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============
``DOWNHILL``
============
The ``downhill`` package provides algorithms for minimizing scalar loss
functions that are defined using Theano_.
Several optimization algorithms are included:
- ADADELTA_
- ADAGRAD_
- Adam_
- `Equilibrated SGD`_
- `Nesterov's Accelerated Gradient`_
- RMSProp_
- `Resilient Backpropagation`_
- `Stochastic Gradient Descent`_
All algorithms permit the use of regular or Nesterov-style momentum as well.
.. _Theano: http://deeplearning.net/software/theano/
.. _Stochastic Gradient Descent: http://downhill.readthedocs.org/en/stable/generated/downhill.first_order.SGD.html
.. _Nesterov's Accelerated Gradient: http://downhill.readthedocs.org/en/stable/generated/downhill.first_order.NAG.html
.. _Resilient Backpropagation: http://downhill.readthedocs.org/en/stable/generated/downhill.adaptive.RProp.html
.. _ADAGRAD: http://downhill.readthedocs.org/en/stable/generated/downhill.adaptive.ADAGRAD.html
.. _RMSProp: http://downhill.readthedocs.org/en/stable/generated/downhill.adaptive.RMSProp.html
.. _ADADELTA: http://downhill.readthedocs.org/en/stable/generated/downhill.adaptive.ADADELTA.html
.. _Adam: http://downhill.readthedocs.org/en/stable/generated/downhill.adaptive.Adam.html
.. _Equilibrated SGD: http://downhill.readthedocs.org/en/stable/generated/downhill.adaptive.ESGD.html
Quick Start: Matrix Factorization
=================================
Let's say you have 100 samples of 1000-dimensional data, and you want to
represent your data as 100 coefficients in a 10-dimensional basis. This is
pretty straightforward to model using Theano: you can use a matrix
multiplication as the data model, a squared-error term for optimization, and a
sparse regularizer to encourage small coefficient values.
Once you have constructed an expression for the loss, you can optimize it with a
single call to ``downhill.minimize``:
.. code:: python
import downhill
import numpy as np
import theano
import theano.tensor as TT
FLOAT = 'df'[theano.config.floatX == 'float32']
def rand(a, b):
return np.random.randn(a, b).astype(FLOAT)
A, B, K = 20, 5, 3
# Set up a matrix factorization problem to optimize.
u = theano.shared(rand(A, K), name='u')
v = theano.shared(rand(K, B), name='v')
z = TT.matrix()
err = TT.sqr(z - TT.dot(u, v))
loss = err.mean() + abs(u).mean() + (v * v).mean()
# Minimize the regularized loss with respect to a data matrix.
y = np.dot(rand(A, K), rand(K, B)) + rand(A, B)
# Monitor during optimization.
monitors = (('err', err.mean()),
('|u|<0.1', (abs(u) < 0.1).mean()),
('|v|<0.1', (abs(v) < 0.1).mean()))
downhill.minimize(
loss=loss,
train=[y],
patience=0,
batch_size=A, # Process y as a single batch.
max_gradient_norm=1, # Prevent gradient explosion!
learning_rate=0.1,
monitors=monitors,
monitor_gradients=True)
# Print out the optimized coefficients u and basis v.
print('u =', u.get_value())
print('v =', v.get_value())
If you prefer to maintain more control over your model during optimization,
downhill provides an iterative optimization interface:
.. code:: python
opt = downhill.build(algo='rmsprop',
loss=loss,
monitors=monitors,
monitor_gradients=True)
for metrics, _ in opt.iterate(train=[[y]],
patience=0,
batch_size=A,
max_gradient_norm=1,
learning_rate=0.1):
print(metrics)
If that's still not enough, you can just plain ask downhill for the updates to
your model variables and do everything else yourself:
.. code:: python
updates = downhill.build('rmsprop', loss).get_updates(
batch_size=A, max_gradient_norm=1, learning_rate=0.1)
func = theano.function([z], loss, updates=list(updates))
for _ in range(100):
print(func(y)) # Evaluate func and apply variable updates.
More Information
================
Source: http://github.com/lmjohns3/downhill
Documentation: http://downhill.readthedocs.org
Mailing list: https://groups.google.com/forum/#!forum/downhill-users