Introduction to Machine Learning

Transcription

1 Introduction to Machine Learning Brown University CSCI 1950-F, Spring 2012 Prof. Erik Sudderth Lecture 25: Markov Chain Monte Carlo (MCMC) Course Review and Advanced Topics Many figures courtesy Kevin Murphy s textbook, Machine Learning: A Probabilistic Perspective

2 E[f] = Monte Carlo Methods f(z)p(z) dz 1 L Provably good if L sufficiently large:! Unbiased for any sample size! Variance inversely proportional to sample size (and independent of dimension of space)! Weak law of large numbers! Strong law of large numbers! Problem: Drawing samples from complex distributions! L l=1 Alternatives for hard problems: f(z (l) )! Importance sampling! Markov chain Monte Carlo (MCMC) z (l) p(z) Estimation of expected model properties via simulation

3 Markov Chain Monte Carlo z (0) z (1) z (2) z (t+1) q(z z (t) ) z (t) q(z z (t) )! At each time point, state is a configuration of all the variables in the model: parameters, hidden variables, etc.! We design the transition distribution so that the chain is irreducible and ergodic, with a unique stationary distribution p (z) p (z) = Z q(z z )p (z ) dz! For learning, the target equilibrium distribution is usually the posterior distribution given data x: p (z) =p(z x)! Popular recipes: Metropolis-Hastings and Gibbs samplers

4 Gibbs Sampler for a 2D Gaussian z 2 L General Gibbs Sampler z (t) i p(z i z (t 1) \i ) i = i(t) z (t) j = z (t 1) j j i(t) l Under mild conditions, converges assuming all variables are resampled infinitely often (order can be fixed or random) z 1 C. Bishop, Pattern Recognition & Machine Learning

5 Probabilistic Mixture Models 1 (a) 0.5 θ k = {µ k, Σ k } p(z i π) = Cat(z i π) p(x i z i, µ, Σ) =N (x i µ zi, Σ zi )

6 Mixture Sampler Pseudocode

7 Snapshots of Mixture Gibbs Sampler Initialization B Initialization A 2 Iterations 10 Iterations 50 Iterations

8 Collapsed Sampling Algorithms π Dir(α) z i Cat(π) x i F (θ zi ) θ k G(β) Conjugate priors allow exact marginalization of parameters, to make an equivalent model with fewer variables

9 Gibbs: Representation and Mixing Multiple Initializations Quantiles of 100 Chains Standard Gibbs: Alternatively sample assignments, parameters Collapsed Gibbs: Marginalize parameters, sample assignments

10 MCMC & Computational Resources Best practical option: A few (> 1) initializations for as many iterations as possible

11 End of New Material Next Slides: Some review and some advertisement of advanced topics

12 The Main Learning Problems Supervised Learning Unsupervised Learning Discrete classification or categorization clustering Continuous regression dimensionality reduction!supervised: Learn to approximate a function from examples!unsupervised: Learn a representation which compresses data!probabilistic learning: Learn by maximizing probability, or minimizing an expected loss

13 Supervised Learning Generative ML or MAP Learning: Naïve Bayes N log p(π)+ logp(θ)+ [log p(y i π)+logp(x i y i, θ)] max π,θ π y i π y t i=1 y i y t θ Train x i N θ Test x t Discriminative ML or MAP Learning: Logistic regression max θ log p(θ)+ θ N log p(y i x i, θ) i=1 x i Train N θ Test x t

14 Learning via Optimization ML Estimate: MAP Estimate: Gradient vectors: f : R M R ŵ = arg min w ŵ = arg min w w f : R M R M log p(y i x i,w) log i p(w) log p(y i x i,w) i ( w f(w)) k = f(w) w k Hessian matrices: 2 wf : R M R M M Optimization of Smooth Functions: ( w f(w)) k,l = 2 f(w) w k w l! Closed form: Find zero gradient points, check curvature! Iterative: Initialize somewhere, use gradients to take steps towards better (by convention, smaller) values

15 Clustering: max π,θ Unsupervised Learning log p(π)+ log p(θ)+ Dimensionality Reduction: max π,θ log p(π)+ log p(θ)+ [ ] N log p(z i π)p(x i z i, θ) z i i=1 N [ ] log p(z i π)p(x i z i, θ) dz i z i i=1! No notion of training and test data: labels are never observed! As before, maximize posterior probability of model parameters! For hidden variables associated with each observation, we marginalize over possible values rather than estimating! Fully accounts for uncertainty in these variables! There is one hidden variable per observation, so cannot perfectly estimate even with infinite data! Must use generative model (discriminative degenerates) π θ z i x i N

16 π Expectation Maximization (EM) y i π y t π z i θ x i N Supervised Training θ x t Supervised Testing θ x i N Unsupervised Learning π, θ parameters (define low-dimensional manifold) z 1,...,z N hidden data (locate observations on manifold)! Initialization: Randomly select starting parameters! E-Step: Given parameters, find posterior of hidden data! Equivalent to test inference of full posterior distribution! M-Step: Given posterior distributions, find likely parameters! Similar to supervised ML/MAP training! Iteration: Alternate E-step & M-step until convergence

17 EM as Lower Bound Maximization ln p(x θ) = ln ln p(x θ) z ( z! Initialization: Randomly select starting parameters! E-Step: Given parameters, find posterior of hidden data q (t) = arg max q L(q, θ (t 1) )! M-Step: Given posterior distributions, find likely parameters θ (t) = arg max ) p(x, z θ) dz q(z) ln p(x, z θ) dz θ L(q (t), θ)! Iteration: Alternate E-step & M-step until convergence z q(z) ln q(z) dz L(q, θ) θ (0)

18 Gaussian Mixture Models vs. HMMs Mixture Model z 1 z 2 z 3 z 4 z 5 x 1 x 2 x 3 x 4 x 5 p(z i π, µ, Σ) = Cat(z i π) z i {1,...,K} Hidden Markov Model p(x i z i, π, µ, Σ) = Norm(x i µ zi, Σ zi ) z 1 z 2 z 3 z 4 z 5 x 1 x 2 x 3 x 4 x 5 p(z t π, µ, Σ,z t 1,z t 2,...) = Cat(z t π zt 1 ) p(x t z t, π, µ, Σ) = Norm(x t µ zt, Σ zt ) Recover mixture model when all rows of state transition matrix are equal.

19 Probabilistic PCA & Factor Analysis! Both Models: Data is a linear function of low-dimensional latent coordinates, plus Gaussian noise p(x i z i, θ) =N (x i Wz i + µ, Ψ) p(x i θ) =N (x i µ, W W T + Ψ) Ψ! Factor analysis:! Probabilistic PCA: x 2 is a general diagonal matrix is a multiple of identity matrix Ψ = σ 2 I p(x ẑ) w p(z i θ) =N (z i 0,I) x 2 low rank covariance parameterization µ } ẑ w µ p(z) p(x) ẑ z x 1 C. Bishop, Pattern Recognition & Machine Learning x 1

20 Linear State Space Models! States & observations jointly Gaussian:!! All marginals & conditionals Gaussian!! Linear transformations remain Gaussian

21 Simple Linear Dynamics Brownian Motion Constant Velocity Time Time

22 Kalman Filter! Represent Gaussians by mean & covariance: Prediction: Kalman Gain: Update:

23 Constant Velocity Tracking Kalman Filter Kalman Smoother (K. Murphy, 1998)

24 Nonlinear State Space Models! State dynamics and measurements given by potentially complex nonlinear functions! Noise sampled from non-gaussian distributions

25 Examples of Nonlinear Models Dynamics implicitly determined by geophysical simulations Observed image is a complex function of the 3D pose, other nearby objects & clutter, lighting conditions, camera calibration, etc.

26 Nonlinear Filtering Prediction: Update:

27 Approximate Nonlinear Filters! No direct represention of continuous functions, or closed form for the prediction integral! Big literature on approximate filtering:!! Histogram filters!! Extended & unscented Kalman filters!! Particle filters!!!

28 Nonlinear Filtering Taxonomy Histogram Filter:!!Evaluate on fixed discretization grid!!only feasible in low dimensions!!expensive or inaccurate Extended/Unscented Kalman Filter:!!Approximate posterior as Gaussian via linearization, quadrature,!!!inaccurate for multimodal posterior distributions Particle Filter:!!Dynamically evaluate states with highest probability!!monte Carlo approximation

29 Particle Filters Condensation, Sequential Monte Carlo, Survival of the Fittest,!! Represent state estimates using a set of samples! Propagate over time using importance sampling Sample-based density estimate Weight by observation likelihood Resample & propagate by dynamics

30 Particle Filtering Movie (M. Isard, 1996)

31 Dynamic Bayesian Networks Specify and exploit internal structure in the hidden states underlying a time series Maneuver Mode Spatial Position Noisy Observations

32 DBN Hand Tracking Video Isard et. al., 1998

33 Particle Filtering Caveats! Particle filters are easy to implement, and effective in many applications, BUT!! It can be difficult to know how many samples to use, or to tell when the approximation is poor!! Sometimes suffer catastrophic failures, where NO particles have significant posterior probability!! This is particularly true with peaky observations in high-dimensional spaces: likelihood dynamics

34 The Big Picture Ghahramani & Roweis