November 28 th, Carlos Guestrin. Lower dimensional projections


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1 PCA Machine Learning 070/578 Carlos Guestrin Carnegie Mellon University November 28 th, 2007 Lower dimensional projections Rather than picking a subset of the features, we can new features that are combinations of existing features Let s see this in the unsupervised setting just X, but no Y 2
2 Linear projection and reconstruction x 2 project into dimension z x reconstruction: only know z, what was (x,x 2 ) 3 Linear projections, a review Project a point into a (lower dimensional) space: point: x = (x,,x n ) select a basis set of basis vectors (u,,u k ) we consider orthonormal basis: u i u i =, and u i u j =0 for i j select a center defines offset of space best coordinates in lower dimensional space defined by dotproducts: (z,,z k ), z i = (xx) u i minimum squared error 4
3 PCA finds projection that minimizes reconstruction error Given m data points: x i = (x i,,x ni ), i= m Will represent each point as a projection: where: and PCA: Given k n, find (u,,u k ) minimizing reconstruction error: x 2 x 5 Understanding the reconstruction error Note that x i can be represented exactly by ndimensional projection: Given k n, find (u,,u k ) minimizing reconstruction error: Rewriting error: 6
4 Reconstruction error and covariance matrix 7 Minimizing reconstruction error and eigen vectors Minimizing reconstruction error equivalent to picking orthonormal basis (u,,u n ) minimizing: Eigen vector: Minimizing reconstruction error equivalent to picking (u k+,,u n ) to be eigen vectors with smallest eigen values 8
5 Basic PCA algoritm Start from m by n data matrix X Recenter: subtract mean from each row of X X c X X Compute covariance matrix: Σ /m X c T X c Find eigen vectors and values of Σ Principal components: k eigen vectors with highest eigen values 9 PCA example 0
6 PCA example reconstruction only used first principal component Eigenfaces [Turk, Pentland 9] Input images: Principal components: 2
7 Eigenfaces reconstruction Each image corresponds to adding 8 principal components: 3 Scaling up Covariance matrix can be really big! Σ is n by n 0000 features! Σ finding eigenvectors is very slow Use singular value decomposition (SVD) finds to k eigenvectors great implementations available, e.g., Matlab svd 4
8 SVD Write X = W S V T X data matri one row per datapoint W weight matri one row per datapoint coordinate of x i in eigenspace S singular value matri diagonal matrix in our setting each entry is eigenvalue λ j V T singular vector matrix in our setting each row is eigenvector v j 5 PCA using SVD algoritm Start from m by n data matrix X Recenter: subtract mean from each row of X X c X X Call SVD algorithm on X c ask for k singular vectors Principal components: k singular vectors with highest singular values (rows of V T ) Coefficients become: 6
9 What you need to know Dimensionality reduction why and when it s important Simple feature selection Principal component analysis minimizing reconstruction error relationship to covariance matrix and eigenvectors using SVD 7 Announcements University Course Assessments Please, please, please, please, please, please, please, please, please, please, please, please, please, please, please, please Last lecture: Thursday, 9, 4:406:30pm, Wean
10 Markov Decision Processes (MDPs) Machine Learning 070/578 Carlos Guestrin Carnegie Mellon University November 28 th, Thus far this semester Regression: Classification: Density estimation: 20
11 Learning to act [Ng et al. 05] Reinforcement learning An agent Makes sensor observations Must select action Receives rewards positive for good states negative for bad states 2 Learning to play backgammon [Tesauro 95] Combines reinforcement learning with neural networks Played 300,000 games against itself Achieved grandmaster level! 22
12 Roadmap to learning about reinforcement learning When we learned about Bayes nets: First talked about formal framework: representation inference Then learning for BNs For reinforcement learning: Formal framework Markov decision processes Then learning 23 peasant footman building Realtime Strategy Game Peasants collect resources and build Footmen attack enemies Buildings train peasants and footmen 24
13 States and actions State space: Joint state x of entire system Action space: Joint action a= {a,, a n } for all agents 25 States change over time Like an HMM, state changes over time Next state depends on current state and action selected e.g., action= build castle likely to lead to a state where you have a castle Transition model: Dynamics of the entire system P(x 26
14 Some states and actions are better than others Each state x is associated with a reward positive reward for successful attack negative for loss Reward function: Total reward R(x) 27 Markov Decision Process (MDP) Representation State space: Joint state x of entire system Action space: Joint action a= {a,, a n } for all agents Reward function: Total reward R( sometimes reward can depend on action Transition model: Dynamics of the entire system P(x 28
15 Discounted Rewards An assistant professor gets paid, say, 20K per year. How much, in total, will the A.P. earn in their life? = Infinity $ $ What s wrong with this argument? 29 Discounted Rewards A reward (payment) in the future is not worth quite as much as a reward now. Because of chance of obliteration Because of inflation Example: Being promised $0,000 next year is worth only 90% as much as receiving $0,000 right now. Assuming payment n years in future is worth only (0.9) n of payment now, what is the AP s Future Discounted Sum of Rewards? 30
16 Discount Factors People in economics and probabilistic decisionmaking do this all the time. The Discounted sum of future rewards using discount factor γ is (reward now) + γ (reward in time step) + γ 2 (reward in 2 time steps) + γ 3 (reward in 3 time steps) + : : (infinite sum) 3 The Academic Life Assume Discount Factor γ = A. Assistant Prof B. Assoc. Prof T. Tenured Prof 400 Define: 0.2 S. 0.2 On the Street D. Dead 0 V A = Expected discounted future rewards starting in state A V B = Expected discounted future rewards starting in state B V T = T V S = S V D = D How do we compute V A, V B, V T, V S, V D?
17 Computing the Future Rewards of an Academic 0.6 A. Assistant Prof Assume Discount Factor γ = 0.9 B. Assoc. Prof S. On the Street D. Dead T. Tenured Prof Policy Policy: π(x) = a At state action a for all agents x 0 π(x 0 ) = both peasants get wood x π(x ) = one peasant builds barrack, other gets gold x 2 π(x 2 ) = peasants get gold, footmen attack 34
18 Value of Policy Value: V π (x) Expected longterm reward starting from x Start from x 0 x 0 R(x 0 ) π(x 0 ) x R(x ) V π (x 0 ) = E π [R(x 0 ) + γ R(x ) + γ 2 R(x 2 ) + γ 3 R(x 3 ) + γ 4 R(x 4 ) + L] π(x ) x 2 π(x 2 ) Future rewards discounted by γ 2 [0,) x R(x ) π(x ) x R(x ) π(x ) R(x 2 ) x 3 R(x 3 ) π(x 3 ) x 4 R(x 4 ) 35 Computing the value of a policy Discounted value of a state: V π (x 0 ) = E π [R(x 0 ) + γ R(x ) + γ 2 R(x 2 ) + γ 3 R(x 3 ) + γ 4 R(x 4 ) + L] value of starting from x 0 and continuing with policy π from then on A recursion! 36
19 Simple approach for computing the value of a policy: Iteratively Can solve using a simple convergent iterative approach: (a.k.a. dynamic programming) Start with some guess V 0 Iteratively say: V t+ = R + γ P π V t Stop when V t+ V t ε means that V π V t+ ε/(γ) 37 But we want to learn a Policy So far, told you how good a policy is But how can we choose the best policy??? x 0 Policy: π(x) = a At state action a for all agents π(x 0 ) = both peasants get wood Suppose there was only one time step: world is about to end!!! select action that maximizes reward! x x 2 π(x ) = one peasant builds barrack, other gets gold π(x 2 ) = peasants get gold, footmen attack 38
20 Unrolling the recursion Choose actions that lead to best value in the long run Optimal value policy achieves optimal value V * 39 Bellman equation Evaluating policy π: Computing the optimal value V *  Bellman equation " V ( x) = max R( + # a! x' P( x' V " ( x' ) 40
21 Optimal Longterm Plan Optimal value function V * (x) Optimal Policy: π * (x) " # (x) = argmax a Optimal policy: R( + $ % x' P(x' V # (x') 4 Interesting fact Unique value " V ( x) = max R( + # a! Slightly surprising fact: There is only one V * that solves Bellman equation! there may be many optimal policies that achieve V * Surprising fact: optimal policies are good everywhere!!! x' P( x' V " ( x' ) 42
22 Solve Bellman equation Solving an MDP Optimal value V * (x) " V ( x) = max R( + # a! Many algorithms solve the Bellman equations: x' P( x' V Bellman equation is nonlinear!!! Optimal policy π*(x) " ( x' ) Policy iteration [Howard 60, Bellman 57] Value iteration [Bellman 57] Linear programming [Manne 60] 43 Value iteration (a.k.a. dynamic programming) the simplest of all " V ( x) = max R( + # a! x' P( x' V " ( x' ) Start with some guess V 0 Iteratively say: Stop when V t+ V t ε means that V V t+ ε/(γ)! V t+ ( x) = max R( + " P( x' Vt ( x' ) a x' 44
23 A simple example You run a startup company. In every state you must choose between Saving money or Advertising. Poor & Unknown +0 S S A Rich & Unknown +0 A S Poor & Famous +0 S A Rich & Famous +0 γ = 0.9 A 45 Let s compute V t (x) for our example Poor & Unknown +0 S S A Rich & Unknown +0 A S Poor & Famous +0 S A Rich & Famous +0 γ = 0.9 A t V t (PU) V t (PF) V t (RU) V t (RF)! V t+ ( x) = max R( + " P( x' Vt ( x' ) a x' 46
24 Let s compute V t (x) for our example Poor & Unknown +0 S S A Rich & Unknown +0 A S Poor & Famous +0 S A Rich & Famous +0 γ = 0.9 A t V t (PU) V t (PF) V t (RU) V t (RF) ! V t+ ( x) = max R( + " P( x' Vt ( x' ) a x' 47 What you need to know What s a Markov decision process state, actions, transitions, rewards a policy value function for a policy computing V π Optimal value function and optimal policy Bellman equation Solving Bellman equation with value iteration, policy iteration and linear programming 48
25 Acknowledgment This lecture contains some material from Andrew Moore s excellent collection of ML tutorials: 49
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