Concepts / Training Sequence

Training Sequence

A learning algorithm A receives a training sequence S and returns the hypothesis A(S), which is a particular prediction rule.

  • Programming

From Input to Prediction

A machine-learning learner ultimately needs to produce a way to make predictions. That way is a prediction rule: a function that takes a domain point and maps it to a label. Once the rule exists, it can be applied to new domain points, such as future papayas, to predict their labels.

domain point x₁inputdomain point x₂inputprediction rule hmaps points to labelslabel y₁predictionlabel y₂predictionapply hproducesapply hproduces
How does a prediction rule transform each point in the domain X into a corresponding label in Y?

One Point Through h

What do you think happens?

Suppose a new domain point is passed into a prediction rule h. What kind of result should come out?

  • Another domain point

  • A label in Y

  • The training sequence S

  • The learning algorithm A

Reveal answer
Answer:

A label in Y

A prediction rule takes a domain point as input and maps it to a label. The result is therefore a label in Y, which is used as the prediction for that point.

To follow one prediction from start to finish, begin with a new domain point. Apply the prediction rule h to that point. The result is a label in Y. The rule has transformed the input point into the label that will be used as its prediction.

Worked example

Applying a Rule to a New Papaya

A prediction rule h is used with a new papaya as its domain point. Describe the prediction process without specifying the internal details of h.

  1. 1
    Start with the point

    The new papaya is the domain point supplied to the prediction rule.

  2. 2
    Apply h

    The prediction rule receives that domain point as its input.

  3. 3
    Read the result

    The rule returns a label in Y. That label is the prediction for the new papaya.

Result

A domain point is passed into h, and a label in Y comes out as its predicted label.

Reading h : X → Y

Definition

The notation h : X → Y says that h is a prediction rule whose inputs are domain points from X and whose outputs are labels in Y. The arrow expresses the direction from domain points to labels.

hprediction ruleXdomain points→maps towardYlabelsdescribesinput sideoutput direction
What do X, Y, h, and the arrow represent, and how does a point move from X to its label in Y?

Read the notation from left to right as follows: h is the rule, X is the domain of points that the rule can receive, and Y is the set of labels that the rule produces. The arrow is not describing a learning step. It describes the direction of prediction: a point from X is mapped to a label in Y.

From Sequence to Hypothesis

A learning algorithm produces a prediction rule. The source uses A for the learning algorithm and S for the training sequence. The notation A(S) means the hypothesis returned by algorithm A after it receives training sequence S.

prediction rulea function from points tolabelslearning algorithm Areceives Straining sequence Sinput to Ahypothesis A(S)returned prediction rulereturnsreceived byis a specific instance of
How are the general idea of a prediction rule and the specific hypothesis produced by a learning algorithm related but different?

Prediction rule and hypothesis are closely related, but they emphasize different things. Prediction rule names the kind of object needed to make predictions: a function from domain points to labels. Hypothesis names the particular prediction rule returned by a learning algorithm after processing a training sequence. Thus, A(S) is not the training sequence itself and not the algorithm itself; it is the rule produced from them.

TermWhat it identifiesRole
Prediction ruleA function from domain points to labelsMakes predictions
Learning algorithm AThe algorithm that receives a training sequenceProduces a prediction rule
Training sequence SThe sequence received by the learning algorithmServes as input to A
Hypothesis A(S)The prediction rule returned by A after receiving SMakes predictions for domain points

The terms describe different parts of the learning-and-prediction setup.

Common Mistakes

  • Common mistake

    Treating a prediction rule as a domain point.

    What it looks like

    Calling h the new papaya rather than the rule applied to the new papaya.

    Why it fails

    The domain point is the input, while h is the function that maps the point to a label.

    Do this instead

    Keep the roles separate: supply a domain point to h and interpret the resulting element of Y as the prediction.

  • Common mistake

    Reading h : X → Y as a statement that X changes into Y by itself.

    What it looks like

    Saying that every point in X automatically becomes a label without applying h.

    Why it fails

    The notation identifies h as the rule that maps points from X to labels in Y.

    Do this instead

    Include the rule in the process: a domain point from X is passed through h, producing a label in Y.

  • Common mistake

    Confusing the hypothesis A(S) with the training sequence S.

    What it looks like

    Saying that A(S) is the sequence given to the algorithm.

    Why it fails

    S is the training sequence received by A, whereas A(S) is the hypothesis returned after receiving S.

    Do this instead

    Remember the direction: A receives S and returns A(S), a prediction rule.

  • Common mistake

    Assuming that a hypothesis is separate from a prediction rule.

    What it looks like

    Describing a hypothesis as something that cannot predict labels.

    Why it fails

    A hypothesis is the prediction rule returned by a learning algorithm.

    Do this instead

    Understand hypothesis as the particular prediction rule produced from a training sequence.

Practice

EASY

For the notation h : X → Y, identify the rule, the input side, the output side, and the direction of prediction. Then explain in one sentence what happens when a new domain point is passed to h.

Hints
  • The symbol h names the prediction rule.
  • X contains the domain points and Y contains the labels.
  • The arrow points from the domain side toward the label side.
MEDIUM

Complete the relationship in words: algorithm A receives training sequence S and returns ________. Explain why the blank is a prediction rule rather than the training sequence.

Hints
  • Use the notation introduced for the result of applying A to S.
  • The result is the hypothesis returned by the learning algorithm.

Key Takeaways

  1. 1
    A prediction rule is a function that maps a domain point to a label. In h : X → Y, h is the rule, X is the domain of points, Y is the set of labels, and the arrow shows the prediction direction. Applying h changes the focus from an input point to its predicted label. A learning algorithm A receives a training sequence S and returns the hypothesis A(S), which is a particular prediction rule.

At a glance

  • A prediction rule maps domain points to labels.
  • The notation h : X → Y expresses that h maps points from X to labels in Y.
  • Applying a prediction rule to a new domain point produces a label used as its prediction.
  • A learning algorithm A receives a training sequence S and returns the hypothesis A(S).
  • The hypothesis is the particular prediction rule produced by the learning algorithm.

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