Adam updates
Adam updates implemented as in [1]_.
Parameters
----------
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.
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.
"""
def __init__(self, beta1=0.9, beta2=0.999, epsilon=1e-8, *args, **kwargs):
super(Adam, self).__init__(*args, **kwargs)
self.beta1 = beta1
self.beta2 = beta2
self.epsilon = epsilon
self.ms = None
self.vs = None
[docs] def update(self, params, grads):
# init
self.iterations += 1
a_t = self.lr * np.sqrt(1 - np.power(self.beta2, self.iterations)) / \
(1 - np.power(self.beta1, self.iterations))
if self.ms is None:
self.ms = [_zero(p.shape) for p in params]
if self.vs is None:
self.vs = [_zero(p.shape) for p in params]
# update parameters
for i, (m, v, p, g) in enumerate(zip(self.ms, self.vs, params, grads)):
m = self.beta1 * m + (1 - self.beta1) * g
v = self.beta2 * v + (1 - self.beta2) * np.power(g, 2)
p -= a_t * m / (np.sqrt(v) + self.epsilon)
self.ms[i] = m
self.vs[i] = v
[docs]class Adamax(Optimizer):
"""Adamax updates
Adamax updates implemented as in [1]_. This is a variant of of the Adam
algorithm based on the infinity norm.
Parameters
----------
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.
References
----------
.. [1] Kingma, Diederik, and Jimmy Ba (2014):
Adam: A Method for Stochastic Optimization.
arXiv preprint arXiv:1412.6980.
"""
def __init__(self, beta1=0.9, beta2=0.999, epsilon=1e-8, *args, **kwargs):
super(Adamax, self).__init__(*args, **kwargs)
self.beta1 = beta1
self.beta2 = beta2
self.epsilon = epsilon
self.ms = None
self.vs = None
[docs] def update(self, params, grads):
# init
self.iterations += 1
a_t = self.lr / (1 - np.power(self.beta1, self.iterations))
if self.ms is None:
self.ms = [_zero(p.shape) for p in params]
if self.vs is None:
self.vs = [_zero(p.shape) for p in params]
# update parameters
for i, (m, v, p, g) in enumerate(zip(self.ms, self.vs, params, grads)):
m = self.beta1 * m + (1 - self.beta1) * g
v = np.maximum(self.beta2 * v, np.abs(g))
p -= a_t * m / (v + self.epsilon)
self.ms[i] = m
self.vs[i] = v
def npdl_clip(grad, boundary):
if boundary > 0:
return np.clip(grad, -boundary, boundary)
else:
return grad
def get(optimizer):
if optimizer.__class__.__name__ == 'str':
if optimizer in ['sgd', 'SGD']:
return SGD()
if optimizer in ['momentum', 'Momentum']:
return Momentum()
if optimizer in ['nesterov_momentum', 'NesterovMomentum']:
return NesterovMomentum()
if optimizer in ['adagrad', 'Adagrad']:
return Adagrad()
if optimizer in ['rmsprop', 'RMSprop']:
return RMSprop()
if optimizer in ['adadelta', 'Adadelta']:
return Adadelta()
if optimizer in ['adam', 'Adam']:
return Adam()
if optimizer in ['adamax', 'Adamax']:
return Adamax()
raise ValueError('Unknown optimizer name: {}.'.format(optimizer))
elif isinstance(optimizer, Optimizer):
return copy.deepcopy(optimizer)
else:
raise ValueError("Unknown type: {}.".format(optimizer.__class__.__name__))