regression[1]

regression[1] - 4/8/10 Regression CPS 170 Ron Parr...

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Unformatted text preview: 4/8/10 Regression CPS 170 Ron Parr Regression figures provided by Christopher Bishop and © 2007 Christopher Bishop With content adapted from Lise Getoor, Tom Die>erich, Andrew Moore & Rich Maclin Supervised Learning •  Given: Training Set •  Goal: Good performance on test set •  AssumpHons: –  Training samples are independently drawn, and idenHcally distributed (IID) –  Test set is from same distribuHon as training set 1 4/8/10 FiPng ConHnuous Data (Regression) •  Datum i has feature vector: φ=(φ1(x(i))…φk(x(i))) •  Has real valued target: t(i) •  Concept space: linear combinaHons of features: •  Learning objecHve: Search to find “best” w •  (This is standard “data fiPng” that most people learn in some form or another.) Linearity of Regression •  Regression typically considered a linear method, but… •  Features not necessarily linear •  Features not necessarily linear •  Features not necessarily linear •  Features not necessarily linear •  and, BTW, features not necessarily linear 2 4/8/10 Regression Examples •  PredicHng housing price from: –  House size, lot size, rooms, neighborhood*, etc. •  PredicHng weight from: –  Sex, height, ethnicity, etc. •  PredicHng life expectancy increase from: –  MedicaHon, disease state, etc. •  PredicHng crop yield from: –  PrecipitaHon, ferHlizer, temperature, etc. •  FiPng polynomials –  Features are monomials Features/Basis FuncHons •  Polynomials •  Indicators •  Gaussian densiHes •  Step funcHons or sigmoids •  Sinusoids (Fourier basis) •  Wavelets •  Anything you can imagine… 3 4/8/10 What is “best”? •  No obvious answer to this quesHon •  Three compaHble answers: –  Minimize squared error on training set –  Maximize likelihood of the data (under certain assumpHons) –  Project data into “closest” approximaHon •  Other answers possible Minimizing Squared Training Set Error •  Why is this good? •  How could this be bad? •  Minimize: N E ( w) = ∑ ( wT φ ( x ( i ) ) − t ( i) ) i =1 2 € 4 4/8/10 Minimizing E by Gradient Descent E(w) gradient vector Start with iniHal weight vector w 0 w 2 w1 w0 w Compute the gradient Compute where α is the step size Repeat until convergence (Adapted from Lise Getoor s Slides) Gradient Descent Issues •  For this parHcular problem: –  Global minimum exists –  Convergence “guaranteed” if done in “batch” •  In general –  Local opHmum only –  Batch mode more stable –  Incremental possible •  Can oscillate •  Use decreasing step size (Robbins ­Monro) to stabilize 5 4/8/10 Solving the MinimizaHon Directly n E = ∑ (t ( i) − w T φ ( x ( i) ))2 i =1 n ∇w E ∝ ∑ (t ( i) − w T φ ( x ( i) ))φ ( x ( i) )T i =1 scalar row vector Set gradient to 0 to find min: n € ∑ (t ( i) − wT φ ( x ( i) ))φ ( x ( i) )T = 0 i =1 n ∑ φ(x i =1 T n ( i) T ( i) ) t − wT ∑ φ ( x ( i) )φ ( x ( i ) )T = 0 i =1 T T T t Φ − w Φ Φ = Φ t − ΦT Φw = 0 −1 w = ΦT Φ) ΦT t ( € Ⱥφ ( x (1) ) Ⱥ Ⱥ (2 ) Ⱥ φ ( x ) Ⱥ Φ = Ⱥ Ⱥ Ⱥ Ⱥ ( n) Ⱥ Ⱥφ ( x ) Ⱥ € What is the Best Choice of Features? Noisy Source Data 6 4/8/10 Degree 0 Fit Degree 1 Fit 7 4/8/10 Degree 3 Fit Degree 9 Fit 8 4/8/10 ObservaHons •  Degree 3 is the best match to the source •  Degree 9 is the best match to the samples •  Performance on test data: Bias and Variance •  Bias: How much of our error comes from our choice of hypothesis space? •  Variance: How much of our error comes from noise in the training data? 9 4/8/10 Example: 20 points y = x + 2 sin(1.5x) + N(0,0.2) Noise Hypothesis space = linear in x 50 fits (20 examples each) What are we seeing here? 10 4/8/10 Bias Variance 11 4/8/10 Trade off Between Bias and Variance •  •  •  •  Is the problem a bad choice of polynomial? Is the problem that we don’t have enough data? Answer: Yes For small datasets: –  Lower bias  ­> Higher Variance –  Higher bias  ­> Lower Variance Bias and Variance: Lessons Learned •  When data are scarce relaHve to the “capacity” of our hypothesis space –  Variance can be a problem –  RestricHng hypothesis space can reduce variance at cost of increased bias •  When data are plenHful –  Variance is less of a concern –  May afford to use richer hypothesis space 12 4/8/10 Methods for Choosing Features •  Cross validaHon •  RegularizaHon Cross ValidaHon •  Suppose we have many possible hypothesis spaces, e.g., different degree polynomials •  Recall our empirical performance results: •  Why not use the data to find min of the red curve? 13 4/8/10 ImplemenHng Cross ValidaHon •  Many possible approaches to cross validaHon •  Typical approach divides data into k equally sized chunks: –  –  –  –  –  Do k instances of learning For each instance hold out 1/k of the data Train on (k ­1)/k fracHon of the data Test on held out data Average results •  Can also sample subsets of data with replacement •  Cross validaHon can be used to search range of hypothesis classes to find where overfi'ng starts Problems with Cross ValidaHon •  Cross validaHon is a sound method, but requires a lot of data and/ or is slow •  Must trade off two factors: –  Want enough data within each run –  Want to average over enough trials •  With scarce data: –  Choose k close to n –  Almost as many learning problems as data points •  With abundant data (then why are you doing cross validaHon?) –  Choose k = a small constant, e.g., 10 –  Not too painful if you have a lot of parallel compuHng resources and a lot of data, e.g., if you are Google 14 4/8/10 RegularizaHon •  Cross validaHon may also be impracHcal if range of hypothesis classes is not easily enumerated a searched iteraHvely •  RegularizaHon aims to avoid overfiPng, while –  Avoiding speed penalty of cross validaHon –  Not assuming an ordering on hypothesis spaces RegularizaHon •  Idea: Penalize overly complicated answers •  Ordinary regression minimizes: •  L2 Regularized regression minimizes: •  Note: May exclude constants form the norm 15 4/8/10 L2 RegularizaHon: Why? •  For polynomials, extreme curves typically require extreme values •  In general, encourages use of features only when they lead to a substanHal increase in performance •  Problem: How to choose λ (cross validaHon?) The L2 Regularized SoluHon •  Minimize: •  Set gradient to 0, solve for w for features Φ: •  Compare with unregularized soluHon 16 4/8/10 RegularizaHon Example High regularizaHon produces “flat” soluHons because weights must approach 0. Lower values allow for more curviness in the value funcHon. Concluding Comments •  Regression is the most basic machine learning algorithm for conHnuous targets •  MulHple views are all equivalent: –  Minimize squared loss –  Maximize likelihood –  Orthogonal projecHon •  Big quesHon: Choosing features •  Step towards understanding this: Bias/variance trade off •  Cross validaHon, regularizaHon automate (to some extent) balancing bias and variance 17 ...
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This note was uploaded on 02/17/2012 for the course COMPSCI 170 taught by Professor Parr during the Spring '11 term at Duke.

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