 # Part I: Classification and Bayesian Learning

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Part I: Classification and Bayesian Learning
Machine Learning Part I: Classification and Bayesian Learning Ref: E. Alpaydin, Intro to Machine Learning, MIT 2004

Machine Learning Machine Leaning is programming computers to optimize a perf criteria using example data or past experience Inference from samples There is a process that explains the data we observe. But we don’t know the details about how the data are generated. Internet requests, failure events, etc It’s hard to identify (model) the process completely, we could construct a good and useful approximation that detect certain patterns. Such patterns would help us to understand the process and make predictions about the future.

Types of Machine Learning
Supervised learning is to create a function from training data. The training data consist of pairs of input objects (typically vectors), and desired outputs. Classification: Given an input, the output is Boolean (yes/no) to predict a class label of the input object; Regression: If the label is a numerical value, learn the function f(x) that best explain the input instance; Unsupervised learning: manual labels of inputs are not used. Clustering: partition a data set into subsets (clusters), so that the data in each subset share some common trait Semi-supervised learning: make use of both labeled and unlabeled data for training Reinforcement Learning Learning a policy: A sequence of outputs; No supervised output but delayed reward Examples: game playing, robot navigation

Supervised Learning Use of Supervised Learning Classification
Regression Evaluation Methodology Bayesian Learning for Classification

Why Supervised Learning?
Prediction of future cases: Use the rule to predict the output for future inputs Knowledge extraction: The rule is easy to understand Compression: The rule is simpler than the data it explains Outlier detection: Exceptions that are not covered by the rule, e.g., fraud

THEN low-risk ELSE high-risk
Classification E.g: Credit scoring Differentiating between low-risk and high-risk customers from their income and savings Rule-based prediction Discriminant: IF income > θ1 AND savings > θ2 THEN low-risk ELSE high-risk

Learning a Class from Examples
Given a set of examples of cars, with a label of “family car” or not according to a survey, class learning is to find a description that is shared by all positive examples. Use of the class info Prediction: Is car x a family car? Knowledge extraction: What do people expect from a family car?

Training set X Input representation Attributes: price & engine power
Label of each instance

Most specific hypothesis, S Most general hypothesis, G
Hypothesis Class: C Most specific hypothesis, S Most general hypothesis, G Learning is to find a particular hypothesis h to approximate C

Hypothesis h and Empirical Error
Error of h:

Model Selection & Generalization
Learning is an ill-posed problem: data is not sufficient to find a unique solution Limited number of sample data Some data might be noise due to imprecision in recording, labeling, or hidden (latent, unobservable) attributes that affect the label of instances The need for inductive bias: assumptions about class structureH Why rectangle, not circle or irregular shape? What’s degree of tightness of fitting? Generalization: How well a model performs on new data

Noise and Model Complexity
Simple model is preferred Easy to use (check) (lower time complexity) Easy to train (lower space complexity) Easyto explain (more interpretable) Easy to generalize (lower variance ) Noise: any anomaly in the data which leads it infeasible to reach a zero-error classification with a simple hypothesis class

Probably Approximately Correct (PAC) Learning
How many training examples N should we have, such that with probability at least 1 ‒ δ, h has error at most ε ? Each strip is at most ε/4 Pr that we miss a strip 1‒ ε/4 Pr that N instances miss a strip (1 ‒ ε/4)N Pr that N instances miss 4 strips 4(1 ‒ ε/4)N 4(1 ‒ ε/4)N ≤ δ and (1 ‒ x)≤exp( ‒ x) 4exp(‒ εN/4) ≤ δ and N ≥ (4/ε)log(4/δ)

2-Class vs K-Class K-class problem be viewed as K 2-class problem:
Train hypotheses hi(x), i =1,...,K:

Regression Examples x : car attributes y : price y = g (x | θ )
Price of a used car Speed of Top500 x : car attributes y : price y = g (x | θ ) g ( ) model, θ parameters Linear regression y = wx+w0

Basic Concepts Interpolation Extrapolation Regression
Find a function that best fits a training set with no presence of noise r = f(x) Extrapolation Predict the output for any x, if x is NOT in the training set Regression Noise factor must be considered r = f(x) +  OR there’re hidden variables we couldn’t observe: r = f(x, z)

For a given test set, find g() that minimizes the empirical error
Regression For a given test set, find g() that minimizes the empirical error

Underfitting vs Overfitting
Underfitting: Hypothesis (H) less complex than actual model (C) Using a line to fit data sampled from a 3rd order polynomial Accuracy increases with more sample data; may not enough if the hypothesis is too complex Overfitting: H more complex than C Having more training data helps but only up to a certain point

Complexity of the hypothesisH, c (H): capacity of the hypothesis class Training set size, N, Generalization error, E, on new examples As N­, E¯ As c (H)­, first E¯ and then E­ (The error of an over-complex hypothesis can be kept in check by increasing the amount of training data, but only up to a point)

Cross-Validation To estimate generalization error, we need data unseen during training. Three types of data in cross-validation: Training set (50%) Validation set (25%) Test (publication) set (25%) Resampling when there is few data

Dimensions of a Supervised Learner: Summary
Model g() and parameter  Loss function L(): diff between desired output and approximation Optimization procedure: return the argument that minimizes