Lecture 3: Optimization and Linear Regression

Applied Machine Learning

Volodymyr Kuleshov
Cornell Tech

Part 1: Optimization and Calculus Background

In the previous lecture, we learned what is a supervised machine learning problem.

Before we turn our attention to Linear Regression, we will first dive deeper into the question of optimization.

Review: Components of A Supervised Machine Learning Problem

At a high level, a supervised machine learning problem has the following structure:

$$ \text{Dataset} + \underbrace{\text{Learning Algorithm}}_\text{Model Class + Objective + Optimizer } \to \text{Predictive Model} $$

The predictive model is chosen to model the relationship between inputs and targets. For instance, it can predict future targets.

Optimizer: Notation

At a high-level an optimizer takes

• an objective $J$ (also called a loss function) and
• a model class $\mathcal{M}$ and finds a model $f \in \mathcal{M}$ with the smallest value of the objective $J$.

\begin{align*} \min_{f \in \mathcal{M}} J(f) \end{align*}

Intuitively, this is the function that bests "fits" the data on the training dataset $\mathcal{D} = \{(x^{(i)}, y^{(i)}) \mid i = 1,2,...,n\}$.

We will use the a quadratic function as our running example for an objective $J$.

import numpy as np
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = [8, 4]

def quadratic_function(theta):
    """The cost function, J(theta)."""
    return 0.5*(2*theta-1)**2

We can visualize it.

# First construct a grid of theta1 parameter pairs and their corresponding
# cost function values.
thetas = np.linspace(-0.2,1,10)
f_vals = quadratic_function(thetas[:,np.newaxis])

plt.plot(thetas, f_vals)
plt.xlabel('Theta')
plt.ylabel('Objective value')
plt.title('Simple quadratic function')

Text(0.5, 1.0, 'Simple quadratic function')

Calculus Review: Derivatives

Recall that the derivative $$\frac{d f(\theta_0)}{d \theta}$$ of a univariate function $f : \mathbb{R} \to \mathbb{R}$ is the instantaneous rate of change of the function $f(\theta)$ with respect to its parameter $\theta$ at the point $\theta_0$.

def quadratic_derivative(theta):
    return (2*theta-1)*2

df0 = quadratic_derivative(np.array([[0]])) # derivative at zero
f0 = quadratic_function(np.array([[0]]))
line_length = 0.2

plt.plot(thetas, f_vals)
plt.annotate('', xytext=(0-line_length, f0-line_length*df0), xy=(0+line_length, f0+line_length*df0),
             arrowprops={'arrowstyle': '-', 'lw': 1.5}, va='center', ha='center')
plt.xlabel('Theta')
plt.ylabel('Objective value')
plt.title('Simple quadratic function')

Text(0.5, 1.0, 'Simple quadratic function')

pts = np.array([[0, 0.5, 0.8]]).reshape((3,1))
df0s = quadratic_derivative(pts)
f0s = quadratic_function(pts)

plt.plot(thetas, f_vals)
for pt, f0, df0 in zip(pts.flatten(), f0s.flatten(), df0s.flatten()): 
    plt.annotate('', xytext=(pt-line_length, f0-line_length*df0), xy=(pt+line_length, f0+line_length*df0),
             arrowprops={'arrowstyle': '-', 'lw': 1}, va='center', ha='center')
plt.xlabel('Theta')
plt.ylabel('Objective value')
plt.title('Simple quadratic function')

Text(0.5, 1.0, 'Simple quadratic function')

Calculus Review: Partial Derivatives

The partial derivative $$\frac{\partial f(\theta_0)}{\partial \theta_j}$$ of a multivariate function $f : \mathbb{R}^d \to \mathbb{R}$ is the derivative of $f$ with respect to $\theta_j$ while all othe other inputs $\theta_k$ for $k\neq j$ are fixed.

Calculus Review: The Gradient

The gradient $\nabla_\theta f$ further extends the derivative to multivariate functions $f : \mathbb{R}^d \to \mathbb{R}$, and is defined at a point $\theta_0$ as

$$ \nabla_\theta f (\theta_0) = \begin{bmatrix} \frac{\partial f(\theta_0)}{\partial \theta_1} \\ \frac{\partial f(\theta_0)}{\partial \theta_2} \\ \vdots \\ \frac{\partial f(\theta_0)}{\partial \theta_d} \end{bmatrix}.$$

The $j$-th entry of the vector $\nabla_\theta f (\theta_0)$ is the partial derivative $\frac{\partial f(\theta_0)}{\partial \theta_j}$ of $f$ with respect to the $j$-th component of $\theta$.

We will use a quadratic function as a running example.

def quadratic_function2d(theta0, theta1):
    """Quadratic objective function, J(theta0, theta1).
    
    The inputs theta0, theta1 are 2d arrays and we evaluate
    the objective at each value theta0[i,j], theta1[i,j].
    We implement it this way so it's easier to plot the
    level curves of the function in 2d.

    Parameters:
    theta0 (np.array): 2d array of first parameter theta0
    theta1 (np.array): 2d array of second parameter theta1
    
    Returns:
    fvals (np.array): 2d array of objective function values
        fvals is the same dimension as theta0 and theta1.
        fvals[i,j] is the value at theta0[i,j] and theta1[i,j].
    """
    theta0 = np.atleast_2d(np.asarray(theta0))
    theta1 = np.atleast_2d(np.asarray(theta1))
    return 0.5*((2*theta1-2)**2 + (theta0-3)**2)

Let's visualize this function.

theta0_grid = np.linspace(-4,7,101)
theta1_grid = np.linspace(-1,4,101)
theta_grid = theta0_grid[np.newaxis,:], theta1_grid[:,np.newaxis]
J_grid = quadratic_function2d(theta0_grid[np.newaxis,:], theta1_grid[:,np.newaxis])

X, Y = np.meshgrid(theta0_grid, theta1_grid)
contours = plt.contour(X, Y, J_grid, 10)
plt.clabel(contours)
plt.axis('equal')

(-4.0, 7.0, -1.0, 4.0)

Let's write down the derivative of the quadratic function.

def quadratic_derivative2d(theta0, theta1):
    """Derivative of quadratic objective function.
    
    The inputs theta0, theta1 are 1d arrays and we evaluate
    the derivative at each value theta0[i], theta1[i].

    Parameters:
    theta0 (np.array): 1d array of first parameter theta0
    theta1 (np.array): 1d array of second parameter theta1
    
    Returns:
    grads (np.array): 2d array of partial derivatives
        grads is of the same size as theta0 and theta1
        along first dimension and of size
        two along the second dimension.
        grads[i,j] is the j-th partial derivative 
        at input theta0[i], theta1[i].
    """
    # this is the gradient of 0.5*((2*theta1-2)**2 + (theta0-3)**2)
    grads = np.stack([theta0-3, (2*theta1-2)*2], axis=1)
    grads = grads.reshape([len(theta0), 2])
    return grads

We can visualize the derivative.

theta0_pts, theta1_pts = np.array([2.3, -1.35, -2.3]), np.array([2.4, -0.15, 2.75])
dfs = quadratic_derivative2d(theta0_pts, theta1_pts)
line_length = 0.2

contours = plt.contour(X, Y, J_grid, 10)
for theta0_pt, theta1_pt, df0 in zip(theta0_pts, theta1_pts, dfs):
    plt.annotate('', xytext=(theta0_pt, theta1_pt), 
                     xy=(theta0_pt-line_length*df0[0], theta1_pt-line_length*df0[1]),
                     arrowprops={'arrowstyle': '->', 'lw': 2}, va='center', ha='center')
plt.scatter(theta0_pts, theta1_pts)
plt.clabel(contours)
plt.xlabel('Theta0')
plt.ylabel('Theta1')
plt.title('Gradients of the quadratic function')
plt.axis('equal')

(-4.0, 7.0, -1.0, 4.0)

Part 1b: Gradient Descent

Next, we will use gradients to define an important algorithm called gradient descent.

Calculus Review: The Gradient

The gradient $\nabla_\theta f$ further extends the derivative to multivariate functions $f : \mathbb{R}^d \to \mathbb{R}$, and is defined at a point $\theta_0$ as

$$ \nabla_\theta f (\theta_0) = \begin{bmatrix} \frac{\partial f(\theta_0)}{\partial \theta_1} \\ \frac{\partial f(\theta_0)}{\partial \theta_2} \\ \vdots \\ \frac{\partial f(\theta_0)}{\partial \theta_d} \end{bmatrix}.$$

The $j$-th entry of the vector $\nabla_\theta f (\theta_0)$ is the partial derivative $\frac{\partial f(\theta_0)}{\partial \theta_j}$ of $f$ with respect to the $j$-th component of $\theta$.

theta0_pts, theta1_pts = np.array([2.3, -1.35, -2.3]), np.array([2.4, -0.15, 2.75])
dfs = quadratic_derivative2d(theta0_pts, theta1_pts)
line_length = 0.2

contours = plt.contour(X, Y, J_grid, 10)
for theta0_pt, theta1_pt, df0 in zip(theta0_pts, theta1_pts, dfs):
    plt.annotate('', xytext=(theta0_pt, theta1_pt), 
                     xy=(theta0_pt-line_length*df0[0], theta1_pt-line_length*df0[1]),
                     arrowprops={'arrowstyle': '->', 'lw': 2}, va='center', ha='center')
plt.scatter(theta0_pts, theta1_pts)
plt.clabel(contours)
plt.xlabel('Theta0')
plt.ylabel('Theta1')
plt.title('Gradients of the quadratic function')
plt.axis('equal')

(-4.0, 7.0, -1.0, 4.0)

Gradient Descent: Intuition

Gradient descent is a very common optimization algorithm used in machine learning.

The intuition behind gradient descent is to repeatedly obtain the gradient to determine the direction in which the function decreases most steeply and take a step in that direction.

Gradient Descent: Notation

More formally, if we want to optimize $J(\theta)$, we start with an initial guess $\theta_0$ for the parameters and repeat the following update until $\theta$ is no longer changing: $$ \theta_i := \theta_{i-1} - \alpha \cdot \nabla_\theta J(\theta_{i-1}). $$

As code, this method may look as follows:

theta, theta_prev = random_initialization()
while norm(theta - theta_prev) > convergence_threshold:
    theta_prev = theta
    theta = theta_prev - step_size * gradient(theta_prev)

In the above algorithm, we stop when $||\theta_i - \theta_{i-1}||$ is small.

It's easy to implement this function in numpy.

convergence_threshold = 2e-1
step_size = 2e-1
theta, theta_prev = np.array([[-2], [3]]), np.array([[0], [0]])
opt_pts = [theta.flatten()]
opt_grads = []

while np.linalg.norm(theta - theta_prev) > convergence_threshold:
    # we repeat this while the value of the function is decreasing
    theta_prev = theta
    gradient = quadratic_derivative2d(*theta).reshape([2,1])
    theta = theta_prev - step_size * gradient
    opt_pts += [theta.flatten()]
    opt_grads += [gradient.flatten()]

We can now visualize gradient descent.

opt_pts = np.array(opt_pts)
opt_grads = np.array(opt_grads)

contours = plt.contour(X, Y, J_grid, 10)
plt.clabel(contours)
plt.scatter(opt_pts[:,0], opt_pts[:,1])

for opt_pt, opt_grad in zip(opt_pts, opt_grads):
    plt.annotate('', xytext=(opt_pt[0], opt_pt[1]), 
                 xy=(opt_pt[0]-0.8*step_size*opt_grad[0], opt_pt[1]-0.8*step_size*opt_grad[1]),
                 arrowprops={'arrowstyle': '->', 'lw': 2}, va='center', ha='center')

plt.axis('equal')

(-4.0, 7.0, -1.0, 4.0)

Part 2: Gradient Descent in Linear Models

Let's now use gradient descent to derive a supervised learning algorithm for linear models.

Review: Gradient Descent

If we want to optimize $J(\theta)$, we start with an initial guess $\theta_0$ for the parameters and repeat the following update: $$ \theta_i := \theta_{i-1} - \alpha \cdot \nabla_\theta J(\theta_{i-1}). $$

As code, this method may look as follows:

theta, theta_prev = random_initialization()
while norm(theta - theta_prev) > convergence_threshold:
    theta_prev = theta
    theta = theta_prev - step_size * gradient(theta_prev)

Review: Linear Model Family

Recall that a linear model has the form \begin{align*} y & = \theta_0 + \theta_1 \cdot x_1 + \theta_2 \cdot x_2 + ... + \theta_d \cdot x_d \end{align*} where $x \in \mathbb{R}^d$ is a vector of features and $y$ is the target. The $\theta_j$ are the parameters of the model.

By using the notation $x_0 = 1$, we can represent the model in a vectorized form $$ f_\theta(x) = \sum_{j=0}^d \theta_j \cdot x_j = \theta^\top x. $$

Let's define our model in Python.

def f(X, theta):
    """The linear model we are trying to fit.
    
    Parameters:
    theta (np.array): d-dimensional vector of parameters
    X (np.array): (n,d)-dimensional data matrix
    
    Returns:
    y_pred (np.array): n-dimensional vector of predicted targets
    """
    return X.dot(theta)

An Objective: Mean Squared Error

We pick $\theta$ to minimize the mean squared error (MSE). Slight variants of this objective are also known as the residual sum of squares (RSS) or the sum of squared residuals (SSR). $$J(\theta)= \frac{1}{2n} \sum_{i=1}^n(y^{(i)}-\theta^\top x^{(i)})^2$$ In other words, we are looking for the best compromise in $\theta$ over all the data points.

Let's implement mean squared error.

def mean_squared_error(theta, X, y):
    """The cost function, J, describing the goodness of fit.
    
    Parameters:
    theta (np.array): d-dimensional vector of parameters
    X (np.array): (n,d)-dimensional design matrix
    y (np.array): n-dimensional vector of targets
    """
    return 0.5*np.mean((y-f(X, theta))**2)

Mean Squared Error: Partial Derivatives

Let's work out what a partial derivative is for the MSE error loss for a linear model.

\begin{align*} \frac{\partial J(\theta)}{\partial \theta_j} & = \frac{\partial}{\partial \theta_j} \frac{1}{2} \left( f_\theta(x) - y \right)^2 \\ & = \left( f_\theta(x) - y \right) \cdot \frac{\partial}{\partial \theta_j} \left( f_\theta(x) - y \right) \\ & = \left( f_\theta(x) - y \right) \cdot \frac{\partial}{\partial \theta_j} \left( \sum_{k=0}^d \theta_k \cdot x_k - y \right) \\ & = \left( f_\theta(x) - y \right) \cdot x_j \end{align*}

Mean Squared Error: The Gradient

We can use this derivation to obtain an expression for the gradient of the MSE for a linear model \begin{align*} \nabla_\theta J (\theta) = \begin{bmatrix} \frac{\partial f(\theta)}{\partial \theta_1} \ \frac{\partial f(\theta)}{\partial \theta_2} \ \vdots \ \frac{\partial f(\theta)}{\partial \theta_d}

\end{bmatrix}

\begin{bmatrix} \left( f_\theta(x) - y \right) \cdot x_1 \\ \left( f_\theta(x) - y \right) \cdot x_2 \\ \vdots \\ \left( f_\theta(x) - y \right) \cdot x_d \end{bmatrix}

= \left( f_\theta(x) - y \right) \cdot \bf{x} . \end{align*}

Let's implement the gradient.

def mse_gradient(theta, X, y):
    """The gradient of the cost function.
    
    Parameters:
    theta (np.array): d-dimensional vector of parameters
    X (np.array): (n,d)-dimensional design matrix
    y (np.array): n-dimensional vector of targets
    
    Returns:
    grad (np.array): d-dimensional gradient of the MSE
    """
    return np.mean((f(X, theta) - y) * X.T, axis=1)

The UCI Diabetes Dataset

In this section, we are going to again use the UCI Diabetes Dataset.

• For each patient we have a access to a measurement of their body mass index (BMI) and a quantiative diabetes risk score (from 0-300).
• We are interested in understanding how BMI affects an individual's diabetes risk.

%matplotlib inline
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = [8, 4]

import numpy as np
import pandas as pd
from sklearn import datasets

# Load the diabetes dataset
X, y = datasets.load_diabetes(return_X_y=True, as_frame=True)

# add an extra column of onens
X['one'] = 1

# Collect 20 data points and only use bmi dimension
X_train = X.iloc[-20:].loc[:, ['bmi', 'one']]
y_train = y.iloc[-20:] / 300

plt.scatter(X_train.loc[:,['bmi']], y_train,  color='black')
plt.xlabel('Body Mass Index (BMI)')
plt.ylabel('Diabetes Risk')

Text(0, 0.5, 'Diabetes Risk')

Gradient Descent for Linear Regression

Putting this together with the gradient descent algorithm, we obtain a learning method for training linear models.

theta, theta_prev = random_initialization()
while abs(J(theta) - J(theta_prev)) > conv_threshold:
    theta_prev = theta
    theta = theta_prev - step_size * (f(x, theta)-y) * x

This update rule is also known as the Least Mean Squares (LMS) or Widrow-Hoff learning rule.

threshold = 1e-3
step_size = 4e-1
theta, theta_prev = np.array([2,1]), np.ones(2,)
opt_pts = [theta]
opt_grads = []
iter = 0

while np.linalg.norm(theta - theta_prev) > threshold:
    if iter % 100 == 0:
        print('Iteration %d. MSE: %.6f' % (iter, mean_squared_error(theta, X_train, y_train)))
    theta_prev = theta
    gradient = mse_gradient(theta, X_train, y_train)
    theta = theta_prev - step_size * gradient
    opt_pts += [theta]
    opt_grads += [gradient]
    iter += 1

Iteration 0. MSE: 0.171729
Iteration 100. MSE: 0.014765
Iteration 200. MSE: 0.014349
Iteration 300. MSE: 0.013997
Iteration 400. MSE: 0.013701

x_line = np.stack([np.linspace(-0.1, 0.1, 10), np.ones(10,)])
y_line = opt_pts[-1].dot(x_line)

plt.scatter(X_train.loc[:,['bmi']], y_train,  color='black')
plt.plot(x_line[0], y_line)
plt.xlabel('Body Mass Index (BMI)')
plt.ylabel('Diabetes Risk')

Text(0, 0.5, 'Diabetes Risk')

Part 3: Ordinary Least Squares

In practice, there is a more effective way than gradient descent to find linear model parameters.

We will see this method here, which will lead to our first non-toy algorithm: Ordinary Least Squares.

Review: The Gradient

The gradient $\nabla_\theta f$ further extends the derivative to multivariate functions $f : \mathbb{R}^d \to \mathbb{R}$, and is defined at a point $\theta_0$ as

$$ \nabla_\theta f (\theta_0) = \begin{bmatrix} \frac{\partial f(\theta_0)}{\partial \theta_1} \\ \frac{\partial f(\theta_0)}{\partial \theta_2} \\ \vdots \\ \frac{\partial f(\theta_0)}{\partial \theta_d} \end{bmatrix}.$$

In other words, the $j$-th entry of the vector $\nabla_\theta f (\theta_0)$ is the partial derivative $\frac{\partial f(\theta_0)}{\partial \theta_j}$ of $f$ with respect to the $j$-th component of $\theta$.

The UCI Diabetes Dataset

In this section, we are going to again use the UCI Diabetes Dataset.

• For each patient we have a access to a measurement of their body mass index (BMI) and a quantiative diabetes risk score (from 0-300).
• We are interested in understanding how BMI affects an individual's diabetes risk.

%matplotlib inline
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = [8, 4]

import numpy as np
import pandas as pd
from sklearn import datasets

# Load the diabetes dataset
X, y = datasets.load_diabetes(return_X_y=True, as_frame=True)

# add an extra column of onens
X['one'] = 1

# Collect 20 data points
X_train = X.iloc[-20:]
y_train = y.iloc[-20:]

plt.scatter(X_train.loc[:,['bmi']], y_train,  color='black')
plt.xlabel('Body Mass Index (BMI)')
plt.ylabel('Diabetes Risk')

Text(0, 0.5, 'Diabetes Risk')

Notation: Design Matrix

Machine learning algorithms are most easily defined in the language of linear algebra. Therefore, it will be useful to represent the entire dataset as one matrix $X \in \mathbb{R}^{n \times d}$, of the form: $$ X =