Introduction to Statistical Machine Learning
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Marcus Hutter
Introduction to Statistical Machine Learning Marcus Hutter Canberra, ACT, 0200, Australia http://www.hutter1.net/
ANU
RSISE
NICTA
Machine Learning Summer School MLSS-2009, 26 Janurary – 6 February, Canberra
Introduction to Statistical Machine Learning
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Marcus Hutter
Abstract This course provides a brief overview of the methods and practice of statistical machine learning. It’s purpose is to (a) give a mini-introduction and background to logicians interested in the AI courses, and (b) to summarize the core concepts covered by the machine learning courses during this week. Topics covered include Bayesian inference and maximum likelihood modeling; regression, classification, density estimation, clustering, principal component analysis; parametric, semi-parametric, and non-parametric models; basis functions, neural networks, kernel methods, and graphical models; deterministic and stochastic optimization; overfitting, regularization, and validation.
Introduction to Statistical Machine Learning
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Marcus Hutter
Table of Contents 1. Introduction / Overview / Preliminaries 2. Linear Methods for Regression 3. Nonlinear Methods for Regression 4. Model Assessment & Selection 5. Large Problems 6. Unsupervised Learning 7. Sequential & (Re)Active Settings 8. Summary
Intro/Overview/Preliminaries
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INTRO/OVERVIEW/PRELIMINARIES • What is Machine Learning? Why Learn? • Related Fields • Applications of Machine Learning • Supervised↔Unsupervised↔Reinforcement Learning • Dichotomies in Machine Learning • Mini-Introduction to Probabilities
Intro/Overview/Preliminaries
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What is Machine Learning? Machine Learning is concerned with the development of algorithms and techniques that allow computers to learn Learning in this context is the process of gaining understanding by constructing models of observed data with the intention to use them for prediction.
Related fields
• Artificial Intelligence: smart algorithms • Statistics: inference from a sample • Data Mining: searching through large volumes of data • Computer Science: efficient algorithms and complex models
Intro/Overview/Preliminaries
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Why ‘Learn’ ? There is no need to “learn” to calculate payroll Learning is used when: • Human expertise does not exist (navigating on Mars), • Humans are unable to explain their expertise (speech recognition) • Solution changes in time (routing on a computer network) • Solution needs to be adapted to particular cases (user biometrics) Example: It is easier to write a program that learns to play checkers or backgammon well by self-play rather than converting the expertise of a master player to a program.
Intro/Overview/Preliminaries
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Handwritten Character Recognition an example of a difficult machine learning problem
Task: Learn general mapping from pixel images to digits from examples
Intro/Overview/Preliminaries
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Applications of Machine Learning machine learning has a wide spectrum of applications including: • natural language processing, • search engines, • medical diagnosis, • detecting credit card fraud, • stock market analysis, • bio-informatics, e.g. classifying DNA sequences, • speech and handwriting recognition, • object recognition in computer vision, • playing games – learning by self-play: Checkers, Backgammon. • robot locomotion.
Intro/Overview/Preliminaries
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Some Fundamental Types of Learning • Supervised Learning Classification Regression
• Reinforcement Learning Agents
• Unsupervised Learning Association Clustering Density Estimation
• Others SemiSupervised Learning Active Learning
Intro/Overview/Preliminaries
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Supervised Learning • Prediction of future cases: Use the rule to predict the output for future inputs • Knowledge extraction: The rule is easy to understand • Compression: The rule is simpler than the data it explains • Outlier detection: Exceptions that are not covered by the rule, e.g., fraud
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Intro/Overview/Preliminaries
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Classification Example: Credit scoring
Differentiating between low-risk and high-risk customers from their Income and Savings
Discriminant: IF income > θ1 AND savings > θ2 THEN low-risk ELSE high-risk
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Intro/Overview/Preliminaries
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Regression Example: Price y = f (x)+noise of a used car as function of age x
Intro/Overview/Preliminaries
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Unsupervised Learning • Learning “what normally happens” • No output • Clustering: Grouping similar instances • Example applications: Customer segmentation in CRM Image compression: Color quantization Bioinformatics: Learning motifs
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Intro/Overview/Preliminaries
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Reinforcement Learning • Learning a policy: A sequence of outputs • No supervised output but delayed reward • Credit assignment problem • Game playing • Robot in a maze • Multiple agents, partial observability, ...
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Intro/Overview/Preliminaries
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Dichotomies in Machine Learning (machine) learning / statistical induction ⇔ prediction regression independent identically distributed online learning passive prediction parametric conceptual/mathematical exact/principled supervised learning
⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔
logic/knowledge-based (GOFAI) decision ⇔ action classification sequential / non-iid offline/batch learning active learning non-parametric computational issues heuristic unsupervised ⇔ RL learning
Intro/Overview/Preliminaries
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Probability Basics Probability is used to describe uncertain events; the chance or belief that something is or will be true. Example: Fair Six-Sided Die: • Sample space: Ω = {1, 2, 3, 4, 5, 6} • Events: Even= {2, 4, 6}, Odd= {1, 3, 5}
⊆Ω
• Probability: P(6) = 16 , P(Even) = P(Odd) = • Outcome: 6 ∈ E.
1 2
P (6and Even) 1/6 1 • Conditional probability: P (6|Even) = = = P (Even) 1/2 3 General Axioms: • P({}) = 0 ≤ P(A) ≤ 1 = P(Ω), • P(A ∪ B) + P(A ∩ B) = P(A) + P(B), • P(A ∩ B) = P(A|B)P(B).
Intro/Overview/Preliminaries
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Probability Jargon Example: (Un)fair coin: Ω = {Tail,Head} ' {0, 1}. P(1) = θ ∈ [0, 1]: Likelihood: P(1101|θ) = θ × θ × (1 − θ) × θ Maximum Likelihood (ML) estimate: θˆ = arg maxθ P(1101|θ) = 3 4
Prior: If we are indifferent, then P(θ) =const. R P 1 Evidence: P(1101) = θ P(1101|θ)P(θ) = 20 (actually ) Posterior: P(θ|1101) =
P(1101|θ)P(θ) P(1101)
∝ θ3 (1 − θ) (BAYES RULE!).
Maximum a Posterior (MAP) estimate: θˆ = arg maxθ P(θ|1101) = 2 Predictive distribution: P(1|1101) = P(11011) = P(1101) 3 P Expectation: E[f |...] = θ f (θ)P(θ|...), e.g. E[θ|1101] =
Variance: Var(θ) = E[(θ − Eθ)2 |1101] =
2 63
Probability density: P(θ) = 1ε P([θ, θ + ε]) for ε → 0
2 3
3 4
Linear Methods for Regression
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LINEAR METHODS FOR REGRESSION • Linear Regression • Coefficient Subset Selection • Coefficient Shrinkage • Linear Methods for Classifiction • Linear Basis Function Regression (LBFR) • Piecewise linear, Splines, Wavelets • Local Smoothing & Kernel Regression • Regularization & 1D Smoothing Splines
Linear Methods for Regression
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Linear Regression fitting a linear function to the data • Input “feature” vector x := (1 ≡ x(0) , x(1) , ..., x(d) ) ∈ IRd+1 • Real-valued noisy response y ∈ IR. • Linear regression model: yˆ = fw (x) = w0 x(0) + ... + wd x(d) • Data: D = (x1 , y1 ), ..., (xn , yn ) • Error or loss function: Example: Residual sum of squares: Pn Loss(w) = i=1 (yi − fw (xi ))2 • Least squares (LSQ) regression: w ˆ = arg minw Loss(w) • Example: Person’s weight y as a function of age x1 , height x2 .
Linear Methods for Regression
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Coefficient Subset Selection Problems with least squares regression if d is large: • Overfitting: The plane fits the data well (perfect for d ≥ n), but predicts (generalizes) badly. • Interpretation: We want to identify a small subset of features important/relevant for predicting y. Solution 1: Subset selection: Take those k out of d features that minimize the LSQ error.
Linear Methods for Regression
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Coefficient Shrinkage Solution 2: Shrinkage methods: Shrink the least squares w by penalizing the Loss: Ridge regression: Add ∝ ||w||22 . Lasso: Add ∝ ||w||1 . Bayesian linear regression: Comp. MAP arg maxw P(w|D) from prior P (w) and sampling model P (D|w). Weights of low variance components shrink most.
Linear Methods for Regression
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Linear Methods for Classification Example: Y = {spam,non-spam} ' {−1, 1}
(or {0, 1})
Reduction to regression: Regard y ∈ IR ⇒ w ˆ from linear regression. Binary classification: If fwˆ (x) > 0 then yˆ = 1 else yˆ = −1. Probabilistic classification: Predict probability that new x is in class y. P(y=1|x,D) := fw Log-odds log P(y=0|x,D) ˆ (x) Improvements: • Linear Discriminant Analysis (LDA) • Logistic Regression • Perceptron • Maximum Margin Hyperplane • Support Vector Machine Generalization to non-binary Y possible.
Linear Methods for Regression
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Linear Basis Function Regression (LBFR) = powerful generalization of and reduction to linear regression! Problem: Response y ∈ IR is often not linear in x ∈ IRd . Solution: Transform x ; φ(x) with φ : IRd → IRp . Assume/hope response y ∈ IR is now linear in φ. Examples: • Linear regression: p = d and φi (x) = xi . • Polynomial regression: d = 1 and φi (x) = xi . • Piecewise constant regression: E.g. d = 1 with φi (x) = 1 for i ≤ x < i + 1 and 0 else. • Piecewise polynomials ... • Splines ...
Linear Methods for Regression
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Linear Methods for Regression
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2D Spline LBFR and 1D Symmlet-8 Wavelets
Linear Methods for Regression
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Local Smoothing & Kernel Regression Estimate f (x) by averaging the yi for all xi a-close to x: fˆ(x) =
Pn K(x,xi )yi Pi=1 n i=1
K(x,xi )
Nearest-Neighbor Kernel: i |
ˆ ⇒ f (x) = w ˆ φ(x) = i=1 ai K(xi , x) depends only on φ via Kernel Pd K(xi , x) = b=1 φb (xi )φb (x). ⇒ Huge time savings if d À n Example K(x, x0 ): • polynomial (1 + hx, x0 i)d , • Gaussian exp(−||x − x0 ||22 ), • neural network tanh(hx, x0 i).
Model Assessment & Selection
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MODEL ASSESSMENT & SELECTION • Example: Polynomial Regression • Training=Empirical versus Test=True Error • Empirical Model Selection • Theoretical Model Selection • The Loss Rank Principle for Model Selection
Model Assessment & Selection
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Example: Polynomial Regression • Straight line does not fit data well (large training error) high bias ⇒ poor predictive performance • High order polynomial fits data perfectly (zero training error) high variance (overfitting) ⇒ poor prediction too! • Reasonable polynomial degree d performs well. How to select d? minimizing training error obviously does not work.
Model Assessment & Selection
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Training=Empirical versus Test=True Error • Learn functional relation f for data D = {(x1 , y1 ), ..., (xn , yn )}. • We can compute the empirical error on past data: Pn 1 ErrD (f ) = n i=1 (yi − f (xi ))2 . • Assume data D is sample from some distribution P. • We want to know the expected true error on future examples: ErrP (f ) = EP [(y − f (x))]. • How good an estimate of ErrP (f ) is ErrD (f ) ? • Problem: ErrD (f ) decreases with increasing model complexity, but not ErrP (f ).
Model Assessment & Selection
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Empirical Model Selection How to select complexity parameter • Kernel width a, • penalization constant λ, • number k of nearest neighbors, • the polynomial degree d? Empirical test-set-based methods: Regress on training set and minimize empirical error w.r.t. “complexity” parameter (a, λ, k, d) on a separate test-set. Sophistication: cross-validation, bootstrap, ...
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Model Assessment & Selection
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Theoretical Model Selection How to select complexity or flexibility or smoothness parameter: Kernel width a, penalization constant λ, number k of nearest neighbors, the polynomial degree d? For • • • •
parametric regression with d parameters: Bayesian model selection, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Minimum Description Length (MDL),
They all add a penalty proportional to d to the loss. For non-parametric linear regression: • Add trace of on-data regressor = effective # of parameters to loss. • Loss Rank Principle (LoRP).
Model Assessment & Selection
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The Loss Rank Principle for Model Selection c : X → Y be the (best) regressor of complexity c on data D. Let fˆD c The loss Rank of fˆD is defined as the number of other (fictitious) data c ˆc D0 that are fitted better by fˆD 0 than D is fitted by fD . c • c is small ⇒ fˆD fits D badly ⇒ many other D0 can be fitted better ⇒ Rank is large.
• c is large ⇒ many D0 can be fitted well ⇒ Rank is large. c • c is appropriate ⇒ fˆD fits D well and not too many other D0 can be fitted well ⇒ Rank is small. LoRP: Select model complexity c that has minimal loss Rank Unlike most penalized maximum likelihood variants (AIC,BIC,MDL), • LoRP only depends on the regression and the loss function. • It works without a stochastic noise model, and • is directly applicable to any non-parametric regressor, like kNN.
How to Attack Large Problems
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HOW TO ATTACK LARGE PROBLEMS • Probabilistic Graphical Models (PGM) • Trees Models • Non-Parametric Learning • Approximate (Variational) Inference • Sampling Methods • Combining Models • Boosting
How to Attack Large Problems
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Probabilistic Graphical Models (PGM) Visualize structure of model and (in)dependence ⇒ faster and more comprehensible algorithms • Nodes = random variables. • Edges = stochastic dependencies. • Bayesian network = directed PGM • Markov random field = undirected PGM Example: P(x1 )P(x2 )P(x3 )P(x4 |x1 x2 x3 )P(x5 |x1 x3 )P (x6 |x4 )P(x7 |x4 x5 )
How to Attack Large Problems
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Additive Models & Trees & Related Methods Generalized additive model: f (x) = α + f1 (x1 ) + ... + fd (xd ) Reduces determining f : IRd → IR to d 1d functions fb : IR → IR Classification/decision trees: Outlook: • PRIM, • bump hunting, • How to learn tree structures.
How to Attack Large Problems
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Regression Trees f (x) = cb for x ∈ Rb , and cb = Average[yi |xi ∈ Rb ]
How to Attack Large Problems
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Non-Parametric Learning = prototype methods = instance-based learning = model-free Examples: • K-means: Data clusters around K centers with cluster means µ1 , ...µK . Assign xi to closest cluster center. • K Nearest neighbors regression (kNN): Estimate f (x) by averaging the yi for the k xi closest to x: • Kernel regression:
Pn K(x, xi )yi i=1 ˆ Take a weighted average f (x) = Pn . i=1 K(x, xi )
How to Attack Large Problems
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How to Attack Large Problems
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Approximate (Variational) Inference Approximate full distribution P(z) by q(z) Popular: Factorized distribution: q(z) = q1 (z1 ) × ... × qM (zM ). Measure of fit: Relative entropy R KL(p||q) = p(z) log p(z) q(z) dz Red curves: Left minimizes KL(P||q), Middle and Right are the two local minima of KL(q||P).
Examples: Gaussians
How to Attack Large Problems
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Elementary Sampling Methods How to sample from P : Z → [0, 1]? • Special sampling algorithms for standard distributions P. • Rejection sampling: Sample z uniformly from domain Z, but accept sample only with probability ∝ P(z). • Importance sampling: R PL 1 E[f ] = f (z)p(z)dz ' L l=1 f (z l )p(z l )/q(z l ), where z l are sampled from q. Choose q easy to sample and large where f (z l )p(z l ) is large. • Others: Slice sampling
How to Attack Large Problems
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Markov Chain Monte Carlo (MCMC) Sampling Metropolis: Choose some convenient q with q(z|z 0 ) = q(z 0 |z). Sample z l+1 from q(·|z l ) but accept only with probability min{1, p(zl+1 )/p(z l )}. Gibbs Sampling: Metropolis with q leaving z unchanged from l ; l + 1,except resample coordinate i from P(zi |z \i ). Green lines are accepted and red lines are rejected Metropolis steps.
How to Attack Large Problems
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Combining Models Performance can often be improved by combining multiple models in some way, instead of just using a single model in isolation • Committees: Average the predictions of a set of individual models. P • Bayesian model averaging: P(x) = Models P(x|Model)P(Model) P • Mixture models: P(x|θ, π) = k πk Pk (x|θk ) • Decision tree: Each model is responsible for a different part of the input space. • Boosting: Train many weak classifiers in sequence and combine output to produce a powerful committee.
How to Attack Large Problems
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Boosting Idea: Train many weak classifiers Gm in sequence and combine output to produce a powerful committee G. AdaBoost.M1: [Freund & Schapire (1997) received famous G¨odel-Prize] Initialize observation weights wi uniformly. For m = 1 to M do: (a) Gm classifies x as Gm (x) ∈ {−1, 1}. Train Gm weighing data i with wi . (b) Give Gm high/low weight αi if it performed well/bad. (c) Increase attention=weight wi for obs. xi misclassified by Gm . PM Output weighted majority vote: G(x) = sign( m=1 αm Gm (x))
Unsupervised Learning
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UNSUPERVISED LEARNING • K-Means Clustering • Mixture Models • Expectation Maximization Algorithm
Unsupervised Learning
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Unsupervised Learning Supervised learning: Find functional relationship f : X → Y from I/O data {(xi , yi )} Unsupervised learning: Find pattern in data (x1 , ..., xn ) without an explicit teacher (no y values). Example: Clustering e.g. by K-means Implicit goal: Find simple explanation, i.e. compress data (MDL, Occam’s razor). Density estimation: From which probability distribution P are the xi drawn?
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Unsupervised Learning
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K-means Clustering and EM • Data points seem to cluster around two centers. • Assign each data point i to a cluster ki . • Let µk = center of cluster k. • Distortion measure: Total distance2 of data points from cluster centers: Pn J(k, µ) := i=1 ||xi − µki ||2 • Choose centers µk initially at random. • M-step: Minimize J w.r.t. k: Assign each point to closest center • E-step: Minimize J w.r.t. µ: Let µk be the mean of points belonging to cluster k • Iteration of M-step and E-step converges to local minimum of J.
Unsupervised Learning
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Iterations of K-means EM Algorithm
Unsupervised Learning
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Mixture Models and EM Mixture of Gaussians: PK P(x|πµΣ) = i=1 Gauss(x|µk , Σk )πk Maximize likelihood P(x|...) w.r.t. π, µ, Σ. E-Step: Compute probability γik that data point i belongs to cluster k, based on estimates π, µ, Σ. M-Step: Re-estimate π, µ, Σ (take empirical mean/variance) given γik .
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Non-IID: Sequential & (Re)Active Settings
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NON-IID: SEQUENTIAL & (RE)ACTIVE SETTINGS • Sequential Data • Sequential Decision Theory • Learning Agents • Reinforcement Learning (RL) • Learning in Games
Non-IID: Sequential & (Re)Active Settings
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Sequential Data General stochastic time-series: P(x1 ...xn ) =
Qn i=1
P(xi |x1 ...xi−1 )
Independent identically distributed (roulette,dice,classification): P(x1 ...xn ) = P(x1 )P(x2 )...P(xn ) First order Markov chain (Backgammon): P(x1 ...xn ) = P(x1 )P(x2 |x1 )P(x3 |x2 )...P(xn |xn−1 ) Second order Markov chain (mechanical systems): P(x1 ...xn ) = P(x1 )P(x2 |x1 )P(x3 |x1 x2 )× ... × P(xn |xn−1 xn−2 ) Hidden Markov model (speech recognition): R P(x1 ...xn ) = P(z1 )P(z2 |z1 )...P(zn |zn−1 ) Qn × i=1 P(xi |zi )dz
Non-IID: Sequential & (Re)Active Settings
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Sequential Decision Theory Setup: For t = 1, 2, 3, 4, ... Given sequence x1 , x2 , ..., xt−1 (1) predict/make decision yt , (2) observe xt , (3) suffer loss Loss(xt , yt ), (4) t → t + 1, goto (1)
Example: Weather Forecasting xt ∈ X = {sunny, rainy} yt ∈ Y = {umbrella, sunglasses} Loss
sunny
rainy
umbrella
0.1
0.3
sunglasses
0.0
1.0
Goal: Minimize expected Loss: yˆt = arg minyt E[Loss(xt , yt )|x1 ...xt−1 ] Greedy minimization of expected loss is optimal if: Important: Decision yt does not influence env. (future observations). Examples:
Loss yˆ
= =
square / absolute / 0-1 error mean / median / mode
function of P(xt | · · · )
Non-IID: Sequential & (Re)Active Settings
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Learning Agents 1 sensors percepts ?
environment actions
agent
actuators
Additional complication: Learner can influence environment, and hence what he observes next. ⇒ farsightedness, planning, exploration necessary. Exploration ⇔ Exploitation problem
Non-IID: Sequential & (Re)Active Settings
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Learning Agents 2 Performance standard
Sensors
Critic
changes Learning element
knowledge
Performance element
learning goals Problem generator
Agent
Actuators
Environment
feedback
Non-IID: Sequential & (Re)Active Settings
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Reinforcement Learning (RL) r1 | o1 r2 | o2 r3 | o3 r4 | o4 r5 | o5 r6 | o6 ...
© © © ¼ ©
H Y HH H
Agent p
Environment q
PP P
y1
y2
³ 1 ³ ³ ³
PP P q³³
y3
y4
y5
y6
...
• RL is concerned with how an agent ought to take actions in an environment so as to maximize its long-term reward. • Find policy that maps states of the world to the actions the agent ought to take in those states. • The environment is typically formulated as a finite-state Markov decision process (MDP).
Non-IID: Sequential & (Re)Active Settings
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Learning in Games • Learning though self-play. • Backgammon (TD-Gammon). • Samuel’s checker program.
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Summary
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SUMMARY
• Important Loss Functions • Learning Algorithm Characteristics Comparison • More Learning • Data Sets • Journals • Annual Conferences • Recommended Literature
Summary
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Important Loss Functions
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Summary
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Learning Algorithm Characteristics Comparison
Summary
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More Learning • Concept Learning • Bayesian Learning • Computational Learning Theory (PAC learning) • Genetic Algorithms • Learning Sets of Rules • Analytical Learning
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Summary
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Data Sets • UCI Repository: http://www.ics.uci.edu/ mlearn/MLRepository.html • UCI KDD Archive: http://kdd.ics.uci.edu/summary.data.application.html • Statlib: http://lib.stat.cmu.edu/ • Delve: http://www.cs.utoronto.ca/ delve/
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Summary
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Journals • Journal of Machine Learning Research • Machine Learning • IEEE Transactions on Pattern Analysis and Machine Intelligence • Neural Computation • Neural Networks • IEEE Transactions on Neural Networks • Annals of Statistics • Journal of the American Statistical Association • ...
Summary
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Annual Conferences • Algorithmic Learning Theory (ALT) • Computational Learning Theory (COLT) • Uncertainty in Artificial Intelligence (UAI) • Neural Information Processing Systems (NIPS) • European Conference on Machine Learning (ECML) • International Conference on Machine Learning (ICML) • International Joint Conference on Artificial Intelligence (IJCAI) • International Conference on Artificial Neural Networks (ICANN)
Summary
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Marcus Hutter
Recommended Literature [Bis06]
C. M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006.
[HTF01] T. Hastie, R. Tibshirani, and J. H. Friedman. The Elements of Statistical Learning. Springer, 2001. [Alp04]
E. Alpaydin. Introduction to Machine Learning. MIT Press, 2004.
[Hut05] M. Hutter. Universal Artificial Intelligence: Sequential Decisions based on Algorithmic Probability. Springer, Berlin, 2005. http://www.hutter1.net/ai/uaibook.htm. 
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