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miniflow in Udacity nanodegree
import numpy as np
class Node(object):
"""
Base class for nodes in the network.
Arguments:
`inbound_nodes`: A list of nodes with edges into this node.
"""
def __init__(self, inbound_nodes=[]):
"""
Node's constructor (runs when the object is instantiated). Sets
properties that all nodes need.
"""
# A list of nodes with edges into this node.
self.inbound_nodes = inbound_nodes
# The eventual value of this node. Set by running
# the forward() method.
self.value = None
# A list of nodes that this node outputs to.
self.outbound_nodes = []
# New property! Keys are the inputs to this node and
# their values are the partials of this node with
# respect to that input.
self.gradients = {}
# Sets this node as an outbound node for all of
# this node's inputs.
for node in inbound_nodes:
node.outbound_nodes.append(self)
def forward(self):
"""
Every node that uses this class as a base class will
need to define its own `forward` method.
"""
raise NotImplementedError
def backward(self):
"""
Every node that uses this class as a base class will
need to define its own `backward` method.
"""
raise NotImplementedError
class Input(Node):
"""
A generic input into the network.
"""
def __init__(self):
# The base class constructor has to run to set all
# the properties here.
#
# The most important property on an Input is value.
# self.value is set during `topological_sort` later.
Node.__init__(self)
def forward(self):
# Do nothing because nothing is calculated.
pass
def backward(self):
# An Input node has no inputs so the gradient (derivative)
# is zero.
# The key, `self`, is reference to this object.
self.gradients = {self: 0}
# Weights and bias may be inputs, so you need to sum
# the gradient from output gradients.
for n in self.outbound_nodes:
grad_cost = n.gradients[self]
self.gradients[self] += grad_cost * 1
class Linear(Node):
"""
Represents a node that performs a linear transform.
"""
def __init__(self, X, W, b):
# The base class (Node) constructor. Weights and bias
# are treated like inbound nodes.
Node.__init__(self, [X, W, b])
def forward(self):
"""
Performs the math behind a linear transform.
"""
X = self.inbound_nodes[0].value
W = self.inbound_nodes[1].value
b = self.inbound_nodes[2].value
self.value = np.dot(X, W) + b
def backward(self):
"""
Calculates the gradient based on the output values.
"""
# Initialize a partial for each of the inbound_nodes.
self.gradients = {n: np.zeros_like(n.value) for n in self.inbound_nodes}
# Cycle through the outputs. The gradient will change depending
# on each output, so the gradients are summed over all outputs.
for n in self.outbound_nodes:
# Get the partial of the cost with respect to this node.
grad_cost = n.gradients[self]
# Set the partial of the loss with respect to this node's inputs.
self.gradients[self.inbound_nodes[0]] += np.dot(grad_cost, self.inbound_nodes[1].value.T)
# Set the partial of the loss with respect to this node's weights.
self.gradients[self.inbound_nodes[1]] += np.dot(self.inbound_nodes[0].value.T, grad_cost)
# Set the partial of the loss with respect to this node's bias.
self.gradients[self.inbound_nodes[2]] += np.sum(grad_cost, axis=0, keepdims=False)
class Sigmoid(Node):
"""
Represents a node that performs the sigmoid activation function.
"""
def __init__(self, node):
# The base class constructor.
Node.__init__(self, [node])
def _sigmoid(self, x):
"""
This method is separate from `forward` because it
will be used with `backward` as well.
`x`: A numpy array-like object.
"""
return 1. / (1. + np.exp(-x))
def forward(self):
"""
Perform the sigmoid function and set the value.
"""
input_value = self.inbound_nodes[0].value
self.value = self._sigmoid(input_value)
def backward(self):
"""
Calculates the gradient using the derivative of
the sigmoid function.
"""
# Initialize the gradients to 0.
self.gradients = {n: np.zeros_like(n.value) for n in self.inbound_nodes}
# Cycle through the outputs. The gradient will change depending
# on each output, so the gradients are summed over all outputs.
for n in self.outbound_nodes:
# Get the partial of the cost with respect to this node.
grad_cost = n.gradients[self]
"""
TODO: Your code goes here!
Set the gradients property to the gradients with respect to each input.
NOTE: See the Linear node and MSE node for examples.
"""
sigmoid = self.value
self.gradients[self.inbound_nodes[0]] += sigmoid *(1-sigmoid)*grad_cost
class MSE(Node):
def __init__(self, y, a):
"""
The mean squared error cost function.
Should be used as the last node for a network.
"""
# Call the base class' constructor.
Node.__init__(self, [y, a])
def forward(self):
"""
Calculates the mean squared error.
"""
# NOTE: We reshape these to avoid possible matrix/vector broadcast
# errors.
#
# For example, if we subtract an array of shape (3,) from an array of shape
# (3,1) we get an array of shape(3,3) as the result when we want
# an array of shape (3,1) instead.
#
# Making both arrays (3,1) insures the result is (3,1) and does
# an elementwise subtraction as expected.
y = self.inbound_nodes[0].value.reshape(-1, 1)
a = self.inbound_nodes[1].value.reshape(-1, 1)
self.m = self.inbound_nodes[0].value.shape[0]
# Save the computed output for backward.
self.diff = y - a
self.value = np.mean(self.diff**2)
def backward(self):
"""
Calculates the gradient of the cost.
This is the final node of the network so outbound nodes
are not a concern.
"""
self.gradients[self.inbound_nodes[0]] = (2 / self.m) * self.diff
self.gradients[self.inbound_nodes[1]] = (-2 / self.m) * self.diff
def topological_sort(feed_dict):
"""
Sort the nodes in topological order using Kahn's Algorithm.
`feed_dict`: A dictionary where the key is a `Input` Node and the value is the respective value feed to that Node.
Returns a list of sorted nodes.
"""
input_nodes = [n for n in feed_dict.keys()]
G = {}
nodes = [n for n in input_nodes]
while len(nodes) > 0:
n = nodes.pop(0)
if n not in G:
G[n] = {'in': set(), 'out': set()}
for m in n.outbound_nodes:
if m not in G:
G[m] = {'in': set(), 'out': set()}
G[n]['out'].add(m)
G[m]['in'].add(n)
nodes.append(m)
L = []
S = set(input_nodes)
while len(S) > 0:
n = S.pop()
if isinstance(n, Input):
n.value = feed_dict[n]
L.append(n)
for m in n.outbound_nodes:
G[n]['out'].remove(m)
G[m]['in'].remove(n)
# if no other incoming edges add to S
if len(G[m]['in']) == 0:
S.add(m)
return L
def forward_and_backward(graph):
"""
Performs a forward pass and a backward pass through a list of sorted Nodes.
Arguments:
`graph`: The result of calling `topological_sort`.
"""
# Forward pass
for n in graph:
n.forward()
# Backward pass
# see: https://docs.python.org/2.3/whatsnew/section-slices.html
for n in graph[::-1]:
n.backward()
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