Gradient Descent Step Sizes Lab

Introduction

In this lab, we'll practice applying gradient descent. As we know gradient descent begins with an initial regression line, and moves to a "best fit" regression line by changing values of $m$ and $b$ and evaluating the RSS. So far, we have illustrated this technique by changing the values of $b$ and evaluating the RSS. In this lab, we will work through applying our technique by changing the value of $m$ instead. Let's get started.

Setting up our initial regression line

Once again, we'll take take a look at revenues of movies to predict revenue.

first_show = {'budget': 100, 'revenue': 275}
second_show = {'budget': 200, 'revenue': 300}
third_show = {'budget': 400, 'revenue': 700}

shows = [first_show, second_show, third_show]

We can start with some values for an initial not-so-accurate regression line, $y = .6x + 133.33$.

from linear_equations import build_regression_line

budgets = list(map(lambda show: show['budget'], shows))
revenues = list(map(lambda show: show['revenue'], shows))

def regression_line(x):
    return .6*x + 133.33

Now using the residual_sum_squares, function, we calculate the RSS to measure the accuracy of the regression line to our data. Let's take another look at that function:

def squared_error(x, y, m, b):
    return (y - (m*x + b))**2

def residual_sum_squares(x_values, y_values, m, b):
    data_points = list(zip(x_values, y_values))
    squared_errors = map(lambda data_point: squared_error(data_point[0], data_point[1], m, b), data_points)
    return sum(squared_errors)

from plotly.offline import iplot, init_notebook_mode
from graph import plot, m_b_trace
import plotly.graph_objs as go

init_notebook_mode(connected=True)

from graph import trace_values, plot

data_trace = trace_values(budgets, revenues)
regression_trace = m_b_trace(.6, 133.33, budgets)
plot([data_trace, regression_trace])

Building a cost curve

Now let's use the residual_sum_squares function to build a cost curve. Keeping the $b$ value fixed at $133.33$, write a function called rss_values.

rss_values passes our dataset with the x_values and y_values arguments.
It also takes a list of values of $m$, and an initial $b$ value as arguments.
It outputs a dictionary with keys of m_values and rss_values, with each key pointing to a list of the corresponding values.

def rss_values(x_values, y_values, m_values, b):
    pass

budgets = list(map(lambda show: show['budget'] ,shows))
revenues = list(map(lambda show: show['revenue'] ,shows))
initial_m_values = list(range(8, 19, 1))
scaled_m_values = list(map(lambda initial_m_value: initial_m_value/10,initial_m_values))
b_value = 133.33
rss_values(budgets, revenues, scaled_m_values, b_value)

# {'m_values': [0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8],
#  'rss_values': [64693.76669999998,
#   45559.96669999998,
#   30626.166699999987,
#   19892.36669999999,
#   13358.5667,
#   11024.766700000004,
#   12890.96670000001,
#   18957.166700000016,
#   29223.36670000002,
#   43689.566700000025,
#   62355.76670000004]}

Plotly provides for us a table chart, and we can pass the values generated from our rss_values function to create a table.

from plotly.offline import iplot, init_notebook_mode
from graph import plot
import plotly.graph_objs as go

init_notebook_mode(connected=True)

def plot_table(headers, columns):
    trace_cost_chart = go.Table(
        header=dict(values=headers,
                    line = dict(color='#7D7F80'),
                    fill = dict(color='#a1c3d1'),
                    align = ['left'] * 5),
        cells=dict(values=columns,
                   line = dict(color='#7D7F80'),
                   fill = dict(color='#EDFAFF'),
                   align = ['left'] * 5))
    plot([trace_cost_chart])

cost_chart = rss_values(budgets, revenues, scaled_m_values, b_value)
if cost_chart: 
    column_values = list(cost_chart.values())
    plot_table(headers = ['M values', 'RSS values'], columns=column_values)

And let's plot this out using a a line chart.

from plotly.offline import iplot, init_notebook_mode
init_notebook_mode(connected=True)
from graph import plot, trace_values

initial_m_values = list(range(1, 18, 1))
scaled_m_values = list(map(lambda initial_m_value: initial_m_value/10,initial_m_values))
cost_values = rss_values(budgets, revenues, scaled_m_values, 133.33)

if cost_values:
    rss_trace = trace_values(cost_values['m_values'], cost_values['rss_values'], mode = 'lines')
    plot([rss_trace])

Looking at the slope of our cost curve

In this section, we'll work up to building a gradient descent function that automatically changes our step size. To get you started, we'll provide a function called slope_at that calculates the slope of the cost curve at a given point on the cost curve. Here it is in action:

from helper import slope_at

slope_at(budgets, revenues, .6, 133.33333333333326)

slope_at(budgets, revenues, 1.6, 133.33333333333326)

So the slope_at function takes in our dataset, and returns the slope of the cost curve at that point. So the numbers -296312.33 and 123687.67 reflect the slopes at the cost curve when m is .6 and 1.6 respectively.

As you can see, it seems pretty accurate. When the curve is steeper and downwards at $m = 0.6$, the slope is around -290,000. And at $m = 1.6$ with our cost curve pointing upwards yet flatter, our slope is around 120,000.

Moving towards gradient descent

Now that we are familiar with our slope_at function and how it calculates the slope of our cost curve at a given point, we can begin to use that function with our gradient descent procedure.

Remember that gradient descent works by starting at a regression line with values m, and b, which corresponds to a point on our cost curve. Then we alter our m or b value (here, the m value) by looking to the slope of the cost curve at that point. Then we look to the slope of the cost curve at the new m value to indicate the size and direction of the next step.

So now let's write a function called updated_m. The function will tell us the step size and direction to move along our cost curve. The updated_m function takes as arguments an initial value of $m$, a learning rate, and the slope of the cost curve at that value of $m$. Its return value is the next value of m that it calculates.

from error import residual_sum_squares

def updated_m(m, learning_rate, cost_curve_slope):
    pass

This is what our function returns.

current_slope = slope_at(budgets, revenues, 1.7, 133.33333333333326)['slope']
updated_m(1.7, .000001, current_slope)
# 1.5343123333335096

current_slope = slope_at(budgets, revenues, 1.534, 133.33333333333326)['slope']
updated_m(1.534, .000001, current_slope)
# 1.43803233333338

current_slope = slope_at(budgets, revenues, 1.438, 133.33333333333326)['slope']
updated_m(1.438, .000001, current_slope)
# 1.3823523333332086

Take a careful look at how we use the updated_m function. By using our updated value of $m$ we are quickly converging towards an optimal value of $m$.

Now let's write another function called gradient_descent. The inputs of the function are x_values, y_values, steps, the b we are holding constant, the learning_rate, and the current_m that we are looking at. The steps arguments represents the number of steps the function will take before the function stops. We can get a sense of the return value in the cell below. It is a list of dictionaries, with each dictionary having a key of the current m value, the slope of the cost curve at that m value, and the rss at that m value.

def gradient_descent(x_values, y_values, steps, b, learning_rate, current_m):
    pass

descent_steps = gradient_descent(budgets, revenues, 12, 133.33, learning_rate = .000001, current_m = 0)
descent_steps

# [{'m': 0, 'rss': 368964.16669999994, 'slope': -548316.9999998063},
#  {'m': 0.5483169999998062, 'rss': 131437.9413767516, 'slope': -318023.86000024853},
#  {'m': 0.8663408600000547,  'rss': 51531.31420747324, 'slope': -184453.83880040026}, 
#  {'m': 1.050794698800455,  'rss': 24649.097944855268,  'slope': -106983.22650372575},
#  {'m': 1.1577779253041809,  'rss': 15604.976802103287,  'slope': -62050.271372208954},
#  {'m': 1.2198281966763898,  'rss': 12561.987166284125,  'slope': -35989.15739588847},
#  {'m': 1.2558173540722781,  'rss': 11538.008028425651, 'slope': -20873.711289696075},
#  {'m': 1.2766910653619743,  'rss': 11193.357340315255,  'slope': -12106.752547970245},
#  {'m': 1.2887978179099446,  'rss': 11077.310067278091,  'slope': -7021.916477824561},
#  {'m': 1.295819734387769,  'rss': 11038.209831325046,  'slope': -4072.711557128059},
#  {'m': 1.299892445944897,  'rss': 11025.020590634533,  'slope': -2362.172703124088},
#  {'m': 1.302254618648021,  'rss': 11020.562895703,  'slope': -1370.0601677373925}]

if descent_steps:
    m_values = list(map(lambda step: step['m'],descent_steps))
    rss_result_values = list(map(lambda step: step['rss'], descent_steps))
    text_values = list(map(lambda step: 'cost curve slope: ' + str(step['slope']), descent_steps))
    gradient_trace = trace_values(m_values, rss_result_values, text=text_values)
    plot([gradient_trace])

Taking a look at a plot of our trace, you can get a nice visualization of how our gradient descent function works. It starts far away with $m = 0$, and the step size is relatively large, as is the slope of the cost curve. As the $m$ value updates such that it approaches a minimum of the RSS, the slope of the cost curve and the size of each step both decrease.

Remember that each of these steps indicates a change in our regression line's slope value towards a "fit" that more accurately matches our dataset. Let's plot these various regression lines below, starting with our slope at 0 and moving from there.

from plotly.offline import iplot, init_notebook_mode
from graph import plot, m_b_trace
import plotly.graph_objs as go

init_notebook_mode(connected=True)

from graph import trace_values, plot

data_trace = trace_values(budgets, revenues)
if descent_steps:
    m_values = list(map(lambda step: step['m'] ,descent_steps))
    regression_traces = list(map(lambda m: m_b_trace(m, 133.33, budgets, name = 'm:' + str(round(m, 2))), m_values))
    plot([data_trace, *regression_traces])

As you can see the slope converges towards a slope that better matches our data, around m = 1.3. You can isolate how the line changes clicking on the names of the lines to the right, which toggles the display of the respective lines.

Summary

In this lesson, we learned some more about gradient descent. We saw how gradient descent allows our function to improve to a regression line that better matches our data. We see how to change our regression line, by looking at the Residual Sum of Squares related to current regression line. We update our regression line by looking at the rate of change of our RSS as we adjust our regression line in the right direction -- that is, the slope of our cost curve. The larger the magnitude of our rate of change (or slope of our cost curve) the larger our step size. This way, we take larger steps the further away we are from our minimizing our RSS, and take smaller steps as we converge towards our minimum RSS.

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