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3.1. Linear Regression¶
Regression refers to a set of methods for modeling the relationship between data points \(\mathbf{x}\) and corresponding real-valued targets \(y\). In the natural sciences and social sciences, the purpose of regression is most often to characterize the relationship between the inputs and outputs. Machine learning, on the other hand, is most often concerned with prediction.
Regression problems pop up whenever we want to predict a numerical value. Common examples include predicting prices (of homes, stocks, etc.), predicting length of stay (for patients in the hospital), demand forecasting (for retail sales), among countless others. Not every prediction problem is a classic regression problem. In subsequent sections, we will introduce classification problems, where the goal is to predict membership among a set of categories.
3.1.1. Basic Elements of Linear Regression¶
Linear regression may be both the simplest and most popular among the standard tools to regression. Dating back to the dawn of the 19th century, linear regression flows from a few simple assumptions. First, we assume that the relationship between the features \(\mathbf{x}\) and targets \(y\) is linear, i.e., that \(y\) can be expressed as a weighted sum of the inputs \(\textbf{x}\), give or take some noise on the observations. Second, we assume that any noise is well-behaved (following a Gaussian distribution). To motivate the approach, let us start with a running example. Suppose that we wish to estimate the prices of houses (in dollars) based on their area (in square feet) and age (in years).
To actually fit a model for predicting house prices, we would need to get our hands on a dataset consisting of sales for which we know the sale price, area and age for each home. In the terminology of machine learning, the dataset is called a training data set or training set, and each row (here the data corresponding to one sale) is called an example (or data instance, “data point”, sample). The thing we are trying to predict (here, the price) is called a label (or target). The variables (here age and area) upon which the predictions are based are called features or covariates.
Typically, we will use \(n\) to denote the number of examples in our dataset. We index the data instances by \(i\), denoting each input as \(x^{(i)} = [x_1^{(i)}, x_2^{(i)}]\) and the corresponding label as \(y^{(i)}\).
3.1.1.1. Linear Model¶
The linearity assumption just says that the target (price) can be expressed as a weighted sum of the features (area and age):
Here, \(w_{\mathrm{area}}\) and \(w_{\mathrm{age}}\) are called weights, and \(b\) is called a bias (also called an offset or intercept). The weights determine the influence of each feature on our prediction and the bias just says what value the predicted price should take when all of the features take value \(0\). Even if we will never see any homes with zero area, or that are precisely zero years old, we still need the intercept or else we will limit the expressivity of our linear model.
Given a dataset, our goal is to choose the weights \(w\) and bias \(b\) such that on average, the predictions made according to our model best fit the true prices observed in the data.
In disciplines where it is common to focus on datasets with just a few features, explicitly expressing models long-form like this is common. In ML, we usually work with high-dimensional datasets, so it is more convenient to employ linear algebra notation. When our inputs consist of \(d\) features, we express our prediction \(\hat{y}\) as
Collecting all features into a vector \(\mathbf{x}\) and all weights into a vector \(\mathbf{w}\), we can express our model compactly using a dot product:
Here, the vector \(\mathbf{x}\) corresponds to a single data point. We will often find it convenient to refer to our entire dataset via the design matrix \(\mathbf{X}\). Here, \(\mathbf{X}\) contains one row for every example and one column for every feature.
For a collection of data points \(\mathbf{X}\), the predictions \(\hat{\mathbf{y}}\) can be expressed via the matrix-vector product:
Given a training dataset \(\mathbf{X}\) and corresponding (known) targets \(\mathbf{y}\), the goal of linear regression is to find the weight vector \(w\) and bias term \(b\) that given a new data point \(\mathbf{x}_i\), sampled from the same distribution as the training data will (in expectation) predict the target \(y_i\) with the lowest error.
Even if we believe that the best model for predicting \(y\) given \(\mathbf{x}\) is linear, we would not expect to find real-world data where \(y_i\) exactly equals \(\mathbf{w}^\top \mathbf{x}+b\) for all points (\(\mathbf{x}, y)\). For example, whatever instruments we use to observe the features \(\mathbf{X}\) and labels \(\mathbf{y}\) might suffer small amount of measurement error. Thus, even when we are confident that the underlying relationship is linear, we will incorporate a noise term to account for such errors.
Before we can go about searching for the best parameters \(\mathbf{w}\) and \(b\), we will need two more things: (i) a quality measure for some given model; and (ii) a procedure for updating the model to improve its quality.
3.1.1.2. Loss Function¶
Before we start thinking about how to fit our model, we need to determine a measure of fitness. The loss function quantifies the distance between the real and predicted value of the target. The loss will usually be a non-negative number where smaller values are better and perfect predictions incur a loss of \(0\). The most popular loss function in regression problems is the sum of squared errors. When our prediction for an example \(i\) is \(\hat{y}^{(i)}\) and the corresponding true label is \(y^{(i)}\), the squared error is given by:
The constant \(1/2\) makes no real difference but will prove
notationally convenient, cancelling out when we take the derivative of
the loss. Since the training dataset is given to us, and thus out of our
control, the empirical error is only a function of the model parameters.
To make things more concrete, consider the example below where we plot a
regression problem for a one-dimensional case as shown in
fig_fit_linreg.
.. _fig_fit_linreg:
Note that large differences between estimates \(\hat{y}^{(i)}\) and observations \(y^{(i)}\) lead to even larger contributions to the loss, due to the quadratic dependence. To measure the quality of a model on the entire dataset, we simply average (or equivalently, sum) the losses on the training set.
When training the model, we want to find parameters (\(\mathbf{w}^*, b^*\)) that minimize the total loss across all training examples:
3.1.1.3. Analytic Solution¶
Linear regression happens to be an unusually simple optimization problem. Unlike most other models that we will encounter in this book, linear regression can be solved analytically by applying a simple formula, yielding a global optimum. To start, we can subsume the bias \(b\) into the parameter \(\mathbf{w}\) by appending a column to the design matrix consisting of all \(1s\). Then our prediction problem is to minimize \(||\mathbf{y} - \mathbf{X}\mathbf{w}||\). Because this expression has a quadratic form, it is convex, and so long as the problem is not degenerate (our features are linearly independent), it is strictly convex.
Thus there is just one critical point on the loss surface and it corresponds to the global minimum. Taking the derivative of the loss with respect to \(\mathbf{w}\) and setting it equal to \(0\) yields the analytic solution:
While simple problems like linear regression may admit analytic solutions, you should not get used to such good fortune. Although analytic solutions allow for nice mathematical analysis, the requirement of an analytic solution is so restrictive that it would exclude all of deep learning.
3.1.1.4. Gradient descent¶
Even in cases where we cannot solve the models analytically, and even when the loss surfaces are high-dimensional and nonconvex, it turns out that we can still train models effectively in practice. Moreover, for many tasks, these difficult-to-optimize models turn out to be so much better that figuring out how to train them ends up being well worth the trouble.
The key technique for optimizing nearly any deep learning model, and which we will call upon throughout this book, consists of iteratively reducing the error by updating the parameters in the direction that incrementally lowers the loss function. This algorithm is called gradient descent. On convex loss surfaces, it will eventually converge to a global minimum, and while the same cannot be said for nonconvex surfaces, it will at least lead towards a (hopefully good) local minimum.
The most naive application of gradient descent consists of taking the derivative of the true loss, which is an average of the losses computed on every single example in the dataset. In practice, this can be extremely slow. We must pass over the entire dataset before making a single update. Thus, we will often settle for sampling a random minibatch of examples every time we need to compute the update, a variant called stochastic gradient descent.
In each iteration, we first randomly sample a minibatch \(\mathcal{B}\) consisting of a fixed number of training examples. We then compute the derivative (gradient) of the average loss on the mini batch with regard to the model parameters. Finally, we multiply the gradient by a predetermined step size \(\eta > 0\) and subtract the resulting term from the current parameter values.
We can express the update mathematically as follows (\(\partial\) denotes the partial derivative) :
To summarize, steps of the algorithm are the following: (i) we initialize the values of the model parameters, typically at random; (ii) we iteratively sample random batches from the data (many times), updating the parameters in the direction of the negative gradient.
For quadratic losses and linear functions, we can write this out explicitly as follows: Note that \(\mathbf{w}\) and \(\mathbf{x}\) are vectors. Here, the more elegant vector notation makes the math much more readable than expressing things in terms of coefficients, say \(w_1, w_2, \ldots, w_d\).
In the above equation, \(|\mathcal{B}|\) represents the number of examples in each minibatch (the batch size) and \(\eta\) denotes the learning rate. We emphasize that the values of the batch size and learning rate are manually pre-specified and not typically learned through model training. These parameters that are tunable but not updated in the training loop are called hyper-parameters. Hyperparameter tuning is the process by which these are chosen, and typically requires that we adjust the hyperparameters based on the results of the inner (training) loop as assessed on a separate validation split of the data.
After training for some predetermined number of iterations (or until some other stopping criteria is met), we record the estimated model parameters, denoted \(\hat{\mathbf{w}}, \hat{b}\) (in general the “hat” symbol denotes estimates). Note that even if our function is truly linear and noiseless, these parameters will not be the exact minimizers of the loss because, although the algorithm converges slowly towards a local minimum it cannot achieve it exactly in a finite number of steps.
Linear regression happens to be a convex learning problem, and thus there is only one (global) minimum. However, for more complicated models, like deep networks, the loss surfaces contain many minima. Fortunately, for reasons that are not yet fully understood, deep learning practitioners seldom struggle to find parameters that minimize the loss on training data. The more formidable task is to find parameters that will achieve low loss on data that we have not seen before, a challenge called generalization. We return to these topics throughout the book.
3.1.1.5. Making Predictions with the Learned Model¶
Given the learned linear regression model \(\hat{\mathbf{w}}^\top \mathbf{x} + \hat{b}\), we can now estimate the price of a new house (not contained in the training data) given its area \(x_1\) and age (year) \(x_2\). Estimating targets given features is commonly called prediction and inference.
We will try to stick with prediction because calling this step inference, despite emerging as standard jargon in deep learning, is somewhat of a misnomer. In statistics, inference more often denotes estimating parameters based on a dataset. This misuse of terminology is a common source of confusion when deep learning practitioners talk to statisticians.
3.1.1.6. Vectorization for Speed¶
When training our models, we typically want to process whole minibatches of examples simultaneously. Doing this efficiently requires that we vectorize the calculations and leverage fast linear algebra libraries rather than writing costly for-loops in Java.
To illustrate why this matters so much, we can consider two methods for
adding vectors. To start we instantiate two \(10000\)-dimensional
vectors containing all ones. In one method we will loop over the vectors
with a Java for loop. In the other method we will rely on a single
call to DJL.
We need some utilities such as StopWatch. We can load them using the
%load macro.
%load ../utils/djl-imports
%load ../utils/plot-utils
%load ../utils/StopWatch.java
import java.util.stream.*;
int n = 10000;
NDManager manager = NDManager.newBaseManager();
NDArray a = manager.ones(new Shape(n));
NDArray b = manager.ones(new Shape(n));
Now we can benchmark the workloads. First, we add them, one coordinate
at a time, using a for loop.
NDArray c = manager.zeros(new Shape(n));
StopWatch stopWatch = new StopWatch();
for (int i = 0; i < n; i++) {
c.set(new NDIndex(i), a.getFloat(i) + b.getFloat(i));
}
String.format("%.5f sec", stopWatch.stop());
6.54053 sec
Alternatively, we rely on DJL to compute the elementwise sum:
stopWatch.start();
NDArray d = a.add(b);
String.format("%.5f sec", stopWatch.stop());
0.05434 sec
You probably noticed that the second method is dramatically faster than the first. Vectorizing code often yields order-of-magnitude speedups. Moreover, we push more of the math to the library and need not write as many calculations ourselves, reducing the potential for errors.
3.1.2. The Normal Distribution and Squared Loss¶
While you can already get your hands dirty using only the information above, in the following section we can more formally motivate the square loss objective via assumptions about the distribution of noise.
Recall from the above that the squared loss \(l(y, \hat{y}) = \frac{1}{2} (y - \hat{y})^2\) has many convenient properties. These include a simple derivative \(\partial_{\hat{y}} l(y, \hat{y}) = (\hat{y} - y)\).
As we mentioned earlier, linear regression was invented by Gauss in 1795, who also discovered the normal distribution (also called the Gaussian). It turns out that the connection between the normal distribution and linear regression runs deeper than common parentage. To refresh your memory, the probability density of a normal distribution with mean \(\mu\) and variance \(\sigma^2\) is given as follows:
Below we define a Java function to compute the normal distribution.
public float[] normal(float[] z, float mu, float sigma) {
float[] dist = new float[z.length];
for (int i = 0; i < z.length; i++) {
float p = 1.0f / (float) Math.sqrt(2 * Math.PI * sigma * sigma);
dist[i] = p * (float) Math.pow(Math.E, -0.5 / (sigma * sigma) * (z[i] - mu) * (z[i] - mu));
}
return dist;
}
We can now visualize the normal distributions.
int start = -7;
int end = 14;
float step = 0.01f;
int count = (int) (end / step);
float[] x = new float[count];
for (int i = 0; i < count; i++) {
x[i] = start + i * step;
}
public float[] combine3(float[] x, float[] y, float[] z) {
return ArrayUtils.addAll(ArrayUtils.addAll(x, y), z);
}
import tech.tablesaw.api.*;
import tech.tablesaw.plotly.api.*;
import tech.tablesaw.plotly.components.*;
import tech.tablesaw.api.FloatColumn;
import org.apache.commons.lang3.ArrayUtils;
float[] y1 = normal(x, 0, 1);
float[] y2 = normal(x, 0, 2);
float[] y3 = normal(x, 3, 1);
String[] params = new String[x.length * 3];
Arrays.fill(params, 0, x.length, "mean 0, var 1");
Arrays.fill(params, x.length, x.length * 2, "mean 0, var 2");
Arrays.fill(params, x.length * 2, x.length * 3, "mean 3, var 1");
Table normalDistributions = Table.create("normal")
.addColumns(
FloatColumn.create("z", combine3(x, x, x)),
FloatColumn.create("p(z)", combine3(y1, y2, y3)),
StringColumn.create("params", params)
);
LinePlot.create("Normal Distributions", normalDistributions, "z", "p(z)", "params");
As you can see, changing the mean corresponds to a shift along the x axis, and increasing the variance spreads the distribution out, lowering its peak.
One way to motivate linear regression with the mean squared error loss function is to formally assume that observations arise from noisy observations, where the noise is normally distributed as follows
Thus, we can now write out the likelihood of seeing a particular \(y\) for a given \(\mathbf{x}\) via
Now, according to the maximum likelihood principle, the best values of \(b\) and \(\mathbf{w}\) are those that maximize the likelihood of the entire dataset:
Estimators chosen according to the maximum likelihood principle are called Maximum Likelihood Estimators (MLE). While, maximizing the product of many exponential functions, might look difficult, we can simplify things significantly, without changing the objective, by maximizing the log of the likelihood instead. For historical reasons, optimizations are more often expressed as minimization rather than maximization. So, without changing anything we can minimize the Negative Log-Likelihood (NLL) \(-\log p(\mathbf y|\mathbf X)\). Working out the math gives us:
Now we just need one more assumption: that \(\sigma\) is some fixed constant. Thus we can ignore the first term because it does not depend on \(\mathbf{w}\) or \(b\). Now the second term is identical to the squared error objective introduced earlier, but for the multiplicative constant \(\frac{1}{\sigma^2}\). Fortunately, the solution does not depend on \(\sigma\). It follows that minimizing squared error is equivalent to maximum likelihood estimation of a linear model under the assumption of additive Gaussian noise.
3.1.3. From Linear Regression to Deep Networks¶
So far we only talked about linear functions. While neural networks cover a much richer family of models, we can begin thinking of the linear model as a neural network by expressing it in the language of neural networks. To begin, let us start by rewriting things in a ‘layer’ notation.
3.1.3.1. Neural Network Diagram¶
Deep learning practitioners like to draw diagrams to visualize what is happening in their models. In Section 3.1.3.1, we depict our linear model as a neural network. Note that these diagrams indicate the connectivity pattern (here, each input is connected to the output) but not the values taken by the weights or biases.
Because there is just a single computed neuron (node) in the graph (the input values are not computed but given), we can think of linear models as neural networks consisting of just a single artificial neuron. Since for this model, every input is connected to every output (in this case there is only one output!), we can regard this transformation as a fully-connected layer, also commonly called a dense layer. We will talk a lot more about networks composed of such layers in the next chapter on multilayer perceptrons.
3.1.3.2. Biology¶
Since linear regression (invented in 1795) predates computational
neuroscience, it might seem anachronistic to describe linear regression
as a neural network. To see why linear models were a natural place to
begin when the cyberneticists/neurophysiologists Warren McCulloch and
Walter Pitts began to develop models of artificial neurons, consider the
cartoonish picture of a biological neuron in fig_Neuron,
consisting of dendrites (input terminals), the nucleus (CPU), the
axon (output wire), and the axon terminals (output terminals),
enabling connections to other neurons via synapses.
.. _fig_Neuron:
Information \(x_i\) arriving from other neurons (or environmental sensors such as the retina) is received in the dendrites. In particular, that information is weighted by synaptic weights \(w_i\) determining the effect of the inputs (e.g., activation or inhibition via the product \(x_i w_i\)). The weighted inputs arriving from multiple sources are aggregated in the nucleus as a weighted sum \(y = \sum_i x_i w_i + b\), and this information is then sent for further processing in the axon \(y\), typically after some nonlinear processing via \(\sigma(y)\). From there it either reaches its destination (e.g., a muscle) or is fed into another neuron via its dendrites.
Certainly, the high-level idea that many such units could be cobbled together with the right connectivity and right learning algorithm, to produce far more interesting and complex behavior than any one neuron alone could express owes to our study of real biological neural systems.
At the same time, most research in deep learning today draws little direct inspiration in neuroscience. We invoke Stuart Russell and Peter Norvig who, in their classic AI text book Artificial Intelligence: A Modern Approach [Russell & Norvig, 2016], pointed out that although airplanes might have been inspired by birds, ornithology has not been the primary driver of aeronautics innovation for some centuries. Likewise, inspiration in deep learning these days comes in equal or greater measure from mathematics, statistics, and computer science.
3.1.4. Summary¶
Key ingredients in a machine learning model are training data, a loss function, an optimization algorithm, and quite obviously, the model itself.
Vectorizing makes everything better (mostly math) and faster (mostly code).
Minimizing an objective function and performing maximum likelihood can mean the same thing.
Linear models are neural networks, too.
3.1.5. Exercises¶
Assume that we have some data \(x_1, \ldots, x_n \in \mathbb{R}\). Our goal is to find a constant \(b\) such that \(\sum_i (x_i - b)^2\) is minimized.
Find a closed-form solution for the optimal value of \(b\).
How does this problem and its solution relate to the normal distribution?
Derive the closed-form solution to the optimization problem for linear regression with squared error. To keep things simple, you can omit the bias \(b\) from the problem (we can do this in principled fashion by adding one column to \(X\) consisting of all ones).
Write out the optimization problem in matrix and vector notation (treat all the data as a single matrix, all the target values as a single vector).
Compute the gradient of the loss with respect to \(w\).
Find the closed form solution by setting the gradient equal to zero and solving the matrix equation.
When might this be better than using stochastic gradient descent? When might this method break?
Assume that the noise model governing the additive noise \(\epsilon\) is the exponential distribution. That is, \(p(\epsilon) = \frac{1}{2} \exp(-|\epsilon|)\).
Write out the negative log-likelihood of the data under the model \(-\log P(Y \mid X)\).
Can you find a closed form solution?
Suggest a stochastic gradient descent algorithm to solve this problem. What could possibly go wrong (hint - what happens near the stationary point as we keep on updating the parameters). Can you fix this?