Applying Gradient Descent

Introduction

In the last lesson, we derived the functions that help us descend down cost functions efficiently. Remember that this technique is not so different from what we saw when using the derivative to tell us the next step size and direction in two dimensions.

When descending down our cost curve in two dimensions, we used the slope of the tangent line at each point to tell us how large of a step to take next. Since the cost curve is a function of $m$ and $b$, we used the gradient to determine each step.

But really this approach is analogous to what you have seen before. For a single variable function, $f(x)$, the derivative tells you the slope of the line tangent to the plot of $f(x)$ at a given value of $x$. In turn, this tells you the next step size. In the case of a multivariable function, $J(m, b)$, we calculate the partial derivative with respect to both variables. For a regression line, these variables are our slope and y-intercept. The gradient allows you to calculate how much to move in either direction in order to reach the local minimum.

Reviewing our gradient descent formulas

Luckily for us, we already did the hard work of deriving these formulas. Now we get to see the fruit of our labor. The following formulas tell us how to update regression variables of $m$ and $b$ to approach a "best fit" line.

  • $ \frac{\partial J}{\partial m}J(m,b) = -2\sum_{i = 1}^n x(y_i - (mx_i + b)) = -2\sum_{i = 1}^n x_i*\epsilon_i$
  • $ \frac{\partial J}{\partial b}J(m,b) = -2\sum_{i = 1}^n(y_i - (mx_i + b)) = -2\sum_{i = 1}^n \epsilon_i $

Given the formulas above, we can work with any dataset of $x$ and $y$ values to determine the best fit line. We simply iterate through our dataset and use the formulas above to determine an update to $m$ and $b$ that will bring us closer to the minimum. So ultimately, to descend along the cost function, we will use the calculations:

current_m = old_m $ - (-2*\sum_{i=1}^n x_i*\epsilon_i )$

current_b = old_b $ - ( -2*\sum_{i=1}^n \epsilon_i )$

Ok, let's turn this into code. First, let's initialize some data.

# our data
first_show = {'x': 30, 'y': 45}
second_show = {'x': 40, 'y': 60}
third_show = {'x': 100, 'y': 150}

shows = [first_show, second_show, third_show]

Now we set our initial regression line by initializing $m$ and $b$ to zero. Then to update our regression line, we iterate through each of the points in the dataset, and at each iteration, change our update_to_b by $2*\epsilon$ and change our update_to_m by $2*x*\epsilon$.

# initial variables of our regression line
b_current = 0
m_current = 0

# amount to update our variables for our next step
update_to_b = 0
update_to_m = 0 

def error_at(point, b, m):
    return (m*point['x'] + b - point['y'])

for i in range(0, len(shows)):
    update_to_b += -2*(error_at(shows[i], b_current, m_current))
    update_to_m += -2*(error_at(shows[i], b_current, m_current)*shows[i]['x'])

new_b = b_current - update_to_b
new_m = m_current - update_to_m

In the last two lines of the code above, we calculate our new_b and new_m values by updating our current values and adding our respective updates. We define a function called error_at, which we can use in the error component of our partial derivatives above.

The code above represents just one update to our regression line and, therefore, just one step towards our best fit line. We'll just repeat the process to take multiple steps. But first we have to make a couple other changes.

Tweaking our approach

Ok, the above code is very close to what we want, but we just need to make a few small tweaks before it's perfect.

The first one is obvious if we think about what these formulas are really telling us to do. Look at the graph below, and think about what it means to change each of our $m$ and $b$ variables by at least the sum of all of the errors (the $y_i$ values that our regression line predicts and our actual data). That would be an enormous change. To ensure that we do not drastically update our regression line after each step, we multiply each of these partial derivatives by a learning rate. As we have seen before, the learning rate is just a small number, like $0.0001$, which controls how large our updates to the regression line will be. The learning rate is represented by the Greek letter eta, $\eta$, or alpha $\alpha$. We'll use eta, so $\eta = 0.0001$ means the learning rate is $0.0001$.

Multiplying our step size by our learning rate works fine, so long as we multiply both of the partial derivatives by the same amount. This is because with think of our gradient, $ \nabla J(m,b)$, as steering us in the correct direction. In other words, our derivatives ensure we are making the correct proportional changes to $m$ and $b$. So scaling down these changes to make sure we don't update our regression line too quickly works fine, so long as we keep me moving in the correct direction. While we're at it, we can also get rid of multiplying our partials by 2. As mentioned, so long as our changes are proportional we're in good shape.

Before discussing our second tweak, note that as the size of the dataset increases, the sum of the errors will also increase. But this doesn't mean our formulas are less accurate or that they require larger changes. It just means that the total error is larger. We should really think of accuracy as being proportional to the size of our dataset. We can correct for this effect by dividing the effect of our update by the size of our dataset, $n$.

Making these changes, our formula looks like the following:

#amount to update our variables for our next step
update_to_b = 0
update_to_m = 0 

learning_rate = .0001
n = len(shows)
for i in range(0, n):

    update_to_b += -(1/n)*(error_at(shows[i], b_current, m_current))
    update_to_m += -(1/n)*(error_at(shows[i], b_current, m_current)*shows[i]['x'])

new_b = b_current - (learning_rate*update_to_b)
new_m = m_current - (learning_rate*update_to_m)

So our code now reflects everything we know about our gradient descent process: Start with an initial regression line with values of $m$ and $b$. Then for each point, calculate how the regression line's prediction compares to the actual point (that is, find the error). Update what our next step to each variable should be by using the partial derivative. And after iterating through all of the points, update the values of $b$ and $m$ appropriately, scaled down by a learning rate.

Seeing our gradient descent formulas in action

As mentioned earlier, the code above represents just one update to our regression line, and therefore just one step towards our best fit line. To take multiple steps, we'll wrap the process we want to duplicate in a function called step_gradient so we can call that function as much as we want.

first_show = {'x': 30, 'y': 45}
second_show = {'x': 40, 'y': 60}
third_show = {'x': 100, 'y': 150}

shows = [first_show, second_show, third_show]

def step_gradient(b_current, m_current, points):
    b_gradient = 0
    m_gradient = 0
    learning_rate = .0001
    N = float(len(points))
    for i in range(0, len(points)):
        x = points[i]['x']
        y = points[i]['y']
        b_gradient += -(1/N) * (y - ((m_current * x) + b_current))
        m_gradient += -(1/N) * x * (y - ((m_current * x) + b_current))
    new_b = b_current - (learning_rate * b_gradient)
    new_m = m_current - (learning_rate * m_gradient)
    return {'b': new_b, 'm': new_m}
b = 0
m = 0

step_gradient(b, m, shows) # {'b': 0.0085, 'm': 0.6249999999999999}
{'b': 0.0085, 'm': 0.6249999999999999}

Take a look at the input and output. We begin by setting $b$ and $m$ to 0, 0. Then from our step_gradient function, we receive new values of $b$ and $m$ of .0085 and .6245. Now what we need to do, is take another step in the correct direction by calling our step gradient function with our updated values of $b$ and $m$.

updated_b = 0.0085
updated_m = 0.6249
step_gradient(updated_b, updated_m, shows) # {'b': 0.01345805, 'm': 0.9894768333333332}
{'b': 0.01345805, 'm': 0.9894768333333332}

Let's do this, say, 10 times.

# set our initial step with m and b values, and the corresponding error.
b = 0
m = 0
iterations = []
for i in range(10):
    iteration = step_gradient(b, m, shows)
    # {'b': value, 'm': value}
    b = iteration['b']
    m = iteration['m']
    # update values of b and m
    iterations.append(iteration)

Let's take a look at these iterations.

iterations
[{'b': 0.0085, 'm': 0.6249999999999999},
 {'b': 0.013457483333333336, 'm': 0.9895351666666665},
 {'b': 0.016348771640555558, 'm': 1.20215258815},
 {'b': 0.018034938763874835, 'm': 1.3261630333815368},
 {'b': 0.01901821141416974, 'm': 1.398492904819568},
 {'b': 0.019591516465717437, 'm': 1.4406797579467343},
 {'b': 0.019925705352372706, 'm': 1.4652855068756228},
 {'b': 0.020120428242875608, 'm': 1.4796369666804499},
 {'b': 0.02023380672219544, 'm': 1.4880075481368862},
 {'b': 0.020299740568747532, 'm': 1.4928897448417577}]

As you can see, our $m$ and $b$ values both update with each step. Not only that, but with each step, the size of the changes to $m$ and $b$ decrease. This is because they are approaching a best fit line.

Animating these changes

We can use Plotly to see these changes to our regression line visually. We'll write a method called to_line that takes a dictionary of $m$ and $b$ variables and changes it to produce a line object. We can then see how our line changes over time.

def to_line(m, b):
    initial_x = 0
    ending_x = 100
    initial_y = m*initial_x + b
    ending_y = m*ending_x + b
    return {'data': [{'x': [initial_x, ending_x], 'y': [initial_y, ending_y]}]}

frames = list(map(lambda iteration: to_line(iteration['m'], iteration['b']),iterations))
frames[0]
{'data': [{'x': [0, 100], 'y': [0.0085, 62.508499999999984]}]}

Now we can see how our regression line changes, and approaches a better model of our data, with each iteration.

from plotly.offline import init_notebook_mode, iplot
from IPython.display import display, HTML

init_notebook_mode(connected=True)

x_values_of_shows = list(map(lambda show: show['x'], shows))
y_values_of_shows = list(map(lambda show: show['y'], shows))
figure = {'data': [{'x': [0], 'y': [0]}, {'x': x_values_of_shows, 'y': y_values_of_shows, 'mode': 'markers'}],
          'layout': {'xaxis': {'range': [0, 110], 'autorange': False},
                     'yaxis': {'range': [0,160], 'autorange': False},
                     'title': 'Regression Line',
                     'updatemenus': [{'type': 'buttons',
                                      'buttons': [{'label': 'Play',
                                                   'method': 'animate',
                                                   'args': [None]}]}]
                    },
          'frames': frames}
iplot(figure)

As you can see, our regression line starts off far away. Using our step_gradient function, the regression line moved closer to the line that produces the lowest error.

Summary

In this section, we saw our gradient descent formulas in action. The core of the gradient descent functions are understanding the two lines:

$$ \frac{\partial J}{\partial m}J(m,b) = -2\sum_{i = 1}^n x(y_i - (mx_i + b)) = -2\sum_{i = 1}^n x_i*\epsilon_i$$ $$ \frac{\partial J}{\partial b}J(m,b) = -2\sum_{i = 1}^n(y_i - (mx_i + b)) = -2\sum_{i = 1}^n \epsilon_i $$

Both equations use the errors of the current regression line to determine how to update the regression line next. These formulas came from our cost function, $J(m,b) = \sum_{i = 1}^n(y_i - (mx_i + b))^2 $ and from using the gradient to find the direction of steepest descent. Translating this into code, and seeing how the regression line continued to improve in alignment with the data, we saw the effectiveness of this technique in practice.

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