CMU-Q 15-381 Lecture 23: Supervised Learning 1 Teacher: Gianni A. - - PowerPoint PPT Presentation

CMU-Q 15-381

Lecture 23: Supervised Learning 1

Teacher: Gianni A. Di Caro

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MACHINE LEARNING?

D a t a S c i e n c e ! ?

(Model) (Targets) (Model) (Inductive learning)

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GENERAL ML SCHEME

§ ML Design: Use the right features (description language), to build the right model, that achieve the task according to desired performance § Learning by examples: Look at some data, guess at a general scientific hypothesis, make statements - predictions on test data - based on this hypothesis § Inductive learning (from evidence) ≠ Deductive learning (logical, from facts)

Data in the domain is described in the language of selected Features Task: define an appropriate mapping from data to the Outputs

Learning Problem: Obtaining such a mapping from training data

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GENERAL ML SCHEME

Labeled / Unlabeled Given / Not Given Errors / Rewards Performance criteria Hypotheses space Hypothesis function

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SUPERVISED LEARNING

§ Supervised (inductive) learning (labeled data) § A training data set is given § Training data include target outputs (labels put by a teacher / supervisor) § Using the labels a precise error measure for a prediction can be derived § Aims to find out models that explain and generalize observed data

Labeled Given Errors

Performance criteria Hypotheses space Hypothesis function

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EXAMPLE OF SUPERVISED LEARNING TASK: CLASSIFICATION

§ Classification, categorical target: given ! possible classes/categories, to which class each (new) data item belongs to? § Task mapping is ": ℱ% → {0,1}, § Binary (! = 2): Dog or cat? Rich or poor? Hot or cold? § Multi-class (! > 2): Cloudy, or snowing, or mostly clear? Dog, or cat,

• r fox, or …?

EXAMPLE OF SUPERVISED LEARNING TASK: REGRESSION

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§ Regression, numerical target: which is the function that best describes the (conditional expectation) relation between ! dependent variables (outputs) and " independent variables (predictors)? § Task mapping is #: ℱ& → ℝ) § Univariate (* = ,): What is the expected relation between temperature (predictor) and peak electricity usage in Doha (target output)? What is the expected relation between age and diabetes in Qatar? § Multi-variate (* > ,): What is the expected relation between (temperature, time hour, day of the week) and peak electricity usage in Doha? What is the expected relation between advertising in (TV, radio) and sales of a product?

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UNSUPERVISED LEARNING

§ Unsupervised (associative) learning (unlabeled data) § A (training) data set is given § Training data does not include target outputs / labels § Aims to find hidden structure, association relationships in data

Unlabeled Given Similarity measures

Performance criteria Hypotheses space Hypothesis function

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EXAMPLE OF UNSUPERVISED LEARNING TASK: CLUSTERING

§ Clustering, hidden target: based on some measure of similarity/dissimilarity, group data items in ! clusters, ! is (usually) not known in advance. § Task mapping is ": ℱ% → {0,1}, § Given a set of photos, and similarity features (e.g., colors, geometric objects) how can be the photos clustered? § Group these people in different clusters

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EXAMPLE OF UNSUPERVISED LEARNING TASK: DISCOVER RELATIONS

§ Finding underlying structure, hidden target: discover relations and correlations among data. § Given a set of photos, find possible relations among sub-groups of them. § Given shopping data of customers at gas station spread in the country, discover relations that could suggest what to put in the shops and where to locate the items on display.

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REINFORCEMENT LEARNING

§ Reinforcement (direct interaction) learning (reward model) § Explore, to find the training data by your own § Gathered data does not include target outputs, but are associated to (possibly sparse) rewards/costs (advisory signals) § Aims to learn an optimal action policy or make optimal predictions § Sequential decision-making vs. One-shot decision making

RL is hard(er)!

Unlabeled Not Given Rewards

Performance criteria Hypotheses space Hypothesis function

SUPERVISED LEARNING: CLASSIFICATION

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Data set: cars Labeling: + Family car (positive example) − No family car (negative example) Task: Learn class C ≡ Family car %(car) ? ⁄

' (

c a r s +

• Features?

Price Engine power Color Shape Traction Consumption ….

CLASSIFICATION EXAMPLE

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§ A car is represented as a numeric vector of two features: § The label of a car denotes its type:

x = x1 x2

• ≡

 $ cc

• ,

x ∈ R2

§ In the data set !, each car example " is represented by an ordered pair ($(%), ((%)), and there are ) examples in the data set:

! = {($(%), ((%))}%./

r =    1

if x is a positive example (family car) if x is a negative example

(

CLASSIFICATION EXAMPLE: HYPOTHESIS

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§ Plot of the dataset in the two-dimensional feature space

Which is the relationship between (price, power) and the Class !? Hypothesis about the form of the searched mapping: to which class of functions it belongs to? We have to make a choice: Inductive bias. We will explain the data according to the hypothesis class ℋ that we choose (that sets a bias)

CLASSIFICATION EXAMPLE: HYPOTHESIS CLASS

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Target of learning: find a particular hypothesis ℎ ∈ ℋ to approximate the (true) class $ as closely as possible Hypothesis class from which we believe $ is drawn: ℋ = set of axis-aligned rectangles (&' ≤ price ≤ &)) ⋀ (,' ≤ engine power ≤ ,)) Learning a vector - of four parameters: ℎ = ℎ/(0) Problem: We don’t know $, we only have a set

• f examples drawn from $

How do we evaluate how good: ℎ = ℎ/ 0 is?

EMPIRICAL AND GENERALIZATION ERRORS

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§ The empirical error on the training dataset of ! labeled pairs is: § Loss function: quantifies the prediction/classification error done by our hypothesis function ℎ # on a training or test example

ℓ ∶ ℝ×ℝ×{0,1} → {0,1} ℓ ∶ ℱ/×ℝ0 → ℝ ℓ ℎ1 2 , 3 = 1(ℎ1 2

≠ 3) § In the example:

89:; = <

=>? :

ℓ ℎ1 2(=) , 3(=) § Do we aim to minimize the empirical error? To a certain extent, yes, but we really aim to minimize the generalization error: the loss on new examples, not in the training set!

EMPIRICAL AND GENERALIZATION ERRORS

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§ Fundamental problem: we re looking for the parameter values ! that minimize the prediction error resulting from the hypothesis function ℎ# § This “seems” to be equivalent to find: ! = arg min ∑,-.

/ ℓ ℎ# 1(,) , 5(,)

6789 = :

,-. ;

ℓ ℎ#( 1(,)), 5(,) < 1(,), 5(,)

§ … but actually, what we really care about is loss of prediction on new examples 1= , 5= → Generalization error § Expected loss over all > input-output pairs the learning machine may see § To quantify this expectation, we need to define a prior probability distribution < ?, @ over the examples, which we assume as stationary (< doesn’t change) § The expected generalization loss is:

HOW TO ASSESS GENERALIZATION ERROR?

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§ But ! ", $ is not known! Therefore it is only possible to estimate the generalization error, which is the true error for the considered population

• f data examples given the chosen hypothesis

§ How can we make a sound estimate? § Two general ways: § Theoretical: derive statistical bounds on the difference between true error and expected empirical error (PAC learning, VC dimension) § Empirical (Practical): Compute the expected empirical error on training dataset as a local indicator of performance, then use a separate data set to test the learned model and use the expected empirical error on the test set to estimate the generalization error

EMPIRICAL RISK MINIMIZATION

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! "#$% = 1 ( )

*+, $

ℓ ℎ/ 0(*) , 4(*)

§ In any case, we need to estimate the generalization loss with the expected empirical loss on a set of examples ( ≪ 6, that can be estimated as the sample average of losses: § ! "#$% is an approximation of the risk associated with the use of the hypothesis ℎ/ for the learning task (i.e., the risk of incurring in prediction losses when classifying samples that are not in the training set) § The empirical risk minimization principle states that the learning algorithm should choose the hypothesis ℎ/ that minimizes empirical risk:

7 = arg min 1 ( )

*+, $

ℓ ℎ/ 0(*) , 4(*)

HOW TO CHOOSE THE HYPOTHESIS?

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The true class ! For a choice of ℎ #: Most specific $: Most general Consistent hypotheses

THE CANONICAL SUPERVISED ML PROBLEM

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minimize

&

1 ( )

*+,

• ℓ ℎ0 1(*) , 5(*)

Ø Since ,

• is a constant that depends on the size of the dataset, it may be omitted,

making the problem equivalent to minimize the sum of prediction losses: Ø Given a collection of input features and outputs ℎ0 1(*) , 5(*) , 6 = 1, … , ( and a hypothesis function ℎ0, find parameter values & that minimize the average empirical error: In some cases (e.g., when using quadratic losses), it can be convenient to use

, 9-

minimize

&

)

*+,

• ℓ ℎ0 1(*) , 5(*)

Ø Virtually, all supervised learning algorithms can be described in this form, where we need to specify:

• 1. The hypothesis class :, ;& ∈ :
• 2. The loss function ℓ
• 3. The algorithm for solving the optimization problem (often approximately)

BIG PICTURE (PATTERN RECOGNITION)

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A REGRESSION TASK

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Simple example: Predicting electricity use based on measures of temperature as predictors

PREDICTING ELECTRICITY USE

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Several days of peak demand vs. high temperature in Pittsburgh

LINEAR REGRESSION MODEL

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Hypothesis class ℋ = set of linear functions, " = $%& + $(

LINEAR REGRESSION MODEL

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LINEAR REGRESSION MODEL

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WHAT ABOUT CHOOSING A DIFFERENT HYPOTHESIS?

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In classification:

HYPOTHESIS SPACE

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POWER DEMAND FORECASTING PROBLEM

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HOW DO WE SOLVE IT?

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! ← ! − $ % &

'() *

+(') +('). / ! − 0(')

Ø If the averaging factor 1/2% is used, then the update action becomes:

GRADIENT DESCENT FOR GENERAL ML PROBLEMS

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ANALYTICAL SOLUTION

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• Analytical solution can still face computational issues for large datasets

RECAP: GRADIENT DESCENT

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§ Move in the direction opposite to the gradient vector, that is, ⊥ to isocountours, the direction of maximal change of the function § Isocountours / Sublevel set: "# = % ∶ ' % ≤ )

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RCAP: STEP SIZE (LEARNING RATE)

§ GD run ~ Motion of a mass in a potential field towards the minimum energy

• configuration. At each point the gradient defines the attraction force, while the

step ! scales the force, to define the next point

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RECAP: STEP SIZE (LEARNING RATE)

§ Adapting the step size may be necessary to avoid either a too slow progress, or overshooting the target minimum Small, good !, convergence ! is too large, divergence

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RECAP: STEP SIZE (LEARNING RATE)

! is too low, slow convergence ! is too large, divergence

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RECAP: ILL-CONDITIONED PROBLEMS

§ If the (loss) function is very anisotropic, then the problem is said ill- conditioned, since the gradient vector doesn’t point to the direction of the local minimum, resulting into a zig-zagging trajectory § Ill-conditioning can be determined computing the ratio between the eigenvalues of the Hessian matrix (the matrix of the second partial derivatives)

Ill-conditioned problem Well-conditioned problem

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RESCALING FEATURES

§ If the chosen features have significantly different ranges, then the resulting loss function would be ill-conditioned § For instance, if we want to learn a regression model for house pricing, we could use as predicting features 4-components vectors: ! = (square meters, floor, number of bed rooms) § We can easily have the following value ranges: {[0,500], [1,50], [1,5]}, that would result in an ill-conditioned problem, stretched along the sqm dimension § One way to overcome the problem is by rescaling feature values, accounting for their min/max ranges and possibly making all features ranging in [0,1] § Since feature values are stochastic variables, they can also be rescaled to have 0 mean and fixed range or standard deviation § All these rescaling operations can be performed using the available dataset

STOCHASTIC (SEQUENTIAL) GRADIENT DESCENT

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Very effective with training set with redundant / duplicate data

RESULT OF LEARNING

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STOCHASTIC GRADIENT IN ACTION

Example of the on-line evolution of parameter learning as more examples (black dots) are considered in the course of SGD

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ALTERNATIVE LOSS FUNCTION?

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ALTERNATIVE LOSS FUNCTIONS

Issues with differentiability and, therefore, when computing gradients

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EFFECT IN THE POWER PROBLEM