# Backpropagation - Exercise: Computational Graphs

## Introduction

In this exercise, you will apply the backpropagation algorithm to computational graphs. Backpropagation is the method we use to calculate the gradients of all learnable parameters in an artificial neural network efficiently and conveniently. Approaching the algorithm from the perspective of computational graphs gives a good intuition about its operations.

## Requirements

### Knowledge

You should already be familiar with the backpropagation algorithm, as well as differential calculus. The following resources are recommend if this topic is new to you:

• Chapter 6.5 of Deep Learning by Ian Goodfellow gives a brief introduction into the field
• MOOC Calculus from Khan Academy

### Python and System Packages

To run that Notebook you should have the graphviz package installed on your System. Look at the Graphviz-Website for installation guidance. Further, the Python module Digraph is only working with Jupyter Notebook (not Lab) at the moment.

# External Modules
import numpy as np
import hashlib
from graphviz import Digraph
def round_and_hash(value, precision=4, dtype=np.float32):
"""
Function to round and hash a scalar or numpy array of scalars.
Used to compare results with true solutions without spoiling the solution.
"""
rounded = np.array([value], dtype=dtype).round(decimals=precision)
hashed = hashlib.md5(rounded).hexdigest()
return hashed

## Pen and Paper Backpropagation

Given the function$f(a,b,c)$ as a computational graph with the following values for the parameters:

$a = 2 \\ b = e \; (Euler \;number, \;b \approx 2.7183) \\ c = 3$

• Give an equation that expresses the graph.
• Calculate the partial derivatives$\frac {\partial out} {\partial a}$,$\frac {\partial out} {\partial b}$. and$\frac {\partial out} {\partial c}$ using calculus on pen & paper
• For b use the the true$Euler \;number$ and not the approximation
• Assign your results to variables partial_a,partial_b and partial_c to verify your solution with the software test below
# creating empty graph ad set some attributes
f = Digraph('computat:onal_graph', filename='graph_clean.gv')
f.attr(rankdir='LR')
f.attr('node', shape='circle')

# create the graph
f.node('a')
f.node('b')
f.node('c')
f.edge('a', '+', label=' ')
f.edge('b', 'ln', label=' ')
f.edge('ln', '+', label=' ')
f.edge('+','* ', label=' ')
f.edge('c','* ')
f.edge('* ', '*')
f.edge('1/3 ', '*')
f.edge('*','1/x')
f.edge('1/x','out')

f
partial_a = 42 ### assign what u have calculated
partial_b = 42 ### assign what u have calculated
partial_c = 42 ### assign what u have calculated
assert round_and_hash(partial_a) == '2fb0a82d3fe965c8a0d9ce85970058d8'
assert round_and_hash(partial_b) == 'b6bfa6042c097c90ab76a61300d2429a'
assert round_and_hash(partial_c) == '2fb0a82d3fe965c8a0d9ce85970058d8'

## Implement the Example

In the pen and paper exercise, you calculated the partial derivatives for some specific values of$a$,$b$, and$c$ of the graph$f(a,b,c)$. Your task now is to generalize that solution. Implement the used functions$+$,$*$ and$1/x$ and their derivatives. Chain the functions to calculate$f(a,b,c)$ in a forward pass and$\frac {\partial out} {\partial a}$,$\frac {\partial out} {\partial b}$,$\frac {\partial out} {\partial c}$ in the backward pass for arbitrary values of$a$,$b$, and$c$.

Hint:

Complete function:

$out = powminusone(mul_2(mul_1(add(a, ln(b)), c), \frac{1}{3}))$

So...

Remember chain rule: $\frac{\partial out}{\partial a} = \frac{\partial out}{\partial powminusone} * \frac{\partial powminusone}{\partial mul_2} * \frac{\partial mul_2}{\partial mul_1} * \frac{\partial mul_1}{\partial add} * \frac{\partial add}{\partial a}$

$\frac{\partial out}{\partial b} = \frac{\partial out}{\partial powminusone} * \frac{\partial powminusone}{\partial mul_2} * \frac{\partial mul_2}{\partial mul_1} * \frac{\partial mul_1}{\partial add} * \frac{\partial add}{\partial ln(b)} * \frac{\partial ln(b)}{\partial b}$

$\frac{\partial out}{\partial c} = \frac{\partial out}{\partial powminusone} * \frac{\partial powminusone}{\partial mul_2} * \frac{\partial mul_2}{\partial mul_1} * \frac{\partial mul_1}{\partial c}$

### Your implementation here

# 1/x is the same like "x to the power of minus one"
def pow_minus_one(x, derivative=False):
if derivative:
raise NotImplementedError()
else:
raise NotImplementedError()

if derivative:
raise NotImplementedError()
else:
raise NotImplementedError()

def multi(x,y, derivative=False):
if derivative:
# Derivative with respect to x
raise NotImplementedError()
else:
raise NotImplementedError()

a = 2
b = np.e
c = 3

### complete the forward pass:
# forward = pow_minus_one(multi(...

### complete the backwards pass for "a"
#        * ...

### complete the backwards pass for "b"
#part_b = ...

### complete the backwards pass for "c"
#part_c = ...

### print for self control
#print(part_a)
#print(part_b)
#print(part_b)
assert round_and_hash(forward) == '8565183eaf4b5c6356d6abb81b8e139d'

assert round_and_hash(part_a) == '2fb0a82d3fe965c8a0d9ce85970058d8'
assert round_and_hash(part_b) == 'b6bfa6042c097c90ab76a61300d2429a'
assert round_and_hash(part_c) == '2fb0a82d3fe965c8a0d9ce85970058d8'

## Summary and Outlook

In this exercise you implemented backpropagation in a computational graph. You applied the chain rule to compute the partial derivative of the output with respect to each node in the graph. In the upcoming exercises, you'll use the same backpropagation method to learn the gradients of parameters in a neural network.

The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).

Exercise: Computational Graphs
by Benjamin Voigt