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Unsupervised Learning Clustering K-Means Recall: Key Components of Intelligent Agents Representation Language: Graph, Bayes Nets, Linear functions Inference Mechanism: A*, variable elimination, Gibbs sampling Learning Mechanism: Maximum Likelihood, Laplace Smoothing, gradient descent, perceptron, k-Nearest Neighbor, many more: kmeans, EM, PCA, … ------------------------------------Evaluation Metric: Likelihood, quadratic loss (a.k.a. squared error), regularized loss, margins, many more: 0-1 loss, conditional likelihood, precision/recall, … Supervised vs. Unsupervised Learning Supervised Learning: “Labeled” Data Unsupervised Learning: “Unlabeled” Data X11 X12 … X1N Y1 X11 X12 … X1N ? X21 X22 … X2N Y2 X21 X22 … X2N ? … … … … … … … … … … XM1 XM2 … XMN YM XM1 XM2 … XMN ? In supervised learning, the learning algorithm is given training examples that contain inputs (the X values) and “labels” or “outputs” (the Y values). In unsupervised learning, the learning algorithm is given training examples that contain inputs (the X values), but no “labels” or “outputs” (no Y values). It’s called “unsupervised” because there are no “labels” to help “supervise” the learning algorithm during the learning process, to get it to the right model. Example Unsupervised Problem 1 X2 Are these data points distributed completely randomly, or do you see some structure in them? How many clusters do you see? None 1 2 3 4 5 X1 Example Unsupervised Problem 1 Are these data points distributed completely randomly, or do you see some structure in them? X2 Structured – there are clusters! How many clusters do you see? None 1 2 3 4 5 X1 Example Unsupervised Problem 2 There are 2 input variables, X1 and X2, in this space. So this is called a “2-dimensional space”. X2 How many dimensions are actually needed to describe this data? 0 1 2 3 X1 Example Unsupervised Problem 2 There are 2 input variables, X1 and X2, in this space. So this is called a “2-dimensional space”. X2 How many dimensions are actually needed to describe this data? 1 dimension captures most of the variation in this data. 2 dimensions will capture everything. X1 Types of Unsupervised Learning Density Estimation - Clustering (Example 1) - Dimensionality Reduction (Example 2) Factor Analysis - Blind signal separation Example Open Problem in AI: Unsupervised Image Segmentation (and Registration) Examples taken from (Felzenszwab and Huttenlocher, Int. Journal of Computer Vision, 59:2, 2004). http://cs.brown.edu/~pff/segment/. The K-Means Clustering Algorithm Inputs: 1) Some unlabeled (no outputs) training data 2) A number K, which must be greater than 1 Output: A label between 1 and K for each data point, indicating which cluster the data point belongs to. Visualization of K-Means Data Visualization of K-Means 1. Generate K random initial cluster centers, or “means”. Visualization of K-Means 2. Assign each point to the closest “mean” point. Visualization of K-Means 2. Assign each point to the closest “mean” point. Visually, the mean points divide the space into a Voronoi diagram. Visualization of K-Means 3. Recompute the “mean” (center) of each colored set of data. Notice: “means” do not have to be at the same position as a data point, although some times they might be. Visualization of K-Means 3. Recompute the “mean” (center) of each colored set of data. Notice: “means” do not have to be at the same position as a data point, although some times they might be. Visualization of K-Means 4. Repeat steps 2 & 3 until the “means” stop moving (convergence). a. Repeat step 2 (assign each point to the nearest mean) Visualization of K-Means 4. Repeat steps 2 & 3 until the “means” stop moving (convergence). a. Repeat step 2 (assign each point to the nearest mean) Visualization of K-Means 4. Repeat steps 2 & 3 until the “means” stop moving (convergence). a. Repeat step 2 (assign each point to the nearest mean) b. Repeat step 3 (recompute means) Visualization of K-Means Quiz: Where will the means be after the next iteration? 4. Repeat steps 2 & 3 until the “means” stop moving (convergence). a. Repeat step 2 (assign each point to the nearest mean) b. Repeat step 3 (recompute means) Visualization of K-Means Answer: Where will the means be after the next iteration? 4. Repeat steps 2 & 3 until the “means” stop moving (convergence). a. Repeat step 2 (assign each point to the nearest mean) b. Repeat step 3 (recompute means) Visualization of K-Means Quiz: Where will the means be after the next iteration? 4. Repeat steps 2 & 3 until the “means” stop moving (convergence). a. Repeat step 2 (assign each point to the nearest mean) b. Repeat step 3 (recompute means) Visualization of K-Means Answer: Where will the means be after the next iteration? 4. Repeat steps 2 & 3 until the “means” stop moving (convergence). a. Repeat step 2 (assign each point to the nearest mean) b. Repeat step 3 (recompute means) Formal Description of the Algorithm Input: 1) X11, …, X1N; … ; XM1, …, XMN 2) K Output: Y1; …; YM, where each Yi is in {1, …, K} Formal Description of the Algorithm 1. Init: For each k in {1, …, K}, create a random point Ck 2. Repeat until all Ck remain the same: Assignment (aka Expectation): For each Xi, let C[Xi] the k value for the closest Ck to Xi Update (aka Maximization): For each Ck, let Dk{Xi |C[Xi] = k} (set of Xi assigned to cluster k) if |Dk| = 0, let Ck random new point else let Ck 1 | | ∈ 3. Return C[Xi] for each Xi (average of points in Dk) Evaulation metric for K-means LOSS Function (or Objective function) for Kmeans: Within-cluster-sum-of-squares loss (WCSS): WCSS(X1, …, XM, C1, …, CK) ( , )2 = =1 | = Complexity of K-Means Finding a globally-optimal solution to WCSS is known to be an NP-hard problem. K-means is known to converge to a local minimum of WCSS. K-means is a “heuristic” or “greedy” algorithm, with no guarantee that it will find the global optimum. On real datasets, K-means usually converges very quickly. Often, people run it multiple times with different random initializations, and choose the best result. In some cases, K-means will still take exponential time (assuming P!=NP), even to find a local minimum. However, such cases are rare in practice. Quiz Is K-means Classification or Regression? Generative or Discriminative? Parametric or Nonparametric? Answer Is K-means Classification or Regression? - classification: output is a discrete value (cluster label) for each point Generative or Discriminative? - discriminative: it has fixed input variables and output variables. Parametric or Nonparametric? - parametric: the number of cluster centers (K) does not change with the number of training data points Quiz Is K-means Supervised or Unsupervised? Online or batch? Closed-form or iterative? Answer Is K-means Supervised or Unsupervised? - Unsupervised Online or batch? - batch: if you add a new data point, you need to revisit all the training data to recompute the locally-optimal model Closed-form or iterative? -iterative: training requires many passes through the data Quiz Which of the following problems might be solved using K-Means? Check all that apply. For those that work, explain what the inputs and outputs (X and Y variables) would be. • Segmenting an image • Finding galaxies (dense groups of stars) in a telescope’s image of the night sky • Identify different species of bacteria from DNA samples of bacteria in seawater Answer Which of the following problems might be solved using KMeans? Check all that apply. For those that work, explain what the inputs and outputs (X and Y variables) would be. • Segmenting an image: Yes. Inputs are the pixel intensities, outputs are segment labels. • Finding galaxies (dense groups of stars) in a telescope’s image of the night sky. Yes. Inputs are star locations, outputs are galaxy labels • Identify different species of bacteria from DNA samples of bacteria in seawater. Yes. Inputs are gene sequences, outputs are species labels.