# Introduction¶

Supervised learning:

• Classification (binary/not)
• regression: continuous numeric labels
• ranking

Batch assumption: iid

• Distribution of things and measurements defines some unknown $$P(\mathbf{x}, y)$$ or $$D(\mathbf{x}, y)$$ over domain-label pairs
• We want to find a hypothesis $$f(\mathbf{x})$$ that is close to the truth
• so we need a loss fcn $$L(y, t)$$ where y = prediction, t = truth
• We want to minimize $$\int P(\mathbf{x},y) L(y, f(\mathbf{x}))$$

Overfitting and Underfitting

Overfitting is when your hypothesis is too complex for the data, and

Underfitting is when your hypothesis is too simple.

## Main Steps¶

### Feature Extraction¶

In feature extraction, you need to extract some useful data dimension from your sample. It’s very important to extract good features from your data - there’s a whole field around it (feature engineering).

Most commonly, an instance is represented as a vector of features $$\mathbf{x}$$, along with a ground-truth label $$y$$. This can be represented as $$(\mathbf{x}, y)$$.

They can be categorical/nominal, ordinal, or numeric.

Example

For example, in a binary spam/ham email classifier, some features you might extract might be:

• are there words in all caps?
• is the email long?
• this can be represented as a binary attribute (after a certain length), or a numeric one.
• how many ! are in the email?

### Training¶

In the training step, you choose a hypothesis space/hypothesis class and learn a hypothesis: a function that maps an instance to a class/label.

The hypothesis $$g$$ is a learned model defined by parameters. We use ML when $$g(\mathbf{x})$$ is unknown to us and we can’t think about how to implement it algorithmically.

### Testing¶

Now, given a new instance, we use our classifier to predict a label. Did it predict it correctly?

$y' = g(\mathbf{x'})$

It’s important for classifiers to be able to generalize: to predict correctly when encountering a situation that has never been seen before.

The power for a classifier to generalize often depends just as much on how you’re representing instances as it does the classification algorithm.

Generally, test data and training data should be drawn from the same population.

Note

Independent, Identically Distributed (iid) assumption for same population:

We assume the distribution of instances and labels defines some unknown, but fixed, $$P(\mathbf{x}, y)$$.

We also assume that all training and all test instances/labels are independent and identically distributed.

That is - a collection of random variables is iid if they all have the same probability distribution, and are all mutually independent.

## Supervised Learning¶

Supervised learning is the primary type of ML. In this approach, training instances come with a ground-truth label.

### Classification¶

In a classification problem, labels are nominal (an unordered set). The labels can be either binary, or multiclass (for example, a spam/ham classifier or a MNIST digit classifier).

### Regression¶

In a regression problem, labels are numeric (and continuous). For example, predicting the price of a house might be a regression problem.

### Ranking¶

In a ranking problem, the model is asked to order a set of objects. Usually, in this case, the input to these models are keywords and a prior (e.g. prior data gathered on you), and the output is the objects’ ranking.

### More¶

There’s a lot more examples, too!

• Disease diagnosis
• x: patient properties
• y: disease/recommended therapy
• Part-of-speech tagging
• x: an english sentence
• y: the part of speech of a word in there
• Face recognition
• x: bitmap of a person’s face
• y: identity of a person
• Reinforcement Learning
• output: a sequence of actions (policy). Individual actions aren’t important, but the overall policy is.
• no supervised output, but delayed rewards
• e.g. game playing, robot in a maze
• Online Learning
• train on one instance at a time, as opposed to batch learning
• e.g. perceptron

## Other Forms¶

• unsupervised learning: no labels provided during train time
• clustering
• e.g. image compression, bioinformatics
• semi-supervised learning: use partially labeled data as well as unlabeled data

## Training¶

But what does training mean?

The hypothesis, $$g(\mathbf{x})$$, is defined by parameters. How do we optimize these parameters?

We need to use a loss function.

### Loss Function¶

To learn, we want to minimize the loss function $$L(y, y')$$. This function measures the error of the prediction $$y'$$ on the train set, and tells you how good $$g(\mathbf{x})$$ is at this point.

Choosing the right loss function is important, and different functions will want different functions.

Note

For example, a simple loss function for binary classification is as follows:

def L(y, y_pred):
if y_pred == y:
return 1
else:
return 0


Minimization is done using an optimization algorithm like gradient descent.

aka SGD

In a sense, a function that, at an arbitrary point, finds the steepest slope that moves you in the direction of the local minimum.

\begin{align}\begin{aligned}\begin{split}\text{Given:} \\ t = [1..] \\ \text{samples } (\mathbf{x}_t, y_t) \\ \text{current model } f_t \\ \text{learning rate } \eta_t \\ \\\end{split}\\f_{t+1} = f_t - \eta_t \cdot \nabla_{f_t} L(f(\mathbf{x}_t), y_t)\end{aligned}\end{align}

### Choices¶

Model

So really, our goal is to learn a generalizable $$g(x)$$ that is a good approximation of the truth. You should choose a model that is capable of approximating the truth so.

Hypothesis Space

If we choose a hypothesis space that is too simple, there are fewer parameters to learn, but the model is less powerful.

The model will have less variance - fewer changes with changing training data - but more bias (making more assumptions).

Buuut, choosing a hypothesis space that is too complex may cause high variance. This is called the bias-variance tradeoff.

Variance

Having super high variance means you can perfectly represent the training data - but this is overfitting!

Different training sets may produce wildly different hypotheses - this is what high variance is.

Bias

Similarly, we say a model has high inductive bias when the model makes a lot of assumptions about the data.

For example, when we choose to use a linear regression, that model has a high bias towards a linear relationship between the features and the labels.

Overall, variance represents estimation error (limits due to data) and bias represents approximation error (limits due to model family).

These concepts are closely linked to the concept of overfitting/underfitting.

## Evaluation¶

There are lots of ways to measure predictive performance. The most popular are:

• Accuracy and Error Rate
• Precision Recall and F-measure

### Accuracy¶

Useful in X vs. Y problems, where any classes are equally important.

$accuracy = \frac{\text{# correct predictions}}{\text{# test instances}}$
$error = 1 - accuracy = \frac{\text{# incorrect predictions}}{\text{# test instances}}$

### Confusion Matrix¶

An extension of accuracy - given a matrix of positive and negative instances, it is possible to make a matrix

         Predicted
Y   N
+-------
Actual Y | TP  FN       Where TP = True Positive, FN = False Negative,
N | FP  TN       FP = False Positive, TN = True Negative


Example: Given the result table

         Predicted
Y    N
+--------
Actual Y | 100  5
N | 10   50

Total corpus size: 165
Total predicted yes: 110
Total predicted no: 55
Actual yes: 105
Actual no: 60

Accuracy = (100 + 50) / 160 = 0.91
Error = (5 + 10) / 160 = 0.09


### Precision, Recall, F-measure¶

What about “X vs. non-X” types of problems: spam v. not spam, relevant v. not relevant?

$precision = \frac{\text{# relevant records retrieved}}{\text{total # of records retrieved}}$
$recall = \frac{\text{# relevant records received}}{\text{total # of relevant records}}$
$F measure = \frac{2 * P * R}{P + R}$

So, going back to our example, but now we’re predicting whether or not something is relevant…

         Predicted
Y    N
+--------
Actual Y | 100  5
N | 10   50

Accuracy = (100 + 50) / 160 = 0.91

Precision = TP / (TP + FP) = 100 / 110 = 0.91
Recall = TP / (TP + FN) = 100 / 105 = 0.95
F1 score = (2 * 0.91 * 0.95) / (0.91 + 0.95) = 0.93


TLDR:

         Predicted
Y   N
+-------
Actual Y | TP  FN
N | FP  TN

$accuracy = \frac{TP + TN}{P + N}$
$precision = \frac{TP}{TP + FP}$
$recall = \frac{TP}{TP + FN}$

### Reporting Performance¶

• separate training and test data
• never see test data in training
• usually 80-20 split
• k-fold cross validation
• why just split once?
• run k iterations, in each iteration hold out a different test set
• improves robustness of reported set