Outline

• Recap: Supervised learning and reinforcement learning
• Unsupervised learning
• Clustering: \(k\)-means algorithm
  • Clustering vs. classification
  • Initialization matters
  • \(k\) matters
• Auto-encoder
  • Compact representation
• Unsupervised learning again -- representation learning and beyond

Supervised learning

• explicit supervision via labels \(y\).
• both regression or classification are trying to predict accurate, exact labels.

Reinforcement learning

• implicit(evaluative), sequential, iterative supervision via rewards.

Learner

Outline

• Recap: Supervised learning and reinforcement learning
• Unsupervised learning
• Clustering: \(k\)-means algorithm
  • Clustering vs. classification
  • Initialization matters
  • \(k\) matters
• Auto-encoder
  • Compact representation
• Unsupervised learning again -- representation learning and beyond

Unsupervised learning

• No supervision, i.e., no labels nor rewards.
• try to learn something "interesting" using only the features
  • Clustering: learn "similarity" of the 
  • Autoencoder: learn compression/reconstruction/representation
• useful paradigm on its own; often empowers downstream tasks.

Autoencoder

Outline

• Recap: Supervised learning and reinforcement learning
• Unsupervised learning
• Clustering: \(k\)-means algorithm
  • Clustering vs. classification
  • Initialization matters
  • \(k\) matters
• Auto-encoder
  • Compact representation
• Unsupervised learning again -- representation learning and beyond

Food distribution placement

• \(x_1\): longitude, \(x_2\): latitude
• Person \(i\) location \(x^{(i)}\)
• Food truck \(j\) location \(\mu^{(j)}\)
• Q: where should I have my \(k\) food trucks park?
• want to minimize the "loss" of people we serve
• Loss if \(i\) walks to truck \(j\) : \(\left\|x^{(i)}-\mu^{(j)}\right\|_2^2\)
• Index of the truck where person \(i\) is chosen to walk to: \(y^{(i)}\)

Food distribution placement

• \(x_1\): longitude, \(x_2\): latitude
• Person \(i\) location \(x^{(i)}\)

Food distribution placement

• \(x_1\): longitude, \(x_2\): latitude
• Person \(i\) location \(x^{(i)}\)
• Q: where should I have my \(k\) food trucks park?

Food distribution placement

• \(x_1\): longitude, \(x_2\): latitude
• Person \(i\) location \(x^{(i)}\)
• Q: where should I have my \(k\) food trucks park?
• Food truck \(j\) location \(\mu^{(j)}\)
• Want to minimize the "loss" of people we serve

Food distribution placement

• \(x_1\): longitude, \(x_2\): latitude
• Person \(i\) location \(x^{(i)}\)
• Q: where should I have my \(k\) food trucks park?
• Food truck \(j\) location \(\mu^{(j)}\)
  Want to minimize the "loss" of people we serve
• Loss if \(i\) walks to truck \(j\) : \(\left\|x^{(i)}-\mu^{(j)}\right\|_2^2\)
• Index of the truck where person \(i\) is chosen to walk to: \(y^{(i)}\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu\) random initialization

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu\) random initialization

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

each person \(i\) gets assigned to a food truck \(j\), color-coded.

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu\) random initialization

each person \(i\) gets assigned to a food truck \(j\), color-coded.

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu=\) random initialization

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

6            for \(j=1\) to \(k\)

\(7 \quad \quad\quad\quad \quad \mu^{(j)}=\frac{1}{N_j} \sum_{i=1}^n \mathbb{1}\left(y^{(i)}=\mathfrak{j}\right) x^{(i)}\)

food truck \(j\) gets moved to the "central" location of all ppl assigned to it

\(N_j = \sum_{i=1}^n \mathbf{1}\left\{y^{(i)}=j\right\}\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu=\) random initialization

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

6            for \(j=1\) to \(k\)

\(7 \quad \quad\quad\quad \quad \mu^{(j)}=\frac{1}{N_j} \sum_{i=1}^n \mathbb{1}\left(y^{(i)}=\mathfrak{j}\right) x^{(i)}\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu=\) random initialization

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

6            for \(j=1\) to \(k\)

\(7 \quad \quad\quad\quad \quad \mu^{(j)}=\frac{1}{N_j} \sum_{i=1}^n \mathbb{1}\left(y^{(i)}=\mathfrak{j}\right) x^{(i)}\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu=\) random initialization

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

6            for \(j=1\) to \(k\)

\(7 \quad \quad\quad\quad \quad \mu^{(j)}=\frac{1}{N_j} \sum_{i=1}^n \mathbb{1}\left(y^{(i)}=\mathfrak{j}\right) x^{(i)}\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu=\) random initialization

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

6            for \(j=1\) to \(k\)

\(7 \quad \quad\quad\quad \quad \mu^{(j)}=\frac{1}{N_j} \sum_{i=1}^n \mathbb{1}\left(y^{(i)}=\mathfrak{j}\right) x^{(i)}\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu=\) random initialization

2   for \(t=1\) to \(\tau\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

6            for \(j=1\) to \(k\)

\(7 \quad \quad\quad\quad \quad \mu^{(j)}=\frac{1}{N_j} \sum_{i=1}^n \mathbb{1}\left(y^{(i)}=\mathfrak{j}\right) x^{(i)}\)

8            if \(y==y_{\text {old }}\)

9\(\quad \quad \quad\quad \quad\)break

K-MEANS\((k, \tau, \left\{x^{(i)}\right\}_{i=1}^n)\)

1   \(\mu, y=\) random initialization

2   for \(t=1\) to \(\tau\)

3            \(y_{o l d}=y\)

4            for \(i=1\) to \(n\)

\(5 \quad \quad\quad\quad \quad y^{(i)}=\arg \min _j\left\|x^{(i)}-\mu^{(j)}\right\|^2\)

6            for \(j=1\) to \(k\)

\(7 \quad \quad\quad\quad \quad \mu^{(j)}=\frac{1}{N_j} \sum_{i=1}^n \mathbb{1}\left(y^{(i)}=\mathfrak{j}\right) x^{(i)}\)

8            if \(y==y_{\text {old }}\)

9\(\quad \quad \quad\quad \quad\)break

10    return \(\mu, y\)

Outline

• Recap: Supervised learning and reinforcement learning
• Unsupervised learning
• Clustering: \(k\)-means algorithm
  • Clustering vs. classification
  • Initialization matters
  • \(k\) matters
• Auto-encoder
  • Compact representation
• Unsupervised learning again -- representation learning and beyond

\(k\)-means

Compare to classification

• Did we just do \(k\)-class classification?
• Looks like we assigned label \(y^{(i)}\), which takes \(k\) different values, to each feature vector \(x^{(i)}\)
• But we didn't use any labeled data
• The "labels" here don't have meaning; I could permute them and have the same result
• Output is really a partition of the data

Compare to classification

• So what did we do?
• We clustered the data: we grouped the data by similarity
• Why not just plot the data? We should! Whenever we can!
• But also: Precision, big data, high dimensions, high volume
• An example of unsupervised learning: no labeled data, and we're finding patterns

Outline

• Recap: Supervised learning and reinforcement learning
• Unsupervised learning
• Clustering: \(k\)-means algorithm
  • Clustering vs. classification
  • Initialization matters
  • \(k\) matters
• Auto-encoder
  • Compact representation
• Unsupervised learning again -- representation learning and beyond

Effect of initialization

• A theorem say if run for enough outer iterations (line 2), the k-means algorithm will converge to a local minimum of the k-means objective
• That local minimum could be bad!
• The initialization can make a big difference.
• Some options: random restarts.

Effect of initialization

Effect of \(k\)

Outline

• Recap: Supervised learning and reinforcement learning
• Unsupervised learning
• Clustering: \(k\)-means algorithm
  • Clustering vs. classification
  • Initialization matters
  • \(k\) matters
• Auto-encoder
  • Compact representation
• Unsupervised learning again -- representation learning and beyond

Outline

• Recap: Supervised learning and reinforcement learning
• Unsupervised learning
• Clustering: \(k\)-means algorithm
  • Clustering vs. classification
  • Initialization matters
  • \(k\) matters
• Auto-encoder
  • Compact representation
• Unsupervised learning again -- representation learning and beyond

Self-supervision (masking)

(Density estimation)

Dall-E 2 (UnCLIP):

[https://arxiv.org/pdf/2204.06125.pdf]