Intro to Machine Learning
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More Machine Learning Linear Regression Squared Error L1 and L2 Regularization Gradient Descent Recall: Key Components of Intelligent Agents Representation Language: Graph, Bayes Nets Inference Mechanism: A*, variable elimination, Gibbs sampling Learning Mechanism: Maximum Likelihood, Laplace Smoothing, many more: linear regression, perceptron, kNearest Neighbor, … ------------------------------------Evaluation Metric: Likelihood, many more: squared error, 0-1 loss, conditional likelihood, precision/recall, … Recall: Types of Learning The techniques we have discussed so far are examples of a particular kind of learning: Supervised: the training examples included the correct labels or outputs. Vs. Unsupervised (or semi-supervised, or distantly-supervised, …): None (or some, or only part, …) of the labels in the training data are known. Parameter Estimation: We only tried to learn the parameters in the BN, not the structure of the BN graph. Vs. Structure learning: The BN graph is not given as an input, and the learning algorithm’s job is to figure out what the graph should look like. The distinctions below aren’t actually about the learning algorithm itself, but rather about the type of model being learned: Classification: the output is a discrete value, like Happy or not Happy, or Spam or Ham. Vs. Regression: the output is a real number. Generative: The model of the data represents a full joint distribution over all relevant variables. Vs. Discriminative: The model assumes some fixed subset of the variables will always be “inputs” or “evidence”, and it creates a distribution for the remaining variables conditioned on the evidence variables. Parametric vs. Nonparametric: I will explain this later. We won’t talk much about structure learning, but we will cover some other kinds of learning (regression, unsupervised, discriminative, nonparameteric, …) in later lectures. Regression vs. Classification Our NBC spam detector was a classifier: the output Y was one of two options, Ham or Spam. More generally, classifiers give an output from a (usually small) finite (or countably infinite) set of options. E.g., predicting who will win the presidency in the next election is a classification problem (finite set of possible outcomes: US citizens). Regression models give a real number as output. E.g., predicting what the temperature will be tomorrow is a regression problem. Any real number greater than or equal to 0 (Kelvin) is a possible outcome. Quiz: regression vs. classification For each prediction task below, determine whether regression or classification is more appropriate. Task Predict who will win the Super Bowl next year Predict the gender of a baby when it’s born Predict the weight of a child one year from now Predict the average life expectancy of all babies born today Predict the price of Apple, Inc.’s stock at the close of trading tomorrow. Predict whether Microsoft or Apple will have a higher valuation at the close of trading tomorrow Regression or Classification? Answers: regression vs. classification For each prediction task below, determine whether regression or classification is more appropriate. Task Regression or Classification? Predict who will win the Super Bowl next year C Predict the gender of a baby when it’s born C Predict the weight of a child one year from now R Predict the average life expectancy of all babies born today R Predict the price of Apple, Inc.’s stock at the close of trading tomorrow. R Predict whether Microsoft or Apple will have a higher valuation at the close of trading tomorrow C Concrete Example 250000 House Price, $ 200000 175000 150000 100000 50000 0 0 500 1000 1500 2000 Square Footage 2500 Suppose I want to buy a house that’s 2000 square feet. Predict how much it will cost. 3000 More realistic data Violent Crime per Capita Reported Crime Statistics for U.S. Counties Percentage of the population under the federal poverty level Linear Regression Suppose there are N input variables, X1, …, XN (all real numbers). A linear regression is a function that looks like this: Y = w0 + w1X1 + w2X2 + … + wNXN The wi variables are called weights or parameters. Each one is a real number. The set of all functions that look like this (one function for each choice of weights w0 through wN) is called the Hypothesis Class for linear regression. Hypotheses 55100+900*X1 100+900*X1 250000 House Price, $ 200000 150000 80000+270*X1 100000 50000 0 0 500 1000 1500 2000 Square Footage 2500 3000 In this example, there is only one input variable: X1 is square footage. The hypothesis class is all functions Y = w0 + w1 * (square footage). Several example elements of the hypothesis class are drawn. Learning for Linear Regression Linear regression tells us a whole set of possible functions to use for prediction. How do we choose the best one from this set? This is the learning problem for linear regression: Input: a set of training examples, where each example contains a value for (X1, …, XN, Y) Output: a set of weights (w0, …, wN) for the “best-fitting” linear regression model. Quiz: Learning for Linear Regression X 10 30 15 Y 80 40 70 55 -10 For the data on the left, what’s the best fit linear regression model? Answer: Learning for Linear Regression X 10 30 15 Y 80 40 70 55 -10 For the data on the left, what’s the best fit linear regression model? 80 = w0 + w1 * 10 40 = w0 + w1 * 30 80 = w0 + (-2)*10 100 = w0 80-40 = w0-w0 + w1 * 10-w1*30 40 = w1 * (-20) -2 = w1 Y= 100 + (-2) * X Linear Regression with Noisy Data 250000 House Price, $ 200000 150000 100000 50000 0 0 500 1000 1500 2000 Square Footage 2500 3000 In the previous example, we could use only two points and find a line that passed through all of the remaining points. In this example, points are only “approximately” linear. No single line passes through all points exactly. We’ll need a more complex algorithm to handle this. Quadratic Loss (a.k.a. “Squared Error”) Let’s write our training data D with this notation: X11 X12 … X1N Y1 X21 X22 … X2N Y2 … … … … … XM1 XM2 … XMN YM Define πΏπππ(π, π·) = = π ππ − π(π1 , … , ππ ) 2 ππ − π€0 − π€1 π1 − β― − −π€π ππ 2 π Intuitively, this is how much error the function makes on the training data. Objective Function The goal of a linear regression is to find the best linear function. We’ll say that “best” means the one with the least amount of quadratic loss. Mathematically, we say we want f* that satisfies: π ∗ (π1 , … , ππ )= argmin πΏπππ π€0 + π€1 π1 + β― + π€π ππ , π· π€0 ,…,π€π We call LOSS the objective function for our training algorithm, since it’s the function we’re trying to minimize. Closed-form Solution for 1 input variable π ∗ π1 = argmin πΏπππ π€0 + π€1 π1 , π· π€0 ,π€1 To minimize the LOSS function, we’ll take the partial derivatives, and set them to zero: ππΏπππ ππ€1 π = = π −2 ππ − π€0 − π€1 π1π ππ€1 (ππ − π€0 − π€1 π1π )π1π π Set this expression equal to zero: 2 π1π ππ − π1π π€0 − π€1 π1π =0 π 2 π1π = π€1 π π€1 = π1π ππ − π€0 π π 2 π1π π π1π ππ − π€0 π 2 π1π π π1π Closed-form Solution for 1 input variable π ∗ π1 = argmin πΏπππ π€0 + π€1 π1 , π· π€0 ,π€1 To minimize the LOSS function, we’ll take the partial derivatives, and set them to zero: ππΏπππ ππ€0 π = = ππ − π€0 − π€1 π1π ππ€0 π −2 (ππ − π€0 − π€1 π1π ) π Set this expression equal to zero: ππ − π€0 − π€1 π1π = 0 π 1 π€0 = π π 2 π€1 ππ − π π1π π “Closed-form” Result π€0 = 1 π ππ − π Substitute for w0 in the second equation gives: π€1 = π€1 = π€1 π π 2 π1π − π1π ππ − π1π ππ − π€1 = π 1 π π ππ − ππ π π 2 π1π ππ − 2 π π1π π1π π€1 π π π1π π1π + 2 π1π π = π 2 π1π π 1 π 1 π π π 1π 2 π π1π π 2 π1π π π π1π π1π ππ − π€0 π π€1 = π€1 π π 1 π 1 − π π1π ππ − ππ π 1 π π π1π π 2 π1π π1π π π1π π1π π€1 π π π π 2 ππ 2 π π1π Quiz: Learning for Linear Regression π€1 = π€0 = X 10 30 15 Y 80 40 70 55 -10 π 1 π1π ππ − π 2 π π1π 1 π ππ − π π 1 −π π€1 π ππ π π1π π Using the closed-form solution for Quadratic Loss, compute w0 and w1 for this dataset. π1π π π1π 2 Answer: Learning for Linear Regression π€1 = π€0 = π X 10 30 15 Y 80 40 70 55 -10 1 π1π ππ − π 2 π π1π 1 π ππ − π π 1 −π π€1 π ππ π π1π π π1π π1π = π Using the closed-form solution for Quadratic Loss, compute w0 and w1 for this dataset. 2 1 800 + 1200 + 1050 − 550 − 4 180 110 = = −π 1 100 + 900 + 225 + 3025 − 4 1102 1 −2 180 − 110 = πππ 4 4 Note that w1, w0 match what we calculated before! Overfitting and Regularization It is very common to use a technique called regularization to combat overfitting for linear methods. Regularization changes the objective function for training by adding a penalty for the size of the weights: Parameter loss LOSS(f, D) = π ππ − π(π1 , … , ππ ) 2 + π π€π π When p=1, this is called L1 regularization. When p=2, this is called L2 regularization. 1 and 2 are by far the two most commonly-used values of p. Gradient Descent For more complex loss functions, it is often NOT POSSIBLE to find closed-form solutions. Instead, people resort to “iterative methods” that iteratively find better and better parameter estimates, until they converge to the best setting. We’ll go over one example of this kind of method, called “gradient descent”. Gradient Descent Gradient Descent Algorithm ππ‘ππ Create weights π€πππππ₯ ,iο0 1. (π€00 , π€10 ) ο some initial values (often zero) 2. While |π€1π -π€1π−1 +|π€0π −π€0π−1 > π‘βπππ βπππ: for each j: π€ππ+1 ο π€ππ −πΌ ππΏππ π ππ€ππ i ο i+1 3. Return (π€0π , π€1π ) Learning rate LOSS Quiz: Gradient c a b w ππ³πππ ππ a b c positive About zero ππ³πππ ππ negative Check the boxes that apply. LOSS Answer: Gradient c a b w ππ³πππ ππ positive About zero a negative x b c ππ³πππ ππ x x Check the boxes that apply. LOSS Quiz: Gradient a c b w Where is a b c Equal everywhere ππ³πππ ππ the largest? LOSS Answer: Gradient a c b w Where is ππ³πππ ππ a b c Equal everywhere x the largest? LOSS Quiz: Gradient Descent a c b w Which point will allow gradient descent to reach the global minimum, if it is used as the initialization for parameter w? a b c LOSS Answer: Gradient Descent a c b w Which point will allow gradient descent to reach the global minimum, if it is used as the initialization for parameter w? a b c x
