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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: k-means, 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

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

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

X2

X1

Example Unsupervised Problem 1

Are these data points distributed completely randomly, or do you see some structure in them?

Structured – there are clusters!

How many clusters do you see?

None

1

2

3

4

5

X2

X1

Example Unsupervised Problem 2

There are 2 input variables, X1 and X2, in this space. So this is called a “2-dimensional space”.

How many dimensions are actually needed to describe this data?

0

1

2

3

X2

X1

Example Unsupervised Problem 2

There are 2 input variables, X1 and X2, in this space. So this is called a “2-dimensional space”.

How many dimensions are actually needed to describe this data?

1dimension captures most of the variation in this data.

2 dimensions will capture everything.

X2

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:

- Some unlabeled (no outputs) training data
- 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

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:

- X11, …, X1N; … ; XM1, …, XMN
- K
Output: Y1; …; YM, where each Yi is in {1, …, K}

Formal Description of the Algorithm

- Init: For each k in {1, …, K}, create a random point Ck
- 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 (average of points in Dk)

3. Return C[Xi] for each Xi

Evaulation metric for K-means

LOSS Function (or Objective function) for K-means:

Within-cluster-sum-of-squares loss (WCSS):

WCSS(X1, …, XM, C1, …, CK)

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

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 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: 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.

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