Grundlagen der Künstlichen Intelligenz

Transcription

1 Grundlagen der Künstlichen Intelligenz Unsupervised learning Daniel Hennes (WS 2017/18) University Stuttgart - IPVS - Machine Learning & Robotics 1

2 Today Supervised learning Regression (linear regression) Classification (log. regression, MLP) Reinforcement learning Model-free (Q-learning, SARSA) Unsupervised learning Clustering (k-means, Gaussian mixtures) Dimensionality reduction (PCA, Autoencoders) 2

3 Unsupervised learning What is unsupervised learning? Unsupervised learning generally aims at modelling the distribution of the data P(x) Examples: finding lower-dimensional representation clustering density estimation fitting a graphical model 3

4 Unsupervised learning Unsupervised learning is learning without labels Goal: extracting patterns and structure from given data Clustering: finding inherent similarities and patters Dimensionality reduction: finding underlying structure and features 4

5 Clustering 5

6 Clustering Detect patterns in unlabeled data e.g. group s or search results e.g. find categories of customers e.g. detect anomalous program executions Applications: extract useful data classes outlier detection Useful when we do not know what we are looking for Requires data, but no labels 6

7 Clustering Basic idea: group together similar instances What is similar? Small distance: d dist(x, y) = (x y) T (x y) = (x j y j ) 2 j=1 7

8 k-means Iterative algorithm: 1. Pick k random cluster centers 2. Repeat Assign data instances to closest center Update centers to the mean of assigned points Stop when assignments do not change 8

9 k-means as optimization Given data D = {x i } n i=1, find k centers µ j, and a data assignment c : i j to minimize L = min c,µ n (x i µ c(i) ) 2 i=1 Pick k data points randomly to initialize the centers µ j Iterate adapting the assignments c(i) and the centers µ j : i : c(i) arg min(x i µ j ) 2 j j : µ j 1 c 1 x i (j) i c 1 (j) 9

10 k-means questions Will k-means converge (to a global optimum)? Non-deterministic, converges to local minima Perform restarts! How to pick k? Plot L = min c,µ i (xi µ c(i)) 2 for different k Choose a trade-off between model complexity and data fit Will it always find the true patterns in the data? 10

11 k-means problems 11

12 Gaussian mixture models (GMMs) Idea: approximate the true distribution, from which D = {x i } n i=1 is generated, using a mixture of multivariate Gaussians, i.e. P(x) = k π j N (x µ j, Σ j ) j=1 where 0 π j 1 is the mixing value with k j=1 π j = 1 Maximize log-likelihood: n max ln P(X π, µ, Σ) = π,µ,σ i=1 [ k ] ln π j N (x i µ j, Σ j ) 1. Initalize centers randomly 2. Evaluate the posterior probability that x i belongs to cluster j 3. Update estimates π, µ, Σ j=1 12

13 Gaussian mixture models (GMMs) Bishop (2006) Pattern Recognition and Machine Learning 13

14 Gaussian mixture models (GMMs) Soft-assignment version of k-means More powerful than k-means, returns distribution Like k-means, GMM suffers from local minima; choosing k... More iterations than k-means & more computations per cycle 14

15 Dimensionality reduction 15

16 Principle Component Analysis (PCA) Assume we have data D = {x i } n i=1, x i R d We believe that there is an underlying lower-dimensional space explaining this data. How can we formalize this?

17 PCA: minimizing projection error For each x i R d, we postulate a lower-dimensional latent representation z i R p : x i V p z i + µ Find V p, z i, µ that minimize the projection error: n x i (V p z i + µ) 2 i=1 µ = 1 n n x i, z i = Vp T (x i µ) i=1 17

18 Find optimal projection 1. Compute the covariance matrix X T X, where X R n d is the centered data matrix that contains all x i = x i µ 2. Perform eigenvalue decomposition: X T X = VDV T, D = diag(λ 1, λ 2,..., λ d ) 3. Select the p largest λ i and corresponding eigenvectors: V p = (v 1 v 2... v p ) 4. Map data points to lower-dimensional latent space: z i = V T p (x i µ) 18

19 Examples: Digits (MNIST) 19

20 Examples: Digits (MNIST) Every data point can be expressed as a linear combination of the eigenvectors in V p : x µ + V p z = µ + z 1 v 1 + z 2 v = + z 1 + z

21 Example: Eigenfaces Viola & Jones 21

22 Dimensionality reduction Dimensionality reduction facilitates: Classification Visualization Denoising data Communication Storage PCA finds the directions of greatest variance in the data set Represents each data point by its coordinates along each of these directions Non-linear generalizations: kernel PCA (logistic PCA)... 22

23 Autoencoder Multilayer neural network with small central layer (coding layer) to reconstruct high-dimensional input vectors Encoder network: transforms the high-dimensional data into a low-dimensional code Decoder network: recovers the data from the code Weights trained by back-propagation Unsupervised learning: no labels... What are the targets? Hinton et al. (2006) Reducing the Dimensionality of Data with Neural Networks 23

24 30 W W W W 1 Top RBM RBM RBM Decoder T W T W T W T W 4 30 Code layer W W W W 1 T W 1+ε T W 2+ε T W 3+ε T W 4+ε 5 30 W 4+ε W 3+ε W 2+ε W 1+ε 1 train au the initi gradien initial w algorith time. W for bina and sho data set An ages) c work ca (RBM) are con detecto nection units o observe Bhidden the visi given b Eðv, hþ Pretraining RBM Encoder Unrolling Fine-tuning 24 where v

25 Example: Digits reconstruction random test image from each class Autoencoder ( ) logistic PCA (30-dim) PCA (30-dim) 25

26 Example: Digits (A) PCA (first two principal components) (B) Autoencoder ( ) 26

27 Related concepts Semi-supervised learning: small amount of labeled data large amount of unlabeled data Active learning: algorithm can interactively ask for new labels unlabeled data is abundant labeling is expensive Self-supervised learning: primary sensor cue provides supervised training data model based on secondary sensor cue is used to enhance perceptual capabilities 27

28 Self-supervised learning Thrun et al. (2005) Stanley: The Robot that won the DARPA Grand Challenge 28

29 Summary Inference from unlabeled data Clustering: hidden patterns / groupings of data Dimensionality reduction: discovering the underlying space (latent variable model) Great for data exploration / visualization 29