COMPUTATIONAL INTELLIGENCE (CS) (INTRODUCTION TO MACHINE LEARNING) SS16. Lecture 2: Linear Regression Gradient Descent Non-linear basis functions
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1 COMPUTATIONAL INTELLIGENCE (CS) (INTRODUCTION TO MACHINE LEARNING) SS16 Lecture 2: Linear Regression Gradient Descent Non-linear basis functions
2 LINEAR REGRESSION MOTIVATION
3 Why Linear Regression? Regression = Prediction of real-valued outputs Simplest regression algorithm Easy, and fast Benchmark algorithm Mathematical Concepts introduced Data format and Matrix notation Minimizing a cost function: gradient descent Non-linear features and basis functions
4 Examples: (linear) regression application Social science: relationship between data Brain computer interfaces Neuroprosthetic control
5 Examples: (linear) regression application Social science: relationship between data Brain computer interfaces Neuroprosthetic control
6 LINEAR REGRESSION WITH ONE INPUT
7 Linear regression with one input Training set body height Learning algorithm? knee height Hypothesis Test input x Hypothesis h Prediction Parameters?
8 A regression problem We want to learn to predict a person s height based on his/her knee height and/or arm span This is useful for patients who are bed bound or in a wheelchair and cannot stand to take an accurate measurement of their height Knee Height [cm] Arm span [cm] Height [cm]
9 Example Data body height 180 Knee height [cm] Arm span [cm] Height [cm] m=30 data points body height knee height armspan
10 Example Data 190 Knee Height [cm] Arm span [cm] Height [cm] body height armspan knee height 55 60
11 Linear regression with one input Knee Height [cm] Height [cm] Which hypothesis is better? 190 In what sense is it better? body height knee height Hypothesis Parameters?
12 Formalization of problem Knee Height [cm] Height [cm] m=30 data points Given m training examples Goal: learn parameters such that 190 body height for all training examples i= knee height
13 Least Squares Objective Minimize Error body height knee height
14 Least Squares Objective Minimize Error cost function mean squared error body height knee height
15 Least Squares Objective Minimize Error cost function mean squared error body height knee height
16 Cost function illustrated Properties of cost function: Quadratic function Convex Bowl -shaped Unique local and global minimum (under regular conditions) body height knee height body height knee height
17 Minimizing the cost Two ways to find the parameters minimizing Gradient descent Direct analytical solution (setting derivatives = 0)
18 EXCURSUS: GRADIENT DESCENT
19 Descending in the steepest direction Gradient descent on some arbitrary cost function
20 Gradient descent algorithm Repeat until convergence (simultaneously updating and ) negative gradient = descent learning rate ( eta ) partial derivative of with respect to
21 Gradient is orthogonal to contour lines A contour line is a line along which = const
22 Potential issues with gradient descent May get stuck in local minima Learning rate too small: slow convergence Learning rate too large: oscillations, divergence too small too large
23 LINEAR REGRESSION WITH GRADIENT DESCENT (ONE INPUT)
24 Application of gradient descent Linear regression cost Gradient descent (simultaneous update) learning rate (simultaneous update) error input
25 Predicting height from knee height Optimal fit to training data body height knee height
26 LINEAR REGRESSION MORE GENERAL FORMULATION: MULTIPLE FEATURES
27 Multiple inputs (features) Knee Height x1 Arm span x2 Age x3 Height y = = = 3 Notation: number of training examples number of features input features of i th training example (vector-valued). value of feature j in i th training example
28 Linear hypothesis Hypothesis (one input): Hypothesis (multiple input features): Example: h(x) = *kneeheight + 0.3*armspan + 0.1*age More compact notation: Introduce Why? Notation convenience!
29 Multiple inputs (features) revisited x0 Knee Height x1 Arm span x2 Age x3 Height y = = 3 Notation: number of training examples number of features = 1 = 17 input features of i th training example (vector-valued). value of feature j in i th training example
30 Matrix and vector notation x0 Knee Height x1 Arm span x2 Age x3 Height y features of i th training example design matrix output/target vector (n+1) 1 m (n+1) m 1
31 LINEAR REGRESSION WITH GRADIENT DESCENT (GENERAL FORMULATION)
32 Linear regression problem statement Hypothesis: Cost function: high-dimensional quadratic ( bowl -shaped) function Goal is to find parameters which minimize the cost
33 Gradient descent (multiple features) with one input feature: learning rate (simultaneous update) error input with n input features: learning rate error input (simultaneous update for j=0 n) For j=0: define for convenience
34 LINEAR REGRESSION ANALYTICAL SOLUTION
35 Analytical solution Set all partial derivatives of cost function = 0 Solving system of linear equations yields: design matrix output/target vector Moore-Penrose Pseudoinverse of Note: This analytical solution requires that columns of independent ( regular conditions) are linearly
36 Example: analytical solution applied to problem with one input Knee Height [cm] Height [cm] body height knee height
37 Example: analytical solution applied to problem with one input Knee Height [cm] Height [cm]
38 Predicting height from knee height body height knee height
39 Gradient descent Analytical solution Need to choose learning rate Iterative algorithm (needs many iterations to converge) Works well even when number of input features is large No need to choose Direct solution (no iteration) Slow if is too large (inverting nxn matrix)
40 NON-LINEAR FEATURES (NON-LINEAR BASIS FUNCTIONS)
41 Non-linear trends in data How can we learn non-linear hypotheses? x y ? 2 0???
42 Linear fit to this non-linear data x y standard design matrix Hypothesis: Optimal parameters:
43 Linear fit to this non-linear data
44 Non-linear (quadratic) fit x y design matrix with non-linear features Hypothesis: Optimal parameters:
45 Non-linear (quadratic) fit
46 Non-linear (sinusoid) fit x y design matrix with non-linear features Hypothesis: Optimal parameters:
47 Non-linear (sinusoid) fit
48 Image: JPEG = cosin-basis Each block of 8x8 pixels is represented in a Fourrier basis of cosin filters Better representation of edges and corners Allows for compression
49 Audio: cosin or wavelet basis Good signal representation make a compromise between time and frequency
50 Non-linear input features (in general) feature 2 of all training examples all features of 1st training example Feature 2 for each training example i is computed by applying a non-linear basis function: Allows to learn a variety of non-linear functions with the same technique(s): Gradient descent or
51 Polynomial regression Features are powers of x n = degree of polynome to be learned n=0 n=1 n=3 n=9 What happened here? Next lecture
52 Radial basis functions Gaussian -shaped RBFs: Each basis function j has a center in the input space The width of the basis functions is determined by x
53 Radial basis functions Gaussian -shaped RBFs: Each basis function j has a center in the input space The width of the basis functions is determined by x
54 Radial basis functions Gaussian -shaped RBFs: Each basis function j has a center in the input space The width of the basis functions is determined by x
55 Fitting a single RBF to data RBF with
56 Fitting RBFs to data RBFs with
57 SUMMARY (QUESTIONS)
58 Some questions Hypothesis for linear regression =? Cost function for linear regression =? How many local minima may the cost function for lin. reg. have (under regular conditions)? Name two ways to minimize the cost function? General gradient descent formula? Linear regression with gradient descent formula? What issues can arise during gradient descent? What is the design matrix? What are its dimensions? Analytical solution for linear regression =? What are the components of the solution? Pros and Cons of gradient descent vs. analytical solution? How can one learn non-linear hypotheses with linear regression? What is polynomial regression? What are radial basis functions?
59 What is next? Classification with Logistic Regression Gradient descent tricks & more advanced optimization techniques Underfitting & Overfitting Model selection (Training- & Validation- & Testset)
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