Quiz 1 on Wednesday ~20 multiple choice or short answer questions In class, full period Only covers material from lecture, with a bias towards topics not covered by projects Study strategy: Review the slides and consult textbook to clarify confusing parts. Project 3 preview Machine Learning Photo: CMU Machine Learning Department protests G20 Computer Vision James Hays, Brown

Slides: Isabelle Guyon, Erik Sudderth, Mark Johnson, Derek Hoiem, Lana Lazebnik Clustering Strategies K-means Iteratively re-assign points to the nearest cluster center Agglomerative clustering Start with each point as its own cluster and iteratively merge the closest clusters

Mean-shift clustering Estimate modes of pdf Spectral clustering Split the nodes in a graph based on assigned links with similarity weights As we go down this chart, the clustering strategies have more tendency to transitively group points even if they are not nearby in feature space The machine learning framework Apply a prediction function to a feature representation of the image to get the desired output:

f( f( f( ) = apple ) = tomato ) = cow Slide credit: L. Lazebnik The machine learning framework y = f(x) output prediction

function Image feature Training: given a training set of labeled examples {(x1,y1), , (xN,yN)}, estimate the prediction function f by minimizing the prediction error on the training set Testing: apply f to a never before seen test example x and output the predicted value y = f(x) Slide credit: L. Lazebnik Steps Training Training Labels

Training Images Image Features Training Learned model Learned model Prediction Testing

Image Features Test Image Slide credit: D. Hoiem and L. Lazebnik Features Raw pixels Histograms GIST descriptors Slide credit: L. Lazebnik Classifiers: Nearest neighbor

Training examples from class 1 Test example Training examples from class 2 f(x) = label of the training example nearest to x All we need is a distance function for our inputs No training required! Slide credit: L. Lazebnik

Classifiers: Linear Find a linear function to separate the classes: f(x) = sgn(w x + b) Slide credit: L. Lazebnik Many classifiers to choose from

SVM Neural networks Nave Bayes Bayesian network Logistic regression Randomized Forests Boosted Decision Trees K-nearest neighbor RBMs Etc. Which is the best one? Slide credit: D. Hoiem Recognition task and supervision

Images in the training set must be annotated with the correct answer that the model is expected to produce Contains a motorbike Slide credit: L. Lazebnik Unsupervised Weakly supervised Fully supervised Definition depends on task Slide credit: L. Lazebnik Generalization

Training set (labels known) Test set (labels unknown) How well does a learned model generalize from the data it was trained on to a new test set? Slide credit: L. Lazebnik Generalization Components of generalization error Bias: how much the average model over all training sets differ from the true model? Error due to inaccurate assumptions/simplifications made by the model Variance: how much models estimated from different training sets differ from each other

Underfitting: model is too simple to represent all the relevant class characteristics High bias and low variance High training error and high test error Overfitting: model is too complex and fits irrelevant characteristics (noise) in the data Low bias and high variance Low training error and high test error Slide credit: L. Lazebnik Bias-Variance Trade-off Models with too few parameters are inaccurate because of a

large bias (not enough flexibility). Models with too many parameters are inaccurate because of a large variance (too much sensitivity to the sample). Slide credit: D. Hoiem Bias-Variance Trade-off E(MSE) = noise2 + bias2 + variance Unavoidable error Error due to incorrect assumptions

Error due to variance of training samples See the following for explanations of bias-variance (also Bishops Neural Networks book): http://www.inf.ed.ac.uk/teaching/courses/mlsc/Notes/Lecture4/BiasVariance.pdf Slide credit: D. Hoiem Bias-variance tradeoff Overfitting Error Underfitting

Test error Training error High Bias Low Variance Complexity Low Bias High Variance Slide credit: D. Hoiem Bias-variance tradeoff Test Error

Few training examples High Bias Low Variance Many training examples Complexity Low Bias High Variance Slide credit: D. Hoiem Effect of Training Size

Error Fixed prediction model Testing Generalization Error Training Number of Training Examples Slide credit: D. Hoiem Remember No classifier is inherently better than any other: you need to make assumptions to generalize Three kinds of error Inherent: unavoidable

Bias: due to over-simplifications Variance: due to inability to perfectly estimate parameters from limited data Slide Slide credit: credit: D. D. Hoiem Hoiem How to reduce variance? Choose a simpler classifier Regularize the parameters Get more training data

Slide credit: D. Hoiem Very brief tour of some classifiers K-nearest neighbor SVM Boosted Decision Trees

Neural networks Nave Bayes Bayesian network Logistic regression Randomized Forests RBMs Etc. Generative vs. Discriminative Classifiers Generative Models Discriminative Models Represent both the data and Learn to directly predict the the labels labels from the data Often, makes use of

Often, assume a simple conditional independence boundary (e.g., linear) and priors Examples Examples Logistic regression Nave Bayes classifier Bayesian network SVM Boosted decision trees Models of data may apply to Often easier to predict a label from the data than to future prediction problems model the data

Slide credit: D. Hoiem Classification Assign input vector to one of two or more classes Any decision rule divides input space into decision regions separated by decision boundaries Slide credit: L. Lazebnik Nearest Neighbor Classifier Assign label of nearest training data point to each test data point from Duda et al.

Voronoi partitioning of feature space for two-category 2D and 3D data Source: D. Lowe K-nearest neighbor x x o x o x + o o o

o x2 x1 x o+ x x x 1-nearest neighbor x x

o x o x + o o o o x2 x1 x o+

x x x 3-nearest neighbor x x o x o x + o o

o o x2 x1 x o+ x x x 5-nearest neighbor x

x o x o x + o o o o x2 x1 x

o+ x x x Using K-NN Simple, a good one to try first With infinite examples, 1-NN provably has error that is at most twice Bayes optimal error Classifiers: Linear SVM x x o

x x x o o o x x x o

x2 x1 Find a linear function to separate the classes: f(x) = sgn(w x + b) Classifiers: Linear SVM x x o x x x o

o o x x x o x2 x1 Find a linear function to separate the classes: f(x) = sgn(w x + b)

Classifiers: Linear SVM x x o x o x o o x o x

x x o x2 x1 Find a linear function to separate the classes: f(x) = sgn(w x + b) Nonlinear SVMs Datasets that are linearly separable work out great: x 0

But what if the dataset is just too hard? x 0 We can map it to a higher-dimensional space: x2 0 x Slide credit: Andrew Moore Nonlinear SVMs General idea: the original input space can

always be mapped to some higher-dimensional feature space where the training set is separable: : x (x) Slide credit: Andrew Moore Nonlinear SVMs The kernel trick: instead of explicitly computing the lifting transformation (x), define a kernel function K such that K(xi , xj) = (xi ) (xj) (to be valid, the kernel function must satisfy Mercers condition) This gives a nonlinear decision boundary in the original feature space:

y ( x ) ( x ) b y K ( x , x) b i i i i i i i i

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998 Nonlinear kernel: Example Consider the mapping ( x) ( x, x 2 ) x2 ( x) ( y ) ( x, x 2 ) ( y, y 2 ) xy x 2 y 2 K ( x, y ) xy x 2 y 2 Kernels for bags of features Histogram intersection kernel: N I (h1 , h2 ) min(h1 (i ), h2 (i )) i 1

Generalized Gaussian kernel: 1 2 K (h1 , h2 ) exp D(h1 , h2 ) A D can be (inverse) L1 distance, Euclidean distance, 2 distance, etc. J. Zhang, M. Marszalek, S. Lazebnik, and C. Schmid, Local Features and Kernels for Classifcation of Texture and Object Categories: A Compr ehensive Study Summary: SVMs for image classification 1. Pick an image representation (in our case, bag of features)

2. Pick a kernel function for that representation 3. Compute the matrix of kernel values between every pair of training examples 4. Feed the kernel matrix into your favorite SVM solver to obtain support vectors and weights 5. At test time: compute kernel values for your test example and each support vector, and combine them with the learned weights to get the value of the decision function Slide credit: L. Lazebnik What about multi-class SVMs? Unfortunately, there is no definitive multiclass SVM formulation In practice, we have to obtain a multi-class SVM by combining multiple two-class SVMs One vs. others Traning: learn an SVM for each class vs. the others

Testing: apply each SVM to test example and assign to it the class of the SVM that returns the highest decision value One vs. one Training: learn an SVM for each pair of classes Testing: each learned SVM votes for a class to assign to the test example Slide credit: L. Lazebnik SVMs: Pros and cons Pros Many publicly available SVM packages: http://www.kernel-machines.org/software Kernel-based framework is very powerful, flexible SVMs work very well in practice, even with very small training sample sizes

Cons No direct multi-class SVM, must combine two-class SVMs Computation, memory During training time, must compute matrix of kernel values for every pair of examples Learning can take a very long time for large-scale problems What to remember about classifiers No free lunch: machine learning algorithms are tools, not dogmas Try simple classifiers first Better to have smart features and simple classifiers than simple features and smart classifiers Use increasingly powerful classifiers with more training data (bias-variance tradeoff) Slide credit: D. Hoiem

Some Machine Learning References General Tom Mitchell, Machine Learning, McGraw Hill, 1997 Christopher Bishop, Neural Networks for Pattern Recognition, Oxford University Press, 1995 Adaboost Friedman, Hastie, and Tibshirani, Additive logistic regression: a statistical view of boosting, Annals of Statistics, 2000 SVMs http://www.support-vector.net/icml-tutorial.pdf Slide credit: D. Hoiem