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AntreasAntoniou 2017-09-29 17:54:05 +01:00
parent aced4b3e7d
commit 78668638a5
10 changed files with 1547 additions and 28 deletions
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97
mlp/data_providers.py
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@ -75,6 +75,9 @@ class DataProvider(object):
self.inputs = self.inputs[new_order]
self.targets = self.targets[new_order]
def __next__(self):
return self.next()
def next(self):
"""Returns next data batch or raises `StopIteration` if at end."""
if self._curr_batch + 1 > self.num_batches:
@ -133,13 +136,10 @@ class MNISTDataProvider(DataProvider):
super(MNISTDataProvider, self).__init__(
inputs, targets, batch_size, max_num_batches, shuffle_order, rng)
# def next(self):
# """Returns next data batch or raises `StopIteration` if at end."""
# inputs_batch, targets_batch = super(MNISTDataProvider, self).next()
# return inputs_batch, self.to_one_of_k(targets_batch)
#
def __next__(self):
return self.next()
def next(self):
"""Returns next data batch or raises `StopIteration` if at end."""
inputs_batch, targets_batch = super(MNISTDataProvider, self).next()
return inputs_batch, self.to_one_of_k(targets_batch)
def to_one_of_k(self, int_targets):
"""Converts integer coded class target to 1 of K coded targets.
@ -156,7 +156,9 @@ class MNISTDataProvider(DataProvider):
to zero except for the column corresponding to the correct class
which is equal to one.
"""
raise NotImplementedError()
one_of_k_targets = np.zeros((int_targets.shape[0], self.num_classes))
one_of_k_targets[range(int_targets.shape[0]), int_targets] = 1
return one_of_k_targets
class MetOfficeDataProvider(DataProvider):
@ -164,7 +166,7 @@ class MetOfficeDataProvider(DataProvider):
def __init__(self, window_size, batch_size=10, max_num_batches=-1,
shuffle_order=True, rng=None):
"""Create a new Met Offfice data provider object.
"""Create a new Met Office data provider object.
Args:
window_size (int): Size of windows to split weather time series
@ -180,27 +182,74 @@ class MetOfficeDataProvider(DataProvider):
the data before each epoch.
rng (RandomState): A seeded random number generator.
"""
self.window_size = window_size
assert window_size > 1, 'window_size must be at least 2.'
data_path = os.path.join(
os.environ['MLP_DATA_DIR'], 'HadSSP_daily_qc.txt')
assert os.path.isfile(data_path), (
'Data file does not exist at expected path: ' + data_path
)
# load raw data from text file
# ...
raw = np.loadtxt(data_path, skiprows=3, usecols=range(2, 32))
assert window_size > 1, 'window_size must be at least 2.'
self.window_size = window_size
# filter out all missing datapoints and flatten to a vector
# ...
filtered = raw[raw >= 0].flatten()
# normalise data to zero mean, unit standard deviation
# ...
# convert from flat sequence to windowed data
# ...
mean = np.mean(filtered)
std = np.std(filtered)
normalised = (filtered - mean) / std
# create a view on to array corresponding to a rolling window
shape = (normalised.shape[-1] - self.window_size + 1, self.window_size)
strides = normalised.strides + (normalised.strides[-1],)
windowed = np.lib.stride_tricks.as_strided(
normalised, shape=shape, strides=strides)
# inputs are first (window_size - 1) entries in windows
# inputs = ...
inputs = windowed[:, :-1]
# targets are last entry in windows
# targets = ...
# initialise base class with inputs and targets arrays
# super(MetOfficeDataProvider, self).__init__(
# inputs, targets, batch_size, max_num_batches, shuffle_order, rng)
def __next__(self):
return self.next()
targets = windowed[:, -1]
super(MetOfficeDataProvider, self).__init__(
inputs, targets, batch_size, max_num_batches, shuffle_order, rng)
class CCPPDataProvider(DataProvider):
def __init__(self, which_set='train', input_dims=None, batch_size=10,
max_num_batches=-1, shuffle_order=True, rng=None):
"""Create a new Combined Cycle Power Plant data provider object.
Args:
which_set: One of 'train' or 'valid'. Determines which portion of
data this object should provide.
input_dims: Which of the four input dimension to use. If `None` all
are used. If an iterable of integers are provided (consisting
of a subset of {0, 1, 2, 3}) then only the corresponding
input dimensions are included.
batch_size (int): Number of data points to include in each batch.
max_num_batches (int): Maximum number of batches to iterate over
in an epoch. If `max_num_batches * batch_size > num_data` then
only as many batches as the data can be split into will be
used. If set to -1 all of the data will be used.
shuffle_order (bool): Whether to randomly permute the order of
the data before each epoch.
rng (RandomState): A seeded random number generator.
"""
data_path = os.path.join(
os.environ['MLP_DATA_DIR'], 'ccpp_data.npz')
assert os.path.isfile(data_path), (
'Data file does not exist at expected path: ' + data_path
)
# check a valid which_set was provided
assert which_set in ['train', 'valid'], (
'Expected which_set to be either train or valid '
'Got {0}'.format(which_set)
)
# check input_dims are valid
if not input_dims is not None:
input_dims = set(input_dims)
assert input_dims.issubset({0, 1, 2, 3}), (
'input_dims should be a subset of {0, 1, 2, 3}'
)
loaded = np.load(data_path)
inputs = loaded[which_set + '_inputs']
if input_dims is not None:
inputs = inputs[:, input_dims]
targets = loaded[which_set + '_targets']
super(CCPPDataProvider, self).__init__(
inputs, targets, batch_size, max_num_batches, shuffle_order, rng)

44
mlp/errors.py Normal file
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@ -0,0 +1,44 @@
# -*- coding: utf-8 -*-
"""Error functions.
This module defines error functions, with the aim of model training being to
minimise the error function given a set of inputs and target outputs.
The error functions will typically measure some concept of distance between the
model outputs and target outputs, averaged over all data points in the data set
or batch.
"""
import numpy as np
class SumOfSquaredDiffsError(object):
"""Sum of squared differences (squared Euclidean distance) error."""
def __call__(self, outputs, targets):
"""Calculates error function given a batch of outputs and targets.
Args:
outputs: Array of model outputs of shape (batch_size, output_dim).
targets: Array of target outputs of shape (batch_size, output_dim).
Returns:
Scalar error function value.
"""
raise NotImplementedError()
def grad(self, outputs, targets):
"""Calculates gradient of error function with respect to outputs.
Args:
outputs: Array of model outputs of shape (batch_size, output_dim).
targets: Array of target outputs of shape (batch_size, output_dim).
Returns:
Gradient of error function with respect to outputs. This should be
an array of shape (batch_size, output_dim).
"""
raise NotImplementedError()
def __repr__(self):
return 'SumOfSquaredDiffsError'

65
mlp/initialisers.py Normal file
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@ -0,0 +1,65 @@
# -*- coding: utf-8 -*-
"""Parameter initialisers.
This module defines classes to initialise the parameters in a layer.
"""
import numpy as np
from mlp import DEFAULT_SEED
class ConstantInit(object):
"""Constant parameter initialiser."""
def __init__(self, value):
"""Construct a constant parameter initialiser.
Args:
value: Value to initialise parameter to.
"""
self.value = value
def __call__(self, shape):
return np.ones(shape=shape) * self.value
class UniformInit(object):
"""Random uniform parameter initialiser."""
def __init__(self, low, high, rng=None):
"""Construct a random uniform parameter initialiser.
Args:
low: Lower bound of interval to sample from.
high: Upper bound of interval to sample from.
rng (RandomState): Seeded random number generator.
"""
self.low = low
self.high = high
if rng is None:
rng = np.random.RandomState(DEFAULT_SEED)
self.rng = rng
def __call__(self, shape):
return self.rng.uniform(low=self.low, high=self.high, size=shape)
class NormalInit(object):
"""Random normal parameter initialiser."""
def __init__(self, mean, std, rng=None):
"""Construct a random uniform parameter initialiser.
Args:
mean: Mean of distribution to sample from.
std: Standard deviation of distribution to sample from.
rng (RandomState): Seeded random number generator.
"""
self.mean = mean
self.std = std
if rng is None:
rng = np.random.RandomState(DEFAULT_SEED)
self.rng = rng
def __call__(self, shape):
return self.rng.normal(loc=self.mean, scale=self.std, size=shape)

139
mlp/layers.py Normal file
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@ -0,0 +1,139 @@
# -*- coding: utf-8 -*-
"""Layer definitions.
This module defines classes which encapsulate a single layer.
These layers map input activations to output activation with the `fprop`
method and map gradients with repsect to outputs to gradients with respect to
their inputs with the `bprop` method.
Some layers will have learnable parameters and so will additionally define
methods for getting and setting parameter and calculating gradients with
respect to the layer parameters.
"""
import numpy as np
import mlp.initialisers as init
class Layer(object):
"""Abstract class defining the interface for a layer."""
def fprop(self, inputs):
"""Forward propagates activations through the layer transformation.
Args:
inputs: Array of layer inputs of shape (batch_size, input_dim).
Returns:
outputs: Array of layer outputs of shape (batch_size, output_dim).
"""
raise NotImplementedError()
def bprop(self, inputs, outputs, grads_wrt_outputs):
"""Back propagates gradients through a layer.
Given gradients with respect to the outputs of the layer calculates the
gradients with respect to the layer inputs.
Args:
inputs: Array of layer inputs of shape (batch_size, input_dim).
outputs: Array of layer outputs calculated in forward pass of
shape (batch_size, output_dim).
grads_wrt_outputs: Array of gradients with respect to the layer
outputs of shape (batch_size, output_dim).
Returns:
Array of gradients with respect to the layer inputs of shape
(batch_size, input_dim).
"""
raise NotImplementedError()
class LayerWithParameters(Layer):
"""Abstract class defining the interface for a layer with parameters."""
def grads_wrt_params(self, inputs, grads_wrt_outputs):
"""Calculates gradients with respect to layer parameters.
Args:
inputs: Array of inputs to layer of shape (batch_size, input_dim).
grads_wrt_to_outputs: Array of gradients with respect to the layer
outputs of shape (batch_size, output_dim).
Returns:
List of arrays of gradients with respect to the layer parameters
with parameter gradients appearing in same order in tuple as
returned from `get_params` method.
"""
raise NotImplementedError()
@property
def params(self):
"""Returns a list of parameters of layer.
Returns:
List of current parameter values.
"""
raise NotImplementedError()
class AffineLayer(LayerWithParameters):
"""Layer implementing an affine tranformation of its inputs.
This layer is parameterised by a weight matrix and bias vector.
"""
def __init__(self, input_dim, output_dim,
weights_initialiser=init.UniformInit(-0.1, 0.1),
biases_initialiser=init.ConstantInit(0.),
weights_cost=None, biases_cost=None):
"""Initialises a parameterised affine layer.
Args:
input_dim (int): Dimension of inputs to the layer.
output_dim (int): Dimension of the layer outputs.
weights_initialiser: Initialiser for the weight parameters.
biases_initialiser: Initialiser for the bias parameters.
"""
self.input_dim = input_dim
self.output_dim = output_dim
self.weights = weights_initialiser((self.output_dim, self.input_dim))
self.biases = biases_initialiser(self.output_dim)
def fprop(self, inputs):
"""Forward propagates activations through the layer transformation.
For inputs `x`, outputs `y`, weights `W` and biases `b` the layer
corresponds to `y = W.dot(x) + b`.
Args:
inputs: Array of layer inputs of shape (batch_size, input_dim).
Returns:
outputs: Array of layer outputs of shape (batch_size, output_dim).
"""
raise NotImplementedError()
def grads_wrt_params(self, inputs, grads_wrt_outputs):
"""Calculates gradients with respect to layer parameters.
Args:
inputs: array of inputs to layer of shape (batch_size, input_dim)
grads_wrt_to_outputs: array of gradients with respect to the layer
outputs of shape (batch_size, output_dim)
Returns:
list of arrays of gradients with respect to the layer parameters
`[grads_wrt_weights, grads_wrt_biases]`.
"""
raise NotImplementedError()
@property
def params(self):
"""A list of layer parameter values: `[weights, biases]`."""
return [self.weights, self.biases]
def __repr__(self):
return 'AffineLayer(input_dim={0}, output_dim={1})'.format(
self.input_dim, self.output_dim)

162
mlp/learning_rules.py Normal file
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@ -0,0 +1,162 @@
# -*- coding: utf-8 -*-
"""Learning rules.
This module contains classes implementing gradient based learning rules.
"""
import numpy as np
class GradientDescentLearningRule(object):
"""Simple (stochastic) gradient descent learning rule.
For a scalar error function `E(p[0], p_[1] ... )` of some set of
potentially multidimensional parameters this attempts to find a local
minimum of the loss function by applying updates to each parameter of the
form
p[i] := p[i] - learning_rate * dE/dp[i]
With `learning_rate` a positive scaling parameter.
The error function used in successive applications of these updates may be
a stochastic estimator of the true error function (e.g. when the error with
respect to only a subset of data-points is calculated) in which case this
will correspond to a stochastic gradient descent learning rule.
"""
def __init__(self, learning_rate=1e-3):
"""Creates a new learning rule object.
Args:
learning_rate: A postive scalar to scale gradient updates to the
parameters by. This needs to be carefully set - if too large
the learning dynamic will be unstable and may diverge, while
if set too small learning will proceed very slowly.
"""
assert learning_rate > 0., 'learning_rate should be positive.'
self.learning_rate = learning_rate
def initialise(self, params):
"""Initialises the state of the learning rule for a set or parameters.
This must be called before `update_params` is first called.
Args:
params: A list of the parameters to be optimised. Note these will
be updated *in-place* to avoid reallocating arrays on each
update.
"""
self.params = params
def reset(self):
"""Resets any additional state variables to their intial values.
For this learning rule there are no additional state variables so we
do nothing here.
"""
pass
def update_params(self, grads_wrt_params):
"""Applies a single gradient descent update to all parameters.
All parameter updates are performed using in-place operations and so
nothing is returned.
Args:
grads_wrt_params: A list of gradients of the scalar loss function
with respect to each of the parameters passed to `initialise`
previously, with this list expected to be in the same order.
"""
for param, grad in zip(self.params, grads_wrt_params):
param -= self.learning_rate * grad
class MomentumLearningRule(GradientDescentLearningRule):
"""Gradient descent with momentum learning rule.
This extends the basic gradient learning rule by introducing extra
momentum state variables for each parameter. These can help the learning
dynamic help overcome shallow local minima and speed convergence when
making multiple successive steps in a similar direction in parameter space.
For parameter p[i] and corresponding momentum m[i] the updates for a
scalar loss function `L` are of the form
m[i] := mom_coeff * m[i] - learning_rate * dL/dp[i]
p[i] := p[i] + m[i]
with `learning_rate` a positive scaling parameter for the gradient updates
and `mom_coeff` a value in [0, 1] that determines how much 'friction' there
is the system and so how quickly previous momentum contributions decay.
"""
def __init__(self, learning_rate=1e-3, mom_coeff=0.9):
"""Creates a new learning rule object.
Args:
learning_rate: A postive scalar to scale gradient updates to the
parameters by. This needs to be carefully set - if too large
the learning dynamic will be unstable and may diverge, while
if set too small learning will proceed very slowly.
mom_coeff: A scalar in the range [0, 1] inclusive. This determines
the contribution of the previous momentum value to the value
after each update. If equal to 0 the momentum is set to exactly
the negative scaled gradient each update and so this rule
collapses to standard gradient descent. If equal to 1 the
momentum will just be decremented by the scaled gradient at
each update. This is equivalent to simulating the dynamic in
a frictionless system. Due to energy conservation the loss
of 'potential energy' as the dynamics moves down the loss
function surface will lead to an increasingly large 'kinetic
energy' and so speed, meaning the updates will become
increasingly large, potentially unstably so. Typically a value
less than but close to 1 will avoid these issues and cause the
dynamic to converge to a local minima where the gradients are
by definition zero.
"""
super(MomentumLearningRule, self).__init__(learning_rate)
assert mom_coeff >= 0. and mom_coeff <= 1., (
'mom_coeff should be in the range [0, 1].'
)
self.mom_coeff = mom_coeff
def initialise(self, params):
"""Initialises the state of the learning rule for a set or parameters.
This must be called before `update_params` is first called.
Args:
params: A list of the parameters to be optimised. Note these will
be updated *in-place* to avoid reallocating arrays on each
update.
"""
super(MomentumLearningRule, self).initialise(params)
self.moms = []
for param in self.params:
self.moms.append(np.zeros_like(param))
def reset(self):
"""Resets any additional state variables to their intial values.
For this learning rule this corresponds to zeroing all the momenta.
"""
for mom in zip(self.moms):
mom *= 0.
def update_params(self, grads_wrt_params):
"""Applies a single update to all parameters.
All parameter updates are performed using in-place operations and so
nothing is returned.
Args:
grads_wrt_params: A list of gradients of the scalar loss function
with respect to each of the parameters passed to `initialise`
previously, with this list expected to be in the same order.
"""
for param, mom, grad in zip(self.params, self.moms, grads_wrt_params):
mom *= self.mom_coeff
mom -= self.learning_rate * grad
param += mom

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mlp/models.py Normal file
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@ -0,0 +1,67 @@
# -*- coding: utf-8 -*-
"""Model definitions.
This module implements objects encapsulating learnable models of input-output
relationships. The model objects implement methods for forward propagating
the inputs through the transformation(s) defined by the model to produce
outputs (and intermediate states) and for calculating gradients of scalar
functions of the outputs with respect to the model parameters.
"""
from mlp.layers import LayerWithParameters
class SingleLayerModel(object):
"""A model consisting of a single transformation layer."""
def __init__(self, layer):
"""Create a new single layer model instance.
Args:
layer: The layer object defining the model architecture.
"""
self.layer = layer
@property
def params(self):
"""A list of all of the parameters of the model."""
return self.layer.params
def fprop(self, inputs):
"""Calculate the model outputs corresponding to a batch of inputs.
Args:
inputs: Batch of inputs to the model.
Returns:
List which is a concatenation of the model inputs and model
outputs, this being done for consistency of the interface with
multi-layer models for which `fprop` returns a list of
activations through all immediate layers of the model and including
the inputs and outputs.
"""
activations = [inputs, self.layer.fprop(inputs)]
return activations
def grads_wrt_params(self, activations, grads_wrt_outputs):
"""Calculates gradients with respect to the model parameters.
Args:
activations: List of all activations from forward pass through
model using `fprop`.
grads_wrt_outputs: Gradient with respect to the model outputs of
the scalar function parameter gradients are being calculated
for.
Returns:
List of gradients of the scalar function with respect to all model
parameters.
"""
return self.layer.grads_wrt_params(activations[0], grads_wrt_outputs)
def params_cost(self):
"""Calculates the parameter dependent cost term of the model."""
return self.layer.params_cost()
def __repr__(self):
return 'SingleLayerModel(' + str(layer) + ')'

134
mlp/optimisers.py Normal file
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@ -0,0 +1,134 @@
# -*- coding: utf-8 -*-
"""Model optimisers.
This module contains objects implementing (batched) stochastic gradient descent
based optimisation of models.
"""
import time
import logging
from collections import OrderedDict
import numpy as np
logger = logging.getLogger(__name__)
class Optimiser(object):
"""Basic model optimiser."""
def __init__(self, model, error, learning_rule, train_dataset,
valid_dataset=None, data_monitors=None):
"""Create a new optimiser instance.
Args:
model: The model to optimise.
error: The scalar error function to minimise.
learning_rule: Gradient based learning rule to use to minimise
error.
train_dataset: Data provider for training set data batches.
valid_dataset: Data provider for validation set data batches.
data_monitors: Dictionary of functions evaluated on targets and
model outputs (averaged across both full training and
validation data sets) to monitor during training in addition
to the error. Keys should correspond to a string label for
the statistic being evaluated.
"""
self.model = model
self.error = error
self.learning_rule = learning_rule
self.learning_rule.initialise(self.model.params)
self.train_dataset = train_dataset
self.valid_dataset = valid_dataset
self.data_monitors = OrderedDict([('error', error)])
if data_monitors is not None:
self.data_monitors.update(data_monitors)
def do_training_epoch(self):
"""Do a single training epoch.
This iterates through all batches in training dataset, for each
calculating the gradient of the estimated error given the batch with
respect to all the model parameters and then updates the model
parameters according to the learning rule.
"""
for inputs_batch, targets_batch in self.train_dataset:
activations = self.model.fprop(inputs_batch)
grads_wrt_outputs = self.error.grad(activations[-1], targets_batch)
grads_wrt_params = self.model.grads_wrt_params(
activations, grads_wrt_outputs)
self.learning_rule.update_params(grads_wrt_params)
def eval_monitors(self, dataset, label):
"""Evaluates the monitors for the given dataset.
Args:
dataset: Dataset to perform evaluation with.
label: Tag to add to end of monitor keys to identify dataset.
Returns:
OrderedDict of monitor values evaluated on dataset.
"""
data_mon_vals = OrderedDict([(key + label, 0.) for key
in self.data_monitors.keys()])
for inputs_batch, targets_batch in dataset:
activations = self.model.fprop(inputs_batch)
for key, data_monitor in self.data_monitors.items():
data_mon_vals[key + label] += data_monitor(
activations[-1], targets_batch)
for key, data_monitor in self.data_monitors.items():
data_mon_vals[key + label] /= dataset.num_batches
return data_mon_vals
def get_epoch_stats(self):
"""Computes training statistics for an epoch.
Returns:
An OrderedDict with keys corresponding to the statistic labels and
values corresponding to the value of the statistic.
"""
epoch_stats = OrderedDict()
epoch_stats.update(self.eval_monitors(self.train_dataset, '(train)'))
if self.valid_dataset is not None:
epoch_stats.update(self.eval_monitors(
self.valid_dataset, '(valid)'))
return epoch_stats
def log_stats(self, epoch, epoch_time, stats):
"""Outputs stats for a training epoch to a logger.
Args:
epoch (int): Epoch counter.
epoch_time: Time taken in seconds for the epoch to complete.
stats: Monitored stats for the epoch.
"""
logger.info('Epoch {0}: {1:.1f}s to complete\n {2}'.format(
epoch, epoch_time,
', '.join(['{0}={1:.2e}'.format(k, v) for (k, v) in stats.items()])
))
def train(self, num_epochs, stats_interval=5):
"""Trains a model for a set number of epochs.
Args:
num_epochs: Number of epochs (complete passes through trainin
dataset) to train for.
stats_interval: Training statistics will be recorded and logged
every `stats_interval` epochs.
Returns:
Tuple with first value being an array of training run statistics
and the second being a dict mapping the labels for the statistics
recorded to their column index in the array.
"""
run_stats = [list(self.get_epoch_stats().values())]
for epoch in range(1, num_epochs + 1):
start_time = time.clock()
self.do_training_epoch()
epoch_time = time.clock() - start_time
if epoch % stats_interval == 0:
stats = self.get_epoch_stats()
self.log_stats(epoch, epoch_time, stats)
run_stats.append(list(stats.values()))
return np.array(run_stats), {k: i for i, k in enumerate(stats.keys())}

34
notebooks/01_Introduction.ipynb
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@ -323,11 +323,37 @@
},
"outputs": [],
"source": [
"# write your code here for iterating over five batches of \n",
"# 100 data points each and displaying as 10x10 grids\n",
"def show_batch_of_images(img_batch, fig_size=(3, 3)):\n",
" fig = plt.figure(figsize=fig_size)\n",
" batch_size, im_height, im_width = img_batch.shape\n",
" # calculate no. columns per grid row to give square grid\n",
" grid_size = int(batch_size**0.5)\n",
" # intialise empty array to tile image grid into\n",
" tiled = np.empty((im_height * grid_size, \n",
" im_width * batch_size // grid_size))\n",
" # iterate over images in batch + indexes within batch\n",
" for i, img in enumerate(img_batch):\n",
" # calculate grid row and column indices\n",
" r, c = i % grid_size, i // grid_size\n",
" tiled[r * im_height:(r + 1) * im_height, \n",
" c * im_height:(c + 1) * im_height] = img\n",
" ax = fig.add_subplot(111)\n",
" ax.imshow(tiled, cmap='Greys')\n",
" ax.axis('off')\n",
" fig.tight_layout()\n",
" plt.show()\n",
" return fig, ax\n",
"\n",
"def show_batch_of_images(img_batch):\n",
" raise NotImplementedError('Write me!')"
"batch_size = 100\n",
"num_batches = 5\n",
"\n",
"mnist_dp = data_providers.MNISTDataProvider(\n",
" which_set='valid', batch_size=batch_size, \n",
" max_num_batches=num_batches, shuffle_order=True)\n",
"\n",
"for inputs, target in mnist_dp:\n",
" # reshape inputs from batch of vectors to batch of 2D arrays (images)\n",
" _ = show_batch_of_images(inputs.reshape((batch_size, 28, 28)))"
]
},
{

833
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@ -0,0 +1,833 @@
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Single layer models\n",
"\n",
"In this lab we will implement a single-layer network model consisting of solely of an affine transformation of the inputs. The relevant material for this was covered in [the slides of the first lecture](http://www.inf.ed.ac.uk/teaching/courses/mlp/2016/mlp01-intro.pdf). \n",
"\n",
"We will first implement the forward propagation of inputs to the network to produce predicted outputs. We will then move on to considering how to use gradients of an error function evaluated on the outputs to compute the gradients with respect to the model parameters to allow us to perform an iterative gradient-descent training procedure. In the final exercise you will use an interactive visualisation to explore the role of some of the different hyperparameters of gradient-descent based training methods.\n",
"\n",
"#### A note on random number generators\n",
"\n",
"It is generally a good practice (for machine learning applications **not** for cryptography!) to seed a pseudo-random number generator once at the beginning of each experiment. This makes it easier to reproduce results as the same random draws will produced each time the experiment is run (e.g. the same random initialisations used for parameters). Therefore generally when we need to generate random values during this course, we will create a seeded random number generator object as we do in the cell below."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"seed = 27092016 \n",
"rng = np.random.RandomState(seed)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 1: linear and affine transforms\n",
"\n",
"Any *linear transform* (also called a linear map) of a finite-dimensional vector space can be parametrised by a matrix. So for example if we consider $\\boldsymbol{x} \\in \\mathbb{R}^{D}$ as the input space of a model with $D$ dimensional real-valued inputs, then a matrix $\\mathbf{W} \\in \\mathbb{R}^{K\\times D}$ can be used to define a prediction model consisting solely of a linear transform of the inputs\n",
"\n",
"\\begin{equation}\n",
" \\boldsymbol{y} = \\mathbf{W} \\boldsymbol{x}\n",
" \\qquad\n",
" \\Leftrightarrow\n",
" \\qquad\n",
" y_k = \\sum_{d=1}^D \\left( W_{kd} x_d \\right) \\quad \\forall k \\in \\left\\lbrace 1 \\dots K\\right\\rbrace\n",
"\\end{equation}\n",
"\n",
"with here $\\boldsymbol{y} \\in \\mathbb{R}^K$ the $K$-dimensional real-valued output of the model. Geometrically we can think of a linear transform doing some combination of rotation, scaling, reflection and shearing of the input.\n",
"\n",
"An *affine transform* consists of a linear transform plus an additional translation parameterised by a vector $\\boldsymbol{b} \\in \\mathbb{R}^K$. A model consisting of an affine transformation of the inputs can then be defined as\n",
"\n",
"\\begin{equation}\n",
" \\boldsymbol{y} = \\mathbf{W}\\boldsymbol{x} + \\boldsymbol{b}\n",
" \\qquad\n",
" \\Leftrightarrow\n",
" \\qquad\n",
" y_k = \\sum_{d=1}^D \\left( W_{kd} x_d \\right) + b_k \\quad \\forall k \\in \\left\\lbrace 1 \\dots K\\right\\rbrace\n",
"\\end{equation}\n",
"\n",
"In machine learning we will usually refer to the matrix $\\mathbf{W}$ as a *weight matrix* and the vector $\\boldsymbol{b}$ as a *bias vector*.\n",
"\n",
"Generally rather than working with a single data vector $\\boldsymbol{x}$ we will work with batches of datapoints $\\left\\lbrace \\boldsymbol{x}^{(b)}\\right\\rbrace_{b=1}^B$. We could calculate the outputs for each input in the batch sequentially\n",
"\n",
"\\begin{align}\n",
" \\boldsymbol{y}^{(1)} &= \\mathbf{W}\\boldsymbol{x}^{(1)} + \\boldsymbol{b}\\\\\n",
" \\boldsymbol{y}^{(2)} &= \\mathbf{W}\\boldsymbol{x}^{(2)} + \\boldsymbol{b}\\\\\n",
" \\dots &\\\\\n",
" \\boldsymbol{y}^{(B)} &= \\mathbf{W}\\boldsymbol{x}^{(B)} + \\boldsymbol{b}\\\\\n",
"\\end{align}\n",
"\n",
"by looping over each input in the batch and calculating the output. However in general loops in Python are slow (particularly compared to compiled and typed languages such as C). This is due at least in part to the large overhead in dynamically inferring variable types. In general therefore wherever possible we want to avoid having loops in which such overhead will become the dominant computational cost.\n",
"\n",
"For array based numerical operations, one way of overcoming this bottleneck is to *vectorise* operations. NumPy `ndarrays` are typed arrays for which operations such as basic elementwise arithmetic and linear algebra operations such as computing matrix-matrix or matrix-vector products are implemented by calls to highly-optimised compiled libraries. Therefore if you can implement code directly using NumPy operations on arrays rather than by looping over array elements it is often possible to make very substantial performance gains.\n",
"\n",
"As a simple example we can consider adding up two arrays `a` and `b` and writing the result to a third array `c`. First lets initialise `a` and `b` with arbitrary values by running the cell below."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"size = 1000\n",
"a = np.arange(size)\n",
"b = np.ones(size)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's time how long it takes to add up each pair of values in the two array and write the results to a third array using a loop-based implementation. We will use the `%%timeit` magic briefly mentioned in the previous lab notebook specifying the number of times to loop the code as 100 and to give the best of 3 repeats. Run the cell below to get a print out of the average time taken."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"%%timeit -n 100 -r 3\n",
"c = np.empty(size)\n",
"for i in range(size):\n",
" c[i] = a[i] + b[i]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And now we will perform the corresponding summation with the overloaded addition operator of NumPy arrays. Again run the cell below to get a print out of the average time taken."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"%%timeit -n 100 -r 3\n",
"c = a + b"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The first loop-based implementation should have taken on the order of milliseconds ($10^{-3}$s) while the vectorised implementation should have taken on the order of microseconds ($10^{-6}$s), i.e. a $\\sim1000\\times$ speedup. Hopefully this simple example should make it clear why we want to vectorise operations whenever possible!\n",
"\n",
"Getting back to our affine model, ideally rather than individually computing the output corresponding to each input we should compute the outputs for all inputs in a batch using a vectorised implementation. As you saw last week, data providers return batches of inputs as arrays of shape `(batch_size, input_dim)`. In the mathematical notation used earlier we can consider this as a matrix $\\mathbf{X}$ of dimensionality $B \\times D$, and in particular\n",
"\n",
"\\begin{equation}\n",
" \\mathbf{X} = \\left[ \\boldsymbol{x}^{(1)} ~ \\boldsymbol{x}^{(2)} ~ \\dots ~ \\boldsymbol{x}^{(B)} \\right]^\\mathrm{T}\n",
"\\end{equation}\n",
"\n",
"i.e. the $b^{\\textrm{th}}$ input vector $\\boldsymbol{x}^{(b)}$ corresponds to the $b^{\\textrm{th}}$ row of $\\mathbf{X}$. If we define the $B \\times K$ matrix of outputs $\\mathbf{Y}$ similarly as\n",
"\n",
"\\begin{equation}\n",
" \\mathbf{Y} = \\left[ \\boldsymbol{y}^{(1)} ~ \\boldsymbol{y}^{(2)} ~ \\dots ~ \\boldsymbol{y}^{(B)} \\right]^\\mathrm{T}\n",
"\\end{equation}\n",
"\n",
"then we can express the relationship between $\\mathbf{X}$ and $\\mathbf{Y}$ using [matrix multiplication](https://en.wikipedia.org/wiki/Matrix_multiplication) and addition as\n",
"\n",
"\\begin{equation}\n",
" \\mathbf{Y} = \\mathbf{X} \\mathbf{W}^\\mathrm{T} + \\mathbf{B}\n",
"\\end{equation}\n",
"\n",
"where $\\mathbf{B} = \\left[ \\boldsymbol{b} ~ \\boldsymbol{b} ~ \\dots ~ \\boldsymbol{b} \\right]^\\mathrm{T}$ i.e. a $B \\times K$ matrix with each row corresponding to the bias vector. The weight matrix needs to be transposed here as the inner dimensions of a matrix multiplication must match i.e. for $\\mathbf{C} = \\mathbf{A} \\mathbf{B}$ then if $\\mathbf{A}$ is of dimensionality $K \\times L$ and $\\mathbf{B}$ is of dimensionality $M \\times N$ then it must be the case that $L = M$ and $\\mathbf{C}$ will be of dimensionality $K \\times N$.\n",
"\n",
"The first exercise for this lab is to implement *forward propagation* for a single-layer model consisting of an affine transformation of the inputs in the `fprop` function given as skeleton code in the cell below. This should work for a batch of inputs of shape `(batch_size, input_dim)` producing a batch of outputs of shape `(batch_size, output_dim)`.\n",
" \n",
"You will probably want to use the NumPy `dot` function and [broadcasting features](http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html) to implement this efficiently. If you are not familiar with either / both of these you may wish to read the [hints](#Hints:-Using-the-dot-function-and-broadcasting) section below which gives some details on these before attempting the exercise."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def fprop(inputs, weights, biases):\n",
" \"\"\"Forward propagates activations through the layer transformation.\n",
"\n",
" For inputs `x`, outputs `y`, weights `W` and biases `b` the layer\n",
" corresponds to `y = W x + b`.\n",
"\n",
" Args:\n",
" inputs: Array of layer inputs of shape (batch_size, input_dim).\n",
" weights: Array of weight parameters of shape \n",
" (output_dim, input_dim).\n",
" biases: Array of bias parameters of shape (output_dim, ).\n",
"\n",
" Returns:\n",
" outputs: Array of layer outputs of shape (batch_size, output_dim).\n",
" \"\"\"\n",
" raise NotImplementedError('Delete this and write your code here instead.')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Once you have implemented `fprop` in the cell above you can test your implementation by running the cell below."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"inputs = np.array([[0., -1., 2.], [-6., 3., 1.]])\n",
"weights = np.array([[2., -3., -1.], [-5., 7., 2.]])\n",
"biases = np.array([5., -3.])\n",
"true_outputs = np.array([[6., -6.], [-17., 50.]])\n",
"\n",
"if not np.allclose(fprop(inputs, weights, biases), true_outputs):\n",
" print('Wrong outputs computed.')\n",
"else:\n",
" print('All outputs correct!')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Hints: Using the `dot` function and broadcasting\n",
"\n",
"For those new to NumPy below are some details on the `dot` function and broadcasting feature of NumPy that you may want to use for implementing the first exercise. If you are already familiar with these and have already completed the first exercise you can move on straight to [second exercise](#Exercise-2:-visualising-random-models).\n",
"\n",
"#### `numpy.dot` function\n",
"\n",
"Matrix-matrix, matrix-vector and vector-vector (dot) products can all be computed in NumPy using the [`dot`](http://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html) function. For example if `A` and `B` are both two dimensional arrays, then `C = np.dot(A, B)` or equivalently `C = A.dot(B)` will both compute the matrix product of `A` and `B` assuming `A` and `B` have compatible dimensions. Similarly if `a` and `b` are one dimensional arrays then `c = np.dot(a, b)` / `c = a.dot(b)` will compute the [scalar / dot product](https://en.wikipedia.org/wiki/Dot_product) of the two arrays. If `A` is a two-dimensional array and `b` a one-dimensional array `np.dot(A, b)` / `A.dot(b)` will compute the matrix-vector product of `A` and `b`. Examples of all three of these product types are shown in the cell below:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# Initiliase arrays with arbitrary values\n",
"A = np.arange(9).reshape((3, 3))\n",
"B = np.ones((3, 3)) * 2\n",
"a = np.array([-1., 0., 1.])\n",
"b = np.array([0.1, 0.2, 0.3])\n",
"print(A.dot(B)) # Matrix-matrix product\n",
"print(B.dot(A)) # Reversed product of above A.dot(B) != B.dot(A) in general\n",
"print(A.dot(b)) # Matrix-vector product\n",
"print(b.dot(A)) # Again A.dot(b) != b.dot(A) unless A is symmetric i.e. A == A.T\n",
"print(a.dot(b)) # Vector-vector scalar product"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Broadcasting\n",
"\n",
"Another NumPy feature it will be helpful to get familiar with is [broadcasting](http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html). Broadcasting allows you to apply operations to arrays of different shapes, for example to add a one-dimensional array to a two-dimensional array or multiply a multidimensional array by a scalar. The complete set of rules for broadcasting as explained in the official documentation page just linked to can sound a bit complex: you might find the [visual explanation on this page](http://www.scipy-lectures.org/intro/numpy/operations.html#broadcasting) more intuitive. The cell below gives a few examples:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# Initiliase arrays with arbitrary values\n",
"A = np.arange(6).reshape((3, 2))\n",
"b = np.array([0.1, 0.2])\n",
"c = np.array([-1., 0., 1.])\n",
"print(A + b) # Add b elementwise to all rows of A\n",
"print((A.T + c).T) # Add b elementwise to all columns of A\n",
"print(A * b) # Multiply each row of A elementise by b "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 2: visualising random models"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this exercise you will use your `fprop` implementation to visualise the outputs of a single-layer affine transform model with two-dimensional inputs and a one-dimensional output. In this simple case we can visualise the joint input-output space on a 3D axis.\n",
"\n",
"For this task and the learning experiments later in the notebook we will use a regression dataset from the [UCI machine learning repository](http://archive.ics.uci.edu/ml/index.html). In particular we will use a version of the [Combined Cycle Power Plant dataset](http://archive.ics.uci.edu/ml/datasets/Combined+Cycle+Power+Plant), where the task is to predict the energy output of a power plant given observations of the local ambient conditions (e.g. temperature, pressure and humidity).\n",
"\n",
"The original dataset has four input dimensions and a single target output dimension. We have preprocessed the dataset by [whitening](https://en.wikipedia.org/wiki/Whitening_transformation) it, a common preprocessing step. We will only use the first two dimensions of the whitened inputs (corresponding to the first two principal components of the inputs) so we can easily visualise the joint input-output space.\n",
"\n",
"The dataset has been wrapped in the `CCPPDataProvider` class in the `mlp.data_providers` module and the data included as a compressed file in the data directory as `ccpp_data.npz`. Running the cell below will initialise an instance of this class, get a single batch of inputs and outputs and import the necessary `matplotlib` objects."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"from mpl_toolkits.mplot3d import Axes3D\n",
"from mlp.data_providers import CCPPDataProvider\n",
"%matplotlib notebook\n",
"\n",
"data_provider = CCPPDataProvider(\n",
" which_set='train',\n",
" input_dims=[0, 1],\n",
" batch_size=5000, \n",
" max_num_batches=1, \n",
" shuffle_order=False\n",
")\n",
"\n",
"input_dim, output_dim = 2, 1\n",
"\n",
"inputs, targets = data_provider.next()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we used the `%matplotlib notebook` magic command rather than the `%matplotlib inline` we used in the previous lab as this allows us to produce interactive 3D plots which you can rotate and zoom in/out by dragging with the mouse and scrolling the mouse-wheel respectively. Once you have finished interacting with a plot you can close it to produce a static inline plot using the <i class=\"fa fa-power-off\"></i> button in the top-right corner.\n",
"\n",
"Now run the cell below to plot the predicted outputs of a randomly initialised model across the two dimensional input space as well as the true target outputs. This sort of visualisation can be a useful method (in low dimensions) to assess how well the model is likely to be able to fit the data and to judge appropriate initialisation scales for the parameters. Each time you re-run the cell a new set of random parameters will be sampled\n",
"\n",
"Some questions to consider:\n",
"\n",
" * How do the weights and bias initialisation scale affect the sort of predicted input-output relationships?\n",
" * Does the linear form of the model seem appropriate for the data here?"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"weights_init_range = 0.5\n",
"biases_init_range = 0.1\n",
"\n",
"# Randomly initialise weights matrix\n",
"weights = rng.uniform(\n",
" low=-weights_init_range, \n",
" high=weights_init_range, \n",
" size=(output_dim, input_dim)\n",
")\n",
"\n",
"# Randomly initialise biases vector\n",
"biases = rng.uniform(\n",
" low=-biases_init_range, \n",
" high=biases_init_range, \n",
" size=output_dim\n",
")\n",
"# Calculate predicted model outputs\n",
"outputs = fprop(inputs, weights, biases)\n",
"\n",
"# Plot target and predicted outputs against inputs on same axis\n",
"fig = plt.figure(figsize=(8, 8))\n",
"ax = fig.add_subplot(111, projection='3d')\n",
"ax.plot(inputs[:, 0], inputs[:, 1], targets[:, 0], 'r.', ms=2)\n",
"ax.plot(inputs[:, 0], inputs[:, 1], outputs[:, 0], 'b.', ms=2)\n",
"ax.set_xlabel('Input dim 1')\n",
"ax.set_ylabel('Input dim 2')\n",
"ax.set_zlabel('Output')\n",
"ax.legend(['Targets', 'Predictions'], frameon=False)\n",
"fig.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 3: computing the error function and its gradient\n",
"\n",
"Here we will consider the task of regression as covered in the first lecture slides. The aim in a regression problem is given inputs $\\left\\lbrace \\boldsymbol{x}^{(n)}\\right\\rbrace_{n=1}^N$ to produce outputs $\\left\\lbrace \\boldsymbol{y}^{(n)}\\right\\rbrace_{n=1}^N$ that are as 'close' as possible to a set of target outputs $\\left\\lbrace \\boldsymbol{t}^{(n)}\\right\\rbrace_{n=1}^N$. The measure of 'closeness' or distance between target and predicted outputs is a design choice. \n",
"\n",
"A very common choice is the squared Euclidean distance between the predicted and target outputs. This can be computed as the sum of the squared differences between each element in the target and predicted outputs. A common convention is to multiply this value by $\\frac{1}{2}$ as this gives a slightly nicer expression for the error gradient. The error for the $n^{\\textrm{th}}$ training example is then\n",
"\n",
"\\begin{equation}\n",
" E^{(n)} = \\frac{1}{2} \\sum_{k=1}^K \\left\\lbrace \\left( y^{(n)}_k - t^{(n)}_k \\right)^2 \\right\\rbrace.\n",
"\\end{equation}\n",
"\n",
"The overall error is then the *average* of this value across all training examples\n",
"\n",
"\\begin{equation}\n",
" \\bar{E} = \\frac{1}{N} \\sum_{n=1}^N \\left\\lbrace E^{(n)} \\right\\rbrace. \n",
"\\end{equation}\n",
"\n",
"*Note here we are using a slightly different convention from the lectures. There the overall error was considered to be the sum of the individual error terms rather than the mean. To differentiate between the two we will use $\\bar{E}$ to represent the average error here as opposed to sum of errors $E$ as used in the slides with $\\bar{E} = \\frac{E}{N}$. Normalising by the number of training examples is helpful to do in practice as this means we can more easily compare errors across data sets / batches of different sizes, and more importantly it means the size of our gradient updates will be independent of the number of training examples summed over.*\n",
"\n",
"The regression problem is then to find parameters of the model which minimise $\\bar{E}$. For our simple single-layer affine model here that corresponds to finding weights $\\mathbf{W}$ and biases $\\boldsymbol{b}$ which minimise $\\bar{E}$. \n",
"\n",
"As mentioned in the lecture, for this simple case there is actually a closed form solution for the optimal weights and bias parameters. This is the linear least-squares solution those doing MLPR will have come across.\n",
"\n",
"However in general we will be interested in models where closed form solutions do not exist. We will therefore generally use iterative, gradient descent based training methods to find parameters which (locally) minimise the error function. A basic requirement of being able to do gradient-descent based training is (unsuprisingly) the ability to evaluate gradients of the error function.\n",
"\n",
"In the next exercise we will consider how to calculate gradients of the error function with respect to the model parameters $\\mathbf{W}$ and $\\boldsymbol{b}$, but as a first step here we will consider the gradient of the error function with respect to the model outputs $\\left\\lbrace \\boldsymbol{y}^{(n)}\\right\\rbrace_{n=1}^N$. This can be written\n",
"\n",
"\\begin{equation}\n",
" \\frac{\\partial \\bar{E}}{\\partial \\boldsymbol{y}^{(n)}} = \\frac{1}{N} \\left( \\boldsymbol{y}^{(n)} - \\boldsymbol{t}^{(n)} \\right)\n",
" \\qquad \\Leftrightarrow \\qquad\n",
" \\frac{\\partial \\bar{E}}{\\partial y^{(n)}_k} = \\frac{1}{N} \\left( y^{(n)}_k - t^{(n)}_k \\right) \\quad \\forall k \\in \\left\\lbrace 1 \\dots K\\right\\rbrace\n",
"\\end{equation}\n",
"\n",
"i.e. the gradient of the error function with respect to the $n^{\\textrm{th}}$ model output is just the difference between the $n^{\\textrm{th}}$ model and target outputs, corresponding to the $\\boldsymbol{\\delta}^{(n)}$ terms mentioned in the lecture slides.\n",
"\n",
"The third exercise is, using the equations given above, to implement functions computing the mean sum of squared differences error and its gradient with respect to the model outputs. You should implement the functions using the provided skeleton definitions in the cell below."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def error(outputs, targets):\n",
" \"\"\"Calculates error function given a batch of outputs and targets.\n",
"\n",
" Args:\n",
" outputs: Array of model outputs of shape (batch_size, output_dim).\n",
" targets: Array of target outputs of shape (batch_size, output_dim).\n",
"\n",
" Returns:\n",
" Scalar error function value.\n",
" \"\"\"\n",
" raise NotImplementedError('Delete this and write your code here instead.')\n",
" \n",
"def error_grad(outputs, targets):\n",
" \"\"\"Calculates gradient of error function with respect to model outputs.\n",
"\n",
" Args:\n",
" outputs: Array of model outputs of shape (batch_size, output_dim).\n",
" targets: Array of target outputs of shape (batch_size, output_dim).\n",
"\n",
" Returns:\n",
" Gradient of error function with respect to outputs.\n",
" This will be an array of shape (batch_size, output_dim).\n",
" \"\"\"\n",
" raise NotImplementedError('Delete this and write your code here instead.')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Check your implementation by running the test cell below."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"outputs = np.array([[1., 2.], [-1., 0.], [6., -5.], [-1., 1.]])\n",
"targets = np.array([[0., 1.], [3., -2.], [7., -3.], [1., -2.]])\n",
"true_error = 5.\n",
"true_error_grad = np.array([[0.25, 0.25], [-1., 0.5], [-0.25, -0.5], [-0.5, 0.75]])\n",
"\n",
"if not error(outputs, targets) == true_error:\n",
" print('Error calculated incorrectly.')\n",
"elif not np.allclose(error_grad(outputs, targets), true_error_grad):\n",
" print('Error gradient calculated incorrectly.')\n",
"else:\n",
" print('Error function and gradient computed correctly!')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 4: computing gradients with respect to the parameters\n",
"\n",
"In the previous exercise you implemented a function computing the gradient of the error function with respect to the model outputs. For gradient-descent based training, we need to be able to evaluate the gradient of the error function with respect to the model parameters.\n",
"\n",
"Using the [chain rule for derivatives](https://en.wikipedia.org/wiki/Chain_rule#Higher_dimensions) we can write the partial deriviative of the error function with respect to single elements of the weight matrix and bias vector as\n",
"\n",
"\\begin{equation}\n",
" \\frac{\\partial \\bar{E}}{\\partial W_{kj}} = \\sum_{n=1}^N \\left\\lbrace \\frac{\\partial \\bar{E}}{\\partial y^{(n)}_k} \\frac{\\partial y^{(n)}_k}{\\partial W_{kj}} \\right\\rbrace\n",
" \\quad \\textrm{and} \\quad\n",
" \\frac{\\partial \\bar{E}}{\\partial b_k} = \\sum_{n=1}^N \\left\\lbrace \\frac{\\partial \\bar{E}}{\\partial y^{(n)}_k} \\frac{\\partial y^{(n)}_k}{\\partial b_k} \\right\\rbrace.\n",
"\\end{equation}\n",
"\n",
"From the definition of our model at the beginning we have \n",
"\n",
"\\begin{equation}\n",
" y^{(n)}_k = \\sum_{d=1}^D \\left\\lbrace W_{kd} x^{(n)}_d \\right\\rbrace + b_k\n",
" \\quad \\Rightarrow \\quad\n",
" \\frac{\\partial y^{(n)}_k}{\\partial W_{kj}} = x^{(n)}_j\n",
" \\quad \\textrm{and} \\quad\n",
" \\frac{\\partial y^{(n)}_k}{\\partial b_k} = 1.\n",
"\\end{equation}\n",
"\n",
"Putting this together we get that\n",
"\n",
"\\begin{equation}\n",
" \\frac{\\partial \\bar{E}}{\\partial W_{kj}} = \n",
" \\sum_{n=1}^N \\left\\lbrace \\frac{\\partial \\bar{E}}{\\partial y^{(n)}_k} x^{(n)}_j \\right\\rbrace\n",
" \\quad \\textrm{and} \\quad\n",
" \\frac{\\partial \\bar{E}}{\\partial b_{k}} = \n",
" \\sum_{n=1}^N \\left\\lbrace \\frac{\\partial \\bar{E}}{\\partial y^{(n)}_k} \\right\\rbrace.\n",
"\\end{equation}\n",
"\n",
"Although this may seem a bit of a roundabout way to get to these results, this method of decomposing the error gradient with respect to the parameters in terms of the gradient of the error function with respect to the model outputs and the derivatives of the model outputs with respect to the model parameters, will be key when calculating the parameter gradients of more complex models later in the course.\n",
"\n",
"Your task in this exercise is to implement a function calculating the gradient of the error function with respect to the weight and bias parameters of the model given the already computed gradient of the error function with respect to the model outputs. You should implement this in the `grads_wrt_params` function in the cell below."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def grads_wrt_params(inputs, grads_wrt_outputs):\n",
" \"\"\"Calculates gradients with respect to model parameters.\n",
"\n",
" Args:\n",
" inputs: array of inputs to model of shape (batch_size, input_dim)\n",
" grads_wrt_to_outputs: array of gradients of with respect to the model\n",
" outputs of shape (batch_size, output_dim).\n",
"\n",
" Returns:\n",
" list of arrays of gradients with respect to the model parameters\n",
" `[grads_wrt_weights, grads_wrt_biases]`.\n",
" \"\"\"\n",
" raise NotImplementedError('Delete this and write your code here instead.')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Check your implementation by running the test cell below."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"inputs = np.array([[1., 2., 3.], [-1., 4., -9.]])\n",
"grads_wrt_outputs = np.array([[-1., 1.], [2., -3.]])\n",
"true_grads_wrt_weights = np.array([[-3., 6., -21.], [4., -10., 30.]])\n",
"true_grads_wrt_biases = np.array([1., -2.])\n",
"\n",
"grads_wrt_weights, grads_wrt_biases = grads_wrt_params(\n",
" inputs, grads_wrt_outputs)\n",
"\n",
"if not np.allclose(true_grads_wrt_weights, grads_wrt_weights):\n",
" print('Gradients with respect to weights incorrect.')\n",
"elif not np.allclose(true_grads_wrt_biases, grads_wrt_biases):\n",
" print('Gradients with respect to biases incorrect.')\n",
"else:\n",
" print('All parameter gradients calculated correctly!')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 5: wrapping the functions into reusable components\n",
"\n",
"In exercises 1, 3 and 4 you implemented methods to compute the predicted outputs of our model, evaluate the error function and its gradient on the outputs and finally to calculate the gradients of the error with respect to the model parameters. Together they constitute all the basic ingredients we need to implement a gradient-descent based iterative learning procedure for the model.\n",
"\n",
"Although you could implement training code which directly uses the functions you defined, this would only be usable for this particular model architecture. In subsequent labs we will want to use the affine transform functions as the basis for more interesting multi-layer models. We will therefore wrap the implementations you just wrote in to reusable components that we can build more complex models with later in the course.\n",
"\n",
" * In the [`mlp.layers`](/edit/mlp/layers.py) module, use your implementations of `fprop` and `grad_wrt_params` above to implement the corresponding methods in the skeleton `AffineLayer` class provided.\n",
" * In the [`mlp.errors`](/edit/mlp/errors.py) module use your implementation of `error` and `error_grad` to implement the `__call__` and `grad` methods respectively of the skeleton `SumOfSquaredDiffsError` class provided. Note `__call__` is a special Python method that allows an object to be used with a function call syntax.\n",
"\n",
"Run the cell below to use your completed `AffineLayer` and `SumOfSquaredDiffsError` implementations to train a single-layer model using batch gradient descent on the CCCP dataset."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"from mlp.layers import AffineLayer\n",
"from mlp.errors import SumOfSquaredDiffsError\n",
"from mlp.models import SingleLayerModel\n",
"from mlp.initialisers import UniformInit, ConstantInit\n",
"from mlp.learning_rules import GradientDescentLearningRule\n",
"from mlp.optimisers import Optimiser\n",
"import logging\n",
"\n",
"# Seed a random number generator\n",
"seed = 27092016 \n",
"rng = np.random.RandomState(seed)\n",
"\n",
"# Set up a logger object to print info about the training run to stdout\n",
"logger = logging.getLogger()\n",
"logger.setLevel(logging.INFO)\n",
"logger.handlers = [logging.StreamHandler()]\n",
"\n",
"# Create data provider objects for the CCPP training set\n",
"train_data = CCPPDataProvider('train', [0, 1], batch_size=100, rng=rng)\n",
"input_dim, output_dim = 2, 1\n",
"\n",
"# Create a parameter initialiser which will sample random uniform values\n",
"# from [-0.1, 0.1]\n",
"param_init = UniformInit(-0.1, 0.1, rng=rng)\n",
"\n",
"# Create our single layer model\n",
"layer = AffineLayer(input_dim, output_dim, param_init, param_init)\n",
"model = SingleLayerModel(layer)\n",
"\n",
"# Initialise the error object\n",
"error = SumOfSquaredDiffsError()\n",
"\n",
"# Use a basic gradient descent learning rule with a small learning rate\n",
"learning_rule = GradientDescentLearningRule(learning_rate=1e-2)\n",
"\n",
"# Use the created objects to initialise a new Optimiser instance.\n",
"optimiser = Optimiser(model, error, learning_rule, train_data)\n",
"\n",
"# Run the optimiser for 5 epochs (full passes through the training set)\n",
"# printing statistics every epoch.\n",
"stats, keys = optimiser.train(num_epochs=10, stats_interval=1)\n",
"\n",
"# Plot the change in the error over training.\n",
"fig = plt.figure(figsize=(8, 4))\n",
"ax = fig.add_subplot(111)\n",
"ax.plot(np.arange(1, stats.shape[0] + 1), stats[:, keys['error(train)']])\n",
"ax.set_xlabel('Epoch number')\n",
"ax.set_ylabel('Error')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using similar code to previously we can now visualise the joint input-output space for the trained model. If you implemented the required methods correctly you should now see a much improved fit between predicted and target outputs when running the cell below."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"data_provider = CCPPDataProvider(\n",
" which_set='train',\n",
" input_dims=[0, 1],\n",
" batch_size=5000, \n",
" max_num_batches=1, \n",
" shuffle_order=False\n",
")\n",
"\n",
"inputs, targets = data_provider.next()\n",
"\n",
"# Calculate predicted model outputs\n",
"outputs = model.fprop(inputs)[-1]\n",
"\n",
"# Plot target and predicted outputs against inputs on same axis\n",
"fig = plt.figure(figsize=(8, 8))\n",
"ax = fig.add_subplot(111, projection='3d')\n",
"ax.plot(inputs[:, 0], inputs[:, 1], targets[:, 0], 'r.', ms=2)\n",
"ax.plot(inputs[:, 0], inputs[:, 1], outputs[:, 0], 'b.', ms=2)\n",
"ax.set_xlabel('Input dim 1')\n",
"ax.set_ylabel('Input dim 2')\n",
"ax.set_zlabel('Output')\n",
"ax.legend(['Targets', 'Predictions'], frameon=False)\n",
"fig.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 6: visualising training trajectories in parameter space"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Running the cell below will display an interactive widget which plots the trajectories of gradient-based training of the single-layer affine model on the CCPP dataset in the three dimensional parameter space (two weights plus bias) from random initialisations. Also shown on the right is a plot of the evolution of the error function (evaluated on the current batch) over training. By moving the sliders you can alter the training hyperparameters to investigate the effect they have on how training procedes.\n",
"\n",
"Some questions to explore:\n",
"\n",
" * Are there multiple local minima in parameter space here? Why?\n",
" * What happens to learning for very small learning rates? And very large learning rates?\n",
" * How does the batch size affect learning?\n",
" \n",
"**Note:** You don't need to understand how the code below works. The idea of this exercise is to help you understand the role of the various hyperparameters involved in gradient-descent based training methods."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": false,
"scrolled": false
},
"outputs": [],
"source": [
"from ipywidgets import interact\n",
"%matplotlib inline\n",
"\n",
"def setup_figure():\n",
" # create figure and axes\n",
" fig = plt.figure(figsize=(12, 6))\n",
" ax1 = fig.add_axes([0., 0., 0.5, 1.], projection='3d')\n",
" ax2 = fig.add_axes([0.6, 0.1, 0.4, 0.8])\n",
" # set axes properties\n",
" ax2.spines['right'].set_visible(False)\n",
" ax2.spines['top'].set_visible(False)\n",
" ax2.yaxis.set_ticks_position('left')\n",
" ax2.xaxis.set_ticks_position('bottom')\n",
" ax2.set_yscale('log')\n",
" ax1.set_xlim((-2, 2))\n",
" ax1.set_ylim((-2, 2))\n",
" ax1.set_zlim((-2, 2))\n",
" #set axes labels and title\n",
" ax1.set_title('Parameter trajectories over training')\n",
" ax1.set_xlabel('Weight 1')\n",
" ax1.set_ylabel('Weight 2')\n",
" ax1.set_zlabel('Bias')\n",
" ax2.set_title('Batch errors over training')\n",
" ax2.set_xlabel('Batch update number')\n",
" ax2.set_ylabel('Batch error')\n",
" return fig, ax1, ax2\n",
"\n",
"def visualise_training(n_epochs=1, batch_size=200, log_lr=-1., n_inits=5,\n",
" w_scale=1., b_scale=1., elev=30., azim=0.):\n",
" fig, ax1, ax2 = setup_figure()\n",
" # create seeded random number generator\n",
" rng = np.random.RandomState(1234)\n",
" # create data provider\n",
" data_provider = CCPPDataProvider(\n",
" input_dims=[0, 1],\n",
" batch_size=batch_size, \n",
" shuffle_order=False,\n",
" )\n",
" learning_rate = 10 ** log_lr\n",
" n_batches = data_provider.num_batches\n",
" weights_traj = np.empty((n_inits, n_epochs * n_batches + 1, 1, 2))\n",
" biases_traj = np.empty((n_inits, n_epochs * n_batches + 1, 1))\n",
" errors_traj = np.empty((n_inits, n_epochs * n_batches))\n",
" # randomly initialise parameters\n",
" weights = rng.uniform(-w_scale, w_scale, (n_inits, 1, 2))\n",
" biases = rng.uniform(-b_scale, b_scale, (n_inits, 1))\n",
" # store initial parameters\n",
" weights_traj[:, 0] = weights\n",
" biases_traj[:, 0] = biases\n",
" # iterate across different initialisations\n",
" for i in range(n_inits):\n",
" # iterate across epochs\n",
" for e in range(n_epochs):\n",
" # iterate across batches\n",
" for b, (inputs, targets) in enumerate(data_provider):\n",
" outputs = fprop(inputs, weights[i], biases[i])\n",
" errors_traj[i, e * n_batches + b] = error(outputs, targets)\n",
" grad_wrt_outputs = error_grad(outputs, targets)\n",
" weights_grad, biases_grad = grads_wrt_params(inputs, grad_wrt_outputs)\n",
" weights[i] -= learning_rate * weights_grad\n",
" biases[i] -= learning_rate * biases_grad\n",
" weights_traj[i, e * n_batches + b + 1] = weights[i]\n",
" biases_traj[i, e * n_batches + b + 1] = biases[i]\n",
" # choose a different color for each trajectory\n",
" colors = plt.cm.jet(np.linspace(0, 1, n_inits))\n",
" # plot all trajectories\n",
" for i in range(n_inits):\n",
" lines_1 = ax1.plot(\n",
" weights_traj[i, :, 0, 0], \n",
" weights_traj[i, :, 0, 1], \n",
" biases_traj[i, :, 0], \n",
" '-', c=colors[i], lw=2)\n",
" lines_2 = ax2.plot(\n",
" np.arange(n_batches * n_epochs),\n",
" errors_traj[i],\n",
" c=colors[i]\n",
" )\n",
" ax1.view_init(elev, azim)\n",
" plt.show()\n",
"\n",
"w = interact(\n",
" visualise_training,\n",
" elev=(-90, 90, 2),\n",
" azim=(-180, 180, 2), \n",
" n_epochs=(1, 5), \n",
" batch_size=(100, 1000, 100),\n",
" log_lr=(-3., 1.),\n",
" w_scale=(0., 2.),\n",
" b_scale=(0., 2.),\n",
" n_inits=(1, 10)\n",
")\n",
"\n",
"for child in w.widget.children:\n",
" child.layout.width = '100%'"
]
}
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