Backward Function - Each operation

In [1]:

from graphviz import Digraph

def trace(root):
    #Builds a set of all nodes and edges in a graph
    nodes, edges = set(), set()
    def build(v):
        if v not in nodes:
            nodes.add(v)
            for child in v._prev:
                edges.add((child, v))
                build(child)
    build(root)
    return nodes, edges

def draw_dot(root):
    dot = Digraph(format='svg', graph_attr={'rankdir': 'LR'}) #LR == Left to Right

    nodes, edges = trace(root)
    for n in nodes:
        uid = str(id(n))
        #For any value in the graph, create a rectangular ('record') node for it
        dot.node(name = uid, label = "{ %s | data %.4f | grad %.4f }" % ( n.label, n.data, n.grad), shape='record')
        if n._op:
            #If this value is a result of some operation, then create an op node for it
            dot.node(name = uid + n._op, label=n._op)
            #and connect this node to it
            dot.edge(uid + n._op, uid)

    for n1, n2 in edges:
        #Connect n1 to the node of n2
        dot.edge(str(id(n1)), str(id(n2)) + n2._op)

    return dot

from graphviz import Digraph def trace(root): #Builds a set of all nodes and edges in a graph nodes, edges = set(), set() def build(v): if v not in nodes: nodes.add(v) for child in v._prev: edges.add((child, v)) build(child) build(root) return nodes, edges def draw_dot(root): dot = Digraph(format='svg', graph_attr={'rankdir': 'LR'}) #LR == Left to Right nodes, edges = trace(root) for n in nodes: uid = str(id(n)) #For any value in the graph, create a rectangular ('record') node for it dot.node(name = uid, label = "{ %s | data %.4f | grad %.4f }" % ( n.label, n.data, n.grad), shape='record') if n._op: #If this value is a result of some operation, then create an op node for it dot.node(name = uid + n._op, label=n._op) #and connect this node to it dot.edge(uid + n._op, uid) for n1, n2 in edges: #Connect n1 to the node of n2 dot.edge(str(id(n1)), str(id(n2)) + n2._op) return dot

In [2]:

import math

import math

In [ ]:

class Value:

    def __init__(self, data, _children=(), _op='', label=''):
        self.data = data
        self.grad = 0.0
        self._prev = set(_children)
        self._op = _op
        self.label = label


    def __repr__(self):   # This basically allows us to print nicer looking expressions for the final output
        return f"Value(data={self.data})"

    def __add__(self, other):
        out = Value(self.data + other.data, (self, other), '+')

        return out

    def __mul__(self, other):
        out = Value(self.data * other.data, (self, other), '*')

        return out

    def tanh(self):
        x = self.data
        t = (math.exp(2*x) - 1)/(math.exp(2*x) + 1)
        out = Value(t, (self, ), 'tanh')

        return out

class Value: def __init__(self, data, _children=(), _op='', label=''): self.data = data self.grad = 0.0 self._prev = set(_children) self._op = _op self.label = label def __repr__(self): # This basically allows us to print nicer looking expressions for the final output return f"Value(data={self.data})" def __add__(self, other): out = Value(self.data + other.data, (self, other), '+') return out def __mul__(self, other): out = Value(self.data * other.data, (self, other), '*') return out def tanh(self): x = self.data t = (math.exp(2*x) - 1)/(math.exp(2*x) + 1) out = Value(t, (self, ), 'tanh') return out

In [ ]:

#What the graph looks like right now, the grad values are set to 0
draw_dot(o)

#What the graph looks like right now, the grad values are set to 0 draw_dot(o)

Out[ ]:

---

We'll be adding a 'backward' function to our Value object and then implement its functionality in each of the operation function.

We are basically converting everything we did manually to calculate the gradients in each operation to code :)

In [29]:

class Value:

    def __init__(self, data, _children=(), _op='', label=''):
        self.data = data
        self.grad = 0.0
        self._backward = lambda: None #Its an empty function by default. This is what will do that gradient calculation at each of the operations.
        self._prev = set(_children)
        self._op = _op
        self.label = label


    def __repr__(self):   # This basically allows us to print nicer looking expressions for the final output
        return f"Value(data={self.data})"

    def __add__(self, other):
        out = Value(self.data + other.data, (self, other), '+')

        def backward():
          self.grad = 1.0 * out.grad  #Remember we are doing chain rule here, hence the product with out.grad
          other.grad = 1.0 * out.grad

        out._backward = backward
        return out

    def __mul__(self, other):
        out = Value(self.data * other.data, (self, other), '*')

        def backward():
          self.grad = other.data * out.grad #Remember we are doing chain rule here, hence the product with out.grad
          other.grad = self.data * out.grad

        out._backward = backward
        return out

    def tanh(self):
        x = self.data
        t = (math.exp(2*x) - 1)/(math.exp(2*x) + 1)
        out = Value(t, (self, ), 'tanh')

        def backward():
          self.grad = 1 - (t**2) * out.grad #Remember we are doing chain rule here, hence the product with out.grad

        out._backward = backward
        return out

class Value: def __init__(self, data, _children=(), _op='', label=''): self.data = data self.grad = 0.0 self._backward = lambda: None #Its an empty function by default. This is what will do that gradient calculation at each of the operations. self._prev = set(_children) self._op = _op self.label = label def __repr__(self): # This basically allows us to print nicer looking expressions for the final output return f"Value(data={self.data})" def __add__(self, other): out = Value(self.data + other.data, (self, other), '+') def backward(): self.grad = 1.0 * out.grad #Remember we are doing chain rule here, hence the product with out.grad other.grad = 1.0 * out.grad out._backward = backward return out def __mul__(self, other): out = Value(self.data * other.data, (self, other), '*') def backward(): self.grad = other.data * out.grad #Remember we are doing chain rule here, hence the product with out.grad other.grad = self.data * out.grad out._backward = backward return out def tanh(self): x = self.data t = (math.exp(2*x) - 1)/(math.exp(2*x) + 1) out = Value(t, (self, ), 'tanh') def backward(): self.grad = 1 - (t**2) * out.grad #Remember we are doing chain rule here, hence the product with out.grad out._backward = backward return out

In [32]:

#Inputs x1, x2 of the neuron
x1 = Value(2.0, label='x1')
x2 = Value(0.0, label='x2')

#Weights w1, w2 of the neuron - The synaptic values
w1 = Value(-3.0, label='w1')
w2 = Value(1.0, label='w2')

#The bias of the neuron
b = Value(6.8813735870195432, label='b')

x1w1 = x1*w1; x1w1.label = 'x1*w1'
x2w2 = x2*w2; x2w2.label = 'x2*w2'

#The summation
x1w1x2w2 = x1w1 + x2w2; x1w1x2w2.label = 'x1*w1 + x2*w2'

#n is basically the cell body, but without the activation function
n = x1w1x2w2 + b; n.label = 'n'

#Now we pass n to the activation function
o = n.tanh(); o.label = 'o'

#Inputs x1, x2 of the neuron x1 = Value(2.0, label='x1') x2 = Value(0.0, label='x2') #Weights w1, w2 of the neuron - The synaptic values w1 = Value(-3.0, label='w1') w2 = Value(1.0, label='w2') #The bias of the neuron b = Value(6.8813735870195432, label='b') x1w1 = x1*w1; x1w1.label = 'x1*w1' x2w2 = x2*w2; x2w2.label = 'x2*w2' #The summation x1w1x2w2 = x1w1 + x2w2; x1w1x2w2.label = 'x1*w1 + x2*w2' #n is basically the cell body, but without the activation function n = x1w1x2w2 + b; n.label = 'n' #Now we pass n to the activation function o = n.tanh(); o.label = 'o'

Now, we call the '_backward' function that we have made one by one in order (backwards through the equation/graph)

But before we could do that, we have to first set the value of o.grad to 1.0
As notice in the Value object code that, we have initialised it to 0

Therefore, we'll start by adding o.grad to 1.0 and then we'll call the '_backward' function rest of them one by one

In [33]:

o.grad = 1.0

o.grad = 1.0

In [34]:

draw_dot(o)

draw_dot(o)

Out[34]:

Now. we go with o

In [35]:

o._backward()

o._backward()

In [36]:

draw_dot(o)

draw_dot(o)

Out[36]:

Now, from n

In [37]:

n._backward()

n._backward()

In [38]:

draw_dot(o)

draw_dot(o)

Out[38]:

Now, b is a leaf node so we just leave it there (Another reason why _backward was initiallised to None function, it's because it won't be called for leaf nodes. Therefore we set it to None for them)

We'll continue with x1w1x1w2

In [39]:

x1w1x2w2._backward()

x1w1x2w2._backward()

In [40]:

draw_dot(o)

draw_dot(o)

Out[40]:

Now finally, to finding the gradient of the intial values

In [41]:

x1w1._backward()
x2w2._backward()

x1w1._backward() x2w2._backward()

In [42]:

draw_dot(o)

draw_dot(o)

Out[42]:

---

Final output^

Hence, we have not only verified the manual backpropagation calculation that we did, but also created funtions directly for each of them!