Machine Learning CUNY Graduate Center Lecture 3: Linear Regression

Today Calculus –Lagrange Multipliers Linear Regression 1

Optimization with constraints What if I want to constrain the parameters of the model. –The mean is less than 10 Find the best likelihood, subject to a constraint. Two functions: –An objective function to maximize –An inequality that must be satisfied 2

Lagrange Multipliers Find maxima of f(x,y) subject to a constraint. 3

General form Maximizing: Subject to: Introduce a new variable, and find a maxima. 4

Example Maximizing: Subject to: Introduce a new variable, and find a maxima. 5

Example 6 Now have 3 equations with 3 unknowns.

Example 7 Eliminate LambdaSubstitute and Solve

Basics of Linear Regression Regression algorithm Supervised technique. In one dimension: –Identify In D-dimensions: –Identify Given: training data: –And targets: 8

Graphical Example of Regression 9 ?

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Graphical Example of Regression 11

Definition In linear regression, we assume that the model that generates the data involved only a linear combination of input variables. 12 Where w is a vector of weights which define the D parameters of the model

Evaluation How can we evaluate the performance of a regression solution? Error Functions (or Loss functions) –Squared Error –Linear Error 13

Regression Error 14

Empirical Risk Empirical risk is the measure of the loss from data. By minimizing risk on the training data, we optimize the fit with respect to the loss function 15

Model Likelihood and Empirical Risk Two related but distinct ways to look at a model. 1.Model Likelihood. 1.“What is the likelihood that a model generated the observed data?” 2.Empirical Risk 1.“How much error does the model have on the training data?” 16

Model Likelihood 17 Assuming Independently Identically Distributed (iid) data. What is the likelihood that a model with some parameters generated an observed sample

Understanding Model Likelihood 18 Substitution for the eqn of a gaussian Apply a log function Let the log dissolve products into sums

Understanding Model Likelihood 19 Optimize the weights. (Maximum Likelihood Estimation) Log Likelihood Empirical Risk w/ Squared Loss Function

Maximizing Log Likelihood (1-D) Find the optimal settings of w. 20

Maximizing Log Likelihood 21 Partial derivative Set to zero Separate the sum to isolate w 0

Maximizing Log Likelihood 22 Partial derivative Set to zero Separate the sum to isolate w 0

Maximizing Log Likelihood 23 From previous partial From prev. slide Substitute Isolate w 1

Maximizing Log Likelihood Clean and easy. Or not… Apply some linear algebra. 24

Likelihood using linear algebra Representing the linear regression function in terms of vectors. 25

Likelihood using linear algebra Stack x T into a matrix of data points, X. 26 Representation as vectors Stack the data into a matrix and use the Norm operation to handle the sum

Likelihood in multiple dimensions This representation of risk has no inherent dimensionality. 27

Maximum Likelihood Estimation redux 28 Decompose the norm FOIL – linear algebra style Differentiate Combine terms Isolate w

Extension to polynomial regression 29

Extension to polynomial regression Polynomial regression is the same as linear regression in D dimensions 30

Generate new features 31 Standard Polynomial with coefficients, w Risk

Generate new features 32 Feature Trick: To fit a D dimensional polynomial, Create a D-element vector from x i Then standard linear regression in D dimensions

How is this still linear regression? The regression is linear in the parameters, despite projecting x i from one dimension to D dimensions. Now we fit a plane (or hyperplane) to a representation of x i in a higher dimensional feature space. This generalizes to any set of functions 33

Basis functions as feature extraction These functions are called basis functions. –They define the bases of the feature space Allows linear decomposition of any type of function to data points Common Choices: –Polynomial –Gaussian –Sigmoids –Wave functions (sine, etc.) 34

Training data vs. Testing Data Evaluating the performance of a classifier on training data is meaningless. With enough parameters, a model can simply memorize (encode) every training point To evaluate performance, data is divided into training and testing (or evaluation) data. –Training data is used to learn model parameters –Testing data is used to evaluate performance 35

Overfitting 36

Overfitting 37

Overfitting performance 38

Definition of overfitting When the model describes the noise, rather than the signal. How can you tell the difference between overfitting, and a bad model? 39

Possible detection of overfitting Stability –An appropriately fit model is stable under different samples of the training data –An overfit model generates inconsistent performance Performance –A good model has low test error –A bad model has high test error 40

What is the optimal model size? The best model size generalizes to unseen data the best. Approximate this by testing error. One way to optimize parameters is to minimize testing error. –This operation uses testing data as tuning or development data Sacrifices training data in favor of parameter optimization Can we do this without explicit evaluation data? 41

Context for linear regression Simple approach Efficient learning Extensible Regularization provides robust models 42