Report

Made by: Maor Levy, Temple University 2012 1 Up until now: how to reason in a give model Machine learning: how to acquire a model on the basis of data / experience ◦ Learning parameters (e.g. probabilities) ◦ Learning structure (e.g. BN graphs) ◦ Learning hidden concepts (e.g. clustering) 2 What? Parameters Structure Hidden concepts What from? Supervised Unsupervised Reinforcement Selfsupervised What for? Prediction Diagnosis Compression Discovery How? Passive Active Online Offline Output? Classification Regression Clustering Details?? Generative Discriminative Smoothing 3 Commonly attributed to William of Ockham (1290-1349). This was formulated about fifteen hundred years after Epicurus. ◦ In sharp contrast to the principle of multiple explanations, it states: Entities should not be multiplied beyond necessity. Commonly explained as: when have choices, choose the simplest theory. Bertrand Russell: “It is vain to do with more what can be done with fewer.” 4 f(x) f(x) f(x) x (a) f(x) x (b) x (c) x (d) Given a training set: (x1, y1), (x2, y2), (x3, y3), … (xn, yn) Where each yi was generated by an unknown y = f (x), Discover a function h that approximates the true function f. 5 Input: x = email Output: y = “spam” or “ham” Setup: ◦ Get a large collection of example emails, each labeled “spam” or “ham” ◦ Note: someone has to hand label all this data! ◦ Want to learn to predict labels of new, future emails Features: The attributes used to make the ham / spam decision ◦ Words: FREE! ◦ Text Patterns: $dd, CAPS ◦ Non-text: SenderInContacts ◦ … Dear Sir. First, I must solicit your confidence in this transaction, this is by virture of its nature as being utterly confidencial and top secret. … TO BE REMOVED FROM FUTURE MAILINGS, SIMPLY REPLY TO THIS MESSAGE AND PUT "REMOVE" IN THE SUBJECT. 99 MILLION EMAIL ADDRESSES FOR ONLY $99 Ok, I know this is blatantly OT but I'm beginning to go insane. Had an old Dell Dimension XPS sitting in the corner and decided to put it to use, I know it was working pre being stuck in the corner, but when I plugged it in, hit the power nothing happened. 6 Naïve Bayes spam filter Data: ◦ Collection of emails, labeled spam or ham ◦ Note: someone has to hand label all this data! ◦ Split into training, heldout, test sets Classifiers ◦ Learn on the training set ◦ (Tune it on a held-out set) ◦ Test it on new emails Dear Sir. First, I must solicit your confidence in this transaction, this is by virture of its nature as being utterly confidencial and top secret. … TO BE REMOVED FROM FUTURE MAILINGS, SIMPLY REPLY TO THIS MESSAGE AND PUT "REMOVE" IN THE SUBJECT. 99 MILLION EMAIL ADDRESSES FOR ONLY $99 Ok, Iknow this is blatantly OT but I'm beginning to go insane. Had an old Dell Dimension XPS sitting in the corner and decided to put it to use, I know it was working pre being stuck in the corner, but when I plugged it in, hit the power nothing happened. 7 SPAM OFFER IS SECRET CLICK SECRET LINK SECRET SPORTS LINK HAM PLAY SPORTS TODAY WENT PLAY SPORTS SECRET SPORTS EVENT SPORT IS TODAY SPORT COSTS MONEY Questions: ◦ Size of Vocabulary? 13 words ◦ P(SPAM) = 3/8 8 SSSHHHHHH = ◦ = 1 − = 11100000 1− p(S) = 3 ◦ = ∗ 1 − ◦ = 8=1 = (=1) ∗ 1 − ◦ 3 ∗ 1 − 5 5 =0 9 SPAM HAM OFFER IS SECRET CLICK SECRET LINK SECRET SPORTS LINK PLAY SPORTS TODAY WENT PLAY SPORTS SECRET SPORTS EVENT SPORT IS TODAY SPORT COSTS MONEY Questions: ◦ P(“SECRET” | SPAM) = 1/3 ◦ P(“SECRET” | HAM) = 1/15 10 Bag-of-Words Naïve Bayes: Generative model Tied distributions and bag-of-words ◦ Predict unknown class label (spam vs. ham) ◦ Assume evidence features (e.g. the words) are independent Word at position i, not ith word in the dictionary! ◦ Usually, each variable gets its own conditional probability distribution P(F|Y) ◦ In a bag-of-words model Each position is identically distributed All positions share the same conditional probs P(W|C) Why make this assumption? 11 General probabilistic model: |Y| x |F|n parameters General naive Bayes model: Y F1 |Y| parameters F2 Fn n x |F| x |Y| parameters We only specify how each feature depends on the class Total number of parameters is linear in n 12 SPAM HAM OFFER IS SECRET CLICK SECRET LINK SECRET SPORTS LINK PLAY SPORTS TODAY WENT PLAY SPORTS SECRET SPORTS EVENT SPORT IS TODAY SPORT COSTS MONEY Questions: ◦ MESSAGE M = “SPORTS” ◦ P(SPAM | M) = 3/18 Applying Bayes’ Rule 13 SPAM HAM OFFER IS SECRET CLICK SECRET LINK SECRET SPORTS LINK PLAY SPORTS TODAY WENT PLAY SPORTS SECRET SPORTS EVENT SPORT IS TODAY SPORT COSTS MONEY Questions: ◦ MESSAGE M = “SECRET IS SECRET” ◦ P(SPAM | M) = 25/26 Applying Bayes’ Rule 14 SPAM HAM OFFER IS SECRET CLICK SECRET LINK SECRET SPORTS LINK PLAY SPORTS TODAY WENT PLAY SPORTS SECRET SPORTS EVENT SPORT IS TODAY SPORT COSTS MONEY Questions: ◦ MESSAGE M = “TODAY IS SECRET” ◦ P(SPAM | M) = 0 Applying Bayes’ Rule 15 Model: What are the parameters? ham : 0.66 spam: 0.33 the : to : and : of : you : a : with: from: ... 0.0156 0.0153 0.0115 0.0095 0.0093 0.0086 0.0080 0.0075 the : to : of : 2002: with: from: and : a : ... 0.0210 0.0133 0.0119 0.0110 0.0108 0.0107 0.0105 0.0100 Where do these tables come from? Counts from examples! 16 Posteriors determined by relative probabilities (odds ratios): south-west nation morally nicely extent seriously ... : : : : : : inf inf inf inf inf inf screens minute guaranteed $205.00 delivery signature ... : : : : : : inf inf inf inf inf inf What went wrong here? 17 Raw counts will overfit the training data! ◦ Unlikely that every occurrence of “minute” is 100% spam ◦ Unlikely that every occurrence of “seriously” is 100% ham ◦ What about all the words that don’t occur in the training set at all? 0/0? ◦ In general, we can’t go around giving unseen events zero probability At the extreme, imagine using the entire email as the only feature ◦ Would get the training data perfect (if deterministic labeling) ◦ Would not generalize at all ◦ Just making the bag-of-words assumption gives us some generalization, but isn’t enough To generalize better: we need to smooth or regularize the estimates 18 Maximum likelihood estimates: r g g Problems with maximum likelihood estimates: ◦ If I flip a coin once, and it’s heads, what’s the estimate for P(heads)? ◦ What if I flip 10 times with 8 heads? ◦ What if I flip 10M times with 8M heads? Basic idea: ◦ We have some prior expectation about parameters (here, the probability of heads) ◦ Given little evidence, we should skew towards our prior ◦ Given a lot of evidence, we should listen to the data 19 Laplace’s estimate (extended): ◦ Pretend you saw every outcome k extra times c (x) is the number of occurrences of this value of the variable x. |x| is the number of values that the variable x can take on. k is a smoothing parameter. N is the total number of occurrences of x (the variable, not the value) in the sample size. ◦ What’s Laplace with k = 0? ◦ k is the strength of the prior Laplace for conditionals: ◦ Smooth each condition independently: 20 In practice, Laplace often performs poorly for P(X|Y): ◦ When |X| is very large ◦ When |Y| is very large Another option: linear interpolation ◦ Also get P(X) from the data ◦ Make sure the estimate of P(X|Y) isn’t too different from P(X) ◦ What if is 0? 1? 21 For real classification problems, smoothing is critical New odds ratios: helvetica seems group ago areas ... : 11.4 : 10.8 : 10.2 : 8.4 : 8.3 verdana Credit ORDER <FONT> money ... : : : : : 28.8 28.4 27.2 26.9 26.5 Do these make more sense? 22 Now we’ve got two kinds of unknowns How to learn? ◦ Parameters: the probabilities P(Y|X), P(Y) ◦ Hyperparameters, like the amount of smoothing to do: k ◦ Learn parameters from training data ◦ Must tune hyperparameters on different data Why? ◦ For each value of the hyperparameters, train and test on the held-out (validation)data ◦ Choose the best value and do a final test on the test data 23 Data: labeled instances, e.g. emails marked spam/ham ◦ Training set ◦ Held out (validation) set ◦ Test set Features: attribute-value pairs which characterize each x Experimentation cycle ◦ Learn parameters (e.g. model probabilities) on training set ◦ Tune hyperparameters on held-out set ◦ Compute accuracy on test set ◦ Very important: never “peek” at the test set! Evaluation Overfitting and generalization ◦ Accuracy: fraction of instances predicted correctly ◦ Want a classifier which does well on test data ◦ Overfitting: fitting the training data very closely, but not generalizing well to test data Training Data Held-Out Data Test Data 24 Need more features– words aren’t enough! Can add these information sources as new variables in the Naïve Bayes model ◦ ◦ ◦ ◦ ◦ ◦ Have you emailed the sender before? Have 1K other people just gotten the same email? Is the sending information consistent? Is the email in ALL CAPS? Do inline URLs point where they say they point? Does the email address you by (your) name? 25 Input: x = pixel grids Output: y = a digit 0-9 26 Input: x = images (pixel grids) Output: y = a digit 0-9 Setup: ◦ Get a large collection of example images, each labeled with a digit ◦ Note: someone has to hand label all this data! ◦ Want to learn to predict labels of new, future digit images 0 1 2 Features: The attributes used to make the digit decision ◦ Pixels: (6,8)=ON ◦ Shape Patterns: NumComponents, AspectRatio, NumLoops ◦ … 1 ?? 27 Simple version: ◦ One feature Fij for each grid position <i,j> ◦ Boolean features ◦ Each input maps to a feature vector, e.g. ◦ Here: lots of features, each is binary valued Naïve Bayes model: 28 1 0.1 1 0.01 1 0.05 2 0.1 2 0.05 2 0.01 3 0.1 3 0.05 3 0.90 4 0.1 4 0.30 4 0.80 5 0.1 5 0.80 5 0.90 6 0.1 6 0.90 6 0.90 7 0.1 7 0.05 7 0.25 8 0.1 8 0.60 8 0.85 9 0.1 9 0.50 9 0.60 0 0.1 0 0.80 0 0.80 29 2 wins!! 30 Start with very simple example ◦ Linear regression What you learned in high school math ◦ From a new perspective Linear model ◦ y=mx+b ◦ hw(x) = y = w1 x + w0 Find best values for parameters ◦ “maximize goodness of fit” ◦ “maximize probability” or “minimize loss” 31 ◦ Assume true function f is given by y = f (x) = m x + b + noise where noise is normally distributed ◦ Then most probable values of parameters found by minimizing squared-error loss: Loss(hw ) = Σj (yj – hw(xj))2 32 House price in $1000 1000 900 800 700 600 500 400 300 500 1000 1500 2000 2500 3000 3500 House size in square feet 33 House price in $1000 1000 900 800 700 600 Loss 500 w0 400 w1 300 500 1000 1500 2000 2500 3000 3500 House size in square feet y = w1 x + w0 Linear algebra gives an exact solution to the minimization problem 34 w1 = M å xi yi - å xi å yi Måx 2 i (å x ) 2 i 1 w1 w0 = å yi - å xi M M 35 36 w = any point loop until convergence do: for each wi in w do: wi = wi – α ∂ Loss(w) ∂ wi Loss w0 w1 37 You learned this in math class too ◦ hw(x) = w ∙ x = w xT = Σi wi xi The most probable set of weights, w* (minimizing squared error): ◦ w* = (XT X)-1 XT y 38 To avoid overfitting, don’t just minimize loss Maximize probability, including prior over w Can be stated as minimization: ◦ Cost(h) = EmpiricalLoss(h) + λ Complexity(h) For linear models, consider ◦ Complexity(hw) = Lq(w) = ∑i | wi |q ◦ L1 regularization minimizes sum of abs. values ◦ L2 regularization minimizes sum of squares 39 w2 w2 w* w* w1 w1 Cost(h) = EmpiricalLoss(h) + λ Complexity(h) L1 regularization L2 regularization 40 41 ìï 1 if w x + w ³ 0 1 0 f (x) = í ïî 0 if w1 x + w0 < 0 42 Start with random w0, w1 Pick training example <x,y> Update (α is learning rate) ◦ w1 w1+α(y-f(x))x ◦ w0 w0+α(y-f(x)) Converges to linear separator (if exists) Picks “a” linear separator (a good one?) 43 44 Maximizes the “margin” Support Vector Machines 45 Not linearly separable for x1, x2 What if we add a feature? x3= x12+x22 See: “Kernel Trick” X2 X1 X3 46 If the process of learning good values for parameters is prone to overfitting, can we do without parameters? Nearest neighbor for digits: Encoding: image is vector of intensities: What’s the similarity function? ◦ Take new image ◦ Compare to all training images ◦ Assign based on closest example ◦ Dot product of two images vectors? ◦ Usually normalize vectors so ||x|| = 1 ◦ min = 0 (when?), max = 1 (when?) 48 x2 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 4.5 5 5.5 6 6.5 7 x1 Using logistic regression (similar to linear regression) to do linear classification 49 x1 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 4.5 5 5.5 6 6.5 7 x2 Using nearest neighbors to do classification 50 x1 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 4.5 5 5.5 6 6.5 7 x2 Even with no parameters, you still have hyperparameters! 51 Edge length of neighborhood Average neighborhood size for 10-nearest neighbors, n dimensions, 1M uniform points 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 25 50 75 100 125 150 175 200 Number of dimensions 52 Proportion of points in exterior shell Proportion of points that are within the outer shell, 1% of thickness of the hypercube 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 25 50 75 100 125 150 175 200 Number of dimensions 53 References: ◦ Peter Norvig and Sebastian Thrun, Artificial Intelligence, Stanford University http://www.stanford.edu/class/cs221/notes/cs221-lecture5fall11.pdf 54